Research ArticleEquilibrium Investment Strategy for DC Pension Plan withInflation and Stochastic Income under Hestonrsquos SV Model
Jingyun Sun12 Zhongfei Li3 and Yongwu Li4
1School of Mathematics Lanzhou City University Lanzhou 730070 China2School of Mathematics and Statistics Lanzhou University Lanzhou 730000 China3Sun Yat-sen Business School Sun Yat-sen University Guangzhou 510275 China4Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China
Correspondence should be addressed to Zhongfei Li lnslzfmailsysueducn
Received 8 December 2015 Revised 2 April 2016 Accepted 4 April 2016
Academic Editor Reza Jazar
Copyright copy 2016 Jingyun Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We consider a portfolio selection problem for a defined contribution (DC) pension plan under the mean-variance criteria We takeinto account the inflation risk and assume that the salary income process of the pension plan member is stochastic Furthermorethe financial market consists of a risk-free asset an inflation-linked bond and a risky asset with Hestonrsquos stochastic volatility (SV)Under the framework of game theory we derive two extended Hamilton-Jacobi-Bellman (HJB) equations systems and give thecorresponding verification theorems in both the periods of accumulation and distribution of the DC pension plan The explicitexpressions of the equilibrium investment strategies corresponding equilibrium value functions and the efficient frontiers are alsoobtained Finally some numerical simulations and sensitivity analysis are presented to verify our theoretical results
1 Introduction
Nowadays the application of stochastic control theory toportfolio selection problems of pension funds is becoming ahot issue in actuarial research The basic pension plans havetwo types the defined benefit (DB) pension plan and thedefined contribution (DC) pension plan In recent years withthe rapid development of the equity market and the lowermortality level of the population compared with the DBpension plan the DC pension plan is more favored by mostcountries in the world As in a DC pension plan the paymentpressure of benefit which is caused by the uncertainty ofinvestment earnings and pension plan memberrsquos longevityrisk is transferred from the sponsoring company to themember himselfherself Thus in the literature related topension funds the majority of the literature focuses on theDC pension plan
As we know for a DC pension plan the contributionis a predetermined constant or a fixed proportion of thememberrsquos income while the benefit is distributed based onthe accumulation value of the contribution and the returnof the pension fund portfolio until retirement Thus many
scholars are devoted to the optimal investment problem ofthe DC pension plan For example [1] considered a discrete-time multiperiod DC pension plan model to minimize theexpected deviation between the pension fund account anda predetermined target by using a quadratic loss functionto maximize the expected utility from the wealth at retire-ment time [2] investigated the optimal investment strategiesfor a DC pension plan both before and after retirementunder the continuous-time framework [3] extended theabove model into the case with stochastic interest rate andstochastic labor income [4] obtained the closed-form of theoptimal investment strategy for a DC pension plan underthe logarithm utility function Besides [5] investigated theoptimal asset allocation problem for a DC pension planwith downside protection under stochastic inflation and[6] studied the same problem under the stochastic interestrate and stochastic volatility framework Under the regimeswitching environment [7] considered an optimal asset-liability management problem for a pension fund
All the literature mentioned above focus on the opti-mal investment strategy under the objective of maximizingthe expected utility or minimizing the expected quadratic
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2391849 18 pageshttpdxdoiorg10115520162391849
2 Mathematical Problems in Engineering
loss In recent years some scholars pay attention to theportfolio selection problem of the DC pension plan underthe mean-variance (MV) criteria This is because of thefact that the optimal investment problems under the MVcriteria in a multiperiod or continuous-time framework aresuccessfully solved only recently It is well known that theMV problems with multiperiod or continuous-time versionare time-inconsistent in the sense that the Bellman optimalityprinciple does not hold Hence the dynamic programmingapproach can not be used In existing literature there areusually two methods that are suggested to deal with thisproblem The first one is to find the precommitment strategy(the breakthrough work for this method was made by [8 9])which means that the manager derives an optimal strategyat the initial time 0 and commits to performing this strategyin the future even if it does not remain optimal in thelater time However in practice this strategy is not easilyperformed since the preference of the MV manager changeswith time and heshe has an incentive to deviate from theprecommitment strategy in the later time In addition findinga time-consistent strategy is a basic requirement for a rationaldecision-maker Thus the second method is finding a time-consistent equilibrium strategy under the framework of gametheory (the most representative work was made by [10ndash12])In this case we regard the decision-making process as anoncooperative game between an infinite number of distinctplayers At each time 119905 we named player 119905 representing thefuture incarnation of the manager at time 119905 We are interestedin finding a subgame perfect Nash equilibrium point forthe game and formulating an equilibrium strategy thisstrategy is time-consistent For the MV problem under thebackground of pension fund [13] considered a DC pensionplan with the return of premiums clauses under the gametheoretic framework and obtained an equilibrium investmentstrategy in the period before retirement Under the sameframework [14] investigated an equilibrium investment andcontribution strategies for a DB pension plan The researchon the precommitment investment strategy can be found in[15ndash17] and references therein
Meanwhile for the portfolio problem many scholarsbegin to pay attention to the influence of the background risksuch as the risk of interest rate inflation and volatility of therisky asset on the optimal decision-making process On theone hand as we all know the inflation might affect the realvalue of the wealth especially for the problem with long timehorizon investment the higher inflation leads to the lowerreal wealth In theDCpension plan [18] obtained the optimalinvestment strategy which maximizes the expected utility ofthe DC pension plan under inflation risk A similar problemwas studied in [5] under the environment of stochasticinterest and inflation rate with the minimized guaranteeUnder the MV criteria [17] investigated the optimal pre-commitment investment strategy of a DC pension plan withinflation risk using Lagrange method Recently the equilib-rium investment strategies for a DC pension plan with theinflation risk are obtained by [19] On the other hand manyempirical studies have shown that the volatility of the riskyasset is stochastic Some scholars have proposed a variety ofSVmodels such as the constant elastic variance (CEV)model
[20] Hestonrsquos SV model (volatility satisfies Cox-Ingersoll-Ross process) [21] and Stein-Stein model (volatility satisfiesOrnstein-Uhlenbeck process) [22] The CEV and HestonrsquosSV models have been widely considered in investment andreinsurance problems such as [23ndash25] and references thereinFor the stochastic volatility model under the DC pensionplan the interested reader can be referred to [6 26 27]
In this paper we consider the MV portfolio problems foraDCpension plan both before and after retirementThemaindifference between this paper and the existing literature isthat both the inflation risk and the stochastic volatility riskare considered in our model To the best of our knowledgeunder the MV framework there is no literature consideringboth of the two risks in the DC pension fund managementWe assume that the financial market consists of a risk-freeasset an inflation-linked bond and a stock with Hestonrsquosstochastic volatility The salary income of the DC pensionplan member is also assumed to be stochastic because ofthe influence of inflation rate Under the framework of gametheory two MV problems are formulated related to theperiods before and after retirement respectively For each ofthe problems by solving an extended HJB equations systemthe equilibrium strategy equilibrium value function and thecorresponding equilibrium efficient frontier are obtainedWefind that the equilibrium investment money in the inflation-linked bond depends on current wealth the inflation riskand the contribution of the salary income have no influenceon the equilibrium investment money of stock Finally usingMonte Carlo method we investigate the evolution process ofthe equilibrium strategy with time before retirement underdifferent parameters and present the sensitivity of the efficientfrontier to corresponding parameters
The remainder of the paper is organized as follows InSection 2 we introduce the financial market and wealthprocesses both before and after retirement In Section 3 aMV portfolio allocation problem is formulated in the periodbefore retirement Under the framework of game theoryan extended HJB equations system and an equilibriuminvestment strategy are also obtained In Section 4 weconsider a MV portfolio problem after retirement and derivethe corresponding verification theorem and the equilibriumstrategy Some numerical simulations and sensitivity analysisfor our results are presented in Section 5 Section 6 concludesthe paper and outlines further research
2 Assumption and Model
Let (ΩF F P) be a filtered probability space with filtrationF = F
119905119905isin[0119879+119873]
satisfying the usual conditions that isF
119905119905isin[0119879+119873]
is right-continuous and P-complete The timehorizon [0 119879] represents the accumulation period of a DCpension plan member and [119879 119879 + 119873] is the distributionperiod of the member after heshe retires Let F
119905represent
the information available until time 119905 Suppose that all ofstochastic processes and random variables are defined on thefiltered probability space (ΩF F P) In addition we assumethat there are no transaction costs or taxes in the financialmarket trading can take place continuously and short sellingis permitted
Mathematical Problems in Engineering 3
21 Financial Market As we all know for a DC pensionplan manager the objective is making the terminal wealthmaximized by investing the pension fund into the marketGenerally speaking the investment time horizon of thepension fund lasts decades The inflation risk as a kind ofimportant background risk has important influence on thereal value of the pension fund In economics CPI (ConsumerPrice Index) as the typical index can represent the inflationlevel of the market Following [28 29] we assume that theprice level 119875(119905) satisfies the following diffusion process
119889119875 (119905)
119875 (119905)= 120583
119901(119905) 119889119905 + 120590
119901119889119882
119901(119905) 119875 (0) = 119901
0 (1)
where 120583119901(119905) represents the instantaneous expected inflation
rate 120590119901gt 0 is the instantaneous volatility of inflation rate
and 119882119901(119905) is a standard Brownian motion which generates
uncertainty of the price levelWe assume that the financial market consists of three
kinds of assets an inflation-linked asset a money marketaccount and a stock
(1) The inflation-linked index bond has the same risksource as the price level process 119875(119905) and can be freely tradedin themarket Following [18 29] its price process 119868(119905) satisfiesthe following stochastic differential equation (SDE)
where 119882119904(119905) and 119882V(119905) are two standard Brownian motions
and120582119904radic119881(119905) and120582
119901(119905) are themarket price of the risk sources
119882119904(119905) and 119882
119901(119905) respectively Since the inflation risk might
have influence on the price evolution process of the stockthrough direct or indirect ways (the correlated empiricalresearch can be found in [30 31] and references therein)hence we assume that the price process of the stock is derivedby not only its own risk source 119882
119904(119905) but also the risk
source119882119901(119905) We further assume that119882
119901(119905) is independent
of 119882119904(119905) and 119882V(119905) respectively while 119882119904
(119905) and 119882V(119905) aredependent and E[119882
119904(119905)119882V(119905)] = 120588119904V119905 where 120588119904V isin [minus1 1] is
the correlation coefficientThe second equation of (4) is a CIR(Cox-Ingersoll-Ross) mean-revert process whichmodels thestochastic volatility of the stock price Here we suppose 2120572120575 gt1205902
V to assure that 119881(119905) gt 0 holdsIn addition we assume that theDCpension planmember
receives salary income with nominal value 119871(119905) at time 119905 untilthe retirement time 119879 Suppose that the income is stochasticand dynamically influenced by the price level 119875(119905) that is thesalary income process is driven by the source of uncertaintyfrom inflation and it satisfies the following process
119889119871 (119905)
119871 (119905)
= 120583119897119889119905 + 120590
119897119889119882
119901(119905)
119871 (0) = 1198970gt 0
(5)
where 120583119897is the average growth rate of the income and 120590
119897is the
volatility rate
Remark 1 To simplify the model achieve tractability andgive detailed analysis about the optimal strategies we assumethat the real interest rate 119903(119905) nominal interest rate 119877(119905) andexpected inflation rate 120583
119901(119905) are all deterministic functions of
time 119905
Remark 2 If we denote Q as the risk neutral measure then(2) can be rewritten as
119889119868 (119905)
119868 (119905)= (119903 (119905) + 120583
119901(119905) minus 120582
119901(119905) 120590
119901) 119889119905 + 120590
119901119889119882
Q119901(119905) (6)
By the pricing theory of the derivative (to avoid arbitrage) weobtain the following relationship
119903 (119905) + 120583119901(119905) minus 120582
119901(119905) 120590
119901= 119877 (119905) (7)
Remark 3 In the empirical research the evolution processof 119878(119905) and 119881(119905) is usually negatively correlated Since ingeneral with declining of the stock price the volatility of theprice gradually increases (cf [32]) based on this reason inSection 5 we assume that the parameter 120588
119904V lt 0
22 Wealth Process In this paper we consider the optimalportfolio problems of aDCpension planmember both beforeand after retirement The member contributes part of hishersalary into the pension fund account before retirement andobtains benefit after retirement So we need to consider thewealth processes in two different periods
(I) Before Retirement During the accumulation period [0 119879]we assume that the DC pension plan member contributes
4 Mathematical Problems in Engineering
continuously to hisher pension fund account with a fixedrate 119896 of hisher salary income that is heshe contributes theamount of money 119896119871(119905) to hisher pension fund account attime 119905 The pension fund is invested in the financial marketby the manager Denote 120587
119868(119905) and 120587
119878(119905) as the propositions
of wealth invested in the inflation-linked index bond and thestock respectively Then 120587
119861(119905) = 1 minus 120587
119868(119905) minus 120587
119878(119905) is the
proposition of the risk-free money market account Denote120587(119905) = (120587
119868(119905) 120587
119878(119905)) 119905 isin [0 119879] as the decision-making
process Now we rewrite the SDEs (2) and (4) as matrix form
where 1 = (1 1)1015840 The last equation holds because of (7) and
120583 (119905) minus 119877 (119905) 1 = [119903 (119905) minus 119877 (119905) + 120583
119901(119905)
120582119904119881 (119905) + 120582
119901(119905) 120590
119904
]
= [
120582119901(119905) 120590
119901
120582119904119881 (119905) + 120582
119901(119905) 120590
119904
] = Σ (119905) 120579 (119905)
(10)
where 120579(119905) = (120582119901(119905) 120582
119904radic119881(119905))
1015840 Let 1199090be the nominal wealth
at time 0Since the real value of the wealth reflects the real purchase
power of the current market the nominal value of the wealthshould be converted into the real value By discounting thevalue of nominal wealth with the price level process thereal value of wealth will be obtained Now denote 119883(119905) =119883(119905)119875(119905) and 119871(119905) = 119871(119905)119875(119905) and by Itorsquos formula we
obtain the real wealth process 119883120587(119905) and real salary income
process 119871(119905) as follows119889119883
120587(119905) = 119883
120587(119905)
sdot [119877 (119905) minus 120583119901(119905) + 120590
2
119901+ 120587 (119905) Σ (119905) (120579 (119905) minus 120588
initial salary income are 119883120587(0) = 119909
0119901
0and 119871(0) = 119897
0119901
0
respectively Note that the dynamic process of119883120587(119905) depends
on the real salary income process 119871(119905) and the stochasticvolatility process 119881(119905) but it is independent of the price levelprocess 119875(119905)
Definition 4 (admissible strategy) LetO = RtimesR+timesR+ and
G = [0 119879] times O A strategy 120587(119905) = 120587119868(119905) 120587
119878(119905)
119905isin[0119879]is said
to be admissible if(1) forall(119909 V 119897) isin O the SDE (11) has unique solution119883
120587(119904)
119904isin[119905119879]with119883120587
(119905) = 119909 119881(119905) = V and 119871(119905) = 119897
(2) forall119904 isin [119905 119879] E[int119879
119905[(120587
119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904]4119889119904] lt infin
(3) forall984858 isin [1 +infin) and forall(119905 119909 V 119897) isin GE
One denotes Π(119905 119909 V 119897) as the set of all admissiblestrategies with respect to the initial condition (119905 119909 V 119897) isin G
(II) After Retirement When the member of the pension planretires the accumulation value of the pension fund usually isused to purchase an annuity In practice the member usuallyhas two ways to purchase the annuity The first way is thatheshe purchases the annuity directly at the retirement timeFor example as being considered in [2 27] the accumulatedpension fund is used to purchase a paid-up annuity oncethe member arrives at the retirement age and the managerhas also to decide the part of the remaining mathematicalreserve to invest in the market The second way is that themember does not purchase the annuity directly but choosesthe income drawdown option at the retirement It means thatthe member withdraws a fixed income and invests the restof the wealth until heshe achieves the age when the time ofpurchasing the annuity is compulsory The interested readercan see [33 34] for details
Here we consider the first way Similar to [2] we assumethat the member purchases a paid-up annuity at the retire-ment time which guarantees the benefit to be given only on afixed time horizon [119879 119879+119873] and invests the rest of the wealthcontinuously in the market We denote119882 as the part of thefund used to purchase an annuity (119882 ⩽ 119883(119879)) Continuousbenefit (real value) paid from 119879 to 119879
1= 119879 + 119873 is 119902 = 119882119886
119873|
where 119886119873|= (1 minus 119890
minus120575119873)120575 and 120575 is the continuous technical
rate
Mathematical Problems in Engineering 5
Let119872(119905) denote the nominal total payment value of thebenefit at time interval [119879 119905] 119905 isin (119879 119879
1] For simplicity the
dynamics of119872(119905) can be modeled as119872(119905) = int119905
119879119902119875(119904)119889119904 or
119889119872(119905) = 119902119875(119905)119889119905 This setting is reasonable and realistic anda similar setting is used in [17] It means that after retirementthe nominal benefit is adjusted dynamically based on theinflation rate of that time that is at time 119905 the nominalbenefit is 119902119875(119905) which keeps the purchasing power level ofthe retirees not decreasing We assume that once the retiredmember dies during [119879 119879
1] the pension fund continues to be
invested and the remaining annuity benefit and the terminalwealth119883(119879
1) are paid off to hisher offspring If themember is
alive until 1198791 the wealth119883(119879
1) is left for hisher later life Let
119868(119905) and
119878(119905) represent the propositions of the pension fund
wealth invested in the inflation-linked index bond and thestock respectively Denote (119905) = (
119868(119905)
119878(119905)) 119905 isin [119879 119879
1]
as the decision-making process Then after retirement thedynamics of nominal wealth process is
119905119909V[sdot] isthe condition expectation given 119883
(119905) = 119909 and
119881(119905) = V
One denotes Π(119905 119909 V) as the set of all admissible strategieswith respect to the initial condition (119905 119909 V) isin G
1
3 Problem Formulation before Retirementand Verification Theorem
In this section we assume that the pension plan managerhopes to maximize the expectation of the total wealth atretirement time119879 meanwhileminimizing the variance of thewealth at retirement So the pension plan manager faces a
continuous-time MV portfolio problem However since theMV objective function does not satisfy the iterated expecta-tion property Bellmanrsquos optimality principle does not holdunder the continuous-time framework Hence this problemis time-inconsistent This means that for the pension fundmanager without precommitment should take into accountthat heshemay have different objective functions in differenttimes We can thus view this problem as a noncooperativegame At each time 119905 there is one player named player 119905representing the future incarnation of the manager at time119905 At any state (119905 119909 V 119897) the manager faces an optimal controlproblem with value function 119881(119905 119909 V 119897) as follows
119869 (119905 119909 V 119897 120587) = E119905119909V119897 [119883
120587(119879)]
minus120574
2Var
119905119909V119897 [119883120587(119879)]
119881 (119905 119909 V 119897) = sup120587isinΠ(119905119909V119897)
119869 (119905 119909 V 119897 120587)
(P1)
where 120574 represents the risk aversion level of the managerand Var
119905119909V119897[sdot] refers to the conditional variance Since theobjective of the pensionmanager updates with different statethemanagerrsquos decision process is like a game process betweenan infinite number of distinct players Thus we need to lookfor a subgame perfect Nash equilibrium point for the gameand determine an equilibrium strategy 120587lowast
If we denote 119910120587(119905 119909 V 119897) = E
119905119909V119897[119883120587(119879)] 119911120587(119905 119909 V 119897) =
E119905119909V119897[(119883
120587(119879))
2] then the value function 119881(119905 119909 V 119897) can be
rewritten as119881 (119905 119909 V 119897)
= sup120587isinΠ(119905119909V119897)
119891 (119905 119909 V 119897 119910120587(119905 119909 V 119897) 119911120587 (119905 119909 V 119897)) (15)
where 119891(119905 119909 V 119897 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
31 Verification Theorem Based on the idea of the Nashequilibrium similarly to [11] we first give the definition ofequilibrium strategy
Definition 6 For any given initial state (119905 119909 V 119897) isin Gconsider an admissible strategy 120587lowast
(119905 119909 V 119897) and choose threereal numbers 120591 gt 0
119868 and
119861 one defines the following
strategy
120587120591(119904 V )
=
(119868
119861) 119905 ⩽ 119904 lt 119905 + 120591 ( V ) isin O
120587lowast(119904 V ) 119905 + 120591 ⩽ 119904 lt 119879 ( V ) isin O
(16)
If lim120591rarr0
inf((119891120587lowast
(119905) minus 119891120587120591(119905))120591) ⩾ 0 forall(
119868
119861) isin R times R
then 120587lowast(119905 119909 V 119897) is called an equilibrium strategy and the
equilibrium value function is defined by
119881 (119905 119909 V 119897)
= 119891 (119905 119909 V 119897 119910120587lowast
(119905 119909 V 119897) 119911120587lowast
(119905 119909 V 119897)) (17)
where 119891120587(119905) = 119891(119905 119909 V 119897 119910120587
(119905 119909 V 119897) 119911120587(119905 119909 V 119897)) for short
6 Mathematical Problems in Engineering
For any 120601(119905 119909 V 119897) isin 1198621222(G) we define a set 119863
119901(G) =
120601(119905 119909 V 119897) | 120601(119905 119909 V 119897) isin 1198621222
(G) and all first-orderpartial derivatives of 120601(sdot 119909 V 119897) satisfy the polynomial growthcondition on O and we define a variational operator asfollows (in the following we will omit the variable 119905 fornotation convenience)
A120587120601 (119905 119909 V 119897)
= 120601119905+ 120601
119909[119896119897 + 119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901))]
+ 120601V120572 (120575 minus V) + 120601119897120572119897119897
+1
21206011199091199091199092(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
+1
2120601VV120590
2
V V
+1
212060111989711989711989721205731015840
119897120573119897+ 120601
119909V119909120590VradicV (120587Σ minus 1205881015840
119901) 120573V
+ 120601119909119897119909119897 (120587Σ minus 120588
1015840
119901) 120573
119897+ 120601V119897119897120590VradicV120573
1015840
119897120573V
(18)
where 120573V = (0 120588119904V)1015840 Now we give the verification theorem of
problem (P1)
Theorem 7 (verification theorem) For problem (P1) if there
exist three real value functions 119865(119905 119909 V 119897) 119866(119905 119909 V 119897) and119867(119905 119909 V 119897) isin 119863
119901(G) satisfying the following conditions
forall(119905 119909 V 119897) isin G
sup120587isinΠ(119905119909V119897)
A120587119865 (119905 119909 V 119897) minus 120585120587 (119905 119909 V 119897) = 0 (19)
119865 (119879 119909 V 119897) = 119891 (119879 119909 V 119897 119909 1199092) (20)
A120587lowast
119866 (119905 119909 V 119897) = 0
119866 (119879 119909 V 119897) = 119909(21)
A120587lowast
119867(119905 119909 V 119897) = 0
119867 (119879 119909 V 119897) = 1199092
(22)
where 120587lowast= arg sup
120587isinΠ(119905119909V119897)A120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897)
then 119881(119905 119909 V 119897) = 119865(119905 119909 V 119897) 119910120587lowast
(119905 119909 V 119897) = 119866(119905 119909 V 119897) and119911120587lowast
(119905 119909 V 119897) = 119867(119905 119909 V 119897) and 120587lowast is the equilibrium strategywhere
120585120587(119905 119909 V 119897)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901)) + 119896119897] 119891
119909
+ [120572 (120575 minus V)] 119891V + 120572119897119897119891119897
+1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092119880
120587
1+1
21205902
V V119880120587
2
+1
211989721205731015840
119897120573119897119880
120587
3+ 120590VradicV119909 (120587Σ minus 120588
1015840
119901) 120573V119880
120587
4
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897119880
120587
5+ 120590VradicV119897120573
1015840
119897120573V119880
120587
6
119880120587
1
= 119891119909119909+ 119891
119910119910(119910
120587
119909)2
+ 119891119911119911(119911
120587
119909)2
+ 2119891119909119910119910120587
119909+ 2119891
119909119911119911120587
119909
+ 2119891119910119911119910120587
119909119911120587
119909
119880120587
2
= 119891VV + 119891119910119910 (119910120587
V )2
+ 119891119911119911(119911
120587
V )2
+ 2119891V119910119910120587
V + 2119891V119911119911120587
V
+ 2119891119910119911119910120587
V 119911120587
V
119880120587
3
= 119891119897119897+ 119891
119910119910(119910
120587
119897)2
+ 119891119911119911(119911
120587
119897)2
+ 2119891119897119910119910120587
119897+ 2119891
119897119911119911120587
119897
+ 2119891119910119911119910120587
119897119911120587
119897
119880120587
4
= 119891119910119910119910120587
119909119910120587
V + 119891119911119911119911120587
119909119911120587
V + 119891119909V + 119891119909119910119910120587
V + 119891119909119911119911120587
V
+ 119891V119910119910120587
119909+ 119891V119911119911
120587
119909+ 119891
119910119911(119910
120587
119909119911120587
V + 119910120587
V 119911120587
119909)
119880120587
5
= 119891119910119910119910120587
119909119910120587
119897+ 119891
119911119911119911120587
119909119911120587
119897+ 119891
119909119897+ 119891
119909119910119910120587
119897+ 119891
119909119911119911120587
119897
+ 119891119897119910119910120587
119909+ 119891
119897119911119911120587
119909+ 119891
119910119911(119910
120587
119909119911120587
119897+ 119910
120587
119897119911120587
119909)
119880120587
6
= 119891119910119910119910120587
V 119910120587
119897+ 119891
119911119911119911120587
V 119911120587
119897+ 119891V119897 + 119891V119910119910
120587
119897+ 119891V119911119911
120587
119897
+ 119891119897119910119910120587
V + 119891119897119911119911120587
V + 119891119910119911 (119910120587
119897119911120587
V + 119910120587
V 119911120587
119897)
(23)
and 119891 = 119891(119905 119909 V 119897 119910 119911) 119910120587= 119910
120587(119905 119909 V 119897) and 119911120587 =
119911120587(119905 119909 V 119897)
The proof of this theorem is given in the AppendixEquations (19) (21) and (22) are called an extended
HJB equations system By solving this equations systemwe can obtain the equilibrium strategy and correspondingequilibrium value function
32 Solution to the Equilibrium Strategy Since 119891(119905 119909 V 119897 119910119911) = 119910 minus (1205742)(119911 minus 119910
2) according to Theorem 7 we have
119880120587lowast
1= 120574119866
2
119909 119880120587
lowast
2= 120574119866
2
V 119880120587lowast
3= 120574119866
2
119897 119880120587
lowast
4= 120574119866
119909119866V 119880
120587lowast
5=
120574119866119909119866
119897 and 119880120587
lowast
6= 120574119866V119866119897
so
120585120587lowast
(119905 119909 V 119897) =1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092120574119866
2
119909
+1
21205902
V V1205741198662
V +1
211989721205731015840
119897120573119897120574119866
2
119897
+ 120590VradicV1198971205731015840
119897120573V120574119866V119866119897
+ 120590VradicV119909 (120587Σ minus 1205881015840
119901) 120573V120574119866V119866119909
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897120574119866
119909119866
119897
(24)
Mathematical Problems in Engineering 7
By differentiatingA120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897) with respect to
120587(119905) we can obtain the equilibrium strategy
120587lowast(119905) = minus
[119865119909(120579 minus 120588
119901)1015840
+ (119865119909V minus 120574119866119909
119866V) 120590VradicV1205731015840
V + (119865119909119897 minus 120574119866119909119866
119897) 119897120573
1015840
119897] Σ
minus1
(119865119909119909minus 120574119866
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(25)
Inspired by [24] we suppose 119865(119905 119909 V 119897) and 119866(119905 119909 V 119897) havethe following linear forms with respect to 119909 V and 119897
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int119879
119905119903(119904)119889119904
+119861 (119905)
120574V + 119862 (119905) 119897 +
119863 (119905)
120574 (40)
where 119861(119905) 119862(119905) and 119863(119905) are given by (34) (35) and (36)
Note that
E119905119909V119897 [119883
120587lowast
(119879)]
= 119909119890int119879
119905119903(119904)119889119904
+1
120574[119861 (119905) V + 119863 (119905)] + 119862 (119905) 119897
(41)
Var119905119909V119897 [119883
120587lowast
(119879)] = 119867 (119905 119909 V 119897) minus 1198662(119905 119909 V 119897)
=2
120574[119866 (119905 119909 V 119897) minus 119865 (119905 119909 V 119897)]
=2
1205742[(119861 (119905) minus 119861 (119905)) V + 119863 (119905) minus 119863 (119905)]
(42)
Thus we obtain the following so-called equilibrium efficientfrontier which reflects the relationship between the invest-ment risk and return under the equilibrium strategy
Corollary 9 For MV problem (P1) based on state (119905 119909 V 119897)
its equilibrium efficient frontier is
E119905119909V119897 [119883
120587lowast
(119879)] = 119909119890int119879
119905119903(119904)119889119904
+ 119862 (119905) 119897
+ 119873 (119905 V 119897) radicVar119905119909V119897 [119883
120587lowast
(119879)]
(43)
where 119873(119905 V 119897) = [119861(119905)V + 119863(119905)]
radic2[(119861(119905) minus 119861(119905))V + 119863(119905) minus 119863(119905)]
Remark 10 By (38) and (39) we find that the equilibriuminvestment proportion in the stock has the similar expressionas in [24] although their study focuses on the optimal invest-ment and reinsurance problem In addition the equilibriuminvestment money in the stock is only the function of time119905 which is independent of the current wealth However forthe inflation-linked bond the equilibrium investmentmoneydepends on current wealth and the more the wealth isthe more the inflation-linked bond is invested This resultis consistent with the economic intuition considering thatwhen the manager has more wealth the sensitivity to theinflation risk becomes higher and the corresponding hedgingdemand is improved
Remark 11 In view of (38) we find that if 120590119901gt 120590
119897 that is the
volatility rate of the inflation is larger than that of the salaryincome the equilibrium investment amount in the inflation-linked bond has positive relationship with the salary levelOtherwise it has negative relationship with the salary level
Remark 12 From (42) and (43) if Var119905119909V119897[119883
120587(119879)] = 0 it
means that the pension fund manager bears no risk at allunder the state (119905 119909 V 119897) this is equivalent to the case of120574 rarr infin that is the pension fund manager is fully averseto risk In this case 120587lowast
119878(119905) = 0 and 120587lowast
119868(119905) = 1 minus ((120590
119897minus
120590119901)119909120590
119901)119897119862(119905)119890
minusint119879
119905119903(119904)119889119904 This indicates that the manager will
only invest all the fund in the inflation-linked bond and therisk-free money market account and obtain the expected realterminal wealth of 119909119890int
119879
119905119903(119904)119889119904
+ 119862(119905)119897 at retirement time 119879 Itconsists of two parts The first part is the accumulation ofthe initial wealth 119909 and the second part can be seen as theaccumulation of the contribution
Mathematical Problems in Engineering 9
Remark 13 If we do not consider the risk of stochasticvolatility of the stock price that is let 120572 = 0 120590V = 0 thenthe equilibrium strategy is simplified as
120587lowast
119868(119905)
= [
120582119901(119905) minus 120590
119901minus 120582
119904120590119904
120574minus 119897 (120590
119897minus 120590
119901) 119862 (119905)]
119890minusint119879
119905119903(119904)119889119904
119909120590119901
+ 1
120587lowast
119878(119905) =
120582119904119890minusint119879
119905119903(119904)119889119904
120574119909
(44)
In this case we find that 120587lowast
119878(119905) is consistent with the
result of the ordinaryMVportfolio problem under the Black-Scholes market (see [11]) This indicates that the contributionof the salary income has no influence on the equilibriuminvestment amount of the stock Furthermore if we do notconsider the risk of inflation that is 120583
119901(119905) = 120590
119901= 120590
119904= 0
then 119877(119905) = 119903(119905) and the inflation-linked bond is degeneratedto the risk-free money market account
4 Problem Formulation after Retirement andVerification Theorem
After retirement we assume that the pension plan memberpurchases a paid-up annuity during the time horizon [119879 119879 +119873] Thus the objective function of the pension plan manageris to maximize the expected surplus after paying off thebenefit to the member and minimize the variance of thesurplus at time 119879
1= 119879 + 119873 So at the state (119905 119909 V) isin G
1
the manager faces the following optimal control problem
(119905 119909 V) = supisinΠ(119905119909V)
E119905119909V [119883
(119879
1)]
minus120574
2Var
119905119909V [119883(119879
1)]
(P2)
Similarly to the method of Section 3 we denote 119910(119905 119909 V) =
E119905119909V[119883
(119879
1)] 119911(119905 119909 V) = E
119905119909V[(119883(119879
1))
2] Then we can
rewrite
(119905 119909 V)
= supisinΠ(119905119909V)
(119905 119909 V 119910(119905 119909 V) 119911 (119905 119909 V))
(45)
where (119905 119909 V 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
41 VerificationTheorem In this subsection we first give thedefinition of equilibrium strategy after retirement and thengive the verification theoremwhich is satisfied by the equilib-rium value function Now for any 120601(119905 119909 V) isin 119862122
(119905 119909 V) = (119905 119909 V) and 119911lowast
(119905 119909
V) = (119905 119909 V) and lowast is the equilibrium strategy where
120593(119905 119909 V)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ Σ (120579 minus 120588
119901)) minus 119902] 119891
119909
+ 119891V120572 (120575 minus V) +1
2(Σ minus 120588
1015840
119901) (Σ minus 120588
1015840
119901)1015840
1199092
1
+1
21205902
V V
2+ 120590VradicV119909 (
1015840Σ minus 120588
1015840
119901) 120573V
3
10 Mathematical Problems in Engineering
1
= 119891119909119909+ 119891
119910119910(119910
119909)2
+ 119891119911119911(119911
119909)2
+ 2119891119909119910119910
119909+ 2119891
119909119911119911
119909
+ 2119891119910119911119910
119909119911
119909
2
= 119891VV + 119891119910119910 (119910
V )2
+ 119891119911119911(119911
V )2
+ 2119891V119910119910
V + 2119891V119911119911
V
+ 2119891119910119911119910
V 119911
V
3
= 119891119910119910119910
119909119910
V + 119891119911119911119911
119909119911
V + 119891119909V + 119891119909119910119910
V + 119891119909119911119911
V
+ 119891V119910119910
119909+ 119891V119911119911
119909+ 119891
119910119911(119910
119909119911
V + 119910
V 119911
119909)
(49)
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
loss In recent years some scholars pay attention to theportfolio selection problem of the DC pension plan underthe mean-variance (MV) criteria This is because of thefact that the optimal investment problems under the MVcriteria in a multiperiod or continuous-time framework aresuccessfully solved only recently It is well known that theMV problems with multiperiod or continuous-time versionare time-inconsistent in the sense that the Bellman optimalityprinciple does not hold Hence the dynamic programmingapproach can not be used In existing literature there areusually two methods that are suggested to deal with thisproblem The first one is to find the precommitment strategy(the breakthrough work for this method was made by [8 9])which means that the manager derives an optimal strategyat the initial time 0 and commits to performing this strategyin the future even if it does not remain optimal in thelater time However in practice this strategy is not easilyperformed since the preference of the MV manager changeswith time and heshe has an incentive to deviate from theprecommitment strategy in the later time In addition findinga time-consistent strategy is a basic requirement for a rationaldecision-maker Thus the second method is finding a time-consistent equilibrium strategy under the framework of gametheory (the most representative work was made by [10ndash12])In this case we regard the decision-making process as anoncooperative game between an infinite number of distinctplayers At each time 119905 we named player 119905 representing thefuture incarnation of the manager at time 119905 We are interestedin finding a subgame perfect Nash equilibrium point forthe game and formulating an equilibrium strategy thisstrategy is time-consistent For the MV problem under thebackground of pension fund [13] considered a DC pensionplan with the return of premiums clauses under the gametheoretic framework and obtained an equilibrium investmentstrategy in the period before retirement Under the sameframework [14] investigated an equilibrium investment andcontribution strategies for a DB pension plan The researchon the precommitment investment strategy can be found in[15ndash17] and references therein
Meanwhile for the portfolio problem many scholarsbegin to pay attention to the influence of the background risksuch as the risk of interest rate inflation and volatility of therisky asset on the optimal decision-making process On theone hand as we all know the inflation might affect the realvalue of the wealth especially for the problem with long timehorizon investment the higher inflation leads to the lowerreal wealth In theDCpension plan [18] obtained the optimalinvestment strategy which maximizes the expected utility ofthe DC pension plan under inflation risk A similar problemwas studied in [5] under the environment of stochasticinterest and inflation rate with the minimized guaranteeUnder the MV criteria [17] investigated the optimal pre-commitment investment strategy of a DC pension plan withinflation risk using Lagrange method Recently the equilib-rium investment strategies for a DC pension plan with theinflation risk are obtained by [19] On the other hand manyempirical studies have shown that the volatility of the riskyasset is stochastic Some scholars have proposed a variety ofSVmodels such as the constant elastic variance (CEV)model
[20] Hestonrsquos SV model (volatility satisfies Cox-Ingersoll-Ross process) [21] and Stein-Stein model (volatility satisfiesOrnstein-Uhlenbeck process) [22] The CEV and HestonrsquosSV models have been widely considered in investment andreinsurance problems such as [23ndash25] and references thereinFor the stochastic volatility model under the DC pensionplan the interested reader can be referred to [6 26 27]
In this paper we consider the MV portfolio problems foraDCpension plan both before and after retirementThemaindifference between this paper and the existing literature isthat both the inflation risk and the stochastic volatility riskare considered in our model To the best of our knowledgeunder the MV framework there is no literature consideringboth of the two risks in the DC pension fund managementWe assume that the financial market consists of a risk-freeasset an inflation-linked bond and a stock with Hestonrsquosstochastic volatility The salary income of the DC pensionplan member is also assumed to be stochastic because ofthe influence of inflation rate Under the framework of gametheory two MV problems are formulated related to theperiods before and after retirement respectively For each ofthe problems by solving an extended HJB equations systemthe equilibrium strategy equilibrium value function and thecorresponding equilibrium efficient frontier are obtainedWefind that the equilibrium investment money in the inflation-linked bond depends on current wealth the inflation riskand the contribution of the salary income have no influenceon the equilibrium investment money of stock Finally usingMonte Carlo method we investigate the evolution process ofthe equilibrium strategy with time before retirement underdifferent parameters and present the sensitivity of the efficientfrontier to corresponding parameters
The remainder of the paper is organized as follows InSection 2 we introduce the financial market and wealthprocesses both before and after retirement In Section 3 aMV portfolio allocation problem is formulated in the periodbefore retirement Under the framework of game theoryan extended HJB equations system and an equilibriuminvestment strategy are also obtained In Section 4 weconsider a MV portfolio problem after retirement and derivethe corresponding verification theorem and the equilibriumstrategy Some numerical simulations and sensitivity analysisfor our results are presented in Section 5 Section 6 concludesthe paper and outlines further research
2 Assumption and Model
Let (ΩF F P) be a filtered probability space with filtrationF = F
119905119905isin[0119879+119873]
satisfying the usual conditions that isF
119905119905isin[0119879+119873]
is right-continuous and P-complete The timehorizon [0 119879] represents the accumulation period of a DCpension plan member and [119879 119879 + 119873] is the distributionperiod of the member after heshe retires Let F
119905represent
the information available until time 119905 Suppose that all ofstochastic processes and random variables are defined on thefiltered probability space (ΩF F P) In addition we assumethat there are no transaction costs or taxes in the financialmarket trading can take place continuously and short sellingis permitted
Mathematical Problems in Engineering 3
21 Financial Market As we all know for a DC pensionplan manager the objective is making the terminal wealthmaximized by investing the pension fund into the marketGenerally speaking the investment time horizon of thepension fund lasts decades The inflation risk as a kind ofimportant background risk has important influence on thereal value of the pension fund In economics CPI (ConsumerPrice Index) as the typical index can represent the inflationlevel of the market Following [28 29] we assume that theprice level 119875(119905) satisfies the following diffusion process
119889119875 (119905)
119875 (119905)= 120583
119901(119905) 119889119905 + 120590
119901119889119882
119901(119905) 119875 (0) = 119901
0 (1)
where 120583119901(119905) represents the instantaneous expected inflation
rate 120590119901gt 0 is the instantaneous volatility of inflation rate
and 119882119901(119905) is a standard Brownian motion which generates
uncertainty of the price levelWe assume that the financial market consists of three
kinds of assets an inflation-linked asset a money marketaccount and a stock
(1) The inflation-linked index bond has the same risksource as the price level process 119875(119905) and can be freely tradedin themarket Following [18 29] its price process 119868(119905) satisfiesthe following stochastic differential equation (SDE)
where 119882119904(119905) and 119882V(119905) are two standard Brownian motions
and120582119904radic119881(119905) and120582
119901(119905) are themarket price of the risk sources
119882119904(119905) and 119882
119901(119905) respectively Since the inflation risk might
have influence on the price evolution process of the stockthrough direct or indirect ways (the correlated empiricalresearch can be found in [30 31] and references therein)hence we assume that the price process of the stock is derivedby not only its own risk source 119882
119904(119905) but also the risk
source119882119901(119905) We further assume that119882
119901(119905) is independent
of 119882119904(119905) and 119882V(119905) respectively while 119882119904
(119905) and 119882V(119905) aredependent and E[119882
119904(119905)119882V(119905)] = 120588119904V119905 where 120588119904V isin [minus1 1] is
the correlation coefficientThe second equation of (4) is a CIR(Cox-Ingersoll-Ross) mean-revert process whichmodels thestochastic volatility of the stock price Here we suppose 2120572120575 gt1205902
V to assure that 119881(119905) gt 0 holdsIn addition we assume that theDCpension planmember
receives salary income with nominal value 119871(119905) at time 119905 untilthe retirement time 119879 Suppose that the income is stochasticand dynamically influenced by the price level 119875(119905) that is thesalary income process is driven by the source of uncertaintyfrom inflation and it satisfies the following process
119889119871 (119905)
119871 (119905)
= 120583119897119889119905 + 120590
119897119889119882
119901(119905)
119871 (0) = 1198970gt 0
(5)
where 120583119897is the average growth rate of the income and 120590
119897is the
volatility rate
Remark 1 To simplify the model achieve tractability andgive detailed analysis about the optimal strategies we assumethat the real interest rate 119903(119905) nominal interest rate 119877(119905) andexpected inflation rate 120583
119901(119905) are all deterministic functions of
time 119905
Remark 2 If we denote Q as the risk neutral measure then(2) can be rewritten as
119889119868 (119905)
119868 (119905)= (119903 (119905) + 120583
119901(119905) minus 120582
119901(119905) 120590
119901) 119889119905 + 120590
119901119889119882
Q119901(119905) (6)
By the pricing theory of the derivative (to avoid arbitrage) weobtain the following relationship
119903 (119905) + 120583119901(119905) minus 120582
119901(119905) 120590
119901= 119877 (119905) (7)
Remark 3 In the empirical research the evolution processof 119878(119905) and 119881(119905) is usually negatively correlated Since ingeneral with declining of the stock price the volatility of theprice gradually increases (cf [32]) based on this reason inSection 5 we assume that the parameter 120588
119904V lt 0
22 Wealth Process In this paper we consider the optimalportfolio problems of aDCpension planmember both beforeand after retirement The member contributes part of hishersalary into the pension fund account before retirement andobtains benefit after retirement So we need to consider thewealth processes in two different periods
(I) Before Retirement During the accumulation period [0 119879]we assume that the DC pension plan member contributes
4 Mathematical Problems in Engineering
continuously to hisher pension fund account with a fixedrate 119896 of hisher salary income that is heshe contributes theamount of money 119896119871(119905) to hisher pension fund account attime 119905 The pension fund is invested in the financial marketby the manager Denote 120587
119868(119905) and 120587
119878(119905) as the propositions
of wealth invested in the inflation-linked index bond and thestock respectively Then 120587
119861(119905) = 1 minus 120587
119868(119905) minus 120587
119878(119905) is the
proposition of the risk-free money market account Denote120587(119905) = (120587
119868(119905) 120587
119878(119905)) 119905 isin [0 119879] as the decision-making
process Now we rewrite the SDEs (2) and (4) as matrix form
where 1 = (1 1)1015840 The last equation holds because of (7) and
120583 (119905) minus 119877 (119905) 1 = [119903 (119905) minus 119877 (119905) + 120583
119901(119905)
120582119904119881 (119905) + 120582
119901(119905) 120590
119904
]
= [
120582119901(119905) 120590
119901
120582119904119881 (119905) + 120582
119901(119905) 120590
119904
] = Σ (119905) 120579 (119905)
(10)
where 120579(119905) = (120582119901(119905) 120582
119904radic119881(119905))
1015840 Let 1199090be the nominal wealth
at time 0Since the real value of the wealth reflects the real purchase
power of the current market the nominal value of the wealthshould be converted into the real value By discounting thevalue of nominal wealth with the price level process thereal value of wealth will be obtained Now denote 119883(119905) =119883(119905)119875(119905) and 119871(119905) = 119871(119905)119875(119905) and by Itorsquos formula we
obtain the real wealth process 119883120587(119905) and real salary income
process 119871(119905) as follows119889119883
120587(119905) = 119883
120587(119905)
sdot [119877 (119905) minus 120583119901(119905) + 120590
2
119901+ 120587 (119905) Σ (119905) (120579 (119905) minus 120588
initial salary income are 119883120587(0) = 119909
0119901
0and 119871(0) = 119897
0119901
0
respectively Note that the dynamic process of119883120587(119905) depends
on the real salary income process 119871(119905) and the stochasticvolatility process 119881(119905) but it is independent of the price levelprocess 119875(119905)
Definition 4 (admissible strategy) LetO = RtimesR+timesR+ and
G = [0 119879] times O A strategy 120587(119905) = 120587119868(119905) 120587
119878(119905)
119905isin[0119879]is said
to be admissible if(1) forall(119909 V 119897) isin O the SDE (11) has unique solution119883
120587(119904)
119904isin[119905119879]with119883120587
(119905) = 119909 119881(119905) = V and 119871(119905) = 119897
(2) forall119904 isin [119905 119879] E[int119879
119905[(120587
119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904]4119889119904] lt infin
(3) forall984858 isin [1 +infin) and forall(119905 119909 V 119897) isin GE
One denotes Π(119905 119909 V 119897) as the set of all admissiblestrategies with respect to the initial condition (119905 119909 V 119897) isin G
(II) After Retirement When the member of the pension planretires the accumulation value of the pension fund usually isused to purchase an annuity In practice the member usuallyhas two ways to purchase the annuity The first way is thatheshe purchases the annuity directly at the retirement timeFor example as being considered in [2 27] the accumulatedpension fund is used to purchase a paid-up annuity oncethe member arrives at the retirement age and the managerhas also to decide the part of the remaining mathematicalreserve to invest in the market The second way is that themember does not purchase the annuity directly but choosesthe income drawdown option at the retirement It means thatthe member withdraws a fixed income and invests the restof the wealth until heshe achieves the age when the time ofpurchasing the annuity is compulsory The interested readercan see [33 34] for details
Here we consider the first way Similar to [2] we assumethat the member purchases a paid-up annuity at the retire-ment time which guarantees the benefit to be given only on afixed time horizon [119879 119879+119873] and invests the rest of the wealthcontinuously in the market We denote119882 as the part of thefund used to purchase an annuity (119882 ⩽ 119883(119879)) Continuousbenefit (real value) paid from 119879 to 119879
1= 119879 + 119873 is 119902 = 119882119886
119873|
where 119886119873|= (1 minus 119890
minus120575119873)120575 and 120575 is the continuous technical
rate
Mathematical Problems in Engineering 5
Let119872(119905) denote the nominal total payment value of thebenefit at time interval [119879 119905] 119905 isin (119879 119879
1] For simplicity the
dynamics of119872(119905) can be modeled as119872(119905) = int119905
119879119902119875(119904)119889119904 or
119889119872(119905) = 119902119875(119905)119889119905 This setting is reasonable and realistic anda similar setting is used in [17] It means that after retirementthe nominal benefit is adjusted dynamically based on theinflation rate of that time that is at time 119905 the nominalbenefit is 119902119875(119905) which keeps the purchasing power level ofthe retirees not decreasing We assume that once the retiredmember dies during [119879 119879
1] the pension fund continues to be
invested and the remaining annuity benefit and the terminalwealth119883(119879
1) are paid off to hisher offspring If themember is
alive until 1198791 the wealth119883(119879
1) is left for hisher later life Let
119868(119905) and
119878(119905) represent the propositions of the pension fund
wealth invested in the inflation-linked index bond and thestock respectively Denote (119905) = (
119868(119905)
119878(119905)) 119905 isin [119879 119879
1]
as the decision-making process Then after retirement thedynamics of nominal wealth process is
119905119909V[sdot] isthe condition expectation given 119883
(119905) = 119909 and
119881(119905) = V
One denotes Π(119905 119909 V) as the set of all admissible strategieswith respect to the initial condition (119905 119909 V) isin G
1
3 Problem Formulation before Retirementand Verification Theorem
In this section we assume that the pension plan managerhopes to maximize the expectation of the total wealth atretirement time119879 meanwhileminimizing the variance of thewealth at retirement So the pension plan manager faces a
continuous-time MV portfolio problem However since theMV objective function does not satisfy the iterated expecta-tion property Bellmanrsquos optimality principle does not holdunder the continuous-time framework Hence this problemis time-inconsistent This means that for the pension fundmanager without precommitment should take into accountthat heshemay have different objective functions in differenttimes We can thus view this problem as a noncooperativegame At each time 119905 there is one player named player 119905representing the future incarnation of the manager at time119905 At any state (119905 119909 V 119897) the manager faces an optimal controlproblem with value function 119881(119905 119909 V 119897) as follows
119869 (119905 119909 V 119897 120587) = E119905119909V119897 [119883
120587(119879)]
minus120574
2Var
119905119909V119897 [119883120587(119879)]
119881 (119905 119909 V 119897) = sup120587isinΠ(119905119909V119897)
119869 (119905 119909 V 119897 120587)
(P1)
where 120574 represents the risk aversion level of the managerand Var
119905119909V119897[sdot] refers to the conditional variance Since theobjective of the pensionmanager updates with different statethemanagerrsquos decision process is like a game process betweenan infinite number of distinct players Thus we need to lookfor a subgame perfect Nash equilibrium point for the gameand determine an equilibrium strategy 120587lowast
If we denote 119910120587(119905 119909 V 119897) = E
119905119909V119897[119883120587(119879)] 119911120587(119905 119909 V 119897) =
E119905119909V119897[(119883
120587(119879))
2] then the value function 119881(119905 119909 V 119897) can be
rewritten as119881 (119905 119909 V 119897)
= sup120587isinΠ(119905119909V119897)
119891 (119905 119909 V 119897 119910120587(119905 119909 V 119897) 119911120587 (119905 119909 V 119897)) (15)
where 119891(119905 119909 V 119897 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
31 Verification Theorem Based on the idea of the Nashequilibrium similarly to [11] we first give the definition ofequilibrium strategy
Definition 6 For any given initial state (119905 119909 V 119897) isin Gconsider an admissible strategy 120587lowast
(119905 119909 V 119897) and choose threereal numbers 120591 gt 0
119868 and
119861 one defines the following
strategy
120587120591(119904 V )
=
(119868
119861) 119905 ⩽ 119904 lt 119905 + 120591 ( V ) isin O
120587lowast(119904 V ) 119905 + 120591 ⩽ 119904 lt 119879 ( V ) isin O
(16)
If lim120591rarr0
inf((119891120587lowast
(119905) minus 119891120587120591(119905))120591) ⩾ 0 forall(
119868
119861) isin R times R
then 120587lowast(119905 119909 V 119897) is called an equilibrium strategy and the
equilibrium value function is defined by
119881 (119905 119909 V 119897)
= 119891 (119905 119909 V 119897 119910120587lowast
(119905 119909 V 119897) 119911120587lowast
(119905 119909 V 119897)) (17)
where 119891120587(119905) = 119891(119905 119909 V 119897 119910120587
(119905 119909 V 119897) 119911120587(119905 119909 V 119897)) for short
6 Mathematical Problems in Engineering
For any 120601(119905 119909 V 119897) isin 1198621222(G) we define a set 119863
119901(G) =
120601(119905 119909 V 119897) | 120601(119905 119909 V 119897) isin 1198621222
(G) and all first-orderpartial derivatives of 120601(sdot 119909 V 119897) satisfy the polynomial growthcondition on O and we define a variational operator asfollows (in the following we will omit the variable 119905 fornotation convenience)
A120587120601 (119905 119909 V 119897)
= 120601119905+ 120601
119909[119896119897 + 119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901))]
+ 120601V120572 (120575 minus V) + 120601119897120572119897119897
+1
21206011199091199091199092(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
+1
2120601VV120590
2
V V
+1
212060111989711989711989721205731015840
119897120573119897+ 120601
119909V119909120590VradicV (120587Σ minus 1205881015840
119901) 120573V
+ 120601119909119897119909119897 (120587Σ minus 120588
1015840
119901) 120573
119897+ 120601V119897119897120590VradicV120573
1015840
119897120573V
(18)
where 120573V = (0 120588119904V)1015840 Now we give the verification theorem of
problem (P1)
Theorem 7 (verification theorem) For problem (P1) if there
exist three real value functions 119865(119905 119909 V 119897) 119866(119905 119909 V 119897) and119867(119905 119909 V 119897) isin 119863
119901(G) satisfying the following conditions
forall(119905 119909 V 119897) isin G
sup120587isinΠ(119905119909V119897)
A120587119865 (119905 119909 V 119897) minus 120585120587 (119905 119909 V 119897) = 0 (19)
119865 (119879 119909 V 119897) = 119891 (119879 119909 V 119897 119909 1199092) (20)
A120587lowast
119866 (119905 119909 V 119897) = 0
119866 (119879 119909 V 119897) = 119909(21)
A120587lowast
119867(119905 119909 V 119897) = 0
119867 (119879 119909 V 119897) = 1199092
(22)
where 120587lowast= arg sup
120587isinΠ(119905119909V119897)A120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897)
then 119881(119905 119909 V 119897) = 119865(119905 119909 V 119897) 119910120587lowast
(119905 119909 V 119897) = 119866(119905 119909 V 119897) and119911120587lowast
(119905 119909 V 119897) = 119867(119905 119909 V 119897) and 120587lowast is the equilibrium strategywhere
120585120587(119905 119909 V 119897)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901)) + 119896119897] 119891
119909
+ [120572 (120575 minus V)] 119891V + 120572119897119897119891119897
+1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092119880
120587
1+1
21205902
V V119880120587
2
+1
211989721205731015840
119897120573119897119880
120587
3+ 120590VradicV119909 (120587Σ minus 120588
1015840
119901) 120573V119880
120587
4
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897119880
120587
5+ 120590VradicV119897120573
1015840
119897120573V119880
120587
6
119880120587
1
= 119891119909119909+ 119891
119910119910(119910
120587
119909)2
+ 119891119911119911(119911
120587
119909)2
+ 2119891119909119910119910120587
119909+ 2119891
119909119911119911120587
119909
+ 2119891119910119911119910120587
119909119911120587
119909
119880120587
2
= 119891VV + 119891119910119910 (119910120587
V )2
+ 119891119911119911(119911
120587
V )2
+ 2119891V119910119910120587
V + 2119891V119911119911120587
V
+ 2119891119910119911119910120587
V 119911120587
V
119880120587
3
= 119891119897119897+ 119891
119910119910(119910
120587
119897)2
+ 119891119911119911(119911
120587
119897)2
+ 2119891119897119910119910120587
119897+ 2119891
119897119911119911120587
119897
+ 2119891119910119911119910120587
119897119911120587
119897
119880120587
4
= 119891119910119910119910120587
119909119910120587
V + 119891119911119911119911120587
119909119911120587
V + 119891119909V + 119891119909119910119910120587
V + 119891119909119911119911120587
V
+ 119891V119910119910120587
119909+ 119891V119911119911
120587
119909+ 119891
119910119911(119910
120587
119909119911120587
V + 119910120587
V 119911120587
119909)
119880120587
5
= 119891119910119910119910120587
119909119910120587
119897+ 119891
119911119911119911120587
119909119911120587
119897+ 119891
119909119897+ 119891
119909119910119910120587
119897+ 119891
119909119911119911120587
119897
+ 119891119897119910119910120587
119909+ 119891
119897119911119911120587
119909+ 119891
119910119911(119910
120587
119909119911120587
119897+ 119910
120587
119897119911120587
119909)
119880120587
6
= 119891119910119910119910120587
V 119910120587
119897+ 119891
119911119911119911120587
V 119911120587
119897+ 119891V119897 + 119891V119910119910
120587
119897+ 119891V119911119911
120587
119897
+ 119891119897119910119910120587
V + 119891119897119911119911120587
V + 119891119910119911 (119910120587
119897119911120587
V + 119910120587
V 119911120587
119897)
(23)
and 119891 = 119891(119905 119909 V 119897 119910 119911) 119910120587= 119910
120587(119905 119909 V 119897) and 119911120587 =
119911120587(119905 119909 V 119897)
The proof of this theorem is given in the AppendixEquations (19) (21) and (22) are called an extended
HJB equations system By solving this equations systemwe can obtain the equilibrium strategy and correspondingequilibrium value function
32 Solution to the Equilibrium Strategy Since 119891(119905 119909 V 119897 119910119911) = 119910 minus (1205742)(119911 minus 119910
2) according to Theorem 7 we have
119880120587lowast
1= 120574119866
2
119909 119880120587
lowast
2= 120574119866
2
V 119880120587lowast
3= 120574119866
2
119897 119880120587
lowast
4= 120574119866
119909119866V 119880
120587lowast
5=
120574119866119909119866
119897 and 119880120587
lowast
6= 120574119866V119866119897
so
120585120587lowast
(119905 119909 V 119897) =1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092120574119866
2
119909
+1
21205902
V V1205741198662
V +1
211989721205731015840
119897120573119897120574119866
2
119897
+ 120590VradicV1198971205731015840
119897120573V120574119866V119866119897
+ 120590VradicV119909 (120587Σ minus 1205881015840
119901) 120573V120574119866V119866119909
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897120574119866
119909119866
119897
(24)
Mathematical Problems in Engineering 7
By differentiatingA120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897) with respect to
120587(119905) we can obtain the equilibrium strategy
120587lowast(119905) = minus
[119865119909(120579 minus 120588
119901)1015840
+ (119865119909V minus 120574119866119909
119866V) 120590VradicV1205731015840
V + (119865119909119897 minus 120574119866119909119866
119897) 119897120573
1015840
119897] Σ
minus1
(119865119909119909minus 120574119866
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(25)
Inspired by [24] we suppose 119865(119905 119909 V 119897) and 119866(119905 119909 V 119897) havethe following linear forms with respect to 119909 V and 119897
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int119879
119905119903(119904)119889119904
+119861 (119905)
120574V + 119862 (119905) 119897 +
119863 (119905)
120574 (40)
where 119861(119905) 119862(119905) and 119863(119905) are given by (34) (35) and (36)
Note that
E119905119909V119897 [119883
120587lowast
(119879)]
= 119909119890int119879
119905119903(119904)119889119904
+1
120574[119861 (119905) V + 119863 (119905)] + 119862 (119905) 119897
(41)
Var119905119909V119897 [119883
120587lowast
(119879)] = 119867 (119905 119909 V 119897) minus 1198662(119905 119909 V 119897)
=2
120574[119866 (119905 119909 V 119897) minus 119865 (119905 119909 V 119897)]
=2
1205742[(119861 (119905) minus 119861 (119905)) V + 119863 (119905) minus 119863 (119905)]
(42)
Thus we obtain the following so-called equilibrium efficientfrontier which reflects the relationship between the invest-ment risk and return under the equilibrium strategy
Corollary 9 For MV problem (P1) based on state (119905 119909 V 119897)
its equilibrium efficient frontier is
E119905119909V119897 [119883
120587lowast
(119879)] = 119909119890int119879
119905119903(119904)119889119904
+ 119862 (119905) 119897
+ 119873 (119905 V 119897) radicVar119905119909V119897 [119883
120587lowast
(119879)]
(43)
where 119873(119905 V 119897) = [119861(119905)V + 119863(119905)]
radic2[(119861(119905) minus 119861(119905))V + 119863(119905) minus 119863(119905)]
Remark 10 By (38) and (39) we find that the equilibriuminvestment proportion in the stock has the similar expressionas in [24] although their study focuses on the optimal invest-ment and reinsurance problem In addition the equilibriuminvestment money in the stock is only the function of time119905 which is independent of the current wealth However forthe inflation-linked bond the equilibrium investmentmoneydepends on current wealth and the more the wealth isthe more the inflation-linked bond is invested This resultis consistent with the economic intuition considering thatwhen the manager has more wealth the sensitivity to theinflation risk becomes higher and the corresponding hedgingdemand is improved
Remark 11 In view of (38) we find that if 120590119901gt 120590
119897 that is the
volatility rate of the inflation is larger than that of the salaryincome the equilibrium investment amount in the inflation-linked bond has positive relationship with the salary levelOtherwise it has negative relationship with the salary level
Remark 12 From (42) and (43) if Var119905119909V119897[119883
120587(119879)] = 0 it
means that the pension fund manager bears no risk at allunder the state (119905 119909 V 119897) this is equivalent to the case of120574 rarr infin that is the pension fund manager is fully averseto risk In this case 120587lowast
119878(119905) = 0 and 120587lowast
119868(119905) = 1 minus ((120590
119897minus
120590119901)119909120590
119901)119897119862(119905)119890
minusint119879
119905119903(119904)119889119904 This indicates that the manager will
only invest all the fund in the inflation-linked bond and therisk-free money market account and obtain the expected realterminal wealth of 119909119890int
119879
119905119903(119904)119889119904
+ 119862(119905)119897 at retirement time 119879 Itconsists of two parts The first part is the accumulation ofthe initial wealth 119909 and the second part can be seen as theaccumulation of the contribution
Mathematical Problems in Engineering 9
Remark 13 If we do not consider the risk of stochasticvolatility of the stock price that is let 120572 = 0 120590V = 0 thenthe equilibrium strategy is simplified as
120587lowast
119868(119905)
= [
120582119901(119905) minus 120590
119901minus 120582
119904120590119904
120574minus 119897 (120590
119897minus 120590
119901) 119862 (119905)]
119890minusint119879
119905119903(119904)119889119904
119909120590119901
+ 1
120587lowast
119878(119905) =
120582119904119890minusint119879
119905119903(119904)119889119904
120574119909
(44)
In this case we find that 120587lowast
119878(119905) is consistent with the
result of the ordinaryMVportfolio problem under the Black-Scholes market (see [11]) This indicates that the contributionof the salary income has no influence on the equilibriuminvestment amount of the stock Furthermore if we do notconsider the risk of inflation that is 120583
119901(119905) = 120590
119901= 120590
119904= 0
then 119877(119905) = 119903(119905) and the inflation-linked bond is degeneratedto the risk-free money market account
4 Problem Formulation after Retirement andVerification Theorem
After retirement we assume that the pension plan memberpurchases a paid-up annuity during the time horizon [119879 119879 +119873] Thus the objective function of the pension plan manageris to maximize the expected surplus after paying off thebenefit to the member and minimize the variance of thesurplus at time 119879
1= 119879 + 119873 So at the state (119905 119909 V) isin G
1
the manager faces the following optimal control problem
(119905 119909 V) = supisinΠ(119905119909V)
E119905119909V [119883
(119879
1)]
minus120574
2Var
119905119909V [119883(119879
1)]
(P2)
Similarly to the method of Section 3 we denote 119910(119905 119909 V) =
E119905119909V[119883
(119879
1)] 119911(119905 119909 V) = E
119905119909V[(119883(119879
1))
2] Then we can
rewrite
(119905 119909 V)
= supisinΠ(119905119909V)
(119905 119909 V 119910(119905 119909 V) 119911 (119905 119909 V))
(45)
where (119905 119909 V 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
41 VerificationTheorem In this subsection we first give thedefinition of equilibrium strategy after retirement and thengive the verification theoremwhich is satisfied by the equilib-rium value function Now for any 120601(119905 119909 V) isin 119862122
(119905 119909 V) = (119905 119909 V) and 119911lowast
(119905 119909
V) = (119905 119909 V) and lowast is the equilibrium strategy where
120593(119905 119909 V)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ Σ (120579 minus 120588
119901)) minus 119902] 119891
119909
+ 119891V120572 (120575 minus V) +1
2(Σ minus 120588
1015840
119901) (Σ minus 120588
1015840
119901)1015840
1199092
1
+1
21205902
V V
2+ 120590VradicV119909 (
1015840Σ minus 120588
1015840
119901) 120573V
3
10 Mathematical Problems in Engineering
1
= 119891119909119909+ 119891
119910119910(119910
119909)2
+ 119891119911119911(119911
119909)2
+ 2119891119909119910119910
119909+ 2119891
119909119911119911
119909
+ 2119891119910119911119910
119909119911
119909
2
= 119891VV + 119891119910119910 (119910
V )2
+ 119891119911119911(119911
V )2
+ 2119891V119910119910
V + 2119891V119911119911
V
+ 2119891119910119911119910
V 119911
V
3
= 119891119910119910119910
119909119910
V + 119891119911119911119911
119909119911
V + 119891119909V + 119891119909119910119910
V + 119891119909119911119911
V
+ 119891V119910119910
119909+ 119891V119911119911
119909+ 119891
119910119911(119910
119909119911
V + 119910
V 119911
119909)
(49)
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
21 Financial Market As we all know for a DC pensionplan manager the objective is making the terminal wealthmaximized by investing the pension fund into the marketGenerally speaking the investment time horizon of thepension fund lasts decades The inflation risk as a kind ofimportant background risk has important influence on thereal value of the pension fund In economics CPI (ConsumerPrice Index) as the typical index can represent the inflationlevel of the market Following [28 29] we assume that theprice level 119875(119905) satisfies the following diffusion process
119889119875 (119905)
119875 (119905)= 120583
119901(119905) 119889119905 + 120590
119901119889119882
119901(119905) 119875 (0) = 119901
0 (1)
where 120583119901(119905) represents the instantaneous expected inflation
rate 120590119901gt 0 is the instantaneous volatility of inflation rate
and 119882119901(119905) is a standard Brownian motion which generates
uncertainty of the price levelWe assume that the financial market consists of three
kinds of assets an inflation-linked asset a money marketaccount and a stock
(1) The inflation-linked index bond has the same risksource as the price level process 119875(119905) and can be freely tradedin themarket Following [18 29] its price process 119868(119905) satisfiesthe following stochastic differential equation (SDE)
where 119882119904(119905) and 119882V(119905) are two standard Brownian motions
and120582119904radic119881(119905) and120582
119901(119905) are themarket price of the risk sources
119882119904(119905) and 119882
119901(119905) respectively Since the inflation risk might
have influence on the price evolution process of the stockthrough direct or indirect ways (the correlated empiricalresearch can be found in [30 31] and references therein)hence we assume that the price process of the stock is derivedby not only its own risk source 119882
119904(119905) but also the risk
source119882119901(119905) We further assume that119882
119901(119905) is independent
of 119882119904(119905) and 119882V(119905) respectively while 119882119904
(119905) and 119882V(119905) aredependent and E[119882
119904(119905)119882V(119905)] = 120588119904V119905 where 120588119904V isin [minus1 1] is
the correlation coefficientThe second equation of (4) is a CIR(Cox-Ingersoll-Ross) mean-revert process whichmodels thestochastic volatility of the stock price Here we suppose 2120572120575 gt1205902
V to assure that 119881(119905) gt 0 holdsIn addition we assume that theDCpension planmember
receives salary income with nominal value 119871(119905) at time 119905 untilthe retirement time 119879 Suppose that the income is stochasticand dynamically influenced by the price level 119875(119905) that is thesalary income process is driven by the source of uncertaintyfrom inflation and it satisfies the following process
119889119871 (119905)
119871 (119905)
= 120583119897119889119905 + 120590
119897119889119882
119901(119905)
119871 (0) = 1198970gt 0
(5)
where 120583119897is the average growth rate of the income and 120590
119897is the
volatility rate
Remark 1 To simplify the model achieve tractability andgive detailed analysis about the optimal strategies we assumethat the real interest rate 119903(119905) nominal interest rate 119877(119905) andexpected inflation rate 120583
119901(119905) are all deterministic functions of
time 119905
Remark 2 If we denote Q as the risk neutral measure then(2) can be rewritten as
119889119868 (119905)
119868 (119905)= (119903 (119905) + 120583
119901(119905) minus 120582
119901(119905) 120590
119901) 119889119905 + 120590
119901119889119882
Q119901(119905) (6)
By the pricing theory of the derivative (to avoid arbitrage) weobtain the following relationship
119903 (119905) + 120583119901(119905) minus 120582
119901(119905) 120590
119901= 119877 (119905) (7)
Remark 3 In the empirical research the evolution processof 119878(119905) and 119881(119905) is usually negatively correlated Since ingeneral with declining of the stock price the volatility of theprice gradually increases (cf [32]) based on this reason inSection 5 we assume that the parameter 120588
119904V lt 0
22 Wealth Process In this paper we consider the optimalportfolio problems of aDCpension planmember both beforeand after retirement The member contributes part of hishersalary into the pension fund account before retirement andobtains benefit after retirement So we need to consider thewealth processes in two different periods
(I) Before Retirement During the accumulation period [0 119879]we assume that the DC pension plan member contributes
4 Mathematical Problems in Engineering
continuously to hisher pension fund account with a fixedrate 119896 of hisher salary income that is heshe contributes theamount of money 119896119871(119905) to hisher pension fund account attime 119905 The pension fund is invested in the financial marketby the manager Denote 120587
119868(119905) and 120587
119878(119905) as the propositions
of wealth invested in the inflation-linked index bond and thestock respectively Then 120587
119861(119905) = 1 minus 120587
119868(119905) minus 120587
119878(119905) is the
proposition of the risk-free money market account Denote120587(119905) = (120587
119868(119905) 120587
119878(119905)) 119905 isin [0 119879] as the decision-making
process Now we rewrite the SDEs (2) and (4) as matrix form
where 1 = (1 1)1015840 The last equation holds because of (7) and
120583 (119905) minus 119877 (119905) 1 = [119903 (119905) minus 119877 (119905) + 120583
119901(119905)
120582119904119881 (119905) + 120582
119901(119905) 120590
119904
]
= [
120582119901(119905) 120590
119901
120582119904119881 (119905) + 120582
119901(119905) 120590
119904
] = Σ (119905) 120579 (119905)
(10)
where 120579(119905) = (120582119901(119905) 120582
119904radic119881(119905))
1015840 Let 1199090be the nominal wealth
at time 0Since the real value of the wealth reflects the real purchase
power of the current market the nominal value of the wealthshould be converted into the real value By discounting thevalue of nominal wealth with the price level process thereal value of wealth will be obtained Now denote 119883(119905) =119883(119905)119875(119905) and 119871(119905) = 119871(119905)119875(119905) and by Itorsquos formula we
obtain the real wealth process 119883120587(119905) and real salary income
process 119871(119905) as follows119889119883
120587(119905) = 119883
120587(119905)
sdot [119877 (119905) minus 120583119901(119905) + 120590
2
119901+ 120587 (119905) Σ (119905) (120579 (119905) minus 120588
initial salary income are 119883120587(0) = 119909
0119901
0and 119871(0) = 119897
0119901
0
respectively Note that the dynamic process of119883120587(119905) depends
on the real salary income process 119871(119905) and the stochasticvolatility process 119881(119905) but it is independent of the price levelprocess 119875(119905)
Definition 4 (admissible strategy) LetO = RtimesR+timesR+ and
G = [0 119879] times O A strategy 120587(119905) = 120587119868(119905) 120587
119878(119905)
119905isin[0119879]is said
to be admissible if(1) forall(119909 V 119897) isin O the SDE (11) has unique solution119883
120587(119904)
119904isin[119905119879]with119883120587
(119905) = 119909 119881(119905) = V and 119871(119905) = 119897
(2) forall119904 isin [119905 119879] E[int119879
119905[(120587
119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904]4119889119904] lt infin
(3) forall984858 isin [1 +infin) and forall(119905 119909 V 119897) isin GE
One denotes Π(119905 119909 V 119897) as the set of all admissiblestrategies with respect to the initial condition (119905 119909 V 119897) isin G
(II) After Retirement When the member of the pension planretires the accumulation value of the pension fund usually isused to purchase an annuity In practice the member usuallyhas two ways to purchase the annuity The first way is thatheshe purchases the annuity directly at the retirement timeFor example as being considered in [2 27] the accumulatedpension fund is used to purchase a paid-up annuity oncethe member arrives at the retirement age and the managerhas also to decide the part of the remaining mathematicalreserve to invest in the market The second way is that themember does not purchase the annuity directly but choosesthe income drawdown option at the retirement It means thatthe member withdraws a fixed income and invests the restof the wealth until heshe achieves the age when the time ofpurchasing the annuity is compulsory The interested readercan see [33 34] for details
Here we consider the first way Similar to [2] we assumethat the member purchases a paid-up annuity at the retire-ment time which guarantees the benefit to be given only on afixed time horizon [119879 119879+119873] and invests the rest of the wealthcontinuously in the market We denote119882 as the part of thefund used to purchase an annuity (119882 ⩽ 119883(119879)) Continuousbenefit (real value) paid from 119879 to 119879
1= 119879 + 119873 is 119902 = 119882119886
119873|
where 119886119873|= (1 minus 119890
minus120575119873)120575 and 120575 is the continuous technical
rate
Mathematical Problems in Engineering 5
Let119872(119905) denote the nominal total payment value of thebenefit at time interval [119879 119905] 119905 isin (119879 119879
1] For simplicity the
dynamics of119872(119905) can be modeled as119872(119905) = int119905
119879119902119875(119904)119889119904 or
119889119872(119905) = 119902119875(119905)119889119905 This setting is reasonable and realistic anda similar setting is used in [17] It means that after retirementthe nominal benefit is adjusted dynamically based on theinflation rate of that time that is at time 119905 the nominalbenefit is 119902119875(119905) which keeps the purchasing power level ofthe retirees not decreasing We assume that once the retiredmember dies during [119879 119879
1] the pension fund continues to be
invested and the remaining annuity benefit and the terminalwealth119883(119879
1) are paid off to hisher offspring If themember is
alive until 1198791 the wealth119883(119879
1) is left for hisher later life Let
119868(119905) and
119878(119905) represent the propositions of the pension fund
wealth invested in the inflation-linked index bond and thestock respectively Denote (119905) = (
119868(119905)
119878(119905)) 119905 isin [119879 119879
1]
as the decision-making process Then after retirement thedynamics of nominal wealth process is
119905119909V[sdot] isthe condition expectation given 119883
(119905) = 119909 and
119881(119905) = V
One denotes Π(119905 119909 V) as the set of all admissible strategieswith respect to the initial condition (119905 119909 V) isin G
1
3 Problem Formulation before Retirementand Verification Theorem
In this section we assume that the pension plan managerhopes to maximize the expectation of the total wealth atretirement time119879 meanwhileminimizing the variance of thewealth at retirement So the pension plan manager faces a
continuous-time MV portfolio problem However since theMV objective function does not satisfy the iterated expecta-tion property Bellmanrsquos optimality principle does not holdunder the continuous-time framework Hence this problemis time-inconsistent This means that for the pension fundmanager without precommitment should take into accountthat heshemay have different objective functions in differenttimes We can thus view this problem as a noncooperativegame At each time 119905 there is one player named player 119905representing the future incarnation of the manager at time119905 At any state (119905 119909 V 119897) the manager faces an optimal controlproblem with value function 119881(119905 119909 V 119897) as follows
119869 (119905 119909 V 119897 120587) = E119905119909V119897 [119883
120587(119879)]
minus120574
2Var
119905119909V119897 [119883120587(119879)]
119881 (119905 119909 V 119897) = sup120587isinΠ(119905119909V119897)
119869 (119905 119909 V 119897 120587)
(P1)
where 120574 represents the risk aversion level of the managerand Var
119905119909V119897[sdot] refers to the conditional variance Since theobjective of the pensionmanager updates with different statethemanagerrsquos decision process is like a game process betweenan infinite number of distinct players Thus we need to lookfor a subgame perfect Nash equilibrium point for the gameand determine an equilibrium strategy 120587lowast
If we denote 119910120587(119905 119909 V 119897) = E
119905119909V119897[119883120587(119879)] 119911120587(119905 119909 V 119897) =
E119905119909V119897[(119883
120587(119879))
2] then the value function 119881(119905 119909 V 119897) can be
rewritten as119881 (119905 119909 V 119897)
= sup120587isinΠ(119905119909V119897)
119891 (119905 119909 V 119897 119910120587(119905 119909 V 119897) 119911120587 (119905 119909 V 119897)) (15)
where 119891(119905 119909 V 119897 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
31 Verification Theorem Based on the idea of the Nashequilibrium similarly to [11] we first give the definition ofequilibrium strategy
Definition 6 For any given initial state (119905 119909 V 119897) isin Gconsider an admissible strategy 120587lowast
(119905 119909 V 119897) and choose threereal numbers 120591 gt 0
119868 and
119861 one defines the following
strategy
120587120591(119904 V )
=
(119868
119861) 119905 ⩽ 119904 lt 119905 + 120591 ( V ) isin O
120587lowast(119904 V ) 119905 + 120591 ⩽ 119904 lt 119879 ( V ) isin O
(16)
If lim120591rarr0
inf((119891120587lowast
(119905) minus 119891120587120591(119905))120591) ⩾ 0 forall(
119868
119861) isin R times R
then 120587lowast(119905 119909 V 119897) is called an equilibrium strategy and the
equilibrium value function is defined by
119881 (119905 119909 V 119897)
= 119891 (119905 119909 V 119897 119910120587lowast
(119905 119909 V 119897) 119911120587lowast
(119905 119909 V 119897)) (17)
where 119891120587(119905) = 119891(119905 119909 V 119897 119910120587
(119905 119909 V 119897) 119911120587(119905 119909 V 119897)) for short
6 Mathematical Problems in Engineering
For any 120601(119905 119909 V 119897) isin 1198621222(G) we define a set 119863
119901(G) =
120601(119905 119909 V 119897) | 120601(119905 119909 V 119897) isin 1198621222
(G) and all first-orderpartial derivatives of 120601(sdot 119909 V 119897) satisfy the polynomial growthcondition on O and we define a variational operator asfollows (in the following we will omit the variable 119905 fornotation convenience)
A120587120601 (119905 119909 V 119897)
= 120601119905+ 120601
119909[119896119897 + 119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901))]
+ 120601V120572 (120575 minus V) + 120601119897120572119897119897
+1
21206011199091199091199092(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
+1
2120601VV120590
2
V V
+1
212060111989711989711989721205731015840
119897120573119897+ 120601
119909V119909120590VradicV (120587Σ minus 1205881015840
119901) 120573V
+ 120601119909119897119909119897 (120587Σ minus 120588
1015840
119901) 120573
119897+ 120601V119897119897120590VradicV120573
1015840
119897120573V
(18)
where 120573V = (0 120588119904V)1015840 Now we give the verification theorem of
problem (P1)
Theorem 7 (verification theorem) For problem (P1) if there
exist three real value functions 119865(119905 119909 V 119897) 119866(119905 119909 V 119897) and119867(119905 119909 V 119897) isin 119863
119901(G) satisfying the following conditions
forall(119905 119909 V 119897) isin G
sup120587isinΠ(119905119909V119897)
A120587119865 (119905 119909 V 119897) minus 120585120587 (119905 119909 V 119897) = 0 (19)
119865 (119879 119909 V 119897) = 119891 (119879 119909 V 119897 119909 1199092) (20)
A120587lowast
119866 (119905 119909 V 119897) = 0
119866 (119879 119909 V 119897) = 119909(21)
A120587lowast
119867(119905 119909 V 119897) = 0
119867 (119879 119909 V 119897) = 1199092
(22)
where 120587lowast= arg sup
120587isinΠ(119905119909V119897)A120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897)
then 119881(119905 119909 V 119897) = 119865(119905 119909 V 119897) 119910120587lowast
(119905 119909 V 119897) = 119866(119905 119909 V 119897) and119911120587lowast
(119905 119909 V 119897) = 119867(119905 119909 V 119897) and 120587lowast is the equilibrium strategywhere
120585120587(119905 119909 V 119897)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901)) + 119896119897] 119891
119909
+ [120572 (120575 minus V)] 119891V + 120572119897119897119891119897
+1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092119880
120587
1+1
21205902
V V119880120587
2
+1
211989721205731015840
119897120573119897119880
120587
3+ 120590VradicV119909 (120587Σ minus 120588
1015840
119901) 120573V119880
120587
4
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897119880
120587
5+ 120590VradicV119897120573
1015840
119897120573V119880
120587
6
119880120587
1
= 119891119909119909+ 119891
119910119910(119910
120587
119909)2
+ 119891119911119911(119911
120587
119909)2
+ 2119891119909119910119910120587
119909+ 2119891
119909119911119911120587
119909
+ 2119891119910119911119910120587
119909119911120587
119909
119880120587
2
= 119891VV + 119891119910119910 (119910120587
V )2
+ 119891119911119911(119911
120587
V )2
+ 2119891V119910119910120587
V + 2119891V119911119911120587
V
+ 2119891119910119911119910120587
V 119911120587
V
119880120587
3
= 119891119897119897+ 119891
119910119910(119910
120587
119897)2
+ 119891119911119911(119911
120587
119897)2
+ 2119891119897119910119910120587
119897+ 2119891
119897119911119911120587
119897
+ 2119891119910119911119910120587
119897119911120587
119897
119880120587
4
= 119891119910119910119910120587
119909119910120587
V + 119891119911119911119911120587
119909119911120587
V + 119891119909V + 119891119909119910119910120587
V + 119891119909119911119911120587
V
+ 119891V119910119910120587
119909+ 119891V119911119911
120587
119909+ 119891
119910119911(119910
120587
119909119911120587
V + 119910120587
V 119911120587
119909)
119880120587
5
= 119891119910119910119910120587
119909119910120587
119897+ 119891
119911119911119911120587
119909119911120587
119897+ 119891
119909119897+ 119891
119909119910119910120587
119897+ 119891
119909119911119911120587
119897
+ 119891119897119910119910120587
119909+ 119891
119897119911119911120587
119909+ 119891
119910119911(119910
120587
119909119911120587
119897+ 119910
120587
119897119911120587
119909)
119880120587
6
= 119891119910119910119910120587
V 119910120587
119897+ 119891
119911119911119911120587
V 119911120587
119897+ 119891V119897 + 119891V119910119910
120587
119897+ 119891V119911119911
120587
119897
+ 119891119897119910119910120587
V + 119891119897119911119911120587
V + 119891119910119911 (119910120587
119897119911120587
V + 119910120587
V 119911120587
119897)
(23)
and 119891 = 119891(119905 119909 V 119897 119910 119911) 119910120587= 119910
120587(119905 119909 V 119897) and 119911120587 =
119911120587(119905 119909 V 119897)
The proof of this theorem is given in the AppendixEquations (19) (21) and (22) are called an extended
HJB equations system By solving this equations systemwe can obtain the equilibrium strategy and correspondingequilibrium value function
32 Solution to the Equilibrium Strategy Since 119891(119905 119909 V 119897 119910119911) = 119910 minus (1205742)(119911 minus 119910
2) according to Theorem 7 we have
119880120587lowast
1= 120574119866
2
119909 119880120587
lowast
2= 120574119866
2
V 119880120587lowast
3= 120574119866
2
119897 119880120587
lowast
4= 120574119866
119909119866V 119880
120587lowast
5=
120574119866119909119866
119897 and 119880120587
lowast
6= 120574119866V119866119897
so
120585120587lowast
(119905 119909 V 119897) =1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092120574119866
2
119909
+1
21205902
V V1205741198662
V +1
211989721205731015840
119897120573119897120574119866
2
119897
+ 120590VradicV1198971205731015840
119897120573V120574119866V119866119897
+ 120590VradicV119909 (120587Σ minus 1205881015840
119901) 120573V120574119866V119866119909
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897120574119866
119909119866
119897
(24)
Mathematical Problems in Engineering 7
By differentiatingA120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897) with respect to
120587(119905) we can obtain the equilibrium strategy
120587lowast(119905) = minus
[119865119909(120579 minus 120588
119901)1015840
+ (119865119909V minus 120574119866119909
119866V) 120590VradicV1205731015840
V + (119865119909119897 minus 120574119866119909119866
119897) 119897120573
1015840
119897] Σ
minus1
(119865119909119909minus 120574119866
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(25)
Inspired by [24] we suppose 119865(119905 119909 V 119897) and 119866(119905 119909 V 119897) havethe following linear forms with respect to 119909 V and 119897
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int119879
119905119903(119904)119889119904
+119861 (119905)
120574V + 119862 (119905) 119897 +
119863 (119905)
120574 (40)
where 119861(119905) 119862(119905) and 119863(119905) are given by (34) (35) and (36)
Note that
E119905119909V119897 [119883
120587lowast
(119879)]
= 119909119890int119879
119905119903(119904)119889119904
+1
120574[119861 (119905) V + 119863 (119905)] + 119862 (119905) 119897
(41)
Var119905119909V119897 [119883
120587lowast
(119879)] = 119867 (119905 119909 V 119897) minus 1198662(119905 119909 V 119897)
=2
120574[119866 (119905 119909 V 119897) minus 119865 (119905 119909 V 119897)]
=2
1205742[(119861 (119905) minus 119861 (119905)) V + 119863 (119905) minus 119863 (119905)]
(42)
Thus we obtain the following so-called equilibrium efficientfrontier which reflects the relationship between the invest-ment risk and return under the equilibrium strategy
Corollary 9 For MV problem (P1) based on state (119905 119909 V 119897)
its equilibrium efficient frontier is
E119905119909V119897 [119883
120587lowast
(119879)] = 119909119890int119879
119905119903(119904)119889119904
+ 119862 (119905) 119897
+ 119873 (119905 V 119897) radicVar119905119909V119897 [119883
120587lowast
(119879)]
(43)
where 119873(119905 V 119897) = [119861(119905)V + 119863(119905)]
radic2[(119861(119905) minus 119861(119905))V + 119863(119905) minus 119863(119905)]
Remark 10 By (38) and (39) we find that the equilibriuminvestment proportion in the stock has the similar expressionas in [24] although their study focuses on the optimal invest-ment and reinsurance problem In addition the equilibriuminvestment money in the stock is only the function of time119905 which is independent of the current wealth However forthe inflation-linked bond the equilibrium investmentmoneydepends on current wealth and the more the wealth isthe more the inflation-linked bond is invested This resultis consistent with the economic intuition considering thatwhen the manager has more wealth the sensitivity to theinflation risk becomes higher and the corresponding hedgingdemand is improved
Remark 11 In view of (38) we find that if 120590119901gt 120590
119897 that is the
volatility rate of the inflation is larger than that of the salaryincome the equilibrium investment amount in the inflation-linked bond has positive relationship with the salary levelOtherwise it has negative relationship with the salary level
Remark 12 From (42) and (43) if Var119905119909V119897[119883
120587(119879)] = 0 it
means that the pension fund manager bears no risk at allunder the state (119905 119909 V 119897) this is equivalent to the case of120574 rarr infin that is the pension fund manager is fully averseto risk In this case 120587lowast
119878(119905) = 0 and 120587lowast
119868(119905) = 1 minus ((120590
119897minus
120590119901)119909120590
119901)119897119862(119905)119890
minusint119879
119905119903(119904)119889119904 This indicates that the manager will
only invest all the fund in the inflation-linked bond and therisk-free money market account and obtain the expected realterminal wealth of 119909119890int
119879
119905119903(119904)119889119904
+ 119862(119905)119897 at retirement time 119879 Itconsists of two parts The first part is the accumulation ofthe initial wealth 119909 and the second part can be seen as theaccumulation of the contribution
Mathematical Problems in Engineering 9
Remark 13 If we do not consider the risk of stochasticvolatility of the stock price that is let 120572 = 0 120590V = 0 thenthe equilibrium strategy is simplified as
120587lowast
119868(119905)
= [
120582119901(119905) minus 120590
119901minus 120582
119904120590119904
120574minus 119897 (120590
119897minus 120590
119901) 119862 (119905)]
119890minusint119879
119905119903(119904)119889119904
119909120590119901
+ 1
120587lowast
119878(119905) =
120582119904119890minusint119879
119905119903(119904)119889119904
120574119909
(44)
In this case we find that 120587lowast
119878(119905) is consistent with the
result of the ordinaryMVportfolio problem under the Black-Scholes market (see [11]) This indicates that the contributionof the salary income has no influence on the equilibriuminvestment amount of the stock Furthermore if we do notconsider the risk of inflation that is 120583
119901(119905) = 120590
119901= 120590
119904= 0
then 119877(119905) = 119903(119905) and the inflation-linked bond is degeneratedto the risk-free money market account
4 Problem Formulation after Retirement andVerification Theorem
After retirement we assume that the pension plan memberpurchases a paid-up annuity during the time horizon [119879 119879 +119873] Thus the objective function of the pension plan manageris to maximize the expected surplus after paying off thebenefit to the member and minimize the variance of thesurplus at time 119879
1= 119879 + 119873 So at the state (119905 119909 V) isin G
1
the manager faces the following optimal control problem
(119905 119909 V) = supisinΠ(119905119909V)
E119905119909V [119883
(119879
1)]
minus120574
2Var
119905119909V [119883(119879
1)]
(P2)
Similarly to the method of Section 3 we denote 119910(119905 119909 V) =
E119905119909V[119883
(119879
1)] 119911(119905 119909 V) = E
119905119909V[(119883(119879
1))
2] Then we can
rewrite
(119905 119909 V)
= supisinΠ(119905119909V)
(119905 119909 V 119910(119905 119909 V) 119911 (119905 119909 V))
(45)
where (119905 119909 V 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
41 VerificationTheorem In this subsection we first give thedefinition of equilibrium strategy after retirement and thengive the verification theoremwhich is satisfied by the equilib-rium value function Now for any 120601(119905 119909 V) isin 119862122
(119905 119909 V) = (119905 119909 V) and 119911lowast
(119905 119909
V) = (119905 119909 V) and lowast is the equilibrium strategy where
120593(119905 119909 V)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ Σ (120579 minus 120588
119901)) minus 119902] 119891
119909
+ 119891V120572 (120575 minus V) +1
2(Σ minus 120588
1015840
119901) (Σ minus 120588
1015840
119901)1015840
1199092
1
+1
21205902
V V
2+ 120590VradicV119909 (
1015840Σ minus 120588
1015840
119901) 120573V
3
10 Mathematical Problems in Engineering
1
= 119891119909119909+ 119891
119910119910(119910
119909)2
+ 119891119911119911(119911
119909)2
+ 2119891119909119910119910
119909+ 2119891
119909119911119911
119909
+ 2119891119910119911119910
119909119911
119909
2
= 119891VV + 119891119910119910 (119910
V )2
+ 119891119911119911(119911
V )2
+ 2119891V119910119910
V + 2119891V119911119911
V
+ 2119891119910119911119910
V 119911
V
3
= 119891119910119910119910
119909119910
V + 119891119911119911119911
119909119911
V + 119891119909V + 119891119909119910119910
V + 119891119909119911119911
V
+ 119891V119910119910
119909+ 119891V119911119911
119909+ 119891
119910119911(119910
119909119911
V + 119910
V 119911
119909)
(49)
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
continuously to hisher pension fund account with a fixedrate 119896 of hisher salary income that is heshe contributes theamount of money 119896119871(119905) to hisher pension fund account attime 119905 The pension fund is invested in the financial marketby the manager Denote 120587
119868(119905) and 120587
119878(119905) as the propositions
of wealth invested in the inflation-linked index bond and thestock respectively Then 120587
119861(119905) = 1 minus 120587
119868(119905) minus 120587
119878(119905) is the
proposition of the risk-free money market account Denote120587(119905) = (120587
119868(119905) 120587
119878(119905)) 119905 isin [0 119879] as the decision-making
process Now we rewrite the SDEs (2) and (4) as matrix form
where 1 = (1 1)1015840 The last equation holds because of (7) and
120583 (119905) minus 119877 (119905) 1 = [119903 (119905) minus 119877 (119905) + 120583
119901(119905)
120582119904119881 (119905) + 120582
119901(119905) 120590
119904
]
= [
120582119901(119905) 120590
119901
120582119904119881 (119905) + 120582
119901(119905) 120590
119904
] = Σ (119905) 120579 (119905)
(10)
where 120579(119905) = (120582119901(119905) 120582
119904radic119881(119905))
1015840 Let 1199090be the nominal wealth
at time 0Since the real value of the wealth reflects the real purchase
power of the current market the nominal value of the wealthshould be converted into the real value By discounting thevalue of nominal wealth with the price level process thereal value of wealth will be obtained Now denote 119883(119905) =119883(119905)119875(119905) and 119871(119905) = 119871(119905)119875(119905) and by Itorsquos formula we
obtain the real wealth process 119883120587(119905) and real salary income
process 119871(119905) as follows119889119883
120587(119905) = 119883
120587(119905)
sdot [119877 (119905) minus 120583119901(119905) + 120590
2
119901+ 120587 (119905) Σ (119905) (120579 (119905) minus 120588
initial salary income are 119883120587(0) = 119909
0119901
0and 119871(0) = 119897
0119901
0
respectively Note that the dynamic process of119883120587(119905) depends
on the real salary income process 119871(119905) and the stochasticvolatility process 119881(119905) but it is independent of the price levelprocess 119875(119905)
Definition 4 (admissible strategy) LetO = RtimesR+timesR+ and
G = [0 119879] times O A strategy 120587(119905) = 120587119868(119905) 120587
119878(119905)
119905isin[0119879]is said
to be admissible if(1) forall(119909 V 119897) isin O the SDE (11) has unique solution119883
120587(119904)
119904isin[119905119879]with119883120587
(119905) = 119909 119881(119905) = V and 119871(119905) = 119897
(2) forall119904 isin [119905 119879] E[int119879
119905[(120587
119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904]4119889119904] lt infin
(3) forall984858 isin [1 +infin) and forall(119905 119909 V 119897) isin GE
One denotes Π(119905 119909 V 119897) as the set of all admissiblestrategies with respect to the initial condition (119905 119909 V 119897) isin G
(II) After Retirement When the member of the pension planretires the accumulation value of the pension fund usually isused to purchase an annuity In practice the member usuallyhas two ways to purchase the annuity The first way is thatheshe purchases the annuity directly at the retirement timeFor example as being considered in [2 27] the accumulatedpension fund is used to purchase a paid-up annuity oncethe member arrives at the retirement age and the managerhas also to decide the part of the remaining mathematicalreserve to invest in the market The second way is that themember does not purchase the annuity directly but choosesthe income drawdown option at the retirement It means thatthe member withdraws a fixed income and invests the restof the wealth until heshe achieves the age when the time ofpurchasing the annuity is compulsory The interested readercan see [33 34] for details
Here we consider the first way Similar to [2] we assumethat the member purchases a paid-up annuity at the retire-ment time which guarantees the benefit to be given only on afixed time horizon [119879 119879+119873] and invests the rest of the wealthcontinuously in the market We denote119882 as the part of thefund used to purchase an annuity (119882 ⩽ 119883(119879)) Continuousbenefit (real value) paid from 119879 to 119879
1= 119879 + 119873 is 119902 = 119882119886
119873|
where 119886119873|= (1 minus 119890
minus120575119873)120575 and 120575 is the continuous technical
rate
Mathematical Problems in Engineering 5
Let119872(119905) denote the nominal total payment value of thebenefit at time interval [119879 119905] 119905 isin (119879 119879
1] For simplicity the
dynamics of119872(119905) can be modeled as119872(119905) = int119905
119879119902119875(119904)119889119904 or
119889119872(119905) = 119902119875(119905)119889119905 This setting is reasonable and realistic anda similar setting is used in [17] It means that after retirementthe nominal benefit is adjusted dynamically based on theinflation rate of that time that is at time 119905 the nominalbenefit is 119902119875(119905) which keeps the purchasing power level ofthe retirees not decreasing We assume that once the retiredmember dies during [119879 119879
1] the pension fund continues to be
invested and the remaining annuity benefit and the terminalwealth119883(119879
1) are paid off to hisher offspring If themember is
alive until 1198791 the wealth119883(119879
1) is left for hisher later life Let
119868(119905) and
119878(119905) represent the propositions of the pension fund
wealth invested in the inflation-linked index bond and thestock respectively Denote (119905) = (
119868(119905)
119878(119905)) 119905 isin [119879 119879
1]
as the decision-making process Then after retirement thedynamics of nominal wealth process is
119905119909V[sdot] isthe condition expectation given 119883
(119905) = 119909 and
119881(119905) = V
One denotes Π(119905 119909 V) as the set of all admissible strategieswith respect to the initial condition (119905 119909 V) isin G
1
3 Problem Formulation before Retirementand Verification Theorem
In this section we assume that the pension plan managerhopes to maximize the expectation of the total wealth atretirement time119879 meanwhileminimizing the variance of thewealth at retirement So the pension plan manager faces a
continuous-time MV portfolio problem However since theMV objective function does not satisfy the iterated expecta-tion property Bellmanrsquos optimality principle does not holdunder the continuous-time framework Hence this problemis time-inconsistent This means that for the pension fundmanager without precommitment should take into accountthat heshemay have different objective functions in differenttimes We can thus view this problem as a noncooperativegame At each time 119905 there is one player named player 119905representing the future incarnation of the manager at time119905 At any state (119905 119909 V 119897) the manager faces an optimal controlproblem with value function 119881(119905 119909 V 119897) as follows
119869 (119905 119909 V 119897 120587) = E119905119909V119897 [119883
120587(119879)]
minus120574
2Var
119905119909V119897 [119883120587(119879)]
119881 (119905 119909 V 119897) = sup120587isinΠ(119905119909V119897)
119869 (119905 119909 V 119897 120587)
(P1)
where 120574 represents the risk aversion level of the managerand Var
119905119909V119897[sdot] refers to the conditional variance Since theobjective of the pensionmanager updates with different statethemanagerrsquos decision process is like a game process betweenan infinite number of distinct players Thus we need to lookfor a subgame perfect Nash equilibrium point for the gameand determine an equilibrium strategy 120587lowast
If we denote 119910120587(119905 119909 V 119897) = E
119905119909V119897[119883120587(119879)] 119911120587(119905 119909 V 119897) =
E119905119909V119897[(119883
120587(119879))
2] then the value function 119881(119905 119909 V 119897) can be
rewritten as119881 (119905 119909 V 119897)
= sup120587isinΠ(119905119909V119897)
119891 (119905 119909 V 119897 119910120587(119905 119909 V 119897) 119911120587 (119905 119909 V 119897)) (15)
where 119891(119905 119909 V 119897 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
31 Verification Theorem Based on the idea of the Nashequilibrium similarly to [11] we first give the definition ofequilibrium strategy
Definition 6 For any given initial state (119905 119909 V 119897) isin Gconsider an admissible strategy 120587lowast
(119905 119909 V 119897) and choose threereal numbers 120591 gt 0
119868 and
119861 one defines the following
strategy
120587120591(119904 V )
=
(119868
119861) 119905 ⩽ 119904 lt 119905 + 120591 ( V ) isin O
120587lowast(119904 V ) 119905 + 120591 ⩽ 119904 lt 119879 ( V ) isin O
(16)
If lim120591rarr0
inf((119891120587lowast
(119905) minus 119891120587120591(119905))120591) ⩾ 0 forall(
119868
119861) isin R times R
then 120587lowast(119905 119909 V 119897) is called an equilibrium strategy and the
equilibrium value function is defined by
119881 (119905 119909 V 119897)
= 119891 (119905 119909 V 119897 119910120587lowast
(119905 119909 V 119897) 119911120587lowast
(119905 119909 V 119897)) (17)
where 119891120587(119905) = 119891(119905 119909 V 119897 119910120587
(119905 119909 V 119897) 119911120587(119905 119909 V 119897)) for short
6 Mathematical Problems in Engineering
For any 120601(119905 119909 V 119897) isin 1198621222(G) we define a set 119863
119901(G) =
120601(119905 119909 V 119897) | 120601(119905 119909 V 119897) isin 1198621222
(G) and all first-orderpartial derivatives of 120601(sdot 119909 V 119897) satisfy the polynomial growthcondition on O and we define a variational operator asfollows (in the following we will omit the variable 119905 fornotation convenience)
A120587120601 (119905 119909 V 119897)
= 120601119905+ 120601
119909[119896119897 + 119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901))]
+ 120601V120572 (120575 minus V) + 120601119897120572119897119897
+1
21206011199091199091199092(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
+1
2120601VV120590
2
V V
+1
212060111989711989711989721205731015840
119897120573119897+ 120601
119909V119909120590VradicV (120587Σ minus 1205881015840
119901) 120573V
+ 120601119909119897119909119897 (120587Σ minus 120588
1015840
119901) 120573
119897+ 120601V119897119897120590VradicV120573
1015840
119897120573V
(18)
where 120573V = (0 120588119904V)1015840 Now we give the verification theorem of
problem (P1)
Theorem 7 (verification theorem) For problem (P1) if there
exist three real value functions 119865(119905 119909 V 119897) 119866(119905 119909 V 119897) and119867(119905 119909 V 119897) isin 119863
119901(G) satisfying the following conditions
forall(119905 119909 V 119897) isin G
sup120587isinΠ(119905119909V119897)
A120587119865 (119905 119909 V 119897) minus 120585120587 (119905 119909 V 119897) = 0 (19)
119865 (119879 119909 V 119897) = 119891 (119879 119909 V 119897 119909 1199092) (20)
A120587lowast
119866 (119905 119909 V 119897) = 0
119866 (119879 119909 V 119897) = 119909(21)
A120587lowast
119867(119905 119909 V 119897) = 0
119867 (119879 119909 V 119897) = 1199092
(22)
where 120587lowast= arg sup
120587isinΠ(119905119909V119897)A120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897)
then 119881(119905 119909 V 119897) = 119865(119905 119909 V 119897) 119910120587lowast
(119905 119909 V 119897) = 119866(119905 119909 V 119897) and119911120587lowast
(119905 119909 V 119897) = 119867(119905 119909 V 119897) and 120587lowast is the equilibrium strategywhere
120585120587(119905 119909 V 119897)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901)) + 119896119897] 119891
119909
+ [120572 (120575 minus V)] 119891V + 120572119897119897119891119897
+1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092119880
120587
1+1
21205902
V V119880120587
2
+1
211989721205731015840
119897120573119897119880
120587
3+ 120590VradicV119909 (120587Σ minus 120588
1015840
119901) 120573V119880
120587
4
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897119880
120587
5+ 120590VradicV119897120573
1015840
119897120573V119880
120587
6
119880120587
1
= 119891119909119909+ 119891
119910119910(119910
120587
119909)2
+ 119891119911119911(119911
120587
119909)2
+ 2119891119909119910119910120587
119909+ 2119891
119909119911119911120587
119909
+ 2119891119910119911119910120587
119909119911120587
119909
119880120587
2
= 119891VV + 119891119910119910 (119910120587
V )2
+ 119891119911119911(119911
120587
V )2
+ 2119891V119910119910120587
V + 2119891V119911119911120587
V
+ 2119891119910119911119910120587
V 119911120587
V
119880120587
3
= 119891119897119897+ 119891
119910119910(119910
120587
119897)2
+ 119891119911119911(119911
120587
119897)2
+ 2119891119897119910119910120587
119897+ 2119891
119897119911119911120587
119897
+ 2119891119910119911119910120587
119897119911120587
119897
119880120587
4
= 119891119910119910119910120587
119909119910120587
V + 119891119911119911119911120587
119909119911120587
V + 119891119909V + 119891119909119910119910120587
V + 119891119909119911119911120587
V
+ 119891V119910119910120587
119909+ 119891V119911119911
120587
119909+ 119891
119910119911(119910
120587
119909119911120587
V + 119910120587
V 119911120587
119909)
119880120587
5
= 119891119910119910119910120587
119909119910120587
119897+ 119891
119911119911119911120587
119909119911120587
119897+ 119891
119909119897+ 119891
119909119910119910120587
119897+ 119891
119909119911119911120587
119897
+ 119891119897119910119910120587
119909+ 119891
119897119911119911120587
119909+ 119891
119910119911(119910
120587
119909119911120587
119897+ 119910
120587
119897119911120587
119909)
119880120587
6
= 119891119910119910119910120587
V 119910120587
119897+ 119891
119911119911119911120587
V 119911120587
119897+ 119891V119897 + 119891V119910119910
120587
119897+ 119891V119911119911
120587
119897
+ 119891119897119910119910120587
V + 119891119897119911119911120587
V + 119891119910119911 (119910120587
119897119911120587
V + 119910120587
V 119911120587
119897)
(23)
and 119891 = 119891(119905 119909 V 119897 119910 119911) 119910120587= 119910
120587(119905 119909 V 119897) and 119911120587 =
119911120587(119905 119909 V 119897)
The proof of this theorem is given in the AppendixEquations (19) (21) and (22) are called an extended
HJB equations system By solving this equations systemwe can obtain the equilibrium strategy and correspondingequilibrium value function
32 Solution to the Equilibrium Strategy Since 119891(119905 119909 V 119897 119910119911) = 119910 minus (1205742)(119911 minus 119910
2) according to Theorem 7 we have
119880120587lowast
1= 120574119866
2
119909 119880120587
lowast
2= 120574119866
2
V 119880120587lowast
3= 120574119866
2
119897 119880120587
lowast
4= 120574119866
119909119866V 119880
120587lowast
5=
120574119866119909119866
119897 and 119880120587
lowast
6= 120574119866V119866119897
so
120585120587lowast
(119905 119909 V 119897) =1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092120574119866
2
119909
+1
21205902
V V1205741198662
V +1
211989721205731015840
119897120573119897120574119866
2
119897
+ 120590VradicV1198971205731015840
119897120573V120574119866V119866119897
+ 120590VradicV119909 (120587Σ minus 1205881015840
119901) 120573V120574119866V119866119909
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897120574119866
119909119866
119897
(24)
Mathematical Problems in Engineering 7
By differentiatingA120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897) with respect to
120587(119905) we can obtain the equilibrium strategy
120587lowast(119905) = minus
[119865119909(120579 minus 120588
119901)1015840
+ (119865119909V minus 120574119866119909
119866V) 120590VradicV1205731015840
V + (119865119909119897 minus 120574119866119909119866
119897) 119897120573
1015840
119897] Σ
minus1
(119865119909119909minus 120574119866
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(25)
Inspired by [24] we suppose 119865(119905 119909 V 119897) and 119866(119905 119909 V 119897) havethe following linear forms with respect to 119909 V and 119897
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int119879
119905119903(119904)119889119904
+119861 (119905)
120574V + 119862 (119905) 119897 +
119863 (119905)
120574 (40)
where 119861(119905) 119862(119905) and 119863(119905) are given by (34) (35) and (36)
Note that
E119905119909V119897 [119883
120587lowast
(119879)]
= 119909119890int119879
119905119903(119904)119889119904
+1
120574[119861 (119905) V + 119863 (119905)] + 119862 (119905) 119897
(41)
Var119905119909V119897 [119883
120587lowast
(119879)] = 119867 (119905 119909 V 119897) minus 1198662(119905 119909 V 119897)
=2
120574[119866 (119905 119909 V 119897) minus 119865 (119905 119909 V 119897)]
=2
1205742[(119861 (119905) minus 119861 (119905)) V + 119863 (119905) minus 119863 (119905)]
(42)
Thus we obtain the following so-called equilibrium efficientfrontier which reflects the relationship between the invest-ment risk and return under the equilibrium strategy
Corollary 9 For MV problem (P1) based on state (119905 119909 V 119897)
its equilibrium efficient frontier is
E119905119909V119897 [119883
120587lowast
(119879)] = 119909119890int119879
119905119903(119904)119889119904
+ 119862 (119905) 119897
+ 119873 (119905 V 119897) radicVar119905119909V119897 [119883
120587lowast
(119879)]
(43)
where 119873(119905 V 119897) = [119861(119905)V + 119863(119905)]
radic2[(119861(119905) minus 119861(119905))V + 119863(119905) minus 119863(119905)]
Remark 10 By (38) and (39) we find that the equilibriuminvestment proportion in the stock has the similar expressionas in [24] although their study focuses on the optimal invest-ment and reinsurance problem In addition the equilibriuminvestment money in the stock is only the function of time119905 which is independent of the current wealth However forthe inflation-linked bond the equilibrium investmentmoneydepends on current wealth and the more the wealth isthe more the inflation-linked bond is invested This resultis consistent with the economic intuition considering thatwhen the manager has more wealth the sensitivity to theinflation risk becomes higher and the corresponding hedgingdemand is improved
Remark 11 In view of (38) we find that if 120590119901gt 120590
119897 that is the
volatility rate of the inflation is larger than that of the salaryincome the equilibrium investment amount in the inflation-linked bond has positive relationship with the salary levelOtherwise it has negative relationship with the salary level
Remark 12 From (42) and (43) if Var119905119909V119897[119883
120587(119879)] = 0 it
means that the pension fund manager bears no risk at allunder the state (119905 119909 V 119897) this is equivalent to the case of120574 rarr infin that is the pension fund manager is fully averseto risk In this case 120587lowast
119878(119905) = 0 and 120587lowast
119868(119905) = 1 minus ((120590
119897minus
120590119901)119909120590
119901)119897119862(119905)119890
minusint119879
119905119903(119904)119889119904 This indicates that the manager will
only invest all the fund in the inflation-linked bond and therisk-free money market account and obtain the expected realterminal wealth of 119909119890int
119879
119905119903(119904)119889119904
+ 119862(119905)119897 at retirement time 119879 Itconsists of two parts The first part is the accumulation ofthe initial wealth 119909 and the second part can be seen as theaccumulation of the contribution
Mathematical Problems in Engineering 9
Remark 13 If we do not consider the risk of stochasticvolatility of the stock price that is let 120572 = 0 120590V = 0 thenthe equilibrium strategy is simplified as
120587lowast
119868(119905)
= [
120582119901(119905) minus 120590
119901minus 120582
119904120590119904
120574minus 119897 (120590
119897minus 120590
119901) 119862 (119905)]
119890minusint119879
119905119903(119904)119889119904
119909120590119901
+ 1
120587lowast
119878(119905) =
120582119904119890minusint119879
119905119903(119904)119889119904
120574119909
(44)
In this case we find that 120587lowast
119878(119905) is consistent with the
result of the ordinaryMVportfolio problem under the Black-Scholes market (see [11]) This indicates that the contributionof the salary income has no influence on the equilibriuminvestment amount of the stock Furthermore if we do notconsider the risk of inflation that is 120583
119901(119905) = 120590
119901= 120590
119904= 0
then 119877(119905) = 119903(119905) and the inflation-linked bond is degeneratedto the risk-free money market account
4 Problem Formulation after Retirement andVerification Theorem
After retirement we assume that the pension plan memberpurchases a paid-up annuity during the time horizon [119879 119879 +119873] Thus the objective function of the pension plan manageris to maximize the expected surplus after paying off thebenefit to the member and minimize the variance of thesurplus at time 119879
1= 119879 + 119873 So at the state (119905 119909 V) isin G
1
the manager faces the following optimal control problem
(119905 119909 V) = supisinΠ(119905119909V)
E119905119909V [119883
(119879
1)]
minus120574
2Var
119905119909V [119883(119879
1)]
(P2)
Similarly to the method of Section 3 we denote 119910(119905 119909 V) =
E119905119909V[119883
(119879
1)] 119911(119905 119909 V) = E
119905119909V[(119883(119879
1))
2] Then we can
rewrite
(119905 119909 V)
= supisinΠ(119905119909V)
(119905 119909 V 119910(119905 119909 V) 119911 (119905 119909 V))
(45)
where (119905 119909 V 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
41 VerificationTheorem In this subsection we first give thedefinition of equilibrium strategy after retirement and thengive the verification theoremwhich is satisfied by the equilib-rium value function Now for any 120601(119905 119909 V) isin 119862122
(119905 119909 V) = (119905 119909 V) and 119911lowast
(119905 119909
V) = (119905 119909 V) and lowast is the equilibrium strategy where
120593(119905 119909 V)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ Σ (120579 minus 120588
119901)) minus 119902] 119891
119909
+ 119891V120572 (120575 minus V) +1
2(Σ minus 120588
1015840
119901) (Σ minus 120588
1015840
119901)1015840
1199092
1
+1
21205902
V V
2+ 120590VradicV119909 (
1015840Σ minus 120588
1015840
119901) 120573V
3
10 Mathematical Problems in Engineering
1
= 119891119909119909+ 119891
119910119910(119910
119909)2
+ 119891119911119911(119911
119909)2
+ 2119891119909119910119910
119909+ 2119891
119909119911119911
119909
+ 2119891119910119911119910
119909119911
119909
2
= 119891VV + 119891119910119910 (119910
V )2
+ 119891119911119911(119911
V )2
+ 2119891V119910119910
V + 2119891V119911119911
V
+ 2119891119910119911119910
V 119911
V
3
= 119891119910119910119910
119909119910
V + 119891119911119911119911
119909119911
V + 119891119909V + 119891119909119910119910
V + 119891119909119911119911
V
+ 119891V119910119910
119909+ 119891V119911119911
119909+ 119891
119910119911(119910
119909119911
V + 119910
V 119911
119909)
(49)
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
Let119872(119905) denote the nominal total payment value of thebenefit at time interval [119879 119905] 119905 isin (119879 119879
1] For simplicity the
dynamics of119872(119905) can be modeled as119872(119905) = int119905
119879119902119875(119904)119889119904 or
119889119872(119905) = 119902119875(119905)119889119905 This setting is reasonable and realistic anda similar setting is used in [17] It means that after retirementthe nominal benefit is adjusted dynamically based on theinflation rate of that time that is at time 119905 the nominalbenefit is 119902119875(119905) which keeps the purchasing power level ofthe retirees not decreasing We assume that once the retiredmember dies during [119879 119879
1] the pension fund continues to be
invested and the remaining annuity benefit and the terminalwealth119883(119879
1) are paid off to hisher offspring If themember is
alive until 1198791 the wealth119883(119879
1) is left for hisher later life Let
119868(119905) and
119878(119905) represent the propositions of the pension fund
wealth invested in the inflation-linked index bond and thestock respectively Denote (119905) = (
119868(119905)
119878(119905)) 119905 isin [119879 119879
1]
as the decision-making process Then after retirement thedynamics of nominal wealth process is
119905119909V[sdot] isthe condition expectation given 119883
(119905) = 119909 and
119881(119905) = V
One denotes Π(119905 119909 V) as the set of all admissible strategieswith respect to the initial condition (119905 119909 V) isin G
1
3 Problem Formulation before Retirementand Verification Theorem
In this section we assume that the pension plan managerhopes to maximize the expectation of the total wealth atretirement time119879 meanwhileminimizing the variance of thewealth at retirement So the pension plan manager faces a
continuous-time MV portfolio problem However since theMV objective function does not satisfy the iterated expecta-tion property Bellmanrsquos optimality principle does not holdunder the continuous-time framework Hence this problemis time-inconsistent This means that for the pension fundmanager without precommitment should take into accountthat heshemay have different objective functions in differenttimes We can thus view this problem as a noncooperativegame At each time 119905 there is one player named player 119905representing the future incarnation of the manager at time119905 At any state (119905 119909 V 119897) the manager faces an optimal controlproblem with value function 119881(119905 119909 V 119897) as follows
119869 (119905 119909 V 119897 120587) = E119905119909V119897 [119883
120587(119879)]
minus120574
2Var
119905119909V119897 [119883120587(119879)]
119881 (119905 119909 V 119897) = sup120587isinΠ(119905119909V119897)
119869 (119905 119909 V 119897 120587)
(P1)
where 120574 represents the risk aversion level of the managerand Var
119905119909V119897[sdot] refers to the conditional variance Since theobjective of the pensionmanager updates with different statethemanagerrsquos decision process is like a game process betweenan infinite number of distinct players Thus we need to lookfor a subgame perfect Nash equilibrium point for the gameand determine an equilibrium strategy 120587lowast
If we denote 119910120587(119905 119909 V 119897) = E
119905119909V119897[119883120587(119879)] 119911120587(119905 119909 V 119897) =
E119905119909V119897[(119883
120587(119879))
2] then the value function 119881(119905 119909 V 119897) can be
rewritten as119881 (119905 119909 V 119897)
= sup120587isinΠ(119905119909V119897)
119891 (119905 119909 V 119897 119910120587(119905 119909 V 119897) 119911120587 (119905 119909 V 119897)) (15)
where 119891(119905 119909 V 119897 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
31 Verification Theorem Based on the idea of the Nashequilibrium similarly to [11] we first give the definition ofequilibrium strategy
Definition 6 For any given initial state (119905 119909 V 119897) isin Gconsider an admissible strategy 120587lowast
(119905 119909 V 119897) and choose threereal numbers 120591 gt 0
119868 and
119861 one defines the following
strategy
120587120591(119904 V )
=
(119868
119861) 119905 ⩽ 119904 lt 119905 + 120591 ( V ) isin O
120587lowast(119904 V ) 119905 + 120591 ⩽ 119904 lt 119879 ( V ) isin O
(16)
If lim120591rarr0
inf((119891120587lowast
(119905) minus 119891120587120591(119905))120591) ⩾ 0 forall(
119868
119861) isin R times R
then 120587lowast(119905 119909 V 119897) is called an equilibrium strategy and the
equilibrium value function is defined by
119881 (119905 119909 V 119897)
= 119891 (119905 119909 V 119897 119910120587lowast
(119905 119909 V 119897) 119911120587lowast
(119905 119909 V 119897)) (17)
where 119891120587(119905) = 119891(119905 119909 V 119897 119910120587
(119905 119909 V 119897) 119911120587(119905 119909 V 119897)) for short
6 Mathematical Problems in Engineering
For any 120601(119905 119909 V 119897) isin 1198621222(G) we define a set 119863
119901(G) =
120601(119905 119909 V 119897) | 120601(119905 119909 V 119897) isin 1198621222
(G) and all first-orderpartial derivatives of 120601(sdot 119909 V 119897) satisfy the polynomial growthcondition on O and we define a variational operator asfollows (in the following we will omit the variable 119905 fornotation convenience)
A120587120601 (119905 119909 V 119897)
= 120601119905+ 120601
119909[119896119897 + 119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901))]
+ 120601V120572 (120575 minus V) + 120601119897120572119897119897
+1
21206011199091199091199092(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
+1
2120601VV120590
2
V V
+1
212060111989711989711989721205731015840
119897120573119897+ 120601
119909V119909120590VradicV (120587Σ minus 1205881015840
119901) 120573V
+ 120601119909119897119909119897 (120587Σ minus 120588
1015840
119901) 120573
119897+ 120601V119897119897120590VradicV120573
1015840
119897120573V
(18)
where 120573V = (0 120588119904V)1015840 Now we give the verification theorem of
problem (P1)
Theorem 7 (verification theorem) For problem (P1) if there
exist three real value functions 119865(119905 119909 V 119897) 119866(119905 119909 V 119897) and119867(119905 119909 V 119897) isin 119863
119901(G) satisfying the following conditions
forall(119905 119909 V 119897) isin G
sup120587isinΠ(119905119909V119897)
A120587119865 (119905 119909 V 119897) minus 120585120587 (119905 119909 V 119897) = 0 (19)
119865 (119879 119909 V 119897) = 119891 (119879 119909 V 119897 119909 1199092) (20)
A120587lowast
119866 (119905 119909 V 119897) = 0
119866 (119879 119909 V 119897) = 119909(21)
A120587lowast
119867(119905 119909 V 119897) = 0
119867 (119879 119909 V 119897) = 1199092
(22)
where 120587lowast= arg sup
120587isinΠ(119905119909V119897)A120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897)
then 119881(119905 119909 V 119897) = 119865(119905 119909 V 119897) 119910120587lowast
(119905 119909 V 119897) = 119866(119905 119909 V 119897) and119911120587lowast
(119905 119909 V 119897) = 119867(119905 119909 V 119897) and 120587lowast is the equilibrium strategywhere
120585120587(119905 119909 V 119897)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901)) + 119896119897] 119891
119909
+ [120572 (120575 minus V)] 119891V + 120572119897119897119891119897
+1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092119880
120587
1+1
21205902
V V119880120587
2
+1
211989721205731015840
119897120573119897119880
120587
3+ 120590VradicV119909 (120587Σ minus 120588
1015840
119901) 120573V119880
120587
4
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897119880
120587
5+ 120590VradicV119897120573
1015840
119897120573V119880
120587
6
119880120587
1
= 119891119909119909+ 119891
119910119910(119910
120587
119909)2
+ 119891119911119911(119911
120587
119909)2
+ 2119891119909119910119910120587
119909+ 2119891
119909119911119911120587
119909
+ 2119891119910119911119910120587
119909119911120587
119909
119880120587
2
= 119891VV + 119891119910119910 (119910120587
V )2
+ 119891119911119911(119911
120587
V )2
+ 2119891V119910119910120587
V + 2119891V119911119911120587
V
+ 2119891119910119911119910120587
V 119911120587
V
119880120587
3
= 119891119897119897+ 119891
119910119910(119910
120587
119897)2
+ 119891119911119911(119911
120587
119897)2
+ 2119891119897119910119910120587
119897+ 2119891
119897119911119911120587
119897
+ 2119891119910119911119910120587
119897119911120587
119897
119880120587
4
= 119891119910119910119910120587
119909119910120587
V + 119891119911119911119911120587
119909119911120587
V + 119891119909V + 119891119909119910119910120587
V + 119891119909119911119911120587
V
+ 119891V119910119910120587
119909+ 119891V119911119911
120587
119909+ 119891
119910119911(119910
120587
119909119911120587
V + 119910120587
V 119911120587
119909)
119880120587
5
= 119891119910119910119910120587
119909119910120587
119897+ 119891
119911119911119911120587
119909119911120587
119897+ 119891
119909119897+ 119891
119909119910119910120587
119897+ 119891
119909119911119911120587
119897
+ 119891119897119910119910120587
119909+ 119891
119897119911119911120587
119909+ 119891
119910119911(119910
120587
119909119911120587
119897+ 119910
120587
119897119911120587
119909)
119880120587
6
= 119891119910119910119910120587
V 119910120587
119897+ 119891
119911119911119911120587
V 119911120587
119897+ 119891V119897 + 119891V119910119910
120587
119897+ 119891V119911119911
120587
119897
+ 119891119897119910119910120587
V + 119891119897119911119911120587
V + 119891119910119911 (119910120587
119897119911120587
V + 119910120587
V 119911120587
119897)
(23)
and 119891 = 119891(119905 119909 V 119897 119910 119911) 119910120587= 119910
120587(119905 119909 V 119897) and 119911120587 =
119911120587(119905 119909 V 119897)
The proof of this theorem is given in the AppendixEquations (19) (21) and (22) are called an extended
HJB equations system By solving this equations systemwe can obtain the equilibrium strategy and correspondingequilibrium value function
32 Solution to the Equilibrium Strategy Since 119891(119905 119909 V 119897 119910119911) = 119910 minus (1205742)(119911 minus 119910
2) according to Theorem 7 we have
119880120587lowast
1= 120574119866
2
119909 119880120587
lowast
2= 120574119866
2
V 119880120587lowast
3= 120574119866
2
119897 119880120587
lowast
4= 120574119866
119909119866V 119880
120587lowast
5=
120574119866119909119866
119897 and 119880120587
lowast
6= 120574119866V119866119897
so
120585120587lowast
(119905 119909 V 119897) =1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092120574119866
2
119909
+1
21205902
V V1205741198662
V +1
211989721205731015840
119897120573119897120574119866
2
119897
+ 120590VradicV1198971205731015840
119897120573V120574119866V119866119897
+ 120590VradicV119909 (120587Σ minus 1205881015840
119901) 120573V120574119866V119866119909
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897120574119866
119909119866
119897
(24)
Mathematical Problems in Engineering 7
By differentiatingA120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897) with respect to
120587(119905) we can obtain the equilibrium strategy
120587lowast(119905) = minus
[119865119909(120579 minus 120588
119901)1015840
+ (119865119909V minus 120574119866119909
119866V) 120590VradicV1205731015840
V + (119865119909119897 minus 120574119866119909119866
119897) 119897120573
1015840
119897] Σ
minus1
(119865119909119909minus 120574119866
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(25)
Inspired by [24] we suppose 119865(119905 119909 V 119897) and 119866(119905 119909 V 119897) havethe following linear forms with respect to 119909 V and 119897
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int119879
119905119903(119904)119889119904
+119861 (119905)
120574V + 119862 (119905) 119897 +
119863 (119905)
120574 (40)
where 119861(119905) 119862(119905) and 119863(119905) are given by (34) (35) and (36)
Note that
E119905119909V119897 [119883
120587lowast
(119879)]
= 119909119890int119879
119905119903(119904)119889119904
+1
120574[119861 (119905) V + 119863 (119905)] + 119862 (119905) 119897
(41)
Var119905119909V119897 [119883
120587lowast
(119879)] = 119867 (119905 119909 V 119897) minus 1198662(119905 119909 V 119897)
=2
120574[119866 (119905 119909 V 119897) minus 119865 (119905 119909 V 119897)]
=2
1205742[(119861 (119905) minus 119861 (119905)) V + 119863 (119905) minus 119863 (119905)]
(42)
Thus we obtain the following so-called equilibrium efficientfrontier which reflects the relationship between the invest-ment risk and return under the equilibrium strategy
Corollary 9 For MV problem (P1) based on state (119905 119909 V 119897)
its equilibrium efficient frontier is
E119905119909V119897 [119883
120587lowast
(119879)] = 119909119890int119879
119905119903(119904)119889119904
+ 119862 (119905) 119897
+ 119873 (119905 V 119897) radicVar119905119909V119897 [119883
120587lowast
(119879)]
(43)
where 119873(119905 V 119897) = [119861(119905)V + 119863(119905)]
radic2[(119861(119905) minus 119861(119905))V + 119863(119905) minus 119863(119905)]
Remark 10 By (38) and (39) we find that the equilibriuminvestment proportion in the stock has the similar expressionas in [24] although their study focuses on the optimal invest-ment and reinsurance problem In addition the equilibriuminvestment money in the stock is only the function of time119905 which is independent of the current wealth However forthe inflation-linked bond the equilibrium investmentmoneydepends on current wealth and the more the wealth isthe more the inflation-linked bond is invested This resultis consistent with the economic intuition considering thatwhen the manager has more wealth the sensitivity to theinflation risk becomes higher and the corresponding hedgingdemand is improved
Remark 11 In view of (38) we find that if 120590119901gt 120590
119897 that is the
volatility rate of the inflation is larger than that of the salaryincome the equilibrium investment amount in the inflation-linked bond has positive relationship with the salary levelOtherwise it has negative relationship with the salary level
Remark 12 From (42) and (43) if Var119905119909V119897[119883
120587(119879)] = 0 it
means that the pension fund manager bears no risk at allunder the state (119905 119909 V 119897) this is equivalent to the case of120574 rarr infin that is the pension fund manager is fully averseto risk In this case 120587lowast
119878(119905) = 0 and 120587lowast
119868(119905) = 1 minus ((120590
119897minus
120590119901)119909120590
119901)119897119862(119905)119890
minusint119879
119905119903(119904)119889119904 This indicates that the manager will
only invest all the fund in the inflation-linked bond and therisk-free money market account and obtain the expected realterminal wealth of 119909119890int
119879
119905119903(119904)119889119904
+ 119862(119905)119897 at retirement time 119879 Itconsists of two parts The first part is the accumulation ofthe initial wealth 119909 and the second part can be seen as theaccumulation of the contribution
Mathematical Problems in Engineering 9
Remark 13 If we do not consider the risk of stochasticvolatility of the stock price that is let 120572 = 0 120590V = 0 thenthe equilibrium strategy is simplified as
120587lowast
119868(119905)
= [
120582119901(119905) minus 120590
119901minus 120582
119904120590119904
120574minus 119897 (120590
119897minus 120590
119901) 119862 (119905)]
119890minusint119879
119905119903(119904)119889119904
119909120590119901
+ 1
120587lowast
119878(119905) =
120582119904119890minusint119879
119905119903(119904)119889119904
120574119909
(44)
In this case we find that 120587lowast
119878(119905) is consistent with the
result of the ordinaryMVportfolio problem under the Black-Scholes market (see [11]) This indicates that the contributionof the salary income has no influence on the equilibriuminvestment amount of the stock Furthermore if we do notconsider the risk of inflation that is 120583
119901(119905) = 120590
119901= 120590
119904= 0
then 119877(119905) = 119903(119905) and the inflation-linked bond is degeneratedto the risk-free money market account
4 Problem Formulation after Retirement andVerification Theorem
After retirement we assume that the pension plan memberpurchases a paid-up annuity during the time horizon [119879 119879 +119873] Thus the objective function of the pension plan manageris to maximize the expected surplus after paying off thebenefit to the member and minimize the variance of thesurplus at time 119879
1= 119879 + 119873 So at the state (119905 119909 V) isin G
1
the manager faces the following optimal control problem
(119905 119909 V) = supisinΠ(119905119909V)
E119905119909V [119883
(119879
1)]
minus120574
2Var
119905119909V [119883(119879
1)]
(P2)
Similarly to the method of Section 3 we denote 119910(119905 119909 V) =
E119905119909V[119883
(119879
1)] 119911(119905 119909 V) = E
119905119909V[(119883(119879
1))
2] Then we can
rewrite
(119905 119909 V)
= supisinΠ(119905119909V)
(119905 119909 V 119910(119905 119909 V) 119911 (119905 119909 V))
(45)
where (119905 119909 V 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
41 VerificationTheorem In this subsection we first give thedefinition of equilibrium strategy after retirement and thengive the verification theoremwhich is satisfied by the equilib-rium value function Now for any 120601(119905 119909 V) isin 119862122
(119905 119909 V) = (119905 119909 V) and 119911lowast
(119905 119909
V) = (119905 119909 V) and lowast is the equilibrium strategy where
120593(119905 119909 V)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ Σ (120579 minus 120588
119901)) minus 119902] 119891
119909
+ 119891V120572 (120575 minus V) +1
2(Σ minus 120588
1015840
119901) (Σ minus 120588
1015840
119901)1015840
1199092
1
+1
21205902
V V
2+ 120590VradicV119909 (
1015840Σ minus 120588
1015840
119901) 120573V
3
10 Mathematical Problems in Engineering
1
= 119891119909119909+ 119891
119910119910(119910
119909)2
+ 119891119911119911(119911
119909)2
+ 2119891119909119910119910
119909+ 2119891
119909119911119911
119909
+ 2119891119910119911119910
119909119911
119909
2
= 119891VV + 119891119910119910 (119910
V )2
+ 119891119911119911(119911
V )2
+ 2119891V119910119910
V + 2119891V119911119911
V
+ 2119891119910119911119910
V 119911
V
3
= 119891119910119910119910
119909119910
V + 119891119911119911119911
119909119911
V + 119891119909V + 119891119909119910119910
V + 119891119909119911119911
V
+ 119891V119910119910
119909+ 119891V119911119911
119909+ 119891
119910119911(119910
119909119911
V + 119910
V 119911
119909)
(49)
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
For any 120601(119905 119909 V 119897) isin 1198621222(G) we define a set 119863
119901(G) =
120601(119905 119909 V 119897) | 120601(119905 119909 V 119897) isin 1198621222
(G) and all first-orderpartial derivatives of 120601(sdot 119909 V 119897) satisfy the polynomial growthcondition on O and we define a variational operator asfollows (in the following we will omit the variable 119905 fornotation convenience)
A120587120601 (119905 119909 V 119897)
= 120601119905+ 120601
119909[119896119897 + 119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901))]
+ 120601V120572 (120575 minus V) + 120601119897120572119897119897
+1
21206011199091199091199092(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
+1
2120601VV120590
2
V V
+1
212060111989711989711989721205731015840
119897120573119897+ 120601
119909V119909120590VradicV (120587Σ minus 1205881015840
119901) 120573V
+ 120601119909119897119909119897 (120587Σ minus 120588
1015840
119901) 120573
119897+ 120601V119897119897120590VradicV120573
1015840
119897120573V
(18)
where 120573V = (0 120588119904V)1015840 Now we give the verification theorem of
problem (P1)
Theorem 7 (verification theorem) For problem (P1) if there
exist three real value functions 119865(119905 119909 V 119897) 119866(119905 119909 V 119897) and119867(119905 119909 V 119897) isin 119863
119901(G) satisfying the following conditions
forall(119905 119909 V 119897) isin G
sup120587isinΠ(119905119909V119897)
A120587119865 (119905 119909 V 119897) minus 120585120587 (119905 119909 V 119897) = 0 (19)
119865 (119879 119909 V 119897) = 119891 (119879 119909 V 119897 119909 1199092) (20)
A120587lowast
119866 (119905 119909 V 119897) = 0
119866 (119879 119909 V 119897) = 119909(21)
A120587lowast
119867(119905 119909 V 119897) = 0
119867 (119879 119909 V 119897) = 1199092
(22)
where 120587lowast= arg sup
120587isinΠ(119905119909V119897)A120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897)
then 119881(119905 119909 V 119897) = 119865(119905 119909 V 119897) 119910120587lowast
(119905 119909 V 119897) = 119866(119905 119909 V 119897) and119911120587lowast
(119905 119909 V 119897) = 119867(119905 119909 V 119897) and 120587lowast is the equilibrium strategywhere
120585120587(119905 119909 V 119897)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ 120587Σ (120579 minus 120588
119901)) + 119896119897] 119891
119909
+ [120572 (120575 minus V)] 119891V + 120572119897119897119891119897
+1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092119880
120587
1+1
21205902
V V119880120587
2
+1
211989721205731015840
119897120573119897119880
120587
3+ 120590VradicV119909 (120587Σ minus 120588
1015840
119901) 120573V119880
120587
4
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897119880
120587
5+ 120590VradicV119897120573
1015840
119897120573V119880
120587
6
119880120587
1
= 119891119909119909+ 119891
119910119910(119910
120587
119909)2
+ 119891119911119911(119911
120587
119909)2
+ 2119891119909119910119910120587
119909+ 2119891
119909119911119911120587
119909
+ 2119891119910119911119910120587
119909119911120587
119909
119880120587
2
= 119891VV + 119891119910119910 (119910120587
V )2
+ 119891119911119911(119911
120587
V )2
+ 2119891V119910119910120587
V + 2119891V119911119911120587
V
+ 2119891119910119911119910120587
V 119911120587
V
119880120587
3
= 119891119897119897+ 119891
119910119910(119910
120587
119897)2
+ 119891119911119911(119911
120587
119897)2
+ 2119891119897119910119910120587
119897+ 2119891
119897119911119911120587
119897
+ 2119891119910119911119910120587
119897119911120587
119897
119880120587
4
= 119891119910119910119910120587
119909119910120587
V + 119891119911119911119911120587
119909119911120587
V + 119891119909V + 119891119909119910119910120587
V + 119891119909119911119911120587
V
+ 119891V119910119910120587
119909+ 119891V119911119911
120587
119909+ 119891
119910119911(119910
120587
119909119911120587
V + 119910120587
V 119911120587
119909)
119880120587
5
= 119891119910119910119910120587
119909119910120587
119897+ 119891
119911119911119911120587
119909119911120587
119897+ 119891
119909119897+ 119891
119909119910119910120587
119897+ 119891
119909119911119911120587
119897
+ 119891119897119910119910120587
119909+ 119891
119897119911119911120587
119909+ 119891
119910119911(119910
120587
119909119911120587
119897+ 119910
120587
119897119911120587
119909)
119880120587
6
= 119891119910119910119910120587
V 119910120587
119897+ 119891
119911119911119911120587
V 119911120587
119897+ 119891V119897 + 119891V119910119910
120587
119897+ 119891V119911119911
120587
119897
+ 119891119897119910119910120587
V + 119891119897119911119911120587
V + 119891119910119911 (119910120587
119897119911120587
V + 119910120587
V 119911120587
119897)
(23)
and 119891 = 119891(119905 119909 V 119897 119910 119911) 119910120587= 119910
120587(119905 119909 V 119897) and 119911120587 =
119911120587(119905 119909 V 119897)
The proof of this theorem is given in the AppendixEquations (19) (21) and (22) are called an extended
HJB equations system By solving this equations systemwe can obtain the equilibrium strategy and correspondingequilibrium value function
32 Solution to the Equilibrium Strategy Since 119891(119905 119909 V 119897 119910119911) = 119910 minus (1205742)(119911 minus 119910
2) according to Theorem 7 we have
119880120587lowast
1= 120574119866
2
119909 119880120587
lowast
2= 120574119866
2
V 119880120587lowast
3= 120574119866
2
119897 119880120587
lowast
4= 120574119866
119909119866V 119880
120587lowast
5=
120574119866119909119866
119897 and 119880120587
lowast
6= 120574119866V119866119897
so
120585120587lowast
(119905 119909 V 119897) =1
2(120587Σ minus 120588
1015840
119901) (120587Σ minus 120588
1015840
119901)1015840
1199092120574119866
2
119909
+1
21205902
V V1205741198662
V +1
211989721205731015840
119897120573119897120574119866
2
119897
+ 120590VradicV1198971205731015840
119897120573V120574119866V119866119897
+ 120590VradicV119909 (120587Σ minus 1205881015840
119901) 120573V120574119866V119866119909
+ 119909119897 (120587Σ minus 1205881015840
119901) 120573
119897120574119866
119909119866
119897
(24)
Mathematical Problems in Engineering 7
By differentiatingA120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897) with respect to
120587(119905) we can obtain the equilibrium strategy
120587lowast(119905) = minus
[119865119909(120579 minus 120588
119901)1015840
+ (119865119909V minus 120574119866119909
119866V) 120590VradicV1205731015840
V + (119865119909119897 minus 120574119866119909119866
119897) 119897120573
1015840
119897] Σ
minus1
(119865119909119909minus 120574119866
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(25)
Inspired by [24] we suppose 119865(119905 119909 V 119897) and 119866(119905 119909 V 119897) havethe following linear forms with respect to 119909 V and 119897
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int119879
119905119903(119904)119889119904
+119861 (119905)
120574V + 119862 (119905) 119897 +
119863 (119905)
120574 (40)
where 119861(119905) 119862(119905) and 119863(119905) are given by (34) (35) and (36)
Note that
E119905119909V119897 [119883
120587lowast
(119879)]
= 119909119890int119879
119905119903(119904)119889119904
+1
120574[119861 (119905) V + 119863 (119905)] + 119862 (119905) 119897
(41)
Var119905119909V119897 [119883
120587lowast
(119879)] = 119867 (119905 119909 V 119897) minus 1198662(119905 119909 V 119897)
=2
120574[119866 (119905 119909 V 119897) minus 119865 (119905 119909 V 119897)]
=2
1205742[(119861 (119905) minus 119861 (119905)) V + 119863 (119905) minus 119863 (119905)]
(42)
Thus we obtain the following so-called equilibrium efficientfrontier which reflects the relationship between the invest-ment risk and return under the equilibrium strategy
Corollary 9 For MV problem (P1) based on state (119905 119909 V 119897)
its equilibrium efficient frontier is
E119905119909V119897 [119883
120587lowast
(119879)] = 119909119890int119879
119905119903(119904)119889119904
+ 119862 (119905) 119897
+ 119873 (119905 V 119897) radicVar119905119909V119897 [119883
120587lowast
(119879)]
(43)
where 119873(119905 V 119897) = [119861(119905)V + 119863(119905)]
radic2[(119861(119905) minus 119861(119905))V + 119863(119905) minus 119863(119905)]
Remark 10 By (38) and (39) we find that the equilibriuminvestment proportion in the stock has the similar expressionas in [24] although their study focuses on the optimal invest-ment and reinsurance problem In addition the equilibriuminvestment money in the stock is only the function of time119905 which is independent of the current wealth However forthe inflation-linked bond the equilibrium investmentmoneydepends on current wealth and the more the wealth isthe more the inflation-linked bond is invested This resultis consistent with the economic intuition considering thatwhen the manager has more wealth the sensitivity to theinflation risk becomes higher and the corresponding hedgingdemand is improved
Remark 11 In view of (38) we find that if 120590119901gt 120590
119897 that is the
volatility rate of the inflation is larger than that of the salaryincome the equilibrium investment amount in the inflation-linked bond has positive relationship with the salary levelOtherwise it has negative relationship with the salary level
Remark 12 From (42) and (43) if Var119905119909V119897[119883
120587(119879)] = 0 it
means that the pension fund manager bears no risk at allunder the state (119905 119909 V 119897) this is equivalent to the case of120574 rarr infin that is the pension fund manager is fully averseto risk In this case 120587lowast
119878(119905) = 0 and 120587lowast
119868(119905) = 1 minus ((120590
119897minus
120590119901)119909120590
119901)119897119862(119905)119890
minusint119879
119905119903(119904)119889119904 This indicates that the manager will
only invest all the fund in the inflation-linked bond and therisk-free money market account and obtain the expected realterminal wealth of 119909119890int
119879
119905119903(119904)119889119904
+ 119862(119905)119897 at retirement time 119879 Itconsists of two parts The first part is the accumulation ofthe initial wealth 119909 and the second part can be seen as theaccumulation of the contribution
Mathematical Problems in Engineering 9
Remark 13 If we do not consider the risk of stochasticvolatility of the stock price that is let 120572 = 0 120590V = 0 thenthe equilibrium strategy is simplified as
120587lowast
119868(119905)
= [
120582119901(119905) minus 120590
119901minus 120582
119904120590119904
120574minus 119897 (120590
119897minus 120590
119901) 119862 (119905)]
119890minusint119879
119905119903(119904)119889119904
119909120590119901
+ 1
120587lowast
119878(119905) =
120582119904119890minusint119879
119905119903(119904)119889119904
120574119909
(44)
In this case we find that 120587lowast
119878(119905) is consistent with the
result of the ordinaryMVportfolio problem under the Black-Scholes market (see [11]) This indicates that the contributionof the salary income has no influence on the equilibriuminvestment amount of the stock Furthermore if we do notconsider the risk of inflation that is 120583
119901(119905) = 120590
119901= 120590
119904= 0
then 119877(119905) = 119903(119905) and the inflation-linked bond is degeneratedto the risk-free money market account
4 Problem Formulation after Retirement andVerification Theorem
After retirement we assume that the pension plan memberpurchases a paid-up annuity during the time horizon [119879 119879 +119873] Thus the objective function of the pension plan manageris to maximize the expected surplus after paying off thebenefit to the member and minimize the variance of thesurplus at time 119879
1= 119879 + 119873 So at the state (119905 119909 V) isin G
1
the manager faces the following optimal control problem
(119905 119909 V) = supisinΠ(119905119909V)
E119905119909V [119883
(119879
1)]
minus120574
2Var
119905119909V [119883(119879
1)]
(P2)
Similarly to the method of Section 3 we denote 119910(119905 119909 V) =
E119905119909V[119883
(119879
1)] 119911(119905 119909 V) = E
119905119909V[(119883(119879
1))
2] Then we can
rewrite
(119905 119909 V)
= supisinΠ(119905119909V)
(119905 119909 V 119910(119905 119909 V) 119911 (119905 119909 V))
(45)
where (119905 119909 V 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
41 VerificationTheorem In this subsection we first give thedefinition of equilibrium strategy after retirement and thengive the verification theoremwhich is satisfied by the equilib-rium value function Now for any 120601(119905 119909 V) isin 119862122
(119905 119909 V) = (119905 119909 V) and 119911lowast
(119905 119909
V) = (119905 119909 V) and lowast is the equilibrium strategy where
120593(119905 119909 V)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ Σ (120579 minus 120588
119901)) minus 119902] 119891
119909
+ 119891V120572 (120575 minus V) +1
2(Σ minus 120588
1015840
119901) (Σ minus 120588
1015840
119901)1015840
1199092
1
+1
21205902
V V
2+ 120590VradicV119909 (
1015840Σ minus 120588
1015840
119901) 120573V
3
10 Mathematical Problems in Engineering
1
= 119891119909119909+ 119891
119910119910(119910
119909)2
+ 119891119911119911(119911
119909)2
+ 2119891119909119910119910
119909+ 2119891
119909119911119911
119909
+ 2119891119910119911119910
119909119911
119909
2
= 119891VV + 119891119910119910 (119910
V )2
+ 119891119911119911(119911
V )2
+ 2119891V119910119910
V + 2119891V119911119911
V
+ 2119891119910119911119910
V 119911
V
3
= 119891119910119910119910
119909119910
V + 119891119911119911119911
119909119911
V + 119891119909V + 119891119909119910119910
V + 119891119909119911119911
V
+ 119891V119910119910
119909+ 119891V119911119911
119909+ 119891
119910119911(119910
119909119911
V + 119910
V 119911
119909)
(49)
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
By differentiatingA120587119865(119905 119909 V 119897) minus 120585120587(119905 119909 V 119897) with respect to
120587(119905) we can obtain the equilibrium strategy
120587lowast(119905) = minus
[119865119909(120579 minus 120588
119901)1015840
+ (119865119909V minus 120574119866119909
119866V) 120590VradicV1205731015840
V + (119865119909119897 minus 120574119866119909119866
119897) 119897120573
1015840
119897] Σ
minus1
(119865119909119909minus 120574119866
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(25)
Inspired by [24] we suppose 119865(119905 119909 V 119897) and 119866(119905 119909 V 119897) havethe following linear forms with respect to 119909 V and 119897
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int119879
119905119903(119904)119889119904
+119861 (119905)
120574V + 119862 (119905) 119897 +
119863 (119905)
120574 (40)
where 119861(119905) 119862(119905) and 119863(119905) are given by (34) (35) and (36)
Note that
E119905119909V119897 [119883
120587lowast
(119879)]
= 119909119890int119879
119905119903(119904)119889119904
+1
120574[119861 (119905) V + 119863 (119905)] + 119862 (119905) 119897
(41)
Var119905119909V119897 [119883
120587lowast
(119879)] = 119867 (119905 119909 V 119897) minus 1198662(119905 119909 V 119897)
=2
120574[119866 (119905 119909 V 119897) minus 119865 (119905 119909 V 119897)]
=2
1205742[(119861 (119905) minus 119861 (119905)) V + 119863 (119905) minus 119863 (119905)]
(42)
Thus we obtain the following so-called equilibrium efficientfrontier which reflects the relationship between the invest-ment risk and return under the equilibrium strategy
Corollary 9 For MV problem (P1) based on state (119905 119909 V 119897)
its equilibrium efficient frontier is
E119905119909V119897 [119883
120587lowast
(119879)] = 119909119890int119879
119905119903(119904)119889119904
+ 119862 (119905) 119897
+ 119873 (119905 V 119897) radicVar119905119909V119897 [119883
120587lowast
(119879)]
(43)
where 119873(119905 V 119897) = [119861(119905)V + 119863(119905)]
radic2[(119861(119905) minus 119861(119905))V + 119863(119905) minus 119863(119905)]
Remark 10 By (38) and (39) we find that the equilibriuminvestment proportion in the stock has the similar expressionas in [24] although their study focuses on the optimal invest-ment and reinsurance problem In addition the equilibriuminvestment money in the stock is only the function of time119905 which is independent of the current wealth However forthe inflation-linked bond the equilibrium investmentmoneydepends on current wealth and the more the wealth isthe more the inflation-linked bond is invested This resultis consistent with the economic intuition considering thatwhen the manager has more wealth the sensitivity to theinflation risk becomes higher and the corresponding hedgingdemand is improved
Remark 11 In view of (38) we find that if 120590119901gt 120590
119897 that is the
volatility rate of the inflation is larger than that of the salaryincome the equilibrium investment amount in the inflation-linked bond has positive relationship with the salary levelOtherwise it has negative relationship with the salary level
Remark 12 From (42) and (43) if Var119905119909V119897[119883
120587(119879)] = 0 it
means that the pension fund manager bears no risk at allunder the state (119905 119909 V 119897) this is equivalent to the case of120574 rarr infin that is the pension fund manager is fully averseto risk In this case 120587lowast
119878(119905) = 0 and 120587lowast
119868(119905) = 1 minus ((120590
119897minus
120590119901)119909120590
119901)119897119862(119905)119890
minusint119879
119905119903(119904)119889119904 This indicates that the manager will
only invest all the fund in the inflation-linked bond and therisk-free money market account and obtain the expected realterminal wealth of 119909119890int
119879
119905119903(119904)119889119904
+ 119862(119905)119897 at retirement time 119879 Itconsists of two parts The first part is the accumulation ofthe initial wealth 119909 and the second part can be seen as theaccumulation of the contribution
Mathematical Problems in Engineering 9
Remark 13 If we do not consider the risk of stochasticvolatility of the stock price that is let 120572 = 0 120590V = 0 thenthe equilibrium strategy is simplified as
120587lowast
119868(119905)
= [
120582119901(119905) minus 120590
119901minus 120582
119904120590119904
120574minus 119897 (120590
119897minus 120590
119901) 119862 (119905)]
119890minusint119879
119905119903(119904)119889119904
119909120590119901
+ 1
120587lowast
119878(119905) =
120582119904119890minusint119879
119905119903(119904)119889119904
120574119909
(44)
In this case we find that 120587lowast
119878(119905) is consistent with the
result of the ordinaryMVportfolio problem under the Black-Scholes market (see [11]) This indicates that the contributionof the salary income has no influence on the equilibriuminvestment amount of the stock Furthermore if we do notconsider the risk of inflation that is 120583
119901(119905) = 120590
119901= 120590
119904= 0
then 119877(119905) = 119903(119905) and the inflation-linked bond is degeneratedto the risk-free money market account
4 Problem Formulation after Retirement andVerification Theorem
After retirement we assume that the pension plan memberpurchases a paid-up annuity during the time horizon [119879 119879 +119873] Thus the objective function of the pension plan manageris to maximize the expected surplus after paying off thebenefit to the member and minimize the variance of thesurplus at time 119879
1= 119879 + 119873 So at the state (119905 119909 V) isin G
1
the manager faces the following optimal control problem
(119905 119909 V) = supisinΠ(119905119909V)
E119905119909V [119883
(119879
1)]
minus120574
2Var
119905119909V [119883(119879
1)]
(P2)
Similarly to the method of Section 3 we denote 119910(119905 119909 V) =
E119905119909V[119883
(119879
1)] 119911(119905 119909 V) = E
119905119909V[(119883(119879
1))
2] Then we can
rewrite
(119905 119909 V)
= supisinΠ(119905119909V)
(119905 119909 V 119910(119905 119909 V) 119911 (119905 119909 V))
(45)
where (119905 119909 V 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
41 VerificationTheorem In this subsection we first give thedefinition of equilibrium strategy after retirement and thengive the verification theoremwhich is satisfied by the equilib-rium value function Now for any 120601(119905 119909 V) isin 119862122
(119905 119909 V) = (119905 119909 V) and 119911lowast
(119905 119909
V) = (119905 119909 V) and lowast is the equilibrium strategy where
120593(119905 119909 V)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ Σ (120579 minus 120588
119901)) minus 119902] 119891
119909
+ 119891V120572 (120575 minus V) +1
2(Σ minus 120588
1015840
119901) (Σ minus 120588
1015840
119901)1015840
1199092
1
+1
21205902
V V
2+ 120590VradicV119909 (
1015840Σ minus 120588
1015840
119901) 120573V
3
10 Mathematical Problems in Engineering
1
= 119891119909119909+ 119891
119910119910(119910
119909)2
+ 119891119911119911(119911
119909)2
+ 2119891119909119910119910
119909+ 2119891
119909119911119911
119909
+ 2119891119910119911119910
119909119911
119909
2
= 119891VV + 119891119910119910 (119910
V )2
+ 119891119911119911(119911
V )2
+ 2119891V119910119910
V + 2119891V119911119911
V
+ 2119891119910119911119910
V 119911
V
3
= 119891119910119910119910
119909119910
V + 119891119911119911119911
119909119911
V + 119891119909V + 119891119909119910119910
V + 119891119909119911119911
V
+ 119891V119910119910
119909+ 119891V119911119911
119909+ 119891
119910119911(119910
119909119911
V + 119910
V 119911
119909)
(49)
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int119879
119905119903(119904)119889119904
+119861 (119905)
120574V + 119862 (119905) 119897 +
119863 (119905)
120574 (40)
where 119861(119905) 119862(119905) and 119863(119905) are given by (34) (35) and (36)
Note that
E119905119909V119897 [119883
120587lowast
(119879)]
= 119909119890int119879
119905119903(119904)119889119904
+1
120574[119861 (119905) V + 119863 (119905)] + 119862 (119905) 119897
(41)
Var119905119909V119897 [119883
120587lowast
(119879)] = 119867 (119905 119909 V 119897) minus 1198662(119905 119909 V 119897)
=2
120574[119866 (119905 119909 V 119897) minus 119865 (119905 119909 V 119897)]
=2
1205742[(119861 (119905) minus 119861 (119905)) V + 119863 (119905) minus 119863 (119905)]
(42)
Thus we obtain the following so-called equilibrium efficientfrontier which reflects the relationship between the invest-ment risk and return under the equilibrium strategy
Corollary 9 For MV problem (P1) based on state (119905 119909 V 119897)
its equilibrium efficient frontier is
E119905119909V119897 [119883
120587lowast
(119879)] = 119909119890int119879
119905119903(119904)119889119904
+ 119862 (119905) 119897
+ 119873 (119905 V 119897) radicVar119905119909V119897 [119883
120587lowast
(119879)]
(43)
where 119873(119905 V 119897) = [119861(119905)V + 119863(119905)]
radic2[(119861(119905) minus 119861(119905))V + 119863(119905) minus 119863(119905)]
Remark 10 By (38) and (39) we find that the equilibriuminvestment proportion in the stock has the similar expressionas in [24] although their study focuses on the optimal invest-ment and reinsurance problem In addition the equilibriuminvestment money in the stock is only the function of time119905 which is independent of the current wealth However forthe inflation-linked bond the equilibrium investmentmoneydepends on current wealth and the more the wealth isthe more the inflation-linked bond is invested This resultis consistent with the economic intuition considering thatwhen the manager has more wealth the sensitivity to theinflation risk becomes higher and the corresponding hedgingdemand is improved
Remark 11 In view of (38) we find that if 120590119901gt 120590
119897 that is the
volatility rate of the inflation is larger than that of the salaryincome the equilibrium investment amount in the inflation-linked bond has positive relationship with the salary levelOtherwise it has negative relationship with the salary level
Remark 12 From (42) and (43) if Var119905119909V119897[119883
120587(119879)] = 0 it
means that the pension fund manager bears no risk at allunder the state (119905 119909 V 119897) this is equivalent to the case of120574 rarr infin that is the pension fund manager is fully averseto risk In this case 120587lowast
119878(119905) = 0 and 120587lowast
119868(119905) = 1 minus ((120590
119897minus
120590119901)119909120590
119901)119897119862(119905)119890
minusint119879
119905119903(119904)119889119904 This indicates that the manager will
only invest all the fund in the inflation-linked bond and therisk-free money market account and obtain the expected realterminal wealth of 119909119890int
119879
119905119903(119904)119889119904
+ 119862(119905)119897 at retirement time 119879 Itconsists of two parts The first part is the accumulation ofthe initial wealth 119909 and the second part can be seen as theaccumulation of the contribution
Mathematical Problems in Engineering 9
Remark 13 If we do not consider the risk of stochasticvolatility of the stock price that is let 120572 = 0 120590V = 0 thenthe equilibrium strategy is simplified as
120587lowast
119868(119905)
= [
120582119901(119905) minus 120590
119901minus 120582
119904120590119904
120574minus 119897 (120590
119897minus 120590
119901) 119862 (119905)]
119890minusint119879
119905119903(119904)119889119904
119909120590119901
+ 1
120587lowast
119878(119905) =
120582119904119890minusint119879
119905119903(119904)119889119904
120574119909
(44)
In this case we find that 120587lowast
119878(119905) is consistent with the
result of the ordinaryMVportfolio problem under the Black-Scholes market (see [11]) This indicates that the contributionof the salary income has no influence on the equilibriuminvestment amount of the stock Furthermore if we do notconsider the risk of inflation that is 120583
119901(119905) = 120590
119901= 120590
119904= 0
then 119877(119905) = 119903(119905) and the inflation-linked bond is degeneratedto the risk-free money market account
4 Problem Formulation after Retirement andVerification Theorem
After retirement we assume that the pension plan memberpurchases a paid-up annuity during the time horizon [119879 119879 +119873] Thus the objective function of the pension plan manageris to maximize the expected surplus after paying off thebenefit to the member and minimize the variance of thesurplus at time 119879
1= 119879 + 119873 So at the state (119905 119909 V) isin G
1
the manager faces the following optimal control problem
(119905 119909 V) = supisinΠ(119905119909V)
E119905119909V [119883
(119879
1)]
minus120574
2Var
119905119909V [119883(119879
1)]
(P2)
Similarly to the method of Section 3 we denote 119910(119905 119909 V) =
E119905119909V[119883
(119879
1)] 119911(119905 119909 V) = E
119905119909V[(119883(119879
1))
2] Then we can
rewrite
(119905 119909 V)
= supisinΠ(119905119909V)
(119905 119909 V 119910(119905 119909 V) 119911 (119905 119909 V))
(45)
where (119905 119909 V 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
41 VerificationTheorem In this subsection we first give thedefinition of equilibrium strategy after retirement and thengive the verification theoremwhich is satisfied by the equilib-rium value function Now for any 120601(119905 119909 V) isin 119862122
(119905 119909 V) = (119905 119909 V) and 119911lowast
(119905 119909
V) = (119905 119909 V) and lowast is the equilibrium strategy where
120593(119905 119909 V)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ Σ (120579 minus 120588
119901)) minus 119902] 119891
119909
+ 119891V120572 (120575 minus V) +1
2(Σ minus 120588
1015840
119901) (Σ minus 120588
1015840
119901)1015840
1199092
1
+1
21205902
V V
2+ 120590VradicV119909 (
1015840Σ minus 120588
1015840
119901) 120573V
3
10 Mathematical Problems in Engineering
1
= 119891119909119909+ 119891
119910119910(119910
119909)2
+ 119891119911119911(119911
119909)2
+ 2119891119909119910119910
119909+ 2119891
119909119911119911
119909
+ 2119891119910119911119910
119909119911
119909
2
= 119891VV + 119891119910119910 (119910
V )2
+ 119891119911119911(119911
V )2
+ 2119891V119910119910
V + 2119891V119911119911
V
+ 2119891119910119911119910
V 119911
V
3
= 119891119910119910119910
119909119910
V + 119891119911119911119911
119909119911
V + 119891119909V + 119891119909119910119910
V + 119891119909119911119911
V
+ 119891V119910119910
119909+ 119891V119911119911
119909+ 119891
119910119911(119910
119909119911
V + 119910
V 119911
119909)
(49)
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
Remark 13 If we do not consider the risk of stochasticvolatility of the stock price that is let 120572 = 0 120590V = 0 thenthe equilibrium strategy is simplified as
120587lowast
119868(119905)
= [
120582119901(119905) minus 120590
119901minus 120582
119904120590119904
120574minus 119897 (120590
119897minus 120590
119901) 119862 (119905)]
119890minusint119879
119905119903(119904)119889119904
119909120590119901
+ 1
120587lowast
119878(119905) =
120582119904119890minusint119879
119905119903(119904)119889119904
120574119909
(44)
In this case we find that 120587lowast
119878(119905) is consistent with the
result of the ordinaryMVportfolio problem under the Black-Scholes market (see [11]) This indicates that the contributionof the salary income has no influence on the equilibriuminvestment amount of the stock Furthermore if we do notconsider the risk of inflation that is 120583
119901(119905) = 120590
119901= 120590
119904= 0
then 119877(119905) = 119903(119905) and the inflation-linked bond is degeneratedto the risk-free money market account
4 Problem Formulation after Retirement andVerification Theorem
After retirement we assume that the pension plan memberpurchases a paid-up annuity during the time horizon [119879 119879 +119873] Thus the objective function of the pension plan manageris to maximize the expected surplus after paying off thebenefit to the member and minimize the variance of thesurplus at time 119879
1= 119879 + 119873 So at the state (119905 119909 V) isin G
1
the manager faces the following optimal control problem
(119905 119909 V) = supisinΠ(119905119909V)
E119905119909V [119883
(119879
1)]
minus120574
2Var
119905119909V [119883(119879
1)]
(P2)
Similarly to the method of Section 3 we denote 119910(119905 119909 V) =
E119905119909V[119883
(119879
1)] 119911(119905 119909 V) = E
119905119909V[(119883(119879
1))
2] Then we can
rewrite
(119905 119909 V)
= supisinΠ(119905119909V)
(119905 119909 V 119910(119905 119909 V) 119911 (119905 119909 V))
(45)
where (119905 119909 V 119910 119911) = 119910 minus (1205742)(119911 minus 1199102)
41 VerificationTheorem In this subsection we first give thedefinition of equilibrium strategy after retirement and thengive the verification theoremwhich is satisfied by the equilib-rium value function Now for any 120601(119905 119909 V) isin 119862122
(119905 119909 V) = (119905 119909 V) and 119911lowast
(119905 119909
V) = (119905 119909 V) and lowast is the equilibrium strategy where
120593(119905 119909 V)
= 119891119905+ [119909 (119877 minus 120583
119901+ 120590
2
119901+ Σ (120579 minus 120588
119901)) minus 119902] 119891
119909
+ 119891V120572 (120575 minus V) +1
2(Σ minus 120588
1015840
119901) (Σ minus 120588
1015840
119901)1015840
1199092
1
+1
21205902
V V
2+ 120590VradicV119909 (
1015840Σ minus 120588
1015840
119901) 120573V
3
10 Mathematical Problems in Engineering
1
= 119891119909119909+ 119891
119910119910(119910
119909)2
+ 119891119911119911(119911
119909)2
+ 2119891119909119910119910
119909+ 2119891
119909119911119911
119909
+ 2119891119910119911119910
119909119911
119909
2
= 119891VV + 119891119910119910 (119910
V )2
+ 119891119911119911(119911
V )2
+ 2119891V119910119910
V + 2119891V119911119911
V
+ 2119891119910119911119910
V 119911
V
3
= 119891119910119910119910
119909119910
V + 119891119911119911119911
119909119911
V + 119891119909V + 119891119909119910119910
V + 119891119909119911119911
V
+ 119891V119910119910
119909+ 119891V119911119911
119909+ 119891
119910119911(119910
119909119911
V + 119910
V 119911
119909)
(49)
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
and 119891 = 119891(119905 119909 V 119910 119911) 119910= 119910
(119905 119909 V) and 119911 = 119911(119905 119909 V)
The proof is similar to Theorem 7 and here we omit it
42 Solution to the Equilibrium Strategy Using Theorem 15nowwe calculate the equilibrium strategy and correspondingequilibrium value function By the similar method to that inSection 3 we can obtain the equilibrium strategy as followslowast(119905)
= minus
[119909(120579 minus 120588
119901)1015840
+ (119909V minus 120574119909
V) 120590VradicV1205731015840
V] Σminus1
(119909119909minus 120574
2
119909) 119909
+ 1205881015840
119901Σ
minus1
(50)
We suppose (119905 119909 V) and (119905 119909 V) have the following linearforms with respect to 119909 and V
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
and the corresponding equilibrium value function is
119881 (119905 119909 V 119897) = 119909119890int1198791
119905119903(119904)119889119904
+119887 (119905)
120574V +
119888 (119905)
120574 (63)
where 119887(119905) and 119888(119905) are given by (59) and (60)After some calculations we have the following corollary
Corollary 17 Based on state (119905 119909 V) for the MV problem(P
2) the equilibrium efficient frontier is
E119905119909V [119883
lowast
(1198791)] = 119909119890
int1198791
119905119903(119904)119889119904
minus 119902int
1198791
119905
119890int1198791
119904119903(119906)119889119906
119889119904
+ (119905 V) radicVar119905119909V [119883
lowast
(1198791)]
(64)
where
(119905 V) =119887 (119905) V + 120572120575int
1198791
119905[119887 (119904) + (120582
119901(119904) minus 120590
119901)2
] 119889119904
radic2 [(119887 (119905) minus 119887 (119905)) V + 120572120575int1198791
119905(119887 (119904) minus 119887 (119904) + (12) (120582
119901(119904) minus 120590
119901)2
) 119889119904]
(65)
Remark 18 From (62) and (64) we find that at the state(119905 119909 V) if manager does not undertake any risk in the latertime that is let Var
119905119909V[119883120587lowast
(1198791)] = 0 or 120574 rarr infin heshe will
only invest all the surplus into the inflation-linked bond afterpaying off a continuous annuity with real value of 119902 for eachtime Then at terminal time 119879
1the expected real net surplus
is 119909119890int1198791
119905119903(119904)119889119904
minus 119902 int1198791
119905119890int1198791
119904119903(119906)119889119906
119889119904 It includes two parts wherethe first part is the accumulation of the initial real wealth 119909and the second part is the accumulated payment of the realbenefit until time 119879
1
5 Sensitivity Analysis
In this section we analyze the influence of the differentparameters on the equilibrium strategy and the equilibriumefficient frontier Here we only analyze the case beforeretirement and suppose that 119877(119905) 119903(119905) and 120583
119901(119905) are all deter-
ministic constants for simplicity and the basic parameters areset in Table 1
In addition the initial values are1199090= 2119901
0= 1 V
0= 015
and 1198970= 05 Note that the parameter 120582
119901= 0015 based
on (7) To obtain the evolution process of the investmentstrategy over time using Monte Carlo method we simulate
the trajectory of the optimal wealth process with 300 timesandobtain themean investment proportion of the three assets
51 The Analysis of Equilibrium Strategies This subsectionworks on analyzing the influence of the inflation the salaryincome and the risk aversion level of the manager on theequilibrium strategy in the period before retirement Sincethe sign of the first-order derivative about the investmentstrategies to some parameters could not be obtained directlywe only give the intuitive analysis
Figure 1 depicts the mean equilibrium investment pro-portion of the three assets under the basic parameter settingWe find that the proportion of the stock decreases graduallyfrom 40 to almost 20 during time horizon [0 10] andthe proportion of the money market account also decreaseswith time However the proportion of the inflation-linkedbond is relatively low at the beginning even short sellinghappened at time 0 but after about 10 years the proportionincreases to almost 20 In the following analysis we useFigure 1 as a benchmark Figure 2 shows themean investmentproportion when 120574 = 08 Compared with Figure 1 the wholeequilibrium investment proportion of the stock moves downwhen the risk aversion level of the manager increases from120574 = 06 to 08 (this result also can be found from (39) since
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
under the basic parameter setting of this section 120597120587lowast
119878120597120574 lt
0) On the contrary the proportion of the inflation-linkedbond increases under the higher risk aversion level The casewhen the expectation inflation rate 120583
119901= 0042 is illustrated
in Figure 3 Since 120583119901is higher than that in Figure 1 the
manager invests more wealth in the inflation-linked bondto hedge the inflation risk Similarly Figure 4 shows that ifthe expected salary growth rate 120583
119897increases the investment
proportion in the inflation bond also increases This meansthat the more the wealth contributed to the pension fund thehigher the demand to hedge the inflation risk We note thatthe proportion of the stock is independent of 120583
119901and 120583
119897in
(39) so the investment proportion in the stock is the same asin Figure 1
52 The Analysis of the Equilibrium Efficient Frontier In thissubsection we analyze the influence of different parameterson the equilibrium efficient frontier in the investment periodbefore retirement Without loss of generality we only analyzethe efficient frontier at time 0
Figure 5 shows the efficient frontiers for the differentexpected inflation rate 120583
119901 Generally speaking the larger
the value of 120583119901 the lower the real value of the wealth So
as shown in Figure 5 the expected real terminal wealthhas a negative relationship with the expected inflation ratewhen the variance of the real terminal wealth is fixedFigure 6 reveals the relationship between the efficient frontier
0
01
02
03
04
05
06
07
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 2 120574 = 08
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Inflation bondStockMoney account
Figure 3 120583119901= 0042
and the volatility of the inflation rate 120590119901 We find that the
higher 120590119901leads to the efficient frontier moving upwards The
influence of the 120575 on efficient frontier is illustrated in Figure 7which shows that the value of the expected volatility of thestock has positive relationship with the efficient frontierFigure 8 depicts the influence of 120588
119904V on the efficient frontier
Mathematical Problems in Engineering 13
Inflation bondStockMoney account
minus04
minus02
0
02
04
06
08
1
Mea
n eq
uilib
rium
inve
stmen
t pro
port
ion
1 2 3 4 5 6 7 8 9 100Time (year)
Figure 4 120583119897= 009
E[X
120587lowast
(T)]
120583p = 0033
120583p = 0038
120583p = 0042
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
Figure 5 The effect of 120583119901on efficient frontier
This indicates that the stronger negative relationship betweenthe stock price 119878(119905) and the volatility 119881(119905) will lead to theefficient frontier moving upwards Figure 9 shows that theexpected growth rate of the salary has a positive influenceon the efficient frontier that is with increasing of 120583
119897 the
efficient frontiermoves upwards In addition the intersection
E[X
120587lowast
(T)]
35
4
45
5
55
6
65
05 1 15 2 25 30Var[X120587lowast (T)]
120590p = 02
120590p = 03
120590p = 04
Figure 6 The effect of 120590119901on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120575 = 02
120575 = 03
120575 = 04
Figure 7 The effect of 120575 on efficient frontier
of the efficient frontier and the E[119883120587lowast
(119879)] axis is bigger if 120583119897
is larger This result also can be obtained from Remark 12 InFigure 10 the influence of the salary volatility 120590
119897on efficient
frontier is revealed We find that the larger the 120590119897 the more
the uncertainty about the salary income which leads to theefficient frontier moving downwards
14 Mathematical Problems in EngineeringE[X
120587lowast
(T)]
05 1 15 2 25 30Var[X120587lowast (T)]
36
38
4
42
44
46
48
5
52
54
56
120588s = minus01
120588s = minus05
120588s = minus08
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
Figure 8 The effect of 120588119904V on efficient frontier
E[X
120587lowast
(T)]
35
4
45
5
55
6
05 1 15 2 25 30Var[X120587lowast (T)]
120583l = 005
120583l = 007
120583l = 009
Figure 9 The effect of 120583119897on efficient frontier
6 Conclusion
In this paper we investigate the MV portfolio problems fora DC pension plan both before and after retirement Thebackground risks such as the stochastic volatility of the stockprice and the inflation rate are also considered in our model
E[X
120587lowast
(T)]
36
38
4
42
44
46
48
5
52
54
56
05 1 15 2 25 30Var[X120587lowast (T)]
120590l = 01
120590l = 04
120590l = 07
Figure 10 The effect of 120590119897on efficient frontier
We assume that the pension fund that can be invested inthe financial market consists of an inflation-linked bonda stock and money market account Under the frameworkof game theory we obtain the equilibrium strategies andthe corresponding equilibrium efficient frontiers for bothperiods Finally using Monte Carlo method we give somenumerical analysis and find some interesting results Toobtain the closed-form solution we assume that the riskaversion parameter of the manager is a constant in this paperIn fact it may depend on the current wealth or investorsrsquocurrent investment return (see [35]) in practice In addi-tion in the period of after retirement the investment timehorizon is assumed to be fixed in our paper However sincethe lifetime of the pension plan member is stochastic theportfolio problem with uncertain investment time horizonmay bemore realistic and reasonable and these problemswillbe studied in the future
Appendix
To proveTheorem 7 first we give a lemma
Lemma A1 For the processes 119883120587(119905) 119871(119905) and 119881(119905) defined
in Section 2 if 120601(119905 119909 V 119897) isin 119863119901(G) then forall(119905 119909 V 119897) isin G and
forall120587 isin Π(119905 119909 V 119897) the following integrals are martingales
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
others can be proved similarly Let (119905 119909 V 119897) isin G and 120587 isin
Π(119905 119909 V 119897) We write 120601119909= 120601
119909(119904 119883
120587(119904) 119881(119904) 119871(119904)) for short
Note that
119872120587
2(119905 119909 V 119897)
= int
119879
119905
120601119909119883
120587(119904) 120587 (119904) 119889119882
119901(119904)
+ int
119879
119905
120601119909119883
120587(119904)radic119881 (119904)120587
119878(119904) 119889119882
119904(119904)
(A2)
where 120587(119904) = (120587119868(119904) minus 1)120590
119901+ 120587
119878(119904)120590
119904 Since 120601(119905 119909 V 119897) isin
119863119901(G) there exist two constants 119896
1gt 0 and 119896
2⩾ 1 such that
|120601119909(119905 119909 V 119897)| ⩽ 119896
1(1 + |119909|
1198962 + |V|1198962 + |119897|1198962) According to the
definition of the admissible strategy the linear growth andYamada and Watanabe conditions satisfied by 119881(119905) and 119871(119905)(see [24] for details) for the first part of (A2) we have
Proof of Theorem 7 The proof consists of five steps
Step 1 Consider an admissible strategy 120587 isin Π(119905 119909 V 119897) Wewill claim that if there exists a real value function119884(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119884 (119905 119909 V 119897) = 0
119884 (119879 119909 V 119897) = 119909(A5)
then 119884(119905 119909 V 119897) = 119910120587(119905 119909 V 119897)
In fact if wewrite119884120587(119905) = 119884(119905 119883
120587(119905) 119881(119905) 119871(119905)) for short
120587(119904) (120587 (119904) Σ (119904) minus 120588
1015840
119901)
+ 119884120587
119897(119904) 119871 (119904) 120573
1015840
119897] 119889119882 (119904)
(A6)
Using Lemma A1 and taking the conditional expectation onboth sides of (A6) we have
119884 (119905 119909 V 119897) = E119905119909V119897 [119884
120587(119879)] = E
119905119909V119897 [119883120587(119879)]
= 119910120587(119905 119909 V 119897)
(A7)
Similarly by replacing 119884 and 119910 by 119885 and 119911 respectively wecan claim that if there exists a real value function119885(119905 119909 V 119897) isin119863
119901(G) such that forall(119905 119909 V 119897) isin G
A120587119885 (119905 119909 V 119897) = 0
119885 (119879 119909 V 119897) = 1199092
(A8)
then119885(119905 119909 V 119897) = 119911120587(119905 119909 V 119897)We denote119885120587(119905) = 119885(119905 119883
120587(119905)
119881(119905) 119871(119905)) for short
Step 2 Next we denote two functions 119891120587(119905) = 119891(119905 119883
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
(119905) and taking conditional expectation onboth sides of (A22) we obtain
119891120587lowast
(119905)
⩾ 119891120587120591(119905)
+ E119905119909V119897 [int
119905+120591
119905
120585120587120591(119904) 119889119904 minus int
119905+120591
119905
120585120587lowast
(119904) 119889119904]
(A23)
and then
lim120591rarr0
inf119891
120587lowast
(119905) minus 119891120587120591 (119905)
120591⩾ 0 (A24)
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by grants of the National NaturalScience Foundation of China (nos 71231008 and 71501176)the Project Funded by China Postdoctoral Science Founda-tion (2015M580141) Natural Science Foundation of Guang-dong Province of China (no 2014A030312003) and the Foun-dation of City Development Academy of Gansu Province ofChina (no 2014-GSCFY-KJ02)
References
[1] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001
[2] P Devolder M B Princep and I D Fabian ldquoStochasticoptimal control of annuity contractsrdquo Insurance Mathematicsand Economics vol 33 no 2 pp 227ndash238 2003
[3] P Battocchio and F Menoncin ldquoOptimal pension managementin a stochastic frameworkrdquo Insurance Mathematics and Eco-nomics vol 34 no 1 pp 79ndash95 2004
[4] J W Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance Mathematics and Economics vol 42 no 3 pp 1159ndash1164 2008
[5] N-W Han and M-W Hung ldquoOptimal asset allocation forDC pension plans under inflationrdquo Insurance Mathematics ampEconomics vol 51 no 1 pp 172ndash181 2012
18 Mathematical Problems in Engineering
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
[6] G H Guan and Z X Liang ldquoOptimal management of DCpension plan in a stochastic interest rate and stochastic volatilityframeworkrdquo Insurance Mathematics and Economics vol 57 pp58ndash66 2014
[7] P Chen andH L Yang ldquoPension funding problemwith regime-switching geometric Brownian motion assets and liabilitiesrdquoApplied Stochastic Models in Business and Industry vol 26 no2 pp 125ndash141 2010
[8] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[9] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000
[10] S Basak and G Chabakauri ldquoDynamic mean-variance assetallocationrdquo Review of Financial Studies vol 23 no 8 pp 2970ndash3016 2010
[11] T Bjork and A Murgoci ldquoA general theory of Markoviantime inconsistent stochastic control problemsrdquo Working PaperStockholm School of Economics Stockholm Sweden 2010
[12] T Bjork A Murgoci and X Y Zhou ldquoMean-variance portfoliooptimization with state-dependent risk aversionrdquoMathematicalFinance vol 24 no 1 pp 1ndash24 2014
[13] L He and Z X Liang ldquoOptimal investment strategy for the DCplan with the return of premiums clauses in a mean-varianceframeworkrdquo Insurance Mathematics and Economics vol 53 no3 pp 643ndash649 2013
[14] X Q Liang L H Bai and J Y Guo ldquoOptimal time-consistentportfolio and contribution selection for defined benefit pensionschemes undermean-variance criterionrdquoTheANZIAM Journalvol 56 no 1 pp 66ndash90 2014
[15] F Menoncin and E Vigna ldquoMean-variance target-based opti-misation in DC plan with stochastic interest raterdquo Tech Rep337 Collegio Carlo Alberto 2013
[16] E Vigna ldquoOn efficiency ofmean-variance based portfolio selec-tion in defined contribution pension schemesrdquo QuantitativeFinance vol 14 no 2 pp 237ndash258 2014
[17] H Yao Z Yang and P Chen ldquoMarkowitzrsquos mean-variancedefined contribution pension fund management under infla-tion a continuous-time modelrdquo Insurance Mathematics andEconomics vol 53 no 3 pp 851ndash863 2013
[18] A Zhang and C-O Ewald ldquoOptimal investment for a pensionfund under inflation riskrdquoMathematical Methods of OperationsResearch vol 71 no 2 pp 353ndash369 2010
[19] H L Wu L Zhang and H Chen ldquoNash equilibrium strategiesfor a defined contribution pension managementrdquo InsuranceMathematics and Economics vol 62 pp 202ndash214 2015
[20] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquoThe Journal of Portfolio Management vol 23 pp 15ndash171996
[21] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993
[22] E M Stein and J C Stein ldquoStock price distributions withstochastic volatility an analytic approachrdquo Review of FinancialStudies vol 4 no 4 pp 727ndash752 1991
[23] X Lin and Y Li ldquoOptimal reinsurance and investment fora jump diffusion risk process under the CEV modelrdquo NorthAmerican Actuarial Journal vol 15 no 3 pp 417ndash431 2011
[24] Z Li Y Zeng and Y Lai ldquoOptimal time-consistent investmentand reinsurance strategies for insurers under Hestonrsquos SV
modelrdquo InsuranceMathematics and Economics vol 51 no 1 pp191ndash203 2012
[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurance strategyfor mean-variance insurers with square-root factor processrdquoInsurance Mathematics and Economics vol 62 pp 118ndash1372015
[26] J W Gao ldquoOptimal portfolios for DC pension plans under aCEV modelrdquo Insurance Mathematics amp Economics vol 44 no3 pp 479ndash490 2009
[27] J W Xiao Z Hong and C L Qin ldquoThe constant elasticityof variance (CEV) model and the Legendre transform-dualsolution for annuity contractsrdquo Insurance Mathematics andEconomics vol 40 no 2 pp 302ndash310 2007
[28] M J Brennan and Y Xia ldquoDynamic asset allocation underinflationrdquo Journal of Finance vol 57 no 3 pp 1201ndash1238 2002
[29] M Kwak and B H Lim ldquoOptimal portfolio selection with lifeinsurance under inflation riskrdquo Journal of Banking and Financevol 46 no 1 pp 59ndash71 2014
[30] A Anari and J Kolari ldquoStock prices and inflationrdquo Journal ofFinancial Research vol 24 no 4 pp 587ndash602 2001
[31] B S Lee ldquoStock returns and inflation revisited an evaluationof the inflation illusion hypothesisrdquo Journal of Banking andFinance vol 34 no 6 pp 1257ndash1273 2010
[32] J Liu and J Pan ldquoDynamic derivative strategiesrdquo Journal ofFinancial Economics vol 69 no 3 pp 401ndash430 2003
[33] R Gerrard S Haberman and E Vigna ldquoOptimal investmentchoices post-retirement in a defined contribution pensionschemerdquo Insurance Mathematics and Economics vol 35 no 2pp 321ndash342 2004
[34] M Di Giacinto S Federico F Gozzi and E Vigna ldquoIncomedrawdown option with minimum guaranteerdquo European Journalof Operational Research vol 234 no 3 pp 610ndash624 2014
[35] F H Wen Z F He Z F Dai and X G Yang ldquoCharacteristicsof investorsrsquo risk preference for stock marketsrdquo EconomicComputation and Economic Cybernetics Studies and Researchvol 48 no 3 pp 235ndash254 2014
