Abstract A well-known regulation on the management of a closed-end mutual fundis that the managers’ account cannot invest in risky assets. This paper studies the impactof this regulation under a given management fee structure such that the cumulativemanagement fee rate is described by a fixed RCLL deterministic increasing function.We conclude that the manager’s welfare is approximately the same whether the reg-ulation exists or not. In the expected utility maximization framework, we explicitlyfind the optimal investment-consumption plan when it exists, and get a sequence ofasymptotic near-optimal investment-consumption plans when an optimal one doesnot exist.
Keywords Closed-end mutual fund · Regulation · Welfare of the manager ·Portfolio selection · Expected utility
In the mutual fund industry of many countries, there is a well-known regulation thatany mutual fund company is forbidden to invest its own wealth (including the manage-
Y. Li (B)School of Mathematics, Yunnan Normal University, No. 298, Yi-Er-Yi Street, Kunming,People’s Republic of Chinae-mail: [email protected]; [email protected]
B. YuSchool of International Trade and Economics, University of International Business and Economics,No.10, Huixin Dongjie, 100029 Beijing, Chinae-mail: [email protected]
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246 Y. Li, B. Yu
ment fees that are paid out from fund assets to the company) into the stock market.1 Wecall this regulation as the “entry prohibition” in this paper. The motivation of the entryprohibition is natural to avoid conflict of interest between investors and the mutualfund company. As a bad consequence, the prohibition is expected to reduce the welfareof mutual fund companies. But it is not clear that how much reduction will be resultedby this regulation on the welfare of a fund company. In other words, if the prohibitionwas removed, could a mutual fund company improve its welfare significantly?
To answer this question, we need to fix the management fee for the fund man-ager. Management fee structure, or fund manager’s compensation scheme, is one ofthe most important factors that impact on the fund manager’s decisions, and has beestudied widely and deeply in recent years. Fraction of funds fee, a type of symmetric(“fulcrum”) fee, has been studied in Hugonnier and Kaniel (2010), which is a specialcase of our model. Papers that analyze asymmetric performance fees include Grinblattand Titman (1989), Carpenter (2000) among others.2 Cuoco and Kaniel (2011) ana-lyzed the equilibrium asset pricing implications of both symmetric and asymmetricperformance fees.
Since our paper lays emphasis on the welfare of the closed-end fund manager,we do not involve the welfare of the investors. Therefore, the problem of optimalcompensation contract between the investors and the manager is not touched upon.Literatures on the topic of optimal contract include Starks (1987), Ou-Yang (2003),Sung (2005), Cvitanic et al. (2006) and Chang (2010) among others. For open-endfunds, see Hugonnier and Kaniel (2010).
In this paper, we omit the potential conflict between the fund company and its man-ager. We concentrate our analysis on the impact of the entry prohibition on the welfareof a closed-end mutual fund manager, under a given management fee structure suchthat the cumulative management fee rate is described by a fixed RCLL deterministicincreasing function.
Our conclusions are contrast to the expectation of bad consequence, which claimthat the entry prohibition has little impact on the interest of the managers. Themain results are the following in more detail. If there is a lump sum for the man-agement fee at the terminal time, then the manager’s welfare is unaffected by thetrading constraints. If there is no lump sum for the management fee at the ter-minal time, we get two pieces of results. First, if the manager’s preference onthe consumption plans of her welfare is lower semi-continuous under some topol-ogy defined by a particular norm, then the entry constraint has no impact on themanager’s optimal welfare. If additionally she has an optimal consumption planin the case with the entry prohibition, then this consumption plan must be opti-mal in the case without the constraint. Second, if the manager’s preference relationcan be represented by an expected utility maximization with some mild technicalassumptions, then the manager’s welfare is also not worse off with the entry pro-
1 See, e.g., Section 80b-6, the Investment Company Act of 1940. In USA, mutual funds only represent“open-end” funds, but in general, mutual funds also include closed-end funds in other countries. In ourpaper, we use the second definition.2 Actually, under the Investment Advisers Act of 1940, any performance fee charged to a mutual fund mustbe a “fulcrum fee” in USA.
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Portfolio selection of a closed-end mutual fund 247
hibition. Furthermore, the optimal consumption plan and its supporting portfolio inthe constrained case can be mimicked from the optimal ones in the unconstrainedcase.
The remainder of this paper is organized as follows. We set up the market modeland describe the management of the mutual fund together with its consequence tothe manager in Sect. 2. Then we present the definition and properties of the set ofaffordable consumption plans in Sect. 3. We analyze the manager’s welfare and theproblem of expected utility maximization in Sects. 4 and 5 respectively. Section 6proceeds to a case study under the deterministic coefficients of the market and CRRAutility. Finally, Sect. 7 contains concluding remarks.
2 The model
We shall make use of the following notation: a � superscript stands for the transposi-tion of a vector or a matrix, | · | stands for the usual Euclidean norm, 1 = (1, . . . , 1)�
with a proper dimension, at−�= lims↑t as is the left limit at time t of an RCLL (right
continuous with left limits) process a·, and SDE is the abbreviation of “stochasticdifferential equation”.
2.1 The financial market
We consider the typical setup for a continuous-time market on the finite time span[0, T ]. The financial market consists of a bond and n stocks. The bond price processS0· evolves according to the following equation:
d S0t = S0
t rt dt, S00 = 1,
where the interest rate process r· is a bounded deterministic function. Equivalently,
S0t = e
∫ t0 rτ dτ . The i th stock’s price process Si· satisfies the equation
d Sit = Si
t
⎛
⎝bit dt +
n∑
j=1
σi jt d B j
t
⎞
⎠ , Si0 > 0, 1 ≤ i ≤ n.
Here B· = (B1· , . . . , Bn· )� is an n-dimensional standard Brownian motion definedon a filtered probability space (Ω,F , (Ft ),P) with the information flow Ft beingthe P-augmentation of the filtration generated by B· and F = FT . Denote bt =(b1
t , . . . , bnt )
� and σt = (σi jt )n×n . Assume the matrix σt is non-singular for each t
and all of the processes b·, σ· and σ−1· are uniformly bounded and Ft -progressively
measurable. Define the risk premium process by θt�= σ−1
t (bt − rt 1), then it is Ft -progressively measurable and uniformly bounded. In the above setting, the financialmarket admits a unique equivalent martingale measure, denoted by Q, whose densityprocess is
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248 Y. Li, B. Yu
dQ
dP
∣∣∣∣Ft
= �t�= exp
⎧⎨
⎩−
t∫
0
θ�τ d Bτ −
t∫
0
|θτ |22
dτ
⎫⎬
⎭.
By Girsanov’s theorem,
B∗t
�= Bt +t∫
0
θτdτ
is a standard Brownian motion under Q. Equivalently, Si· satisfies the following equa-tion:
d Sit = Si
t
⎛
⎝rt dt +n∑
j=1
σi jt d B∗ j
t
⎞
⎠ , 1 ≤ i ≤ n.
Remark 2.1 We set up the market by a standard Brownian motion to simplify theterminology. In fact, as pointed out by an anonymous referee, the market in this papercan be more general. The key property we need in this paper is that the market is arbi-trage-free and complete. The completeness is a technical requirement, which easesthe analysis of the utility maximization problems.
2.2 Wealth process of the fund and the management fee
A mutual fund is one type of investment instruments that gather money from investorsand collectively invest the money in the financial market. In general, mutual funds aredivided into open-end and closed-end funds. An open-end fund continuously offersnew shares to the public and will redeem those shares from investors who want toliquidate, while a closed-end fund only needs to sell a fixed number of shares at itsinitial time and is not allowed to buy its shares back from investors before maturity.The fund manager receives the management fees that are paid out of fund assets asher labor income. For simplicity, we only discuss the closed-end fund in this paper.For the study of open-end fund, see Hugonnier and Kaniel (2010). In the following,we model the wealth process of a closed-end fund and its management fee structure.
The initial wealth x0 > 0 of the fund is given and fixed throughout this paper. Werepresent a (self-financing) portfolio by an R
n-valued, Ft -progressively measurableprocess πt = (π1
t , . . . , πnt )
�, where π it is the dollar amount invested in the i th stock
at time t .
Definition 2.1 A portfolio πt is admissible if∫ t
0 π�s σsd B∗
s is a Q-martingale in thetime span [0, T ]. The set of all admissible portfolios is denoted by �.
Remark 2.2 The above definition of admissible portfolio set makes the market arbi-trage-free and complete. The martingale requirement is to rule out extreme portfolioslike doubling strategies.
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Portfolio selection of a closed-end mutual fund 249
Remark 2.3 In the literature, the set of admissible portfolios is often defined by localmartingale and some other regularity. But in most cases, we are only interested inportfolios which meet the martingale property as in our definition. Although the mar-tingale property may not be easy to check, there are some easy-to-check sufficientconditions and necessary conditions for this property.
For any admissible portfolio π· for the fund, which is chosen by the manager, thewealth process Xπ· of the mutual fund evolves according to the wealth equation
d Xπt = Xπt−rt dt + π�t σt d B∗
t − d Lt , Xπ0 = x0,
where L · represents the cumulative management fee paid to the manager from thefund, which is specified at the beginning of the investment by fund investors/princi-ples and fund managers together. In this paper, we suppose the management fee becharged linearly with respected to the total wealth of the portfolio, i.e.
d Lt = dlt + Xπt−d At ,
where l· is a deterministic non-decreasing function representing a (cumulative) fixedmanagement fee; A· is another deterministic non-decreasing function representing a(cumulative) bonus rate for the performance of the manager. Since the interest rate r·is deterministic, the payment corresponding to dlt can be saved in the risk-free assetwithout any uncertainty, we can regarded it as a lump-sum payment at time t = 0,which is transferred from the fund’s account to the manager’s own account. Hencewe can assume that lt ≡ 0 without loss of any generality. Under this assumption, thewealth equation for the fund follows
d Xπt = Xπt−rt dt + π�t σt d B∗
t − Xπt−d At , Xπ0 = x0, (2.1)
where A· is a fixed RCLL non-decreasing function with A0 = 0, representing thecumulative management fee rate credited from the fund account to the manager’s ownaccount.
Let T0 = inf{s ∈ [0, T ] : As = AT }. Then the manager receives no more manage-ment fee after T0. The welfare of the manager is completely determined on the timespan [0, T0], on which we shall focus our attention. For notational conciseness andwithout loss of generality, we assume throughout this paper that T0 = T , that is, At <
AT for all t < T . Furthermore, it is reasonable to assume thatΔAt�= At − At− < 1,
which means the lump sum management fee at any time t cannot exceed the totalwealth. In summary, we make the following standing assumption on A·:
Assumption 2.1 A· is an RCLL deterministic non-decreasing function satisfying
A0 = 0, At < AT < +∞ and ΔAt�= At − At− < 1 for all t ≤ T .
We can discount the wealth equation (2.1) by the interest rate, and get its discountedversion
d[(S0t )
−1 Xπt ] = (S0t )
−1π�t σt d B∗
t − (S0t )
−1 Xπt−d At , Xπ0 = x0.
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250 Y. Li, B. Yu
This equation is equivalent to
∫
(0,t](S0τ )
−1 Xπτ−d Aτ = x0 +t∫
0
(S0τ )
−1π�τ στd B∗
τ − (S0t )
−1 Xπt , ∀t ∈ [0, T ].
(2.2)
Because of the management fee payment, we need a different discounting fact forthe wealth process of the mutual fund, which will involve the process
∫(0,t]
11−ΔAτ
d Aτ .
Lemma 2.1 Under Assumption 2.1, At�= ∫
(0,t]1
1−ΔAτd Aτ is non-decreasing, deter-
ministic and bounded.
Proof It is obvious that At is non-decreasing and deterministic. Here we proveAT<+∞.
AT =∫
(0,T ]
1
1 −ΔAτ1ΔAτ≤1/2d Aτ +
∫
(0,T ]
1
1 −ΔAτ1ΔAτ>1/2d Aτ
≤∫
(0,T ]2 d Aτ +
∑
ΔAτ>1/2
ΔAτ1 −ΔAτ
.
Since∑ΔAτ>1/2ΔAτ ≤ AT < +∞, there are at most finitely many times τ in the
set {τ : ΔAτ > 1/2}, and ΔAτ1−ΔAτ
is finite for any τ . Hence∑ΔAτ>1/2
ΔAτ1−ΔAτ
is finite,
so is AT . ��Now we define the new discounting factor β· by the Doléans exponential3 of the
process − ∫ ·0 rτdτ + A·, that is, the unique solution of the equation
dβt = βt−(−rt dt + d At ), β0 = 1.
Then
βt = (S0t )
−1Et ( A) > 0, for all t ∈ [0, T ]. (2.3)
Since both r· and A· are bounded and deterministic, so is β·. Furthermore, β· is RCLLwith finite variation in [0, T ].Example 2.1 (Periodic management fee) Let δ ∈ (0, 1), K > 0 be an integer. DenoteΔt = T/K . The periodic management fee is described by the cumulative rate ofmanagement fee
At = δ
[t
Δt
]
,
3 See Appendix A.1 for a brief introduction to this notion.
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Portfolio selection of a closed-end mutual fund 251
where [x] stands for the largest integer among those no bigger than x .In this example, At = δ
1−δ[ tΔt
], and
Et ( A) =(
1 + δ
1 − δ
1
K
)[t/Δt].
Example 2.2 (Continuous average management fee) The cumulative rate of manage-ment fee up to time t is At = δt
T , where δ > 0 is the total rate of management fee. Inthis example, At = At and Et ( A) = exp{ δtT }.Lemma 2.2 Given Mt is a continuous Ft -martingale in [0, T ]. If Yt is a determin-istic function on [0, T ] with finite variation, then Zt = ∫ t
0 Ys−d Ms is a continuousFt -martingale in [0, T ].Proof It suffices to prove the result for non-decreasing Yt .
Suppose Yt is non-decreasing. By Ito’s formula, Yt Mt = ∫ t0 Ys−d Ms + ∫ t
0 MsdYs ,hence Zt = Yt Mt − ∫ t
0 MsdYs .Since Mt is a martingale in the finite time span [0, T ], it is uniformly integrable,
so is Yt Mt with the fact that Yt is bounded. On the other hand,
∣∣∣∣∣∣
t∫
0
MsdYs
∣∣∣∣∣∣≤
t∫
0
|Ms | dYs ≤T∫
0
|Ms |dYs,
and E| ∫ T0 |Ms |dYs | = ∫ T
0 E|Ms |dYs ≤ E|MT |(YT − Y0) < +∞, hence∫ t
0 MsdYs isalso uniformly integrable, and so is Zt . By the fact that Zt is a local martingale, weproved the claim. ��Remark 2.4 Lemma 2.2 can be generalized for martingales with jumps subject tosome proper regularity. We leave this to interested readers.
Lemma 2.3 For any admissible portfolio π·, β· Xπ· is a Q-martingale and
EQ
⎡
⎢⎣
∫
(0,T ](S0τ )
−1 Xπτ−d Aτ
⎤
⎥⎦ = x0
(
1 − 1
ET ( A)
)
. (2.4)
Proof By Ito-Doeblin formula, see Shreve (2004), we have
d(βt Xπt ) = βt−d Xπt + Xπt−dβt +ΔXπt Δβt
= βt−[Xπt−rt dt − Xπt−d At + π�t σt d B∗
t ]+Xπt−βt−
[
−rt dt + 1
1 −ΔAtd At
]
−Xπt−ΔAtβt−1
1 −ΔAtΔAt
= βt−π�t σt d B∗
t . (2.5)
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252 Y. Li, B. Yu
Since∫ t
0 π�τ στd B∗
τ is a continuous Q-martingale and β· is deterministic with finitevariation, we have β· Xπ· is a Q-martingale by Lemma 2.2. Moreover, together with(2.3) we have
EQ
⎡
⎢⎣
∫
(0,T ](S0τ )
−1 Xπτ−d Aτ
⎤
⎥⎦=
∫
(0,T ](S0τ βτ−)−1
EQ[βτ− Xπτ−]d Aτ=Xπ0
∫
(0,T ]
d Aτ
Eτ−( A).
Since Eτ ( A) = Eτ−( A)+ Eτ−( A)Δ Aτ = Eτ−( A)(1 + ΔAτ1−ΔAτ
) = Eτ−( A) 11−ΔAτ
,we have
∫
(0,T ]
d Aτ
Eτ−( A)=
∫
(0,T ]
1
Eτ ( A)d Aτ
1 −ΔAτ=
∫
(0,T ]
d Aτ
Eτ ( A)
= 1 − 1
ET ( A),
where the last equality is due to Lemma 7.1. ��
2.3 Wealth and consumption processes of the manager
The initial wealth y0 > 0 of the manager is given and fixed throughout this paper.For any admissible portfolio process π· for the fund, the manager receives the man-agement fee as her labor income, which is modeled by Xπt−d At . The manager canput her own wealth into the bond but is not allowed to invest her own wealth into thestocks. An admissible consumption rate process c· is a non-negative Ft -progressivelymeasurable process. We regard ct as the consumption rate at time t . The set of alladmissible consumption rate processes is denoted by C, i.e.
C �= {{ct }t∈[0,T ] : ct ≥ 0, c· is Ft -progressively measureable}.
For any pair (π, c) of admissible portfolio process for the fund and admissible con-sumption rate process of the manager, the wealth process Y π,c of the manager evolvesaccording to the equation
dY π,ct = Y π,ct− rt dt + Xπt−d At − ct dt, Y π,c0 = y0.
By integration by parts, we have
d[(S0t )
−1Y π,ct ] = (S0t )
−1 Xπt−d At − (S0t )
−1ct dt,
and therefore
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Portfolio selection of a closed-end mutual fund 253
(S0t )
−1Y π,ct +t∫
0
(S0τ )
−1cτdτ = y0 +∫
(0,t](S0τ )
−1 Xπτ−d Aτ , for all t ∈ [0, T ].]
(2.6)
By (2.2) and (2.6) together with (2.5) and (2.3), we have for any π ∈ � and c ∈ Cthat
(S0t )
−1Y π,ct +t∫
0
(S0τ )
−1cτdτ
= y0 + x0 +t∫
0
(S0τ )
−1π�τ στd B∗
τ − (S0t )
−1 Xπt
= y0 + x0 +t∫
0
(S0τ )
−1π�τ στd B∗
τ − (S0t βt )
−1
⎛
⎝x0 +t∫
0
βτ−π�τ στd B∗
τ
⎞
⎠
= y0 + x0
(
1 − 1
Et ( A)
)
+t∫
0
(1
Eτ−( A)− 1
Et ( A)
)
βτ−π�τ στd B∗
τ .
Particularly,
(S0T )
−1Y π,cT +T∫
0
(S0τ )
−1cτdτ = K0 +T∫
0
(1
Eτ−( A)− 1
ET ( A)
)
βτ−π�τ στd B∗
τ ,
(2.7)
where
K0�= y0 + x0
(
1 − 1
ET ( A)
)
. (2.8)
Together with the fact that 1Et ( A)
− 1ET ( A)
is non-increasing, deterministic and bounded,
we have
EQ
⎡
⎣(S0T )
−1Y π,cT +T∫
0
(S0τ )
−1cτdτ
⎤
⎦ = K0. (2.9)
3 Budget-feasible consumption plans
A manager uses her initial wealth y0 and the management fee credited from the fundto finance his consumption plan, which consists of an admissible consumption rate
123
254 Y. Li, B. Yu
process c ∈ C and a nonnegative terminal consumption ξ at time T . Denote by C theset of all consumption plans, i.e.
C�= {(c, ξ) : c ∈ C, ξ ≥ 0, ξ is FT -measurable}.
We call a consumption plan (c, ξ) budget-feasible if there exists an admissible port-folio process π ∈ � such that Y π,cT = ξ a.s. In this case, we say the consumption plan(c, ξ) is financed by the portfolio π . Denote
which is the set of all budget-feasible consumption plans.Equation (2.7) leads to the following easy lemma.
Lemma 3.1 For any (c, ξ) ∈ C , the following two conditions are equivalent.
(a) (c, ξ) ∈ C0.(b) There exists some π ∈ � such that
(S0T )
−1ξ +T∫
0
(S0τ )
−1cτdτ = K0 +T∫
0
(1
Eτ−( A)− 1
ET ( A)
)
βτ−π�τ στd B∗
τ .
From (b) in Lemma 3.1, we can see that (c, ξ) is budget-feasible only if
E
⎡
⎣HT ξ +T∫
0
Hτ cτdτ
⎤
⎦ = K0, (3.10)
where Ht�= (S0
t )−1 dQ
dP|Ft = (S0
t )−1�t is the pricing kernel of the market, K0 is given
as in (2.8). Denote
C1�= {(c, ξ) ∈ C : (c, ξ) satisfies Eq. (3.10)}.
Remark 3.1 We will show later in Theorem 3.3 that C1 is exactly the budget constraintfor an investor in the financial market with initial wealth K0 who is free to do riskyinvestment. In other words, if the manager is allowed to invest her own wealth in thestock market, then Eq. (3.10) characterizes the consumption plans she can choose.
By the implication of Eq. (3.10), we can see that the difference between C0 andC1 characterizes the role of the entry prohibition for managers to the stock market, onwhich the rest of this paper will focus.
Theorem 3.1 If ΔAT > 0, then C0 = C1.
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Portfolio selection of a closed-end mutual fund 255
Proof C0 ⊆ C1 is obvious from Lemma 3.1.(b). Hence it suffices to show C0 ⊇ C1.
For any (c, ξ) ∈ C1, we have EQ
[(S0
T )−1ξ + ∫ T
0 (S0τ )
−1cτdτ]
= K0. Define
Vt = EQ
⎡
⎣(S0T )
−1ξ +T∫
0
(S0τ )
−1cτdτ
∣∣∣∣∣∣Ft
⎤
⎦ ,
then Vt is a Q-martingale, and hence a Q-local martingale. By local martingale rep-resentation (see Revuz and Yor 1991, Theorem 3.4, p. 187) there exists a constant v0and progressively measurable process φt such that
Vt = v0 +t∫
0
φsd B∗s .
Since Vt is a Q-martingale, so is∫ t
0 φsd B∗s , and hence v0 = EQ[VT ] = K0. Define
ψt = σ−1t φt , then
Vt = K0 +t∫
0
ψ�τ στd B∗
τ , for all t ∈ [0, T ]. (3.11)
Take
πt�= ψt
[(Et−( A))−1 − (ET ( A))−1]βt−. (3.12)
Since ΔAT > 0, we know for all t ∈ [0, T ] that
(Et−( A))−1 − (ET ( A))−1 ≥ ET −( A))−1 − (ET ( A))
−1 > 0.
Together with the fact that 1[(Et−( A))−1−(ET ( A))−1]βt
is deterministic with finite var-
iation, we have π ∈ �. Finally, we have from (2.7) and (3.12) that
(S0T )
−1Y π,cT +T∫
0
(S0τ )
−1cτdτ = K0 +T∫
0
ψ�τ στd B∗
τ = VT ,
which implies Y π,cT = ξ . Hence (c, ξ) is budget-feasible. ��Remark 3.2 Theorem 3.1 claims that the entry prohibition of risky investment for themanager’s own wealth does not introduce any difference to the set of manager’s con-sumption plans whenΔAT > 0. Intuitively, when the manager cannot invest her ownwealth to stocks, she can change her income from the management fee by adjustingthe portfolio for the mutual fund. When ΔAT > 0, there is a terminal payoff of the
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256 Y. Li, B. Yu
management fee at time T , the adjustment of the portfolio for the mutual fund canfully compensate the loss of entry permission to the stock market for the manager’swealth. If ΔAT = 0, the adjustment of the mutual fund portfolio may be insufficientto compensate for the trade restriction.
In the real world, ΔAT > 0 does not hold in many interesting cases. If ΔAT = 0,then
ET −( A))−1 − (ET ( A))−1 = 0,
which implies
(Et−( A))−1 − (ET ( A))−1 → 0, as t → T .
Thus portfolio π given by (3.12) cannot guarantee σ�t πt integrable with respect
to Brownian motion B on [0, T ], say nothing of π being admissible. So C0 and C1can be different when ΔAT = 0. However, the difference in between seems not verylarge, for example, in the following sense.
For any consumption plan (c, ξ), define ||(c, ξ)|| �= EQ
[|ξ | + | ∫ T
0 cτdτ |], then
|| · || is a norm on C . Obviously, C1 is a closed subset of C .
Theorem 3.2 Under the norm || · ||, C0 is dense in C1, i.e., C0 = C1, where {. . . }stands for the closure under the norm || · ||.Proof See Appendix A.2. ��
To investigate the effect of the entry prohibition for the manager’s wealth to thestock market, we need also study the management of the manager’s own wealth withoutthe prohibition.
Let φ ∈ � be any portfolio process chosen by the manager for her own account,then for any portfolio process π ∈ � for the fund and any consumption rate processc ∈ C, the wealth process Y π,c,φ of the manager is driven by the following equation:
dY π,c,φt = Y π,c,φt− rt dt + φ�t σt d B∗
t + Xπt−d At − ct dt, Y π,c,φ0 = y0.
If the manager’s wealth can enter into the stock market, a consumption plan (c, ξ) ∈ C
is called attainable if ξ = Y π,c,φT for some π ∈ � and φ ∈ �. In this case, we say(c, ξ) is financed by (π, φ).
Theorem 3.3 When the manager’s wealth can enter into the stock market, the set ofall attainable consumption plans is C1.
Proof See Appendix A.2. ��Remark 3.3 In general, according to Theorems 3.2 and 3.3, any unconstrained attain-able consumption plan (c, ξ) ∈ C1 can be approximated by consumption plans inC0, which are budget-feasible in the constrained situation. Hence it is natural to study
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Portfolio selection of a closed-end mutual fund 257
whether the manager’s optimal welfare without the constraint can be approximatedby those with constraint.
In the specific case that ΔAT > 0, according to Theorems 3.1 and 3.3, the set ofbudget-feasible consumption plans with the entry prohibition, C0, is the same as C1,the set of attainable consumption plans without the entry prohibition. That is to say,the entry prohibition does not make the manager worse off at all.
4 The manager’s welfare
In this section, we consider the manager’s welfare obtained under some preference onher consumption plan. We assume the manager has a rational preference relation � onthe consumption set C . The corresponding strict preference relation is denoted by �.
Definition 4.1 The preference relation � is lower semi-continuous if for every(c, ξ) ∈ C , the strict upper contour set {(c′, ξ ′) ∈ C : (c′, ξ ′) � (c, ξ)} is open,under the norm || · ||.Definition 4.2 Let S be a non-empty subset of C . A consumption plan (c, ξ ) isoptimal in S if (c, ξ ) ∈ S and (c, ξ ) � (c, ξ) for every (c, ξ) ∈ S .
Theorem 4.1 Assume the preference relation � is lower semi-continuous. We have:
(a) For any (c, ξ ) ∈ C1, if it is not optimal in C1, then there exists some (c, ξ) ∈ C0such that (c, ξ) � (c, ξ );
(b) For any (c, ξ ) ∈ C0, it is optimal in C0 if and only if it is optimal in C1.
Proof (b) can be easily proved by (a). So it suffices to prove (a).Suppose (c, ξ ) ∈ C1 is not optimal in C1, then we can find some (c, ξ ) ∈ C1 such
that (c, ξ ) � (c, ξ ). Since � is lower semi-continuous, the strict upper contour set{(c′, ξ ′) ∈ C : (c′, ξ ′) � (c, ξ )} is open under the norm || · ||. By Theorem 3.2, thisopen set contains some consumption plan (c, ξ) ∈ C0, which is what we need. ��Remark 4.1 If the manager’s preference relation is lower semi-continuous and she hasa best budget-feasible consumption plan with the entry prohibition, by Theorem 4.1,we can see that this best consumption plan also makes her best off in the situationwithout the entry prohibition. In other words, the right to enter into the stock marketdoes not improve the manager’s welfare.
5 Expected utility maximization
Expected utility is the most widely-used representation of a rational preference. In thissection, we assume the manager’s welfare is measured by the expected utility of herconsumption plan defined as follows.
For a given consumption plan (c, ξ), the manager’s expected utility is defined as
u(c, ξ)�= E
⎡
⎣T∫
0
U (τ, cτ )dτ + U (T, ξ)
⎤
⎦ , (5.13)
123
258 Y. Li, B. Yu
where the von Neumann–Morgenstern utility functions U (t, ·), t ∈ [0, T ], satisfy thefollowing assumption:
Assumption 5.1 The von Neumann–Morgenstern utility function U (t, ·) : [0,+∞)
�→ [−∞,+∞) satisfy the following conditions.
(1) For each t ∈ [0, T ], U (t, ·) is continuously differentiable, strictly concave andstrictly increasing, U (t, x) > −∞ for any x > 0, and U (t, ·) satisfies the Inadacondition, i.e. ∂U
∂x (t, 0+) = +∞ and ∂U∂x (t,+∞) = 0.
(2) For any x ≥ 0, U (·, x) is a measurable function on t ∈ [0, T ).(3) There exist a deterministic function ct ≥ 0 on t ∈ [0, T ], such that
T∫
0
cddt < +∞ and
T∫
0
U (t, ct )dt > −∞.
Remark 5.1 Assumption 5.1 is standard for the expected utility maximization to beinteresting, in which (1) brings smooth concavity to the objective, (2) and (3) ensurethe well-posedness of the integral therein. This assumption is quite mild and satisfiedby most often-used utility functions.
With the entry prohibition, the manager’s optimization problem for her welfare is
We call problem (5.14) as the manager’s constrained problem. Here “u(c, ξ) iswell-defined” means that the expectation in u(c, ξ) is well-defined, i.e. eitherE[ ∫ T
0 max{U (t, ct ), 0}dt + max{U (T, ξ), 0}] or E[ ∫ T
Theorems 3.1–3.3 indicate the previous problem is inherently related to the man-ager’s welfare optimization problem without the entry prohibition, which is the fol-lowing unconstrained problem:
Generally, the preference represented by u(c, ξ) is not lower semi-continuous. Sowe cannot apply Theorem 4.1 directly to study the equivalence between the constrainedproblem (5.14) and the unconstrained problem (5.15). However, the case ofΔAT > 0is trivial by Theorem 3.1, in which we have the following theorem.
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Portfolio selection of a closed-end mutual fund 259
Theorem 5.1 IfΔAT > 0, then the constrained problem (5.14) and the unconstrainedproblem (5.15) have the same optimal solutions if any of them exists.
In the rest of this section, we will focus on the case ΔAT = 0.The unconstrained problem (5.15), which is referred as the static reformulation of
Merton’s problem in complete markets, has been heavily investigated in the literaturewith slight differences; see, e.g., Pliska (1986), Cox and Huang (1989), Karatzas et al.(1987) and Jin et al. (2008). For extensions to incomplete markets, see He and Pearson(1991), Karatzas et al. (1991), among others. In what follows, we just fit the relevantresults into our problem setting.
For any t ∈ [0, T ], denote I (t, ·) as the inverse of marginal utility function∂U∂x (t, ·), which is a strictly decreasing continuous function from R+ to R+ withI (t, 0+) = +∞, I (t,+∞) = 0. For all λ > 0, define
K(λ) �= E
⎡
⎣T∫
0
Hτ I (τ, λHτ )dτ + HT I (T, λHT )
⎤
⎦ ,
where Ht = (S0t )
−1 dQdP
|Ft is the pricing kernel process of the market. Obviously K(·)is non-increasing. In the rest of this paper, we suppose the following Assumption 5.2holds in addition to Assumption 5.1.
Assumption 5.2 For any λ > 0,
(a) K(λ) < +∞, and
(b) E
[∫ T0 U (t, I (t, λHt ))dt
]< +∞ and E[U (T, I (T, λHT ))] < +∞.
Lemma 5.1 Under Assumption 5.1 and 5.2, we have
(a) K(·) is strictly decreasing, continuous with K(0+) = +∞, K(+∞) = 0;
(b) −∞ < E
[∫ T0 U (t, I (t, λHt ))dt
]< +∞ and −∞ < E[U (T, I (T, λHT ))] <
+∞.
Proof Denote K (λ) = ∫ T0 Hτ I (τ, λHτ )dτ + HT I (T, λHT ), then K (·) is strictly
decreasing and continuous, and K(λ) = E[K (λ)] < +∞ for any λ > 0. By the dom-inance convergence theorem, K is continuous and strictly decreasing in (λ0,+∞) forany K(λ0) < +∞, hence continuous and strictly decreasing in (0,+∞). Furthermore,by the monotone convergence theorem,
limλ↓0
K(λ) = E[limλ↓0
K (λ)] = +∞,
and
limλ→+∞ K(λ) = E[ lim
λ→+∞ K (λ)] = 0.
123
260 Y. Li, B. Yu
Finally, by the concavity of U (t, ·), and recall from Assumption 5.1.(3) with theexistence of ct , we have U (t, I (t, λHt )) ≥ U (t, ct )+ λHt [I (t, λHt )− ct ], hence
E
⎡
⎣T∫
0
U (t, I (t, λHt ))dt
⎤
⎦
≥ E
⎡
⎣T∫
0
U (t, ct )dt
⎤
⎦ + E
⎡
⎣T∫
0
λHt I (t, λHt )dt
⎤
⎦ − λ
T∫
0
(S0t )
−1ct dt
> −∞.
Similarly, we can prove E[U (T, I (T, λHT )] > −∞. ��The following theorem is essentially from Karatzas et al. (1987), and can be proved
in exactly the same way.
Lemma 5.2 Suppose Assumptions 5.1 and 5.2 hold. Then for any k > 0, there existsa unique λ(k) > 0 such that K(λ(k)) = k. If define
{ct (k) = I (t, λ(k)Ht ), t ∈ [0, T ]ξ (k) = I (T, λ(k)HT ),
(5.16)
then (c(K0), ξ (K0)) is the unique optimal solution for the unconstrained problem
Our goal is to solve the constrained problem (5.14). By definition, we have v0 ≤ v1.If v0 = v1, then the strict concavity of the von Neumann–Morgenstern utility functionsleads to the following trivial lemma.
Lemma 5.3 If v0 = v1 ∈ (−∞,+∞), then the constrained problem (5.14) admits anoptimal solution (c∗, ξ∗) if and only if (c∗, ξ∗) uniquely solves both the constrainedproblem (5.14) and the unconstrained problem (5.15).
From Lemma 5.3, we can see that v0 = v1 is a key for solving the constrainedproblem (5.14) through the easier unconstrained problem (5.15).
Theorem 5.2 Under Assumptions 5.1 and 5.2, v0 = v1 ∈ (−∞,+∞).
Proof Since v1 ≥ v0 by their definitions, here we only need to prove v0 ≥ v1.For any ε ∈ (0, K0), define K ε
0 = K0 − ε, then v1 = limε↓0 v1(K ε0 ). Now we fix
any small ε > 0 and prove v0 ≥ v1(K ε0 ), which suffices the conclusion. We prove it
by two steps:
4 Here we abuse the notation v1 to emphasis the dependence of v1 on K0.
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Portfolio selection of a closed-end mutual fund 261
– Step 1: find an asymptotically optimal sequence (cn, ξn) feasible for the uncon-strained problem (5.15) with initial wealth K ε
0 .For any integer n, E[HT (ξ (K ε
0 ) ∧ n)] ≤ E[HT ξ (K ε0 )] < K ε. Similar to
Lemma 5.1, we can prove that the equation of λ
E
⎡
⎣T∫
0
Ht I (t, λHt )dt
⎤
⎦ = K ε0 − E[HT (ξ (K
ε0 ) ∧ n)]
admits a unique solution, which we denote as λn . Furthermore, it is not hard toprove that λn ↑ λ(K ε
0 ) when n ↑ +∞.Define
ξn = ξ (K ε0 ) ∧ n, cn
t = I (t, λn Ht ).
For any n > 0, we can easily see that (cn, ξn) ∈ C1 with initial wealth K0 replacedby K ε
0 . Furthermore,
E[|U (T, ξn)|] ≤ E[|U (T, ξ | + |U (T, n)|] < +∞,
−∞ < E
⎡
⎣T∫
0
U (t, cnt )dt
⎤
⎦ ≤ v1(Kε0 )− E[U (T, ξn)] < +∞.
Hence (cn, ξn) is feasible the unconstrained problem (5.15) with initial wealth K ε0 .
By the monotone convergence theorem, we have
limn→+∞ E
⎡
⎣T∫
0
U (t, cnt )dt
⎤
⎦
= E
⎡
⎣T∫
0
U (t, ct (Kε0 ))dt
⎤
⎦ , limn→+∞ E[U (T, ξn)] = E[U (T, ξ (K ε
0 ))],
and hence limn→+∞ u(cn, ξn) = u(c(K ε0 ), ξ (K
ε0 )) = v1(K ε
0 ).– Step 2: for any fixed n > 0, find a sequence (cn,ε, ξn,ε) feasible for the constrained
problem (5.14) with initial wealth K0, such that limε↓0 u(cn,ε, ξn,ε) ≥ u(cn, ξn),which suffices the claim v0 ≥ v1(K ε
0 ).For any integer n, similar to the proof of Theorem 3.2, we define the martingaleV n
t and its martingale representation ψnt by
V nt = EQ
⎡
⎣(S0T )
−1ξn +T∫
0
(S0τ )
−1cnτ dτ
∣∣∣∣∣∣Ft
⎤
⎦ = K ε0 +
t∫
0
(ψnτ )
�στd B∗τ ,
123
262 Y. Li, B. Yu
and for any t ∈ [0, T ), denote πnt = ψn
t
[(Et−( A))−1−(ET ( A))−1]βt−. Then we have
ξn = S0T
⎛
⎝V nT −
T∫
0
(S0t )
−1cnt dt
⎞
⎠ .
According to Assumption 5.1.(3), when 0 < ε < 2∫ T
0 (S0τ )
−1cτdτ , we can find at < T such that
T∫
t
(S0τ )
−1cτdτ = ε
2and
T∫
t
U (s, cτ )dτ > −∞.
For any δ ∈ (0, T − t), define
πn,δt = πn
t It≤T −δ, cn,δt = cn
t It≤T −δ + ct 1t≥t ,
ξn,δ = S0T
⎛
⎝V nT −δ −
T −δ∫
0
(S0t )
−1cnt dt + ε
2
⎞
⎠ .
We claim that (cn,δ, ξn,δ) is a feasible consumption plan for the constrained prob-lem (5.14), which is proved by the following 3 properties.
(i) Firstly, cn,δt ∈ C, and
ξn,δ = S0T EQ
⎡
⎣V nT −
T −δ∫
0
(S0t )
−1cnt dt
∣∣∣∣∣∣FT −δ
⎤
⎦ + ε
2
≥ S0T EQ
⎡
⎣V nT −
T∫
0
(S0t )
−1cnt dt
∣∣∣∣∣∣FT −δ
⎤
⎦ + ε
2
= E[ξn|FT −δ] + ε
2. (5.17)
By ξn ≥ 0 we have ξn,δ > 0. Hence (cn,δ, ξn,δ) ∈ C .(ii) Secondly, since 1
[(Et−( A))−1−(ET ( A))−1]βt−is bounded and of finite variation
on t ∈ [0, T − δ], πn,δ ∈ �. Furthermore,
(S0T )
−1ξn,δ +T∫
0
(S0τ )
−1cn,δτ dτ
= V nT −δ −
T −δ∫
0
(S0τ )
−1cn,δτ dτ + ε
2+
T −δ∫
0
(S0τ )
−1cnτ dτ +
T∫
t
(S0τ )
−1cnτdτ
= V nT −δ + ε
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Portfolio selection of a closed-end mutual fund 263
= K0 +T∫
0
(1
Eτ−( A)− 1
ET ( A)
)
βτ−(πn,δτ )�στd B∗
τ .
By Lemma 3.1, we have (cn,δ, ξn,δ) ∈ C0 is financed by the portfolio πn,δ .
(iii) Finally, since U (T, ξn,δ) ≥ U (T,S0
T ε
2 ), so E[U (T, ξn,δ)] is well-defined.
On the other hand,∫ T
0 U (τ, cn,δτ )dτ≥ ∫ T −δ
0 U (τ, cnτ )dτ+∫ T
T −δ U (τ, cτ )dτ ,
and E[| ∫ T −δ0 U (τ, cn
τ )dτ |] ≤ E[∫ T0 |U (τ, cn
τ )|dτ ] < +∞. Hence u(cn,δ,
ξn,δ) is well-defined.Now we show that lim infδ↓0 u(cn,δ, ξn,δ) ≥ u(cn, ξn). According to (5.17), (recalldQdP
|Ft = �t ), we have
E[U (T, ξn,δ)] > E
[U
(T,EQ
[ξn + ε
2
∣∣∣FT −δ
])]
≥ E
[EQ
[U
(T, ξn + ε
2
)∣∣∣FT −δ
]]
= E
[
E
[�T
�T −δU
(T, ξn + ε
2
)∣∣∣∣FT −δ
]]
= E
[�T
�T −δU
(T, ξn + ε
2
)]
.
Since U (T, ξn + ε2 ) ≥ U (T, ε2 ), by Fatou’s lemma,
lim infδ→0
E[U (T, ξn,δ)] ≥ lim infδ→0
E
[�T
�T −δU
(T, ξn + ε
2
)]
> E[U (T, ξn)
].
On the other hand, since E∫ T
0
∣∣U (t, cn
t )dt∣∣ < +∞, we have
limδ↓0
E
⎡
⎣T∫
0
U (t, cn,δt )dt
⎤
⎦ ≥ limδ↓0
E
⎡
⎣T −δ∫
0
U (t, ct )dt
+T∫
T −δU (t, ct )dt
⎤
⎦ = E
⎡
⎣T∫
0
U (t, cnt )dt
⎤
⎦ .
Hence lim infδ↓0 u(cn,δ, ξn,δ) ≥ u(cn, ξn). ��
Now we are ready to study the optimal solution for the manager’s welfare optimi-zation problem (5.14) with the entry prohibition.
123
264 Y. Li, B. Yu
With Assumption 5.1 and 5.2, we keep the notation (c(k), ξ (k)) given in Lemma 5.2by (5.16), and define
Vt�= EQ
⎡
⎣ (S0T )
−1ξ (K0)+T∫
0
(S0τ )
−1cτ (K0)dτ
∣∣∣∣∣∣Ft
⎤
⎦ = K0 +t∫
0
ψτ (K0)�στd B∗
τ .
(5.18)
Theorem 5.3 Under Assumption 5.1 and 5.2, the constrained problem (5.14) admits
an optimal solution if and only if ψt (K0)
(Et−( A))−1−(ET ( A))−1 ∈ �, i.e.,
t∫
0
ψτ (K0)�στ
(Eτ−( A))−1 − (ET ( A))−1d B∗
τ is a true martingale in t ∈ [0, T ]. (5.19)
where ψ(K0) is defined in the martingale representation (5.18). Furthermore,
(a) if (5.19) is true, the unique solution (c, ξ ) is given by (5.16) and it is financed bythe portfolio
π�= ψt
[(Et−( A))−1 − (ET ( A))−1]βt−, t ∈ [0, T ]; (5.20)
(b) if (5.19) is not true, then we can find an asymptotic optimal sequence{(cn, ξn)}n=1,2,... according to the proof of Theorem 5.2.
Proof According to Lemma 5.2, (c(K0), ξ (K0)) is the unique optimal consump-tion plan for the unconstrained problem (5.15). Also, it is proved that v0 = v1 ∈(−∞,+∞) in Theorem 5.2. According to Lemma 5.3, the constrained problem (5.14)admits a solution if and only if (c(K0), ξ (K0)) ∈ C0, which is equivalent to condition(5.19).
When the condition (5.19) holds, (c(K0), ξ (K0)) is the unique optimal consump-tion plan.
The last statement (b) is obvious from the proof of Theorem 5.2. ��In the next section, we will study the constrained problem (5.14) in a detailed
example.
6 A case study: deterministic coefficients and CRRA utility
In addition to the assumptions in Sect. 2, we further assume in this section that both(bt ) and (σt ) are deterministic. We also assume the von Neumann–Morgenstern utilityfunctions U (t, ·) are of CRRA (constant relative risk aversion) type: for all t ∈ [0, T ],u(t, 0) = −∞ and for all x > 0,
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Portfolio selection of a closed-end mutual fund 265
U (t, x) ={
e−ρt 11−γ x1−γ , γ ∈ (0, 1) ∪ (1,∞),
e−ρt log x, γ = 1.(6.21)
Here e−ρt is the utility discounting factor with constant ρ, γ is the coefficient ofrelative risk aversion. It is easy to verify that Assumptions 5.1, 5.2 hold.
6.1 Unconstrained situation
The solution of problem (5.15) was reported in Karatzas and Shreve (1998) We justlist the result here. The inverse marginal utility function
I (t, x) = exp
{
−ρt
γ
}
x− 1γ ,
and therefore in the optimal consumption plan (5.16) for the unconstrained problemobtained in Lemma 5.2,
Ht I (t, λHt ) = λ− 1γ · exp
{
−ρt
γ
}
· Hγ−1γ
t .
Recalling the definition of K(λ), we have
K(λ) �= E
⎡
⎣T∫
0
Hτ I (τ, λHτ )dτ + HT I (T, λHT )
⎤
⎦
= λ− 1γ N (T ),
where
N (T )�=
T∫
0
n(τ )dτ + n(T )
with
n(t)�= exp
{
−ρt
γ
}
E
[
Hγ−1γ
t
]
> 0.
Hence the solution to K(λ) = K0 is λ(K0) =(
N (T )K0
)γ. By Lemma 5.2, the unique
solution (c, ξ ) of the unconstrained problem (5.15) is given by
ct = K0
N (T )exp
{
−ρt
γ
}
H− 1γ
t , ξ = K0
N (T )exp
{
−ρT
γ
}
H− 1γ
T . (6.22)
6.2 Constrained situation
Now we can apply the result in Theorem 5.3 to study the constrained problem (5.14).
123
266 Y. Li, B. Yu
Let
Vt = EQ
⎡
⎣ (S0T )
−1ξ +T∫
0
(S0τ )
−1cτdτ
∣∣∣∣∣∣Ft
⎤
⎦ .
It is not hard to calculate that
Vt = K0G(t)J (t)
N (T )+
t∫
0
(S0τ )
−1cτdτ,
where
G(t)�=
T∫
t
n(τ )dτ + n(T ),
J (t)�= exp
⎧⎨
⎩1
γ
t∫
0
θ�τ d B∗
τ − 1
2γ 2
t∫
0
|θτ |2dτ
⎫⎬
⎭,
where θt = σ−1t (bt − rt 1) is the risk premium process of the market. Then we can
calculate that
dVt = K0G(t)J (t)
γ N (T )θ�
t d B∗t = ψ�
t σt d B∗t ,
where
ψt = K0G(t)J (t)
γ N (T )(σ�
t )−1θt . (6.23)
Define
πt = K0G(t)J (t)(σ�
t
)−1θt
γ N (T )[(Et−( A))−1 − (ET ( A))−1]βt−for all t ∈ [0, T ]. (6.24)
By Theorem 5.3, the constrained problem (5.14) has an optimal solution if and only ifπ ∈ �. When it exists, the optimal consumption plan is (c, ξ ) financed by the portfolioπ .
Theorem 6.1 If there exists a t0 ∈ [0, T ) and an ε > 0 such that |θτ | ≥ ε overτ ∈ [t0, T ], then the π in Eq. (6.24) is admissible ( i.e. π ∈ �) if and only if
T∫
0
(1
Et−( A)− 1
ET ( A)
)−2
dt < ∞. (6.25)
Proof By the fact that J (t) is an exponential Q-martingale, we have
1 ≤ EQ[J (t)2] ≤ EQ[J (T )2] < ∞.
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Portfolio selection of a closed-end mutual fund 267
Furthermore, we can find a n > 0 such that Q({inf t∈[0,T ] J (t) > 1n }) > 0.
If (6.25) hold, then together with the fact that 0 < G(t) ≤ G(0) < +∞, βt− ≥ 1,we have
EQ
T∫
0
|σ(t)�πt |2ds = K 20
γ 2 N (T )2
T∫
0
G(t)2
β2t−
|θt |2EQ[J (t)2](
1
Et−( A)− 1
ET ( A)
)−2
dt
< +∞.
Hence π ∈ �.
On the contrary, if (6.25) does not hold, then∫ T
t0
(1
Et−( A)− 1
ET ( A)
)−2dt = +∞.
Hence in the event {ω : J (τ ) ≥ 1n for ∀τ ∈ [t0, T ]}, by the fact that G(t) ≥ G(T )
and βt− ≤ βT ,
T∫
0
|σ(t)�πt |2ds = K 20
γ 2 N (T )2
T∫
0
G(t)2
β2t−
|θt |2 J (t)2(
1
Et−( A)− 1
ET ( A)
)−2
dt
≥ K 20
n2γ 2 N (T )2
T∫
t0
G(t)2
β2t−
|θt |2(
1
Et−( A)− 1
ET ( A)
)−2
dt
≥ K 20
n2γ 2 N (T )2G(T )2ε2
β2T
T∫
t0
(1
Et−( A)− 1
ET ( A)
)−2
dt
= +∞,
which means π /∈ �. ��
Corollary 6.1 For the example in this section, if there exists a t0 ∈ [0, T ) and anε > 0 such that |θτ | ≥ ε over τ ∈ [t0, T ], then the constrained problem (5.15) admitsoptimal solution if and only if the condition (6.25) holds, i.e.
T∫
0
(1
Et−( A)− 1
ET ( A)
)−2
dt < ∞.
When this condition holds, the optimal consumption plan is (c, ξ ) given by (6.22),which is financed by the portfolio π given by (6.24).
It is easy to see that when ΔAT > 0, the condition (6.25) holds. For a counterexample for condition (6.25), let us examine the continuous average management feein Example 2.2, where At = δt
T for some δ > 0, and Et ( A) = eδt/T . It is not hard tosee that
T∫
0
(1
Et−( A)− 1
ET ( A)
)−2
dt ≥T∫
0
{1
eδt/T δ(T − t)/T
}2
= +∞.
123
268 Y. Li, B. Yu
Thus for continuous average management fee in the standard Black-Scholes modelwith constant coefficients, the constrained problem (5.14) admits no solution in gen-eral, and the entry prohibition matters slightly for the manager.
7 Conclusion
In this paper, we study the impact of the regulation that the managers’ account cannotinvest in risky assets on the welfare of a closed-end mutual fund manager under agiven linear management fee structure. We find that the regulation has little effect onthe welfare of the manager. The entry constraint into the stock market does not makethe manager worse off. Specifically, if there is a lump sum for the management fee atthe terminal time, the manager has the same set of decisions on her welfare, whethershe is allowed to put her own investment into the stocks or not. If there is no lump sumfor the management fee at the terminal time, two pieces of results are obtained due tothe different cases of preference. First, if the manager’s preference on the consump-tion plans of her welfare is lower semi-continuous under some topology, the entryprohibition does not make the manager worse off. In particular, if she has an optimalconsumption plan on her welfare with in the entry prohibition, this consumption planmust be the optimal one in the case without the entry prohibition. Second, if the man-ager’s preference relation can be represented by an expected utility function and theutility function satisfies some mild technical assumptions, the manager’s welfare isalso the same whether she is subject to the entry constraint or not.
Acknowledgments Yu owes thanks for the financial support from Program for Innovative Research Teamin UIBE and from Youth Grant of National Science Foundation (No. 11101406/A011002).
A Appendix
A.1 The Doléans Exponential
For any RCLL Ft -adapted finite-variation process a·, there exists a unique RCLLFt -adapted finite-variation process E·(a) which solves the equation
dEt (a) = Et−(a)dat , E0(a) = 1.
The unique solution is called the Doléans exponential of a and is given by (seeRogers and Williams 2000, pp. 29–30)
Et (a) = exp{at − a0} ·∏
0<τ≤t
(1 +Δaτ )e−Δaτ .
If Δat > −1 for any t P-a.s., then Et (a) > 0 for any t P-a.s.
123
Portfolio selection of a closed-end mutual fund 269
Lemma 7.1 Assume Δat > −1 a.s. for any t. We have
d
(1
Et (a)
)
= − dat
Et (a)(A.26)
and∫
(t,T ]
daτEτ (a) = 1
Et (a)− 1
ET (a).
Proof It suffices to prove (A.26). SinceΔat > −1 for any t P-a.s., we have Et (a) > 0for any t P-a.s. By Itô’s formula (see Rogers and Williams 2000, pp. 27–29), we have
d
(1
Et (a)
)
= − dEt (a)
(Et−(a))2+ dVt , (A.27)
where
Vt =∑
0<s≤t
(1
Es(a)− 1
Es−(a)+ ΔEs(a)
(Es−(a))2
)
.
By the definition of Es(a), we have ΔEs(a) = Es−(a)Δas and hence
1
Es(a)− 1
Es−(a)= − ΔEs(a)
Es(a) · Es−(a)= − Δas
Es(a). (A.28)
Then we have
1
Es(a)− 1
Es−(a)+ ΔEs(a)
(Es−(a))2= − Δas
Es(a)+ Δas
Es−(a)= (Δas)
2
Es(a),
which implies
Vt =∫
(0,t]
Δas
Es(a)das
and dVt = ΔatEt (a)
dat . Substituting it into (A.27), we have
d
(1
Et (a)
)
= − dEt (a)
(Et−(a))2+ Δat
Et (a)dat = − dat
Et−(a)+ Δat
Et (a)dat = − dat
Et (a),
where the last equality follows from (A.28). ��
123
270 Y. Li, B. Yu
A.2 Proofs
Proof of Theorem 3.2 Given (c, ξ) ∈ C1, let (V, ψ) be the same as in the proof ofTheorem 3.1, i.e.
Vt = EQ
⎡
⎣(S0T )
−1ξ +T∫
0
(S0τ )
−1cτdτ
∣∣∣∣∣∣Ft
⎤
⎦ = K0 +t∫
0
ψ(τ)�στd B∗τ .
Define πt = ψt
[(Et−( A))−1−(ET ( A))−1]βt−for any t ∈ [0, T ). By the assumption that
At is non-decreasing and At < AT for all t < T , we know (Et−( A))−1 − (ET ( A))−1
is non-increasing and strictly positive in [0, T ). For any ε ∈ (0, T ) and t ∈ [0, T ],let πεt
�= πt I[t≤T −ε] and cεt�= ct I[t≤T −ε], then πε ∈ � and cε ∈ C. According to
(πε, cε), the portfolio for the fund keeps being π and the consumption of the managerkeeps being c until time (T −ε); from time (T −ε) on, the wealth of the fund are all putinto the bond and the manager consumes nothing until the terminal time T . By (2.7),
(S0T )
−1Y πε,cε
T +T∫
0
(S0τ )
−1cετdτ
= K0 +T∫
0
(1
Eτ−( A)− 1
ET ( A)
)
βτ−(πετ )�στd B∗τ
= K0 +T −ε∫
0
ψ�τ στd B∗
τ
= VT −ε.
As a consequence, define ξε�= Y π
ε,cε
T = S0T
(VT −ε − ∫ T
0 (S0τ )
−1cετdτ)
, then
ξεL1(Ω,F ,Q)−−−−−−−→
ε→0S0
T
⎛
⎝VT −T∫
0
(S0τ )
−1cτdτ
⎞
⎠ = ξ.
Obviously,
ξε = EQ[ξ |FT −ε] + S0T EQ
⎡
⎣T∫
T −ε(S0τ )
−1cτdτ
∣∣∣∣∣∣FT −ε
⎤
⎦ ≥ 0 a.s.,
and therefore (cε, ξ ε) ∈ C0 for all ε ∈ (0, T ). Then we finish the proof. ��Proof of Theorem 3.3 Assume (c, ξ) is attainable, then ξ = Y π,c,φT for some π ∈ �and φ ∈ �. The integration by parts implies
d[(S0t )
−1Y π,c,φt ] = (S0t )
−1φ�t σt d B∗
t + (S0t )
−1 Xπt−d At − (S0t )
−1ct dt, (A.29)
123
Portfolio selection of a closed-end mutual fund 271
and therefore
EQ
⎡
⎣(S0T )
−1ξ +T∫
0
(S0τ )
−1cτdτ
⎤
⎦
= EQ
⎡
⎣(S0T )
−1Y π,c,φT +T∫
0
(S0τ )
−1cτdτ
⎤
⎦
= y0 + EQ
⎡
⎣T∫
0
(S0τ )
−1φ�τ στd B∗
τ +T∫
0
(S0τ )
−1 Xπτ−d Aτ
⎤
⎦
= y0 + EQ
⎡
⎣T∫
0
(S0τ )
−1 Xπτ−d Aτ
⎤
⎦
= K0. (by (2.4))
Then we have (c, ξ) ∈ C1.Conversely, assume (c, ξ) ∈ C1, then by (2.4), for any π ∈ �,
EQ
⎡
⎣(S0T )
−1ξ +T∫
0
(S0τ )
−1cτdτ −T∫
0
(S0τ )
−1 Xπτ−d Aτ
⎤
⎦ = y0.
Denote
Vt = EQ
⎡
⎣(S0T )
−1ξ +T∫
0
(S0τ )
−1cτdτ −T∫
0
(S0τ )
−1 Xπτ−d Aτ
∣∣∣∣∣∣Ft
⎤
⎦ ,
then Vt is a Q-martingale, and similarly there exists a ψ(·) ∈ � such that
Vt = y0 +t∫
0
ψ�t σt d B∗
t for all t ∈ [0, T ]. (A.30)
Let φt = S0t ϕt , then by (A.29) and (A.30),
(S0T )
−1Y π,c,φT +T∫
0
(S0τ )
−1cτdτ −T∫
0
(S0τ )
−1 Xπτ−d Aτ
= y0 +T∫
0
(S0τ )
−1φ�τ στd B∗
τ
123
272 Y. Li, B. Yu
= y0 +T∫
0
ϕ�τ στd B∗
τ = VT .
This demonstrates Y π,c,φT = ξ , that is, (c, ξ) is financed by (π, φ), and therefore isattainable. ��
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