A combined analysis of hedge effectiveness and capital efficiency in longevity hedging MatthiasB¨orger Technische Hochschule N¨ urnberg Georg Simon Ohm, N¨ urnberg Keßlerplatz 12, 90489 N¨ urnberg, Germany E-mail: [email protected]; Phone: +49 (911) 5880-1755 Arne Freimann (corresponding author) Institute for Financial and Actuarial Sciences (ifa), Ulm & Institute of Insurance, Ulm University Lise-Meitner-Straße 14, 89081 Ulm, Germany E-mail: [email protected]; Phone: +49 (731) 20 644-253 Jochen Ruß Institute for Financial and Actuarial Sciences (ifa), Ulm & Institute of Insurance, Ulm University Lise-Meitner-Straße 14, 89081 Ulm, Germany E-mail: [email protected]; Phone: +49 (731) 20 644-233 December 20, 2019 Abstract Longevity hedges under risk-based solvency regimes typically provide two main bene- fits: a reduction in the uncertainty regarding future cash flows (measured in terms of hedge effectiveness ) and a reduction in capital charges (measured in terms of capital efficiency ). We argue that a separate analysis of these aspects cannot provide a full picture of the implications of hedging, in particular for index-based instruments. Hence, we propose a stochastic modeling framework for a combined analysis of hedge effectiveness and capital efficiency in longevity hedging. Unlike previous studies, our setup explicitly considers the uncertainty regarding future capital charges. In an economic capital model under Solvency II, a wide selection of customized and index-based hedging solutions is analyzed, which give rise to varying levels of population basis risk. We show that the Solvency II standard formula for longevity risk might generate different and less consistent capital reliefs than an internal model. Furthermore, hedge effectiveness might be misestimated systematically if the uncertainty in future capital charges is ignored. In a simultaneous analysis of hedge effectiveness and capital efficiency, we discuss the trade-off between hedge effectiveness and capital efficiency and find that generally no ’universally superior’ hedging solution can be found. We conclude that different hedging objectives require different instruments on different index populations. While customized hedges naturally outperform their index-based counterparts in terms of hedge effectiveness, we show that in many cases cost-efficient index-based designs provide a higher capital efficiency. JEL classification : G22; G23; G32 Keywords : Hedging longevity risk; Hedge effectiveness, Capital efficiency, Solvency regimes, Popula- tion basis risk
Transcript
A combined analysis of hedge effectiveness and capital efficiency
in longevity hedging
Matthias BorgerTechnische Hochschule Nurnberg Georg Simon Ohm, Nurnberg
Longevity hedges under risk-based solvency regimes typically provide two main bene-fits: a reduction in the uncertainty regarding future cash flows (measured in terms of hedgeeffectiveness) and a reduction in capital charges (measured in terms of capital efficiency).We argue that a separate analysis of these aspects cannot provide a full picture of theimplications of hedging, in particular for index-based instruments. Hence, we propose astochastic modeling framework for a combined analysis of hedge effectiveness and capitalefficiency in longevity hedging. Unlike previous studies, our setup explicitly considers theuncertainty regarding future capital charges.
In an economic capital model under Solvency II, a wide selection of customized andindex-based hedging solutions is analyzed, which give rise to varying levels of populationbasis risk. We show that the Solvency II standard formula for longevity risk might generatedifferent and less consistent capital reliefs than an internal model. Furthermore, hedgeeffectiveness might be misestimated systematically if the uncertainty in future capitalcharges is ignored. In a simultaneous analysis of hedge effectiveness and capital efficiency,we discuss the trade-off between hedge effectiveness and capital efficiency and find thatgenerally no ’universally superior’ hedging solution can be found. We conclude thatdifferent hedging objectives require different instruments on different index populations.While customized hedges naturally outperform their index-based counterparts in terms ofhedge effectiveness, we show that in many cases cost-efficient index-based designs providea higher capital efficiency.
The risk that policyholders live longer than anticipated is commonly referred to as ’longevityrisk’ and poses a major risk for annuity providers and pension funds. One possible approachfor dealing with this risk is longevity hedging. At present, the longevity risk transfer market isstill dominated by insurance-based deals with reinsurers acting as main players. However, inlight of the potential size of the global longevity market, the general consensus among practi-tioners appears to be that the limited capacity of the reinsurance sector cannot permanentlymeet the rising demand for de-risking solutions (cf. Blake et al. (2019)).
While aforementioned insurance-based deals are typically ’customized’, i.e. tailored indi-vidually to the hedger’s liability characteristics, they are not appealing to wider capital marketinvestors as they require knowledge of the mortality characteristics of the specific underlyingpopulation. This can be avoided by linking the hedge payout to transparent mortality indicesof a more general reference population. These index-based contracts can be appealing to awider range of investors and the hedger may also benefit from greater liquidity and lower risktransfer premiums due to a more competitive market. However, the volume of index-basedbusiness has been disappointingly low (cf. Blake et al. (2019)). One main obstacle to marketdevelopment is that, from the hedger’s perspective, standardized hedging solutions give riseto various types of residual basis risk, whose implications are to some extent poorly under-stood. Most prominently, the hedger is exposed to population basis risk since the mortalityexperience of the portfolio population might deviate from that captured by the more generalmortality indices.
To obtain a deeper understanding of population basis risk, several authors have recentlyaddressed its modeling and quantification.1 Commonly, longevity hedge effectiveness is eval-uated prospectively in a model-based simulation approach and quantified by means of riskmeasures as the achieved level of risk reduction. This approach allows to analyze and com-pare different hedging instruments with regard to their risk reduction.2 If the simulationmodel properly accounts for the imperfect correlation between the evolution of the hedger’spopulation and the hedge indices, index-based solutions perform less favorably compared tofully customized contracts. However, the potential cost advantages of standardized contractsdo typically not become apparent in these settings.
Under modern risk-based solvency regimes, such as Solvency II in Europe and the SwissSolvency Test in Switzerland, the economic impact of longevity hedging goes beyond thisrisk-mitigating effect. Insurers are required to back their longevity business with adequateeconomic capital and the expected costs for providing future economic capital have to bereserved as a risk margin in addition to the best estimate liabilities. Longevity hedging reducescapital charges and hence creates value (cf. Borger (2010), Meyricke and Sherris (2014)). Incontrast to the above mentioned hedge effectiveness, capital efficiency also takes this effect intoaccount. Obviously, the capital efficiency of a longevity transaction depends on two opposingeffects: the generated cost of capital saving and the costs of hedging. Meyricke and Sherris(2014) demonstrate that Solvency II capital requirements generate incentives to transfer short-term longevity risk while retaining longevity tail risk since the hedging costs tend to exceedthe capital relief over longer time horizons. However, their analysis is limited to customizedlongevity swaps. While index-based hedges typically result in lower capital reliefs due to
1See e.g. Cairns (2013), Cairns et al. (2014), Cairns and El Boukfaoui (2019), Coughlan et al. (2011),Haberman et al. (2014), Li and Hardy (2011), Li et al. (2017), Li et al. (2019), and Villegas et al. (2017).
2Academics have proposed a variety of longevity-linked instruments, see Blake et al. (2019) for an overview.
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population basis risk, they may at the same time offer cost advantages compared to customizeddeals. Hence, partial index-based transactions may actually represent economically viablealternatives. Therefore, both the achieved level of risk reduction (measured in terms of hedgeeffectiveness) and the generated cost of capital relief net of hedging costs (measured in termsof capital efficiency) should be considered for a well-informed hedging decision.
In the current paper, we propose a stochastic modeling framework for a combined analysisof hedge effectiveness and capital efficiency in longevity hedging. In particular, we provide aclear definition of both measures in a setting that, unlike previous studies, explicitly considersthe uncertainty regarding future capital charges (and the reduction resulting from hedging).In a numerical case study, we consider an annuity provider who evaluates a wide selectionof different customized and index-based hedging instruments. In particular, we also considerinstruments which are specifically designed for the purpose of reducing capital charges forlongevity risk. In a model-based simulation approach, we analyze the impact of each hedginginstrument on the annuity provider’s cash flow profile and quantify this impact in terms ofboth hedge effectiveness and capital efficiency. We simultaneously visualize the risk-reducingeffect and economic benefits (or costs) for all available hedging solutions. This can helphedgers to avoid suboptimal decisions and, more importantly, to identify suitable contractdesigns with regard to the strategic hedging objective.
Such a comprehensive analysis requires an extensive stochastic modeling framework whichaccounts for all relevant components of longevity risk, the costs of hedging, and the impactof hedging on capital charges within a realistic regulatory framework. For the individualsubcomponents of our framework, which can be specified independently of each other, webuild on several previously developed ideas and modeling techniques of different authors.Since a detailed discussion of all applied concepts would go beyond the scope of this paper,we limit ourselves to a concise summary of the higher-level elements and refer the interestedreader to the appendix or the cited works. On the whole, our modeling framework consistsof the following subcomponents:
• Economic capital model : We use an economic capital model under Solvency II. To com-pute longevity Solvency Capital Requirements (SCRs), the hedger either implementsa stochastic (partial) internal model or alternatively applies the Solvency II standardformula for longevity risk. Following the general guide outlined by Cairns and El Bouk-faoui (2019), we then determine the impact of longevity hedging on capital charges,in particular in the presence of population basis risk, under both SCR approaches andcompare the results between them. As pointed out by Borger (2010), a proper imple-mentation of a risk-based internal model requires the simulation of all components oflongevity risk over a one-year horizon and subsequent valuation of liabilities based onpotentially revised mortality assumptions.
• Stochastic mortality model : For the computation of capital charges, the mortality modelneeds to consist of two components: First, it requires a simulation model that jointlycaptures all relevant components of longevity risk, namely long-term morality trend riskof the overall population, potentially differing mortality characteristics of the hedger’sportfolio population, and finally idiosyncratic risk arising from a portfolio of limitedsize. This clear distinction between the different components of longevity risk allows usto evaluate different hedging instruments which give rise to varying levels of residualbasis risk. Second, a valuation model is needed for pathwise derivations of best estimate
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mortality assumptions at future valuation dates.3 We follow Borger et al. (2019b) andclearly distinguish between the unobservable actual mortality trend (AMT) prevailingat at certain point in time and the estimated mortality trend (EMT) that an observerwould estimate given the observed mortality evolution up to that point in time.
• Pricing : We assume that the counterparty demands a risk premium as compensationfor taking longevity risk. By way of illustration, we follow Freimann (2019) and apply arisk-adjusted pricing approach. For comparative studies of different pricing approachesfor longevity-linked securities, we refer to Bauer et al. (2010) or Leung et al. (2018).
The remainder of the paper is organized as follows. In Section 2, we describe our modelsetup and provide a definition of hedge effectiveness and capital efficiency in the presenceof uncertain future capital charges for longevity risk. The full specification of the modelingframework, which particularly includes the calibration of the stochastic mortality model to thehistorical mortality experience of English and Welsh males, is provided in the appendix. InSection 3, we introduce the considered hedging instruments by specifying both the underlyinghedge payout structure and the population the instrument is linked to. Section 4 provides thenumerical results. After a brief investigation of the hedger’s initial unhedged situation, wefirst analyze the benefits of hedging, namely capital relief (as driver for capital efficiency) andrisk reduction (as driver for hedge effectiveness), separately. In particular, we compare capitalreliefs derived under the Solvency II standard formula to those obtained under a risk-basedinternal model. Then, we combine both aspects and address the trade-off between hedgeeffectiveness and capital efficiency which allows us to discuss the suitability of various hedgedesigns for different objectives. This is followed by a sensitivity analysis with respect to themodeling assumption for population basis risk and the structure of the underlying liabilities.Finally, Section 5 concludes.
2 Model setup
As outlined in the introduction, we now establish a setting for a combined analysis of hedgeeffectiveness and capital efficiency in longevity hedging. To focus on longevity risk, we assumethat all market participants are fully hedged against interest rate and investment risk andonly invest at the risk-free interest rate r, which is also used for discounting purposes. Forthe sake of simplicity, we assume that the hedger’s book is closed to new business and weignore any other source of uncertainty (such as operational risk or counterparty credit risk4).Hence, any variation in future cash flows is only due to changes in (expected) mortality.
In Section 2.1, we start by describing our stochastic mortality modeling framework whichaccounts for future changes in actual mortality as well as for the accompanying changes inbest estimate mortality assumptions. Afterwards, we introduce the liability to be hedgedin Section 2.2 and describe the hedger’s economic capital model in Section 2.3. Finally, inSection 2.4, we provide a clear definition of hedge effectiveness and capital efficiency.
3Also Cairns (2013), Cairns et al. (2014), Cairns and El Boukfaoui (2019), and Coughlan et al. (2011)clearly distinguish between a simulation model and a valuation model.
4In reality, longevity hedges inevitably entail counterparty credit risk, which can be mitigated if not fullyeliminated by collateralization. Biffis et al. (2016) find that the cost of posing collaterals for longevity swapsis comparable (and often even much smaller) than that observed in the interest rate swaps market.
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2.1 Stochastic modeling framework for the components of longevity risk
In this section, we present the basic idea of our stochastic mortality modeling framework.We focus on its most important features and refer to Appendix A for the detailed modelspecification and to Appendix B for the model calibration.
The hedger’s book population is interpreted as a subset of a larger reference population R(typically the national population) with potentially differing mortality characteristics e.g. dueto a specific socioeconomic structure or selection effects. Within the reference population, weconsider NSub distinct and sufficiently large subpopulations of different socioeconomic statuswhere all individuals within one subgroup are assumed to experience the same force of mor-tality. To model the individual mortality characteristics of the hedger’s book population, wefollow Haberman et al. (2014) and apply a so-called ’characterization approach’ by assum-ing that the company has sufficient information to uniquely assign each policyholder to theappropriate subpopulation and that they can be identified with this subgroup throughouttheir lives.5 This approach offers several advantages. Most importantly, it can be appliedto portfolios of any size without wrongfully capturing unsystematic variations in the dataarising from a small sample size as systematic mortality differentials. Relying on sufficientlylarge subpopulations ensures a clear and consistent distinction between the two componentsof longevity risk, in particular for small portfolios. Also, the evolution of the socioeconomicbook composition over time is adequately captured.
Within this setting, we model the random future evolution of the book population consis-tently to future mortality in the reference population and its socioeconomic subpopulationsin a multi-population stochastic mortality model which captures the following components oflongevity risk:
• The long-term mortality trend risk of the overall population modeled via a stochastic
process for future probabilities of death q[R]x,t , t ≥ 0 for the reference population.
• The evolution of socioeconomic mortality differentials over time relative to the reference
population resulting in subpopulation-specific death probabilities q[p]x,t, t ≥ 0 for each
subpopulation p ∈ {1, . . . , NSub}.
• Small sample risk due to a limited portfolio size by drawing realizations for survivors
(conditional on realized mortality rates q[p]x,t) from a Binomial distribution.
We refer to the stochastic model for the actual mortality evolution of all (sub-)populationsas the actual mortality trend (AMT) simulation model. This model is described in detail inAppendix A.1. A rigorous differentiation between the three components of longevity risk willbe essential for a clear assessment of population basis risk in index-based hedges which areassociated with the reference population or its subpopulations.
We follow Borger et al. (2019b) and acknowledge that an observer at some point in timeT who intends to project mortality beyond time T cannot observe the prevailing AMT. Inpractice, an observer would calibrate a mortality model to the observed mortality evolutionup to time T . This derivation of best estimate mortality assumptions at future valuation
5Lu et al. (2014) find that migration between subpopulations in England does not significantly distort trendsin socioeconomic mortality inequalities. For simplicity, we also assume that the hedger’s individual character-istics are completely captured by the considered subpopulations and we neglect the risk of misspecifying theappropriate subgroup for individual policyholders.
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dates is modeled via the so-called estimated mortality trend (EMT) valuation model6 whichconsists of two components:7
• A methodology for estimating the reference population’s prevailing mortality level and
trend for pathwise derivations of time-T best estimate mortality rates q[R]x,t (T ), t > T .
• Inspired by Cairns and El Boukfaoui (2019), an additional approach for deriving prevail-ing subpopulation-specific experience ratios to account for differing mortality levels (andtrends) relative to the reference population. For each subpopulation p ∈ {1, . . . , NSub},best estimate mortality rates q
[p]x,t(T ), t > T are then derived by adjusting best estimate
mortality rates of the reference population with the prevailing experience ratios.
This model is described in detail in Appendix A.2. In our setup, the AMT simulation modelis used to generate sample paths of future mortality and to derive, among others, annuitypayments and payouts of cash flow hedges, whereas the computation of best estimate liabilitiesand the derivation of hedge payoffs that build on future best estimate mortality assumptionsneed to be carried out under the EMT valuation model.8
2.2 Liability to be hedged
The liability to be hedged consists of a simplified portfolio of immediate or deferred life an-nuities, where all premiums have been paid upfront to the insurer. Starting at the retirementage xR, these contracts pay one unit of currency at the beginning of each year until the ben-eficiary dies.9 We do not consider fees or any further features (such as death benefits or aprofit sharing feature).
We consider a single cohort of size NBook with starting age x0 at time t = 0. The number of
initial policyholders from subpopulation p ∈ {1, . . . , NSub} is given by B[p]x0,0
:= ηpNBook ∈ N.The time-t random present value of all future unhedged liabilities then reads as
L(t) :=
NSub∑p=1
L[p](t) :=
NSub∑p=1
∑s>t
(1 + r)−(s−t)1{x0+s≥xR}B
[p]x0+s,s, t ≥ 0,
where B[p]x0+s,s denotes the number of survivors in the book population from subpopulation p
aged x0 + s at time s > 0. The time-t best estimate unhedged liabilities are calculated usingbest estimate mortality derived under the the EMT valuation model as
L (t) :=
NSub∑p=1
B[p]x0+t,t
∑s>t
(1 + r)−(s−t)1{x0+s≥xR}
s−1∏u=t
(1− q[p]
x0+u,u+1(t)), t ≥ 0,
which is an Ft-measurable point estimate for L(t), where Ft contains the observed mortalityinformation up to time t. Following a pragmatic deterministic best estimate mortality pro-jection for valuation purposes (instead of deriving the theoretical conditional expected value)reflects practice in many countries.
6Quantities that are derived under the EMT valuation model are denoted by · throughout the paper.7Here, we implicitly assume that the whole data set is publicly available for all (sub-)populations and that
it will be updated directly in each future year according to realized mortality.8For a recent discussion on which of the two models is relevant for which kind of question, we refer to
Borger et al. (2019b).9Assuming no concentration of risk by amounts, small sample risk is diversifiable in large portfolios.
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With a hedge in place, the liability is netted with future hedging instrument cash flowsgiving the time-t random present value of the hedged liabilities
LH(t) := L(t)−H(t), t ≥ 0,
where H(t) denotes the time-t random present value of all future cash flows from the hedge ac-cording to the underlying hedge structure, which will be introduced in Section 3. Analogouslyto the unhedged case, we calculate the time-t best estimate hedged liabilities as
LH (t) := L (t)− H (t) , t ≥ 0,
where H(t) denotes the time-t best estimate of all future hedging instrument cash flows. Wewould like to stress that this quantity is derived on a best estimate basis and is not meant torepresent the instrument’s market value. Following a deterministic best estimate mortalityprojection constitutes a meaningful approach for evaluating symmetric hedge structures sincethe median path generally provides a reasonable approximation for the mean. However, thevaluation of contracts which contain asymmetric structures, such as option-type derivatives,would call for a more sophisticated approach.
2.3 Economic capital model
Under modern risk-based solvency regimes, the reference company has to provide SCRs forlongevity risk. Under Solvency II, the SCR is defined as the 99.5% Value-at-Risk (VaR) ofthe basic own funds over a one-year horizon, where the basic own funds correspond to thedifference between the market value of assets and the market value of liabilities. In principle,the SCR corresponds to the capital required to cover all losses which may occur over thefollowing year at a confidence level of at least 99.5%. Following Borger (2010) and Borgeret al. (2019b), we assume that the evolution of assets is independent of realized mortalityand thus does not contribute to the longevity SCR. Moreover, we assume that there is noloss-absorbing capacity of technical provisions and that the (hedged) liabilities do not includea risk margin when computing the SCR.
In addition to the current SCR, the company will be required to hold SCRs over time.For the determination of longevity SCRs, the company can choose between the followingapproaches:
• Internal model: The company might use a (partial) stochastic internal model to deter-mine the longevity SCR according to the 99.5% VaR concept of Solvency II. As discussedby Borger (2010), longevity risk over a one-year horizon consists of two components:more annuitants than anticipated might survive the year or longevity assumptions mightchange over the year in an unfavorable direction. Typically, the latter is the more rele-vant factor. The SCR in year T , denoted as SCRIML (T ), is then defined as the 99.5thpercentile of the change in best estimate liabilities from time T to T + 1:
L (T + 1) + CF (T + 1)
1 + r− L (T ) ,
where CF (T+1) denotes the company’s cash flows of the longevity prone business (ben-efits paid to the annuitants) between T and T +1. For the hedged position, SCRIMLH (T )
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is defined analogously as
LH (T + 1) + CFH(T + 1)
1 + r− LH (T ) ,
where CFH(T + 1) additionally contains cash flows that might result from the hedgeover the year.
• Standard formula: Alternatively, companies are allowed to use a simplified standardformula as an approximation for the 99.5% VaR approach. Under this approach, theSCR in year T is determined as the change in best estimate liabilities due to a suddenand permanent longevity shock, i.e.
SCRSFL (T ) := L (T |shock(20%))− L (T ) ,
where shock(20%) represents a uniform relative shock of 20% on best estimate mortalityrates for all ages and populations under consideration. For the hedged position, the SCRat time T is defined analogously as
SCRSFLH (T ) := LH (T |shock(20%))− LH (T ) .
The standard formula has come under some criticism in the academic literature for itsunrealistically simple structure, see e.g. Borger (2010). In particular, the structure ofthe uniform one-off shock irrespective of age and maturity does not appropriately reflectthe true nature of longevity risk as a typically slowly accumulating demographic trendrisk. In spite of these shortcomings, many companies still use the standard formula.
Since the SCR at some future point in time T obviously depends on the mortality evolution upto time T , we interpret it as an FT -measurable random variable. Given an outer simulationpath containing realized mortality up to time T , the SCR (with or without a longevity hedgein place) can be computed conditional on this simulation path. The derivation of the 99.5thpercentile in the internal model requires an additional inner Monte Carlo simulation, wheremortality over a one-year horizon is simulated with the AMT simulation model and the bestestimate (hedged) liabilities are reevaluated in the EMT valuation model. Within a two-levelnested Monte Carlo simulation, entire distributions for the company’s SCRs over time can bederived.
2.4 Hedging objectives
As outlined in the introduction, our main objective is to investigate the impact of differenthedging instruments on the annuity provider’s future cash flow profile. In addition to annualbenefit payments to surviving annuitants, the company has to compensate its shareholdersfor providing equity to cover its SCR. Due to the stochastic nature of future SCRs thisadditional cost of regulatory capital is also a random variable. This motivates the definitionof the adjusted unhedged liabilities as
ΠML := L(0) + CoCML , M ∈ {IM, SF},
where CoCML :=∑
t≥0rCoCSCR
ML (t)
(1+r)t+1 denotes the time zero random present value of all costs of
capital for the unhedged liabilities which are either derived with the internal model (M = IM)
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or by applying the standard formula (M = SF ). The cost of capital rate rCoC reflects thereturn in excess of the risk-free rate which shareholders demand for providing equity. Thebest estimate adjusted unhedged liabilities are the company’s initial portfolio reserve plus theexpected cost of regulatory capital, which has to be reserved in addition as a risk margin underSolvency II. For the hedged position, the adjusted hedged liabilities are defined analogouslyas
ΠMLH
:= LH(0) + CoCMLH , M ∈ {IM, SF}.
We quantify the effect of hedging on the adjusted liabilities by means of the following measures:
• Capital efficiency: The expected mitigating impact on the adjusted liabilities is de-noted as net cost of capital relief and defined as
NReCoCM (H) :=E(ΠML
)− E
(ΠMLH
)=E
(CoCML
)− E
(CoCMLH
)+ E(H(0)), M ∈ {IM, SF},
where two opposing effects come into play:
– On the one hand, hedging typically reduces the hedger’s SCRs which in turn gen-erates a positive expected cost of capital relief of
ReCoCM (H) := E(CoCML
)− E
(CoCMLH
)≥ 0, M ∈ {IM, SF}.
– On the other hand, the expected present value of all hedging instrument cashflows E(H(0)) is typically negative reflecting the absolute risk loading on top ofthe objective best estimate value charged by the counterparty for taking risk.
In this setting, a company which is completely hedged against longevity risk wouldnot have to provide any SCRs for longevity risk and the cost of capital would conse-quently reduce to zero. If such a perfect hedge was offered on a best estimate basis(i.e. E(H(0)) = 0), it would obviously provide the maximal net cost of capital relief ofE(CoCML
). With regard to this benchmark, we define the capital efficiency of a hedge
H as
CEM (H) :=NReCoCM (H)
E(CoCML
) , M ∈ {IM, SF}.
We refer to a hedge as being capital efficient if CEM (H) > 0, i.e. if the generated cost ofcapital saving exceeds the hedging costs. Hedge H1 is said to be more capital efficientthan hedge H2 if CEM (H1) > CEM (H2).
• Hedge effectiveness: We define the effectiveness of a hedge as the achieved relativerisk reduction measured under a risk measure ρ in the centralized adjusted liabilities:
HEMρ (H) := 1−ρ(
ΠMLH
)ρ(ΠML
) := 1−ρ(
ΠMLH− E
(ΠMLH
))ρ(ΠML − E
(ΠML
)) , M ∈ {IM, SF}.
Obviously, a perfect hedge offers the maximal hedge effectiveness of one. We refer to ahedge H1 as being more effective than hedge H2 if it removes more uncertainty, i.e. ifHEMρ (H1) > HEMρ (H2). We would like to stress that this definition of hedge effective-ness based on the adjusted liabilities explicitly considers the reduction in uncertaintyregarding future costs of capital.
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Even though both quantities allow for an intuitive interpretation, their relation is not obvi-ous when the counterparty charges a non-zero risk premium. Naturally, the question ariseswhether the most effective hedge also provides the highest capital efficiency and vice versa.In this case, a rational hedger would have no incentive to consider alternative instruments.However, if this is not the case, the company has to find a reasonable trade-off between hedgeeffectiveness and capital efficiency depending on the strategic hedging objective.
3 Hedging instruments
In line with the current state of the market for longevity-linked securities, we assume anincomplete and illiquid market where the hedger does not have the opportunity to resell orterminate a contract prior to maturity. In this setting, we assume that market participantsdemand a risk premium for taking on longevity risk. We follow the widely used approachand define an equivalent risk-adjusted measure Q under which the price of any security isdefined as the expected value of its discounted payoff. This pricing measure assigns higherprobability mass to scenarios which are unfavorable for a longevity hedge provider and isexclusively used for pricing purposes. The basic idea is to adjust the distribution of eachindividual risk driver by the so-called ’market price of longevity risk’. Overall, the underlyingmodel structure is preserved under this change of measure giving a risk-adjusted version ofthe AMT simulation model. For the technical details, we refer to Appendix A.4 and Freimann(2019). For comparability, we apply this pricing approach to all hedging instruments underconsideration.
In this section, we describe the considered hedging instruments which consist of twocomponents: a basic hedge payout structure and an underlying index population (IP). Bylinking the same hedge payout structure to different IPs, we construct different hedginginstruments which give rise to varying levels of population basis risk. First, Section 3.1 givesan overview of the underlying IPs. Then, Section 3.2 describes the hedge payout structures.
3.1 Index populations
Each hedging instrument is linked to one of the following IPs:
• IP = B: The hedge is fully customized and directly linked to the survivors and themorality experience in the hedger’s book population.10
• IP = S: The instrument is index-based linked to the subpopulations. As opposed tothe previous design, this gives rise to small sample risk due to a limited portfolio size.
• IP = R: The hedge is index-based and exclusively covers the randomness originatingfrom the reference population leaving the hedger with small sample risk as well associoeconomic basis risk.
In the spirit of Cairns and El Boukfaoui (2019), we construct the index-based instruments(IP = S,R) by replacing customized quantities in the respective fully customized version by
10Even though we also denote the hedger’s portfolio population as an index population for the sake of aconsistent notation, the associated hedge instruments are meant to be indemnity-based rather than index-based.
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appropriate hedge indices. These index-based proxies are defined to match as closely as possi-ble (in terms of magnitude and sensitivity to mortality) the fully customized quantities whileonly using the observable information from the underlying IP. While all information which isalready known at inception may be included in the hedge contract, new morality informationwhich becomes available over the hedge horizon but is not captured by the underlying IP can-not be included. In particular, the hedge indices linked to the reference population (IP = R)are constructed by embedding the current experience ratios in the hedge contract to adjust foranticipated mortality differences between the book and the reference population. By keepingthem fixed, population basis risk is minimized while all randomness in future hedge payoutsexclusively originates from the reference population. This construction of index-based hedgesis described in detail in Appendix A.3.
3.2 Hedge payout structures
The time-t random present value of all future hedging instrument cash flows is given by
• h[IP ](t), 0 < t ≤ τ represents the Ft-measurable payoff to the hedger at time t accordingto the underlying hedge payout structure, and
• p[IP ](t), 0 ≤ t < τ denotes the Ft-measurable path-dependent premium charged by thecounterparty at time t. These will only become relevant for rolling hedge programs.
For all forward-type hedge structures, the risk premium is directly included in the hedgepayout structure by means of risk-adjusted forward rates which are determined at time zeroso that EQ (H [IP ](0)
)= 0. For hedge payments that are based on current or future best
estimate mortality assumptions, we assume that both parties agree on deriving the requiredvalues with the EMT valuation model.11
In what follows, we give a concise overview of the considered hedge payout structures. Forthe details, in particular for the definition of the underlying hedge indices for the index-baseddesigns, we refer to Appendix A.3.
3.2.1 Longevity swaps
In a longevity swap, the hedger receives a sequence of cashflows corresponding to the under-lying liability cashflows in exchange for a series of forward rates of the form
h[IP ](t) := 1{x0+t≥xR}
(S
[IP ]x0+t,t − EQ
(S
[IP ]x0+t,t
)), 0 < t ≤ τ, IP ∈ {B,S,R},
where S[IP ]x0+t,t either represents the actual number of survivors aged x0 + t at time t (in case
of IP = B) or a survivor index as a proxy (in case of IP = S,R). Note that an unlimited(τ =∞) fully customized (IP = B) longevity swap provides by construction a perfect hedge.
11In practice, the hedge instrument valuation model (which is specified in the hedge contract) may differfrom the liability valuation model. For simplicity, we use the EMT valuation model for both purposes.
10
3.2.2 Annuity forwards
In an annuity forward, the hedger receives the best estimate present value (according to up-to-date mortality assumptions at maturity) of all future annuity payments in exchange for aforward rate resulting in a single hedge payout of the form
h[IP ](τ) := L[IP ](τ)− EQ(L[IP ](τ)
), IP ∈ {B,S,R},
where L[IP ](τ) either represents the actual time-τ best estimate liabilities (in case of IP = B)or a liability index as a proxy (in case of IP = S,R). While this hedge (partly) transferslongevity risk up to time τ , the hedger is still exposed to the risk that longevity after timeτ might not evolve as expected. As discussed by Borger et al. (2019b), when assessingthe effectiveness of this hedge in a stochastic simulation one needs to be aware that thehedge payout can only be based on the then prevailing EMT, which might deviate from theunobservable AMT. Otherwise, the hedge effectiveness might be overestimated.
3.2.3 Q-forwards
In a q-forward, the hedger exchanges a fixed forward rate against realized mortality ratesresulting in a single hedge payout of the form
h[IP ](τ) := EQ(Q
[IP ]x0+τ,τ
)−Q[IP ]
x0+τ,τ , IP ∈ {S,R},
where Q[IP ]x0+τ,τ represents the underlying mortality rates which are either linked to the sub-
populations (in case of IP = S) or to the reference population (in case of IP = R). Sincethe liability consists of a single cohort aged x0 + τ at time τ , we focus on a single q-forwardwith same reference date and age. Since this payoff structure obviously differs from the li-ability, there is no natural choice for the optimal number of single q-forwards. As outlinedin Appendix A.3, we find the optimal hedge ratios by maximizing the hedge effectiveness ina Monte Carlo simulation. For simplicity, we limit our analysis to a static setting where allhedge ratios are calibrated at time zero and remain fixed over the hedge horizon.
3.2.4 Rolling portfolios of one-year call spread options
Even though all instruments introduced so far typically reduce SCRs through their risk-mitigating effect, their payoff structures are not primarily designed for this particular purpose.In line with the one-year view of Solvency II, we now propose a rolling program of one-yearcontracts which are tailored to the hedger’s economic capital model.
Prominent SCR-minimizing hedging instruments are so-called call spread options, whichare discussed and analyzed in the context of a regulatory capital model by Cairns and El Bouk-faoui (2019). Assume that at any point in time 0 ≤ t < τ , the hedger enters into a one-yearcall spread option contract of the following form: If the underlying hedge index X [IP ](t+ 1)at the end of the year exceeds a predefined attachment point AP (t), the hedger receives apayment of the form
h[IP ](t+ 1) := (EP (t)−AP (t)) max
{0; min
{X [IP ](t+ 1)−AP (t)
EP (t)−AP (t); 1
}}, IP ∈ {B,S,R},
11
which increases linearly until the hedge index reaches the exhaustion point EP (t). The hedgeindex is defined to match (in case of IP = B) or to replicate as closely as possible (in case ofIP = S,R) the random driver in the company’s SCR computation from time t to t+1. Hence,the hedge provides an offsetting payment if longevity over the year evolves in an unfavorabledirection. For comparability, we use the same attachment and exhaustion points for all IPs:
• The Ft-measurable attachment points are typically set above the current best estimateliabilities to ensure that the option is only triggered in scenarios of significant mortalityimprovements over the year. Exemplarily, we define the attachment points via a uniformrelative longevity stress of ap ≥ 0 on current best estimate liabilities:
AP (t) := (1 + r)L (t |shock(ap)) .
• The Ft-measurable exhaustion points are defined as
EP (t) :=
{αep-th percentile of X [B](t+ 1), for an internal model design
(1 + r)L (t |shock(ep)) , for a standard formula design,
which are typically chosen equal to (or moderately below) αep = 99.5% or ep = 20%respectively. Hence, the instrument always pays out in full in the Solvency II stressscenarios. From a mere cost of capital perspective, there is no incentive for hedginglongevity tail risk beyond these values.
In return, the hedge provider demands a premium of p[IP ](t) := (1 + r)−1EQ (h[IP ](t+ 1)|Ft)
at time t, which obviously depends on realized mortality up to time t. For the technicaldetails, we refer to Appendix A.4.12 Note that the hedger is subject to ’rolling risk’ since thecosts for the next hedge in the sequence might significantly rise in unfavorable scenarios. Asopposed to forward-type hedge structures, pricing therefore affects the hedge effectiveness ofthis rolling strategy.
4 Numerical results
For the following numerical application, we calibrate the multi-population stochastic mortal-ity model to the historical mortality experience of English and Welsh males up to the year2016. The national population serves as the reference population and we consider five sub-populations of different socioeconomic status (ordered from the most to the least deprivedareas) based on the Index of Multiple Deprivation (IMD) for England. In the base case,we follow Villegas and Haberman (2014) and model socioeconomic mortality differentials bya random walk with drift (RWD) and include both a level and a trend adjustment in thesubpopulation-specific experience ratios within the EMT valuation model. For more detailson the model calibration, we refer to Appendix B.
In the base case, we consider a portfolio of immediate life annuities consisting of 10,000policyholders aged 65 at the beginning of the year 2017, which is set to t = 0. Regarding its
12Note that these instruments can be analyzed in our setup without a third level of nested simulations inspite of their asymmetric payout structure. In each inner simulation path in the internal model’s SCR com-putation, the hedge payout at the end of the year is readily available and the one-year contract is terminated.Furthermore, only the current one-year hedge needs to be considered since the next contract in the sequenceis closed and priced at the beginning of next year according to updated best estimate mortality assumptions.
12
Description Parameter Value
Starting age x0 65Retirement age xR 65Initial book size NBook 10,000Socioeconomic composition η (0, 0, 30%, 30%, 40%)Risk-free interest rate r 2%Cost of capital rate rCoC 6%Market price of longevity risk λ 30%Risk measure ρ TVaR90%
Table 1: Model parameters in the base case.
socioeconomic structure, we assume a rather affluent book population which can be identi-fied exclusively with the three most affluent IMD-subpopulations to provoke a considerableexposure to socioeconomic basis risk. Furthermore, we assume a risk-free interest rate of 2%and a cost of capital rate of 6%. Moreover, we set the constant market price of longevity riskequal to 30% for all systematic risk drivers. This parameter is calibrated so that the costs forfull securitization roughly match the expected cost of capital for keeping the risk under theinternal model as suggested by Borger (2010), Levantesi and Menzietti (2017), and Zeddoukand Devolder (2019). Throughout this work, we use the 90% Tail-Value-at-Risk (TVaR) forthe quantification of hedge effectiveness.13 Table 1 summarizes the model parameters for thebase case.
We perform a two-level nested Monte Carlo simulation with 10,000 outer sample paths.For each path and for every year, we rely on 10,000 inner one-year scenarios for the SCRcomputation in the internal model and for the associated derivation of exhaustion pointsfor the rolling call spread portfolios. For their pathwise pricing, we additionally use 1,000risk-adjusted inner one-year scenarios. For consistency, we rely on the same scenarios for allhedging instruments when comparing their distributions for a chosen model parametrization.
4.1 Unhedged (adjusted) liabilities
We start with a brief investigation of the hedger’s initial situation without hedging. Table2 shows the mean and the 90% TVaR of the company’s (centralized) liabilities with andwithout an adjustment for future cost of regulatory capital. As anticipated, the adjustmentfor future capital charges increases the expected financial obligations. The expected cost ofcapital is more than twice as high under the standard formula than under the internal model.A closer look at the company’s SCRs over time, which are displayed in Figure 1, revealsthat the standard formula produces rather high SCRs compared to the internal model. Thisphenomenon is in line with findings of Borger (2010) and can be traced back to structuralshortcomings of the standard formula’s rather conservative longevity stress.14
Regarding the randomness in future SCRs, the 90% TVaR indicates that the allowance
13This risk measure is also used for the derivation of optimal q-forward hedge ratios. We also performedour analyses under alternative risk measures such as variance, 95% TVaR, and 99.5% VaR and found nostructurally different results.
14Of course, the SCRs derived under the internal model highly depend on the applied weighting in the EMTvaluation model, which controls the adaption of best estimate mortality assumptions to observable changes inmortality patterns. To obtain objective SCRs, it is therefore important to determine the optimal weighting bymeans of a reasonable optimization criterion as presented in Appendix B.
Table 2: Mean and 90% TVaR of the hedger’s unhedged (adjusted) liabilities.
for stochastic future SCRs also increases the company’s overall longevity risk exposure. Theuncertainty in future (cost of) regulatory capital is more pronounced under the internal modelthan under the standard formula since the standard formula’s SCR at any point in time solelydepends on the effect of a prescribed longevity shock on current best estimate liabilities. TheSCR under the internal model on the other side is driven by the variability in best estimatemortality assumptions over a one-year horizon and thus mainly by the variability in one-yearEMT changes, which can be quite significant. The reason for this is twofold. First, thevariability in EMT changes from time t to t + 1 depend on realized mortality up to time t,particularly after trend changes in previous years. Second, as explained in Appendix B, weaccount for parameter uncertainty in the trend change parameters at the start of each outersimulation path. Hence, the overall level of SCRs generally differs between the simulationpaths depending on the drawn model parameters.15
4.2 The drivers of hedge effectiveness and capital efficiency
Now, we introduce the hedges which are summarized in Table 3. For annuity forwards, q-forwards, and rolling call spread portfolios, we consider contract maturities up to 25 years,i.e. until the underlying cohort reaches age 90, and for longevity swaps we focus on longer timesto maturity beyond 25 years. Regarding the rolling call spread portfolios, the hedger furtherneeds to agree with the hedge provider on optimal attachment and exhaustion points. To
15This is also the reason why the internal model’s initial SCR lies noticeably above the median SCRs in thefollowing years.
14
Instrument Description Parameter Value
Longevity swapsindex population IP ∈ {B,S,R}
maturity τ ∈ [25,∞)
Annuity forwardsindex population IP ∈ {B,S,R}
maturity τ ∈ [1, 25]
Q-forwardsindex population IP ∈ {S,R}
maturity τ ∈ [1, 25]Rolling portfolios
of one-yearcall spread options
(standard formula design)
index population IP ∈ {B,S,R}maturity τ ∈ [1, 25]
attachment point ap 5% (shock)exhaustion point ep 20% (shock)
Rolling portfoliosof one-year
call spread options(internal model design)
index population IP ∈ {B, S,R}maturity τ ∈ [1, 25]
attachment point ap 1% (shock)exhaustion point αep 99.5(th percentile)
Table 3: Overview of hedging instruments in the base case.
simplify the discussion in this section, we initially focus on one exemplary hedge structure foreach SCR computation method: a design tailored to the internal model with an exhaustionpoint set at the relevant 99.5th percentile and an alternative parameter set of (ap, ep) =(5%, 20%) for a standard formula design. We deliberately choose a higher attachment pointof ap = 5% for the latter design (compared to ap = 1% for the former) to demonstrate theeffect of higher attachment points. Nevertheless, we address the optimal choice of attachmentand exhaustion points from a wider range of parameters in Section 4.3.
In this section, we start with a separate analysis of the two main effects of hedging: capitalrelief (as driver for capital efficiency) and risk reduction (as driver for hedge effectiveness).Subsequently, we analyze both aspects simultaneously in Section 4.3.
4.2.1 Capital relief: standard formula vs. internal model
As a first step, we analyze how selected hedges impact the hedger’s longevity SCRs overtime. Figure 2 shows percentile plots of the SCRs over time with selected fully customized(IP = B) longevity hedges over τ = 25 years in place under the internal model (upper row)and under the standard formula (lower row). Clearly, hedging only impacts the amount ofregulatory capital over the presumed hedge horizon. Beyond that, the company still has toprovide capital. Interestingly, the structures of the SCRs for the hedged positions clearlydiffer in terms of both level and variability not only among the instruments but also betweenthe SCR computation methods.
First, we notice that the limited longevity swap provides a higher capital relief under thestandard formula than under the internal model. This is due to the fact that the latter’s SCRsare mainly driven by the risk that long-term mortality assumptions need to be revised whichare not covered by a longevity swap over 25 years. The standard formula’s uniform shock onthe other side affects short and long-term mortality likewise resulting in a substantial capitalrelief for a limited longevity swap.
In contrast, the annuity forward covers all observable changes in mortality patterns overthe hedge horizon. In particular, one year prior to maturity the instrument provides protectionagainst all changes in best estimate mortality over the year and the hedger therefore only has
Figure 2: Percentile plots of the company’s SCRs over time derived under the internal model (upperrow) and under the standard formula (lower row) without hedging (gray) and with different fullycustomized (IP = B) longevity hedges over τ = 25 years in place: longevity swap (blue, left panel),annuity forward (green, middle panel), and rolling call spread portfolio designed for the internal model(gold, right panel).
to hold SCRs for the uncertain cash flows that are due at the end of the year. This explainswhy the annuity forward is quite effective in reducing SCRs towards the end of its term. Sincesimilar observations can be made under both economic capital models, they seem to agree onan adequate relative capital relief for this instrument.
Finally, the rolling call spread portfolio which is tailored to the internal model shows themost striking behavior. Clearly, it serves its primary purpose of reducing longevity SCRs inthe internal model. Since the attachment points are set moderately above the current bestestimate liabilities, the hedger is still exposed to residual risk requiring the provision of SCRs.Nevertheless, these lie far below their unhedged counterparts and their variability is reducedsubstantially. However, the standard formula’s SCRs for the hedged position are significantlylarger and much more volatile. The reason for this phenomenon lies in the path-dependentexhaustion points which are calibrated with respect to the relevant 99.5th percentiles in theinternal model and therefore obviously not suited for the standard formula’s shock approach.
Overall, the percentile plots illustrate that the standard formula and the internal modelmight produce discordant capital reliefs. Moreover, it can be seen that the considered hedgeshave structurally different impacts on future SCRs even though they are all directly linkedto the hedger’s book population. The matter becomes further complicated when we extendthe analysis to index-based hedges giving additional rise to population basis risk. The hedger
Figure 3: Overview of relative expected cost of capital relief under the internal model (y-axis) com-pared to the standard formula (x-axis). The dot size increases in the contract maturity.
then further needs to accept an appropriate haircut for the reduction in hedge efficiency dueto population basis risk.
To obtain a concise picture across all instruments, hedge horizons, and IPs under consid-eration, Figure 3 provides an overview of the proportionate expected cost of capital relief
ReCoCM(H [IP ]
)E(CoCML
) , M ∈ {IM, SF}, IP ∈ {B,S,R}
for the hedges in Table 3 under the internal model on the y-axis compared to the standardformula on the x-axis. The dot size increases in the instrument’s time to maturity and theunderlying IP is visualized by the following symbols: circles for IP = B, squares for IP = S,and crosses for IP = R.16 If all points were exactly on the dashed gray diagonal, both SCRcomputation methods would yield concordant relative cost of capital reliefs. As we can clearlysee in Figure 3, this is only the case for the unlimited fully customized longevity swap, whichprovides full cost of capital relief under both models, and approximately for a range of annuityforwards. For the other instruments, discordant capital reliefs between the standard formulaand the internal model are the rule rather than the exception. More precisely, all longevityswaps and q-forwards over a limited hedge horizon provide a higher relative cost of capitalrelief under the standard formula. This effect is most pronounced for the rolling call spreadportfolio which is tailored to the standard formula. Since the attachment points, which arespecified via a 5% longevity shock, still lie far below the standard formula’s prescribed stressof 20%, this hedging strategy provides an attractive cost of capital relief under the standard
16This visualization technique is used throughout the paper.
17
formula. However, the internal model recognizes that rather high attachment points leavea substantial part of longevity risk with the hedger and therefore allows a lower capitalrelief. We conclude that a hedging strategy which is constructed with regard to the simplifiedstandard formula does not necessarily perform equally well under a risk-based internal model.The rolling call spread portfolio tailored to the internal model marks the other extreme forreasons we discussed above.
Regarding the impact of the hedge horizon, we find that the resulting cost of capital reliefincreases in the time to maturity for longevity swaps and for rolling call spread portfolios.However, for annuity forwards and q-forwards there appears to be an optimal hedge horizonbeyond which hedge efficiency is declining. As we will see in the next subsection, this is dueto the fact that the degree of risk reduction increases in the time to maturity for longevityswaps and rolling call spread portfolios while it peaks somewhere around τ = 20 for annuityforwards and q-forwards.
Finally, by focusing in Figure 3 on a single hedge payout structure (i.e. on dots of thesame color) and comparing the results among the IPs (i.e. among different symbols), we obtainvaluable insights into the impact of socioeconomic basis risk and small sample risk. First, wenote that the expected cost of capital relief under the standard formula does practically notdiffer among the IPs. Obviously, a prescribed longevity shock to all (sub-)populations regard-less of the underlying risk drivers cannot detect population basis risk. The risk-based internalmodel however identifies that index-based hedges do not cover all components of longevityrisk and the underlying IP consequently has a significant impact on capital relief. In fact,an intuitive ranking among the IPs can be observed for all instruments: the fully customizeddesign always provides the highest cost of capital relief, followed by the subpopulation-linkeddesign, and the instruments linked to the reference population provide the lowest relief. Sincethe difference between the latter two is much more distinct than between the former twodesigns, socioeconomic basis risk seems to be of higher relevance than small sample risk forour model company. Indeed, the applied RWD for modeling socioeconomic mortality differ-entials in conjunction with a trend adjustment in the subpopulation-specific experience ratiosproduces ’haircuts’ for population basis risk of up to 50% (relative to the fully customizedcounterpart) which might be regarded as rather drastic. We will continue this discussion inSubsection 4.4.1 by considering alternative modeling assumptions. Nevertheless, it shouldbe kept in mind that we deliberately assumed a rather affluent socioeconomic structure toprovoke a noticeable exposure to socioeconomic basis risk.
4.2.2 Risk reduction
We proceed by comparing the considered hedges in terms of the achieved level of risk reduc-tion. The left panel of Figure 4 shows their hedge effectiveness depending on their time tomaturity which is derived based on the adjusted liabilities (including future cost of capital)assuming use of the internal model.
We start by discussing the effectiveness of the fully customized designs. For longevityswaps, hedge effectiveness increases in the time to maturity since each additional year ofcoverage secures the annuity payments for another year. Since late payments at high agesgenerally pose the greatest risk, a long hedge horizon is required to reach a high hedge effec-tiveness. By construction, the unlimited fully customized longevity swap offers the maximalhedge effectiveness of one.
In contrast, mid-term contract maturities are sufficient to reach a substantial hedge ef-
Figure 4: Hedge effectiveness for different times to maturity.
fectiveness with q-forwards (of around 60%) and annuity forwards (of around 75%). In linewith our findings in the previous subsection, we clearly see that the effectiveness of these in-struments peaks somewhere around τ = 20 which indicates the existence of an optimal hedgehorizon. On the one side, short-term contracts leave a substantial part of longevity risk withthe hedger since there is little time for longevity risk to accumulate. On the other side, linkingthe hedge payout to higher ages (and at the same time postponing the single hedge payoutfar into the future) obviously loses the connection to the hedger’s liability structure.
Finally, we find that the rolling portfolios of one-year call spread options underperformin terms of hedge effectiveness. The reason lies in their pricing which is performed in eachsimulation path rather than at time zero. For the sake of illustration, consider a scenarioin which longevity increases over time, for instance due to an unfavorable mortality trendchange. As a consequence, longevity assumptions need to be revised and the current one-yearhedge contract typically provides an offsetting payout. However, the premiums for the nextcontracts in the sequence typically rise as the hedge provider adapts to the updated mortalityassumptions. These path-dependent hedge premiums reduce the effectiveness of this hedgingstrategy reflecting rolling risk. Moreover, it can clearly be observed that higher attachmentpoints lead to a reduction in hedge effectiveness since a greater portion of longevity risk isretained.
Regarding the impact of population basis risk, we again observe a clear and intuitive rank-ing among the IPs for all instruments under consideration. As expected, hedge effectivenessis declining for each component of longevity risk which is not covered. This effect is partic-ularly pronounced for long-term longevity swaps: Starting from an effectiveness of 100% foran unlimited fully customized (IP = B) contract, hedge effectiveness reduces to around 90%when small sample risk is retained (in case of IP = S) and to 75% when also socioeconomicmortality differentials are no longer covered (in case of IP = R). This illustrates the relevanceof each component of longevity risk.
Now, assume for a moment that the hedger measures hedge effectiveness based on the
19
unadjusted liabilities regardless of any future capital charges. Since this commonly usedapproach does not account for the uncertainty in future cost of capital (and the reductionthrough the hedge overlay), the resulting hedge effectiveness may differ. The right panel ofFigure 4 shows the absolute difference between the two quantities for the considered instru-ments, where a positive discrepancy means that hedge effectiveness is higher when derivedbased on the adjusted liabilities (including future cost of capital). It can clearly be observedthat the allowance for stochastic cost of capital has a notable impact on longevity hedge effec-tiveness. If the uncertainty in future cost of capital is ignored, the effectiveness of mid-termlongevity swaps with times to maturity between 25 and 40 years appears to be up to six per-centage points higher. For longer hedge horizons, both approaches again coincide for the fullycustomized design and the effectiveness of long-term index-based longevity swaps is slightlyhigher when measured with respect to the adjusted liabilities. This is due to the fact that theuncertainty in future SCRs is mainly driven by mortality in the reference population, whichis covered by index-based hedges. The q-forwards show a similar picture and the differencebetween the two approaches is moderate for annuity forwards. The largest discrepancy canbe observed for the rolling call spread portfolios whose effectiveness is underestimated by upto seven percentage points when the uncertainty in future SCRs is ignored. As discussed inthe previous subsection, these instruments are quite effective in reducing the magnitude andvariability of future SCRs. Assessing their effectiveness without acknowledging the uncer-tainty in future cost of capital (and the reduction through the hedge overlay) clearly missesa key quality of these instruments and therefore underestimates their actual risk-reducingpotential.
Overall, we conclude that hedge effectiveness might sometimes be underestimated, some-times overestimated when the uncertainty in future cost of capital is ignored. The fact thatmisestimation can occur in both directions underlines the necessity to work with stochasticlongevity SCRs for an objective analysis and comparison of different hedging instruments.
4.3 Combined analysis of hedge effectiveness and capital efficiency
So far, we have looked at the benefits of hedging, namely risk reduction and cost of capitalrelief, separately. In particular, we have seen that an unlimited fully customized longevityswap provides maximal hedge effectiveness and full cost of capital relief. If investors did notdemand compensation for taking risk, such a perfect hedge would obviously outperform anypartial hedging solution in terms of both hedge effectiveness and capital efficiency. We nowallow for a non-zero risk premium and incorporate the resulting costs for the hedger into ouranalysis.17
As outlined in Section 3 and specified in Appendix A.4, we rely on a risk-adjusted pricingapproach which compensates the hedge provider for the risks taken. Obviously, this approachdoes not account for all factors that might have an influence on market prices for longevityhedges. For instance, investors typically also consider administration costs, diversificationbenefits with other risks than longevity, or further strategic aspects. Nevertheless, it seemsappropriate for our purposes to not explicitly deal with these issues.
The first subsection gives a brief overview of the hedging costs for the considered instru-ments. In the second subsection, we discuss the trade-off between hedge effectiveness andcapital efficiency.
17Note that pricing already affected risk reduction and capital relief for rolling hedge programs in the previoussection.
Figure 5: Hedging costs (x-axis) vs. hedge effectiveness (y-axis) in the base case.
4.3.1 The costs of effective hedging
The left panel of Figure 5 shows the hedge effectiveness (y-axis) compared to the hedgingcosts (x-axis) for the considered instruments assuming a market price of longevity risk ofλ = 30% for all systematic risk drivers. For each hedging instrument, the costs of hedging arederived as expected present value of all cash flows and typically have a negative sign reflectingthe absolute risk loading on top of the objective best estimate value. Again, we derive hedgeeffectiveness based on the adjusted liabilities (including future cost of capital) in the internalmodel.
When looking at the hedge payout structures individually, we observe that the appliedrisk-adjusted pricing approach produces reasonable prices in the sense that a higher hedgeeffectiveness generally comes with higher hedging costs. Regarding the considered IPs, index-based instruments linked to the reference population (IP = R) are always offered at a lowerprice compared to their fully customized (IP = B) and subpopulation-linked (IP = S)counterparts. Intuitively, investors require higher compensation for instruments that alsocover socioeconomic mortality differentials since they pose a systematic risk. Technicallyspeaking, this is due to the risk adjustment for socioeconomic mortality differentials withinthe construction of the pricing measure. Furthermore, the costs for subpopulation-linkedinstruments coincide with those for their fully customized counterparts since we refrain froma further risk adjustment for diversifiable small sample risk. In reality, fully customizedcontracts might cause higher administration costs which the hedge provider might eventuallypass on to the hedger.
Comparing all instruments, the costs for short-term annuity forwards and for rolling seriesof one-year call spreads appear disproportionately high given that they partly even exceedthose for full securitization. Both phenomena can be traced back to the risk-adjustment foryearly random fluctuations around the prevailing AMT in the applied pricing approach. Inour setup, an observer is not able to distinguish between a recent change in the long-termtrend and a normal random fluctuation around it. For short-dated contracts which contain a
21
feature that protects the hedger against variations in mortality assumptions, the counterpartyis highly exposed to the risk that noise around the AMT is wrongfully identified as a longevitytrend change and that it triggers a high hedge payout. Assigning higher probability mass toscenarios of unfavorable yearly fluctuations therefore significantly raises the costs for short-dated annuity forwards and particularly for rolling portfolios of one-year contracts.
The right panel of Figure 5 shows the results assuming a reduced market price of riskfor yearly random fluctuations of λε = 15%. We find that the resulting hedging costs forannuity forwards and rolling call spread portfolios are much more in line with the remaininginstruments. We deduce that these contracts are only competitive if hedge providers reducethe risk adjustment for random fluctuations around the long-term mortality trend. Investorsmight be willing to do so if they are confident of writing this business over several consecutiveyears (for instance the hedger might commit to annual renewal of the contract). Over amulti-year horizon, the exposure to yearly random fluctuations diversifies. This is also thereason why the choice of the market price of risk for random fluctuations has a negligibleimpact on prices of long-dated contracts. Also note that the effectiveness of rolling portfoliosslightly increases when reducing the hedging costs since rolling risk is mitigated.
For these reasons, we use the reduced market price of risk for yearly random fluctuationsthroughout the remainder of this paper.
4.3.2 The trade-off between hedge effectiveness and capital efficiency
Figure 6 shows the hedge effectiveness of the considered instruments (y-axis) compared totheir capital efficiency (x-axis) under the standard formula (left panel) and under the internalmodel (right panel). Recall from Section 2.4 that we quantify capital efficiency based on theexpected cost of capital relief net of hedging costs, where the expected cost of capital forthe unhedged liabilities serves as a benchmark. For the rolling call spread portfolios, we nowfocus on a fixed hedge horizon of 25 years and instead vary the attachment and exhaustionpoints. In more detail, the attachment points are increased from 1% to 10% (from orange tored) in steps of 1% for the standard formula design and we consider exhaustion points fromthe set {70%, 75%, 80%, 85%, 90%, 95%, 96%, 97%, 98%, 99%, 99.5%} (from yellow to gold) forthe internal model design.
We start by discussing the overall picture obtained under the internal model. The keyfinding in Figure 6 is that there is no ’universally superior’ hedging solution in the sense thatno instrument simultaneously outperforms the others in terms of both hedge effectiveness andcapital efficiency. Depending on the hedger’s objective, different hedges might be suitable.
To begin with, consider a company which does not want to be exposed to longevity riskanymore. Aiming at maximal hedge effectiveness, it would certainly opt for the unlimited fullycustomized longevity swap. Since pricing is adapted to the internal model’s risk assessment,the costs for this strategy practically offset the generated cost of capital saving resulting in acapital efficiency of approximately zero.
However, full risk reduction might not necessarily present the primary objective for allhedgers. Some might have an appetite to retain some longevity risk, especially if the costs fortransferring certain parts of the risk exceed the expected cost of capital saving. For instance,if a fully customized longevity swap with a term of 35 years is considered, hedge effectivenesswill naturally decline while capital efficiency will increase up to around 10%. These results arein line with findings of Meyricke and Sherris (2014) who show that longevity hedging at highages is generally not capital efficient under Solvency II. For terms below 35 years, however,
Figure 6: Capital efficiency (x-axis) vs. hedge effectiveness (y-axis) in the base case.
capital efficiency is again declining since the cost of capital relief decreases more sharply thanthe hedging costs.
Comparing all instruments once again, we observe that an even higher capital efficiencycan be reached by means of index-based instruments. Generally speaking, we identify afrontier of ’efficient’ instruments which provide the highest capital efficiency for their degreeof risk reduction. This set includes long-dated fully customized longevity swaps (beyond 35years), their index-based counterparts (IP = R), and index-based q-forwards (IP = R) overa hedge horizon of around 20 years. Eventually, the highest capital efficiency can be reachedwith a rolling portfolio of properly engineered one-year call spread options. For this purpose,an index-based strategy linked to the reference population (IP = R) with exhaustion pointsset well below the relevant 99.5th percentiles appears to be most suitable. The reason forthis is twofold. First, lowering the exhaustion points naturally reduces the hedging costswhich in turn has a positive impact on capital efficiency. Second, we find that the negativeimpact of population basis risk on the cost of capital saving is negligible when the exhaustionpoints lie well below the relevant 99.5th percentiles (we will visualize this phenomenon inSubsection 4.4.1 under different modeling assumptions for socioeconomic basis risk). This isin line with findings of Cairns and El Boukfaoui (2019), who demonstrate that index-basedcall spread option contracts may provide the same capital relief as fully customized contractswhen designed properly. Hence, these instruments allow the hedger to benefit from lowerhedging costs for index-based deals without the adverse impact of material population basisrisk on capital relief. Exhaustion points below the 85th percentile, however, do not providesufficient capital relief anymore and are therefore less attractive. We conclude that call spreadoptions are highly suited for reducing capital charges since they reduce longevity tail risk overa one-year horizon in a cost-efficient manner.
In Figure 6, we also recognize that many hedges do not appear to be attractive since alter-native instruments can be found which offer a higher hedge effectiveness at the same capitalefficiency or the other way around. First, we observe that all considered annuity forwards are
23
outperformed by long-term longevity swaps. Certainly, one reason for this phenomenon lies inthe underlying portfolio of immediate annuities. Annuity forwards might be more suitable fora portfolio of deferred annuities which we analyze in Subsection 4.4.2. Second, we find thatfor most fully customized hedges an index-based alternative can be found which does not nec-essarily have the same payout structure but offers a higher capital efficiency at a comparablelevel of risk reduction. In fact, fully customized contracts are only indispensable when aimingat a high hedge effectiveness above 80%. A hedger who is willing to accept population basisrisk might benefit from a higher capital efficiency since the additional costs for eliminatingsmall sample risk and/or socioeconomic basis risk might exceed the generated cost of capitalrelief. Of course, the outcome of this complex interplay highly depends on the market priceof risk for socioeconomic mortality differentials. For lower values of this market price, thecompany would eventually decide in favor of the fully customized contract. A similar argu-ment applies when considering the subpopulation-linked designs which are outperformed bytheir fully customized counterparts since they are offered at the same price but provide lessprotection and therefore also lower cost of capital relief. Again, beyond a certain thresholdvalue for additional customization costs, index-based instruments associated with the sub-populations would become more capital efficient than their fully customized counterparts. Asalready stated, we refrain from explicitly dealing with potential further customization costs.
In the left panel of Figure 6, we obtain a completely different picture under the standardformula. First of all, it should be kept in mind that the assumed market price of longevityrisk is still derived from the internal model’s risk assessment, which generally produces lowercapital charges compared to the standard formula as we have shown in Section 4.1. Forthis reason, capital efficiency generally appears to be higher than it actually is under theinternal model. In fact, the standard formula’s conservative longevity stress gives a strongincentive for transferring longevity risk. However, the simplified standard formula clearlycannot provide a solid picture across all considered instruments. As already discussed inSubsection 4.2.1, it tends to overestimate the efficiency of longevity swaps, does not properlyaccount for population basis risk, and allows for rather high capital reliefs for option-typecontracts with high attachment points. In light of these shortcomings, the effects of hedgingshould also be analyzed under a risk-based internal model to obtain an adequate picture ofthe individual risk profile and to avoid suboptimal hedging decisions.
Overall, we conclude that hedgers might face a trade-off between hedge effectiveness andcapital efficiency. In particular, differing hedging objectives typically require different typesof instruments in terms of hedge horizon, payout structure, and underlying IP. While fullycustomized contracts naturally outperform their index-based counterparts in terms of hedgeeffectiveness, predominantly index-based designs offer the highest capital efficiency (at leastfor our model parametrization). The key finding that no instrument simultaneously outper-forms the others in terms of both hedge effectiveness and capital efficiency underlines theimportance of considering both aspects for a profound hedging decision.
4.4 Sensitivity analysis
We now analyze how sensitive our conclusions on hedge effectiveness and capital efficiencyare with respect to different modeling assumptions.
Figure 7: Relative haircuts for population basis risk under different modeling assumptions.
4.4.1 Socioeconomic mortality differentials
The question of how to appropriately model population basis risk is a topic of ongoing dis-cussions. When evaluating index-based hedges in a stochastic simulation, the results highlydepend on the implied correlation between the mortality index and the mortality experienceof the hedger’s portfolio population. Often, mortality differentials between closely relatedpopulations are modeled by mean-reverting vector autoregressive (VAR) processes to enforce’coherent’, i.e non-diverging, mortality rates in the long run. However, the assumption ofcoherence is not always supported by data and might be criticized for being too strong, seefor instance Hunt and Blake (2018) or Li et al. (2017).
In this subsection, we address these issues in the context of the risk-based internal modelby alternatively assuming that socioeconomic mortality differentials are driven by a first-orderVAR process.18 As opposed to the previously applied difference-stationary RWD, temporarysocioeconomic mortality differentials are now assumed to move around a long-term mean-reversion level.
We start by examining the adverse impact of population basis risk on capital relief inindex-based hedging. To this end, we define the following relative ’haircut’:
1−ReCoCIM
(H [IP ]
)ReCoCIM
(H [B]
) , IP ∈ {S,R},
which measures the proportionate loss in expected cost of capital relief due to populationbasis risk for an index-based hedge relative to its fully customized counterpart.
The left panel of Figure 7 shows the haircuts for the index-based instruments for whichfully customized counterparts are available in the base case. For the sake of clarity, we only
18For the sake of consistency, we also adapt the counterpart in the EMT valuation model by restricting thederivation of subpopulation-specific experience ratios to the recalibration of mortality levels, see AppendixA.2. Again, we adapt the pricing to the internal model’s risk assessment, which requires a market price oflongevity risk of λ = 40% (and λε = 20% respectively).
Figure 8: Capital efficiency (x-axis) vs. hedge effectiveness (y-axis) assuming a VAR(1) process formodeling socioeconomic mortality differentials.
show the effect of lower exhaustion points for a 25-year rolling call spread portfolio linkedto IP = R and refrain from showing the instruments tailored to the standard formula. Asalready indicated in Subsection 4.3.2, exhaustion points close to the relevant 99.5th percentiletypically result in rather large haircuts whereas population basis risk vanishes as the exhaus-tion points are lowered (from gold to yellow). The figure also shows that the relative haircutsfor socioeconomic basis risk are considerably larger than those for small sample risk (see alsoSubsection 4.2.1). In contrast, the right panel shows the haircuts under the assumption oflevel-stationary socioeconomic mortality differentials. Apparently, the haircuts for instru-ments linked to the reference population are significantly smaller compared to the base case.For instance, the haircuts for long-dated longevity swaps reduce from nearly 50% in the basecase to just 20% in the right panel. Regarding the rolling call spread portfolios, populationbasis risk already becomes rather negligible for exhaustion points below the 95th percentilein the right panel, whereas choices below the 90th percentile are required to accomplish thesame under the RWD. For the subpopulation-linked hedges, however, the haircuts remainroughly the same since these instruments do not give rise to socioeconomic basis risk. Thehaircuts among the IPs in the right panel show that socioeconomic basis risk is now lessrelevant than small sample risk. The observation that the order of relevance has changedwhen imposing coherence on the model clearly indicates a significant model risk with regardto modeling socioeconomic mortality differentials.
This raises the question to what extent this issue impacts our previous results on hedgeeffectiveness and capital efficiency. For comparison, Figure 8 shows the corresponding chartsobtained under the VAR(1) process. As expected, we observe that the distances betweeninstruments linked to different IPs are smaller compared to the base case (in Figure 6) withrespect to both dimensions. However, the modeling assumption for socioeconomic mortalitydifferentials hardly affects the ’efficient frontier’ in terms of shape and encased hedging in-struments. In fact, for practically any strategic hedging objective, the hedger would arrive at
26
the same optimal hedging decision as in the base case. While the hedger has to accept largerhaircuts for socioeconomic basis risk under the RWD, the additional costs for customizationare also higher since our pricing approach is directly linked to the underlying simulationmodel. Hence, the hedger can benefit from more attractive prices for index-based hedgeswhich compensates for the larger haircuts. These findings suggest that, as long as the marketcan find a common agreement on how to model population basis risk, the question whichmodeling assumption will eventually prevail might after all be of secondary importance forthe choice of optimal hedging strategies. Nevertheless, it is highly relevant with respect topricing and the computation of capital reliefs.
4.4.2 Starting age
Finally, we change the underlying liabilities by considering a portfolio of deferred annuities.Figure 9 shows the combined pictures of hedge effectiveness and capital efficiency for a lowerstarting age of x0 = 50.19 To account for the longer time horizon, we adjust the contractmaturities by considering hedge horizons beyond 40 years for longevity swaps and up to 40years for the remaining instruments. Since we do not obtain any new insights under thestandard formula, we limit our discussion to the internal model. Compared to the base case,we make the following observations:
• Index-based longevity swaps do no longer offer a higher capital efficiency than fullycustomized contracts. As seen in the previous subsection, the hedger needs to acceptrather large haircuts for population basis risk when considering long-term index-basedlongevity swaps. This effect is even more pronounced for longer-dated contracts, whichare naturally required for a lower starting age.
• With q-forwards, only a substantially lower hedge effectiveness of around 40% can bereached compared to nearly 65% in the base case. Obviously, a hedge portfolio forlong-term obligations requires more than a single q-forward to be effective.
• The hedge effectiveness of annuity forwards slightly increases since these value hedgeagreements are structurally more suitable for deferred annuities.
So far, we have assumed that investors are willing to offer hedge contracts over any timeto maturity. However, long-term investment horizons (for instance over more than half acentury) might not be appealing to institutional investors. For simplicity, we also do notexplicitly deal with interest rate risk and counterparty credit risk, which are typically moreproblematic for long-term contracts. In light of these aspects, it seems at least questionablewhether extremely long-dated contracts are available at the given prices.
If we assumed that risk takers were only interested in investment horizons of at most 40years, all longevity swaps (represented by blue dots) in Figure 9 would no longer be available.The only way to reach a hedge effectiveness of around 80% would be to enter into a customizedannuity forward with a time to maturity of around 30 years, which generates an expected costof capital relief that fully compensates for the hedging costs. In fact, the hedger may evenbenefit from a positive capital efficiency at the cost of some hedge effectiveness by consideringan index-based deal. This type of instrument offers a high level of risk reduction compressedto a manageable contract duration in a capital efficient manner and might hence be suitableto reconcile hedger’s and investor’s interests.
19Note that we still assume that the annuity payout will begin at age 65.
Figure 9: Capital efficiency (x-axis) vs. hedge effectiveness (y-axis) for a portfolio of deferred annuitieswith starting age x0 = 50.
5 Conclusion
In this paper, we discuss the two main benefits of transferring longevity risk under modernrisk-based solvency regimes: reducing the uncertainty in future cash flows arising from uncer-tain future mortality and reducing capital charges for longevity risk cost-efficiently. So far,both aspects have solely been addressed independently of each other. We argue that such aseparate analysis cannot provide a full picture of the implications of hedging, in particularfor index-based hedges.
We propose a stochastic modeling framework for a combined analysis of hedge effectivenessand capital efficiency in longevity hedging and hence jointly analyze the risk-reducing effectand the economic impact of longevity hedging. Moreover, we explicitly acknowledge theuncertainty regarding future cost of regulatory capital and incorporate its reduction resultingfrom hedging into the assessment of hedge effectiveness. For our numerical analyses, weconsider a wide selection of different hedging instruments in terms of hedge payout structure,time to maturity, and underlying population. In particular, a clear distinction between thedifferent components of longevity risk within the underlying simulation model allows us toconstruct different hedge designs which give rise to varying levels of population basis risk.In the context of a regulatory capital model under Solvency II, we analyze and compare theimpact of longevity hedging on capital charges under a risk-based (partial) internal modeland the Solvency II standard formula.
After a brief investigation of the hedger’s initial unhedged situation, we start by analyzingthe cost of capital relief. We find that different hedging instruments have structurally differ-ent impacts on the company’s economic capital in terms of both magnitude and variability.Moreover, we show that the Solvency II standard formula approach generates capital reliefswhich are less consistent than those from a risk-based internal model. In particular, the stan-dard formula’s simplified one-off longevity stress does not properly account for populationbasis risk, overestimates the efficiency of longevity swaps, and gives rise to rather high capital
28
reliefs for option-type contracts with high attachment points.We proceed by comparing the considered instruments in terms of the achieved level of
risk reduction. In particular, we show that hedge effectiveness can change considerably ifthe uncertainty regarding future capital charges for longevity risk is properly accounted for.Analyses ignoring this source of uncertainty might systematically misestimate the effectivenessof longevity hedging. In particular, they overestimate the hedge effectiveness for limitedlongevity swaps, underestimate it for rolling call spread portfolios, and might consequentlylead to suboptimal hedging decisions. The fact that misestimation can occur in both directionsunderlines the necessity to work with stochastic capital charges for an objective analysis andcomparison of different hedging instruments.
Subsequently, we conduct a simultaneous analysis of hedge effectiveness and capital ef-ficiency, where we allow the counterparty to charge a non-zero risk premium. Assuming areasonable market price of longevity risk, we find that generally no ’universally superior’hedging solution can be found since different hedging objectives typically require different in-struments. We visualize the implications of hedging in a bigger two-dimensional picture, fromwhich the most effective and the most capital efficient solution clearly emerge as two extremepositions. Between them, a frontier of ’efficient’ instruments can be identified which providethe highest capital efficiency for a given degree of risk reduction. From this set, the hedgermay choose suitable contract designs that provide, with regard to the strategic hedging ob-jective, an optimal trade-off between hedge effectiveness and capital efficiency. Furthermore,we address the benefits and costs of customization and the accompanying trade-off betweenmitigating (or even fully eliminating) population basis risk and lower costs of hedging. Whilecustomized hedges naturally outperform their index-based counterparts in terms of hedgeeffectiveness, in many cases index-based designs provide the highest capital efficiency. Thesefindings should draw more attention to standardized hedging solutions, which are expectedto be more appealing to institutional investors.
Finally, we conduct sensitivity analyses with respect to the modeling assumption for so-cioeconomic mortality differentials and with respect to the structure of the underlying liabil-ities. Altering the model for socioeconomic mortality differentials can materially impact thehaircut for population basis risk. However, the frontier of ’efficient instruments’ is mostlyunaffected. It is slightly shifted, but includes virtually the same instruments for the differentlevels of desired hedge effectiveness. Thus, as long as all market participants rely on the samemodeling assumption, the hedger would arrive at the same optimal instrument for practicallyany strategic hedging decision.
To conclude, this paper provides several novel and deep insights into the effects of longevityhedging. Our findings along with the utilized modeling techniques should be of interest toanybody who is concerned with longevity risk management and the development of the globallongevity risk transfer market.
29
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Appendices
A Model specification and implementation
In this appendix, we present a multi-population extension of the combined AMT/EMT modelof Borger et al. (2019b) in the stochastic mortality modeling framework outlined in Section2.1. We start by specifying the AMT simulation model in Section A.1 followed by the EMTvaluation model in Section A.2. In Section A.3, we provide further details on the hedginginstruments, and Section A.4 provides the technical details of the applied pricing approach.
A.1 AMT simulation model
We rely on a multi-population extension of the Cairns-Blake-Dowd (CBD) model. First, thelogit of the annual death probabilities of the reference population is modeled as
logit(q
[R]x,t
):= log
(q
[R]x,t
1− q[R]x,t
)= κ
(1)[R]t + (x− x)κ
(2)[R]t ,
where x denotes the average age of the calibration age range, see Cairns et al. (2006). The
evolution of the general level of mortality over time is driven by the time process κ(1)[R]t while
the second period effect κ(2)[R]t accounts for changes in the slope of the logit line over time.
We follow the widely used relative modeling approach20 by modeling the differences be-tween the subpopulations and the reference population as
logit(q
[p]x,t
)− logit
(q
[R]x,t
)= κ
(1)[p]t + (x− x)κ
(2)[p]t , p ∈ {1, . . . , NSub},
where x is the same as for the reference population. While all subpopulations share the time
processes of the reference population as a common mortality trend, the time processes κ(1)[p]t
and κ(2)[p]t capture subpopulation-specific mortality differentials over time.
This rather simple multi-population model structure offers several desirable properties.First, the model implies a flexible correlation structure of mortality improvements acrossages and subpopulations, which is important for assessing hedge effectiveness. Second, theextended CBD framework possesses the ’new-data-invariant property’, see Tan et al. (2014),which is important for the EMT valuation component of our model. Finally, mortality ratescan easily be extrapolated for high ages by following the inherent logit-extrapolation in theCBD model structure. As observed by Hoffmann (2005) and Villegas et al. (2017), socioe-conomic mortality differentials are usually more distinct for younger ages and diminish forhigh ages. When fitted to subpopulations of different socioeconomic status, the CBD modeltypically captures this phenomenon by yielding mortality lines in the logit plot that cross oversomewhere in the highest age range. To avoid this unrealistic reversal of the ranking amongthe subpopulations, we adjust the extrapolation as follows:
q[p]x,t :=
max{q
[R]x,t ; q
[p]x,t
}, for p = 1, 2
min{q
[R]x,t ; q
[p]x,t
}, for p = 4, 5
20See among others Jarner and Kryger (2011) or Villegas et al. (2017).
33
and leave the middle subpopulation unadjusted allowing it to take values above or below thereference population.21 To ensure a meaningful extrapolation, also for scenarios of stronglongevity improvements, we extrapolate mortality rates up to a limiting age of 130.
Mortality trend risk
We use the stochastic trend process originally proposed by Borger and Schupp (2018) andfurther refined by Schupp (2019) to model the long-term mortality trend risk of the reference
population, which is represented by the two time processes κ(i)[R]t , i = 1, 2. Their model builds
on empirical findings that historical mortality trends typically appear to be trend-stationaryaround piecewise linear trends with changing slopes over time. As opposed to the widely usedRWD, this process explicitly allows for random future trend changes in both directions.
Both time processes are modeled as stationary random fluctuations around some unob-
servable underlying continuous trend, denoted by κ(i)[R]t , i = 1, 2, i.e.
κ(i)[R]t = κ
(i)[R]t + ε
(i)[R]t , i = 1, 2.
The noise vector ε[R]t := (ε
(1)[R]t , ε
(2)[R]t )
′is assumed to follow a multivariate normal distribution
with mean zero and covariance matrix Σ[R]. We follow Borger and Schupp (2018) and modelboth underlying trend processes independently. The actual trend processes are assumed tobe piecewise linear with the prevailing AMTs as slopes which experience random changes in
both directions at random future points in time. More precisely, let d(i)[R]t , i = 1, 2 denote the
current AMT at time t and let p(i)[R] > 0, i = 1, 2 denote the probability of a trend changebetween two consecutive years. In each year t, the AMTs are then updated as follows:
d(i)[R]t = d
(i)[R]t−1 + S
(i)[R]t M
(i)[R]t 1{trend change occurs between t−1 and t}, i = 1, 2,
where S(i)[R]t ∈ {−1, 1} and M
(i)[R]t > 0 give the sign and the absolute magnitude of the trend
change respectively. Following Borger and Schupp (2018), we assume the sign of a trendchange to be positive or negative with equal probability. Furthermore, the absolute trend
change magnitudes M(i)[R]t , i = 1, 2 are modeled by lognormal distributions with parameters
µ(i)[R]M and σ
(i)[R]M
2. The actual mortality processes are then projected linearly as
κ(i)[R]t = κ
(i)[R]t−1 + d
(i)[R]t , i = 1, 2.
As argued by Borger and Schupp (2018), the symmetric distribution for the trend changeintensities assures that the prevailing AMT (even though unobservable) is always the bestestimate for the trend at any future point in time. Moreover, the heavy-tailed lognormaldistribution with only very little mass around zero produces rather significant trend changesand especially allows for strong trend changes, which can be observed for some countries inthe past.
Socioeconomic mortality differentials
For the projection of socioeconomic mortality differentials, represented by the vectors of
period effects κ[p]t := (κ
(1)[p]t , κ
(2)[p]t )
′, p ∈ {1, . . . , NSub}, we consider two types of processes:
21Indeed, also more sophisticated extrapolations are conceivable but not justified for our model purpose.
34
• In the base case, we follow Villegas and Haberman (2014) and consider a multivariateRWD of the form
κ[p]t+1 = µ[p] + κ
[p]t + ε
[p]t , p ∈ {1, . . . , NSub},
where µ[p] ∈ R2 denotes the drift and ε[p]t ∈ R2 denotes the annual random inno-
vations for subpopulation p ∈ {1, . . . , NSub}. The joint vector of annual innovationsfollows a multivariate normal distribution with mean zero and constant covariance ma-trix Σ[RWD] ∈ R(2NSub)×(2NSub). Since the RWD has unbounded future variability, theuncertainty regarding future socioeconomic mortality differentials is rather pronounced.If the mortality gap between the subpopulations has been widening over the calibrationperiod, this process will create even further diverging mortality rates.
• Alternatively, we impose the assumption of ’coherent’, i.e. non-diverging, mortalityrates, which is widely used in the field of multi-population mortality modeling.22 To en-sure non-diverging mortality rates in the long-run, we model the vectors of subpopulation-specific spreads by correlated first-order VAR processes of the form
κ[p]t+1 = θ[p] + Θ[p]κ
[p]t + ε
[p]t , p ∈ {1, . . . , NSub},
where θ[p] ∈ R2 denotes the offset vector and Θ[p] ∈ R2×2 denotes the matrix of au-toregressive coefficients for subpopulation p ∈ {1, . . . , NSub}. The joint vector of annualinnovations is modeled as a multivariate normal distribution with mean zero and covari-ance matrix Σ[V AR] ∈ R(2NSub)×(2NSub). Equivalently, this process can be interpretedas a single 2NSub-by-2NSub VAR(1) process with restricted autoregressive matrix. Inparticular, we do not capture autoregressive effects between the time processes of dis-tinct subpopulations to avoid overparametrization. Stationarity for subpopulation p isguaranteed as long as all eigenvalues of Θ[p] are smaller than one in absolute terms.
We follow Villegas et al. (2017) and model socioeconomic mortality differentials independentlyof the stochastic processes for the reference population. The latter serve as the commonmortality trend shared by all subpopulations in the relative modeling setup which ensures areasonable dependence between subpopulation-specific mortality rates.23
Unsystematic small sample risk
Conditional on realized mortality rates q[p]x,t+1, we draw realizations for survivors aged x + 1
at time t+ 1 from subpopulation p from the following Binomial distribution:
B[p]x+1,t+1 ∼ Binom
(B
[p]x,t, 1− q
[p]x,t+1
), p ∈ {1, . . . , NSub},
given B[p]x,t policyholders aged x still alive at time t.
22See among others Cairns et al. (2011), Dowd et al. (2011), Hyndman et al. (2013), Jarner and Kryger(2011), Li and Hardy (2011), Li and Lee (2005), Plat (2009), Villegas et al. (2017), Wan and Bertschi (2015),and Zhou et al. (2014).
23Note that neither the VAR process nor the RWD contains a specific feature which would force the mortalityrates of the reference population to lie exactly in the middle of those of the subpopulations. However, thisis actually not required since the sizes of the subpopulations might change in the future. Nevertheless, thetypically moderate volatility in conjunction with a significant correlation structure ensures that the referencepopulation always lies reasonably between the subpopulations in our numerical applications.
35
A.2 EMT valuation model
In practice, an actuary who intends to derive current best estimate mortality assumptions ata certain point in time T would calibrate a mortality model to (observable) realized mortalityup to that point in time. Even though this recalibrated model could theoretically be chosenindependently of the underlying simulation model, it is convenient from a practical point ofview to recalibrate the same model structure. Due to the new-data-invariant property of the
CBD model, this would yield (for sufficiently large populations) exactly the same κ(i)[R]t , t ≤ T
and κ(i)[p]t , t ≤ T as generated by the underlying simulation model in the first place. Hence,
the process of model recalibration narrows down to the reestimation of prevailing mortalitylevels and trends based on the realized time processes up to time T , which makes the modelingframework highly efficient in practical applications.24
The EMT valuation model consists of two steps: First, the prevailing AMT is estimatedfor the reference population giving the base mortality level and common mortality trend ofall subpopulations. Afterwards, subpopulation-specific adjustments are made to account fordiffering mortality levels (and trends) relative to the reference population.
Reference population
Regarding the reference population, best estimate future mortality rates beyond time T areprojected by following a deterministic central path of mortality, i.e.
logit(q
[R]x,t (T )
)= κ
(1)[R]T + (t− T ) d
(1)[R]T + (x− x)
(κ
(2)[R]T + (t− T ) d
(2)[R]T
), t > T,
where κ(i)[R]T , i = 1, 2 denote the estimated mortality levels and d
(i)[R]T , i = 1, 2 denote the
EMT derived at time T . We follow Borger et al. (2019b) and estimate these quantities forboth processes independently by means of a weighted linear regression on the most recentdata points. To this end, we rely on exponentially declining weights, i.e.
w(i)(t, T ) :=1(
1 + 1/ψ(i))T−t , i = 1, 2, t ≤ T,
where the parameters ψ(i) ≥ 0, i = 1, 2 control the speed of exponential decay. As discussedby Borger et al. (2019b), it is important to find a reasonable trade-off between relying onenough data points to avoid the misinterpretation of noise as a trend change and assigningsufficient weight to the most recent data points to ensure an adequate reaction of the EMTto recent trend changes. By construction, the EMT derived under exponential weighting willremain the same as in the previous year if the new data point is realized exactly as expectedunder last year’s EMT.
Subpopulations
Subsequently, mortality rates for each subpopulation p ∈ {1, . . . , NSub} are projected as
q[p]x,t(T ) := φ
(q
[R]x,t (T ), ξ
[p]x,t(T )
):= logit−1
(logit
(q
[R]x,t (T )
)+ ξ
[p]x,t(T )
), t > T,
24For other models, for instance the model of Lee and Carter (1992), a full model recalibration would berequired in each simulation path, which is associated with a huge computational burden.
36
where the function φ links the subpopulation-specific rates to those of the reference populationwithin the CBD model structure via the prevailing experience ratios
ξ[p]x,t(T ) := κ
(1)[p]T + (t− T ) ν
(1)[p]T + (x− x)
(κ
(2)[p]T + (t− T ) ν
(2)[p]T
), t > T,
which are also derived at time T from observed mortality in previous years. In line withthe underlying CBD model structure, these experience ratios are made up of subpopulation-
specific adjustments for different mortality levels κ(i)[p]T , i = 1, 2 and trends ν
(i)[p]T , i = 1, 2
respectively. For their derivation, we again consider two different approaches:
• The observer recalibrates both subpopulation-specific levels and trends as typically donewhen calibrating a RWD. Analogously to the reference population, these quantities arederived via a weighted linear regression on the most recent data points.
• Under the assumption of ’coherence’, the observer restricts the recalibration to thesubpopulation-specific mortality levels while relying on the mortality trend of the refer-ence population for all subpopulations. In this case, the prevailing mortality levels aredetermined as the weighted average of the most recent data points.
For consistency, we apply the weights used for the reference population also to all subpopu-lations.
A.3 Construction of index-based hedging instruments
In this section, we provide further details on the hedging instruments which are introduced inSection 3.2. In particular, we construct index-based designs based on suitable hedge indices.
Mortality indices and q-forwards
For each subpopulation p ∈ {1, . . . , NSub}, we define the following mortality indices:
q[IP (p)]x,t :=
{q
[p]x,t, IP = Sφ(q
[R]x,t , ξ
[p]x,t(0)
), IP = R,
which either represents the subpopulation-specific mortality rates (in case of IP = S) ora proxy (in case of IP = R) which is constructed by adjusting the mortality rates of thereference population by the initial experience ratios to account for anticipated mortalitydifferences between the populations. The payout of the q-forward contract is then based on
Q[IP ]x0+τ,τ :=
NSub∑p=1
n[p]τ q
[IP (p)]x0+τ,τ , IP ∈ {S,R},
where n[p]τ ≥ 0 represents the hedge ratio for subpopulation p ∈ {1, . . . , NSub}. For a given risk
measure ρ and maturity τ , we determine the optimal hedge ratios via the objective function
argminn
[p]τ
ρ(L[p](0)− (1 + r)−τ n[p]
τ
(EQ(q
[IP (p)]x0+τ,τ
)− q[IP (p)]
x0+τ,τ
)), p ∈ {1, . . . , NSub},
37
which is solved for each subpopulation p ∈ {1, . . . , NSub} numerically over all outer simulationpaths.25 To ensure comparability between the two designs, we determine the optimal hedgeratios for IP = S and choose the same hedge ratios for IP = R.
Survivor indices and longevity swaps
The fully customized longevity swap is based on the actual number of living policyholders
S[B]x0+t,t :=
∑NSubp=1 B
[p]x0+t,t at time t. The index-based designs are then constructed by using
the following survivor indices:
S[IP (p)]x0+t,t := B
[p]x0,0
t−1∏s=0
(1− q[IP (p)]
x0+s,s+1
), IP = S,R,
which build on the previously introduced mortality indices.
Liability indices and annuity forwards
The fully customized annuity forward is based on the actual time-τ best estimate liabilities
L[B] (τ) :=
NSub∑p=1
B[p]x0+τ,τ
∑s>τ
(1 + r)−(s−τ)1{x0+s≥xR}
s−1∏u=τ
(1− q[p]
x0+u,u+1(τ)).
For each subpopulation p ∈ {1, . . . , NSub}, we define the following annuity indices:
a[IP (p)]x0+τ,τ (τ) :=
∑s>τ
(1 + r)−(s−τ)1{x0+s≥xR}
s−1∏u=τ
(1− q[IP (p)]
x0+u,u+1(τ)), IP ∈ {S,R},
which denotes (in case of IP = S) the time-τ best estimate value of a lifelong annuity ofone paid annually in advance starting at the age of xR for an individual from subpopulationp ∈ {1, . . . , NSub} aged x0 + τ at time τ or a suitable proxy (in case of IP = R). The index-based annuity forwards are then constructed by replacing the actual number of survivors bythe survivor indices and the actual annuity value by the annuity indices:
L[IP ] (τ) :=
NSub∑p=1
S[IP (p)]x0+τ,τ a
[IP (p)]x0+τ,τ (τ), IP ∈ {S,R}.
Call spread indices
Recall from Section 2.3 that the SCR at time t is basically determined by the randomness (orchange due to a longevity shock) in
X [B](t+ 1) :=CF (t+ 1) + L(t+ 1) =
NSub∑p=1
B[p]x0+t+1,t+11{x0+t+1≥xR}
+
NSub∑p=1
B[p]x0+t+1,t+1
∑s>t+1
(1 + r)−(s−(t+1))1{x0+s≥xR}
s−1∏u=t+1
(1− q[p]
x0+u,u+1(t+ 1)),
25Note that the hedge ratios are defined based on the unadjusted liabilities instead of the adjusted liabilities,which constitutes a slight inconsistency but makes the derivation of the optimal hedge ratios practically feasible.
38
on which the fully customized call spread contracts are based. The index-based designs arethen constructed as follows:
• For the subpopulation-linked design, X [S](t+ 1) is defined analogously by replacing the
actual number of survivors in the book population B[p]x0+t+1,t+1 by B
[p]x0+t,t
(1− q[p]
x0+t,t+1
)so that small sample risk over the one-year hedge horizon is no longer covered.
• Finally, X [R](t + 1) is defined structurally similarly to X [S](t + 1) by fixing the time-texperience ratios to also remove the uncertainty originating from the subpopulationover the year. More precisely, for each subpopulation p ∈ {1, . . . , NSub}, realized
mortality q[p]x0+t,t+1 is replaced by φ
(q
[R]x0+t,t+1, ξ
[p]x0+t,t+1(t)
)and best estimate mortality
q[p]x0+u,u+1(t+ 1) is replaced by φ
(q
[R]x0+u,u+1(t+ 1), ξ
[p]x0+u,u+1(t)
)for u ≥ t+ 1.
A.4 Pricing
In our setup, longevity risk is a cumulative risk which is made up of several secondary riskfactors, namely the unpredictable occurrence, sign, and magnitude of future trend changes,random noise around the underlying AMT, and socioeconomic mortality differentials.26 Fol-lowing Boyer and Stentoft (2013), Chen and Cox (2009), and Freimann (2019), we applythe technique of multivariate normalized exponential tilting, which is (given independent riskdrivers) equivalent to applying the Wang transform to each risk driver individually, to con-struct an equivalent measure Q for pricing purposes.27 Compared to the objective measure,this pricing measure is constructed to assign a higher probability mass to scenarios which areunfavorable for a longevity risk taker. Note that in the CBD model structure lower realiza-tions of the first period effect imply decreasing overall mortality while lower realizations of thesecond period effect alleviate the slope of the mortality curve in the logit plot. Since longevityat high ages generally presents the greatest risk, the hedge provider’s risk adjustment aimsat shifting the distributions of both time processes downwards via the distortion parameters
λ(i)Risk > 0, i = 1, 2 for the risk drivers Risk ∈ {O,S,M, ε, Sub}. This change of measure is
summarized in Table 4. For a more detailed discussion of this pricing technique in the contextof our mortality trend model, we refer to Freimann (2019).
Theoretically, infinitely many equivalent measures can be found in our discrete time modelsetup. For the sake of simplicity, we start our analysis by assuming that the hedge providerdemands the same risk premium for all risk drivers and consequently applies the same distor-tion parameter λ, also referred to as ’market price of longevity risk’, to both time processesand all risk drivers. Nevertheless, as long as the market for longevity-linked securities lackscompleteness, a unique risk-adjusted measure cannot be deduced from market data and thechoice of the market price of longevity risk becomes a modeling assumption.
We apply this risk-adjusted pricing approach for several reasons. First, since this changeof measure retains the Bernoulli, Normal, and Log-Normal distribution as well as the inde-
26Assuming hedge providers are able to diversify small sample risk, they refrain from demanding a riskpremium for it. Moreover, we follow most of the existing literature and do not price parameter uncertainty.More precisely, we account for parameter uncertainty in selected model parameters by sampling them at thestart of each outer simulation path from suitable distributions as outlined in Appendix B. Subsequently, thechange of measure for pricing purposes is conducted conditional on this parameter set.
27In particular, the independence among all risk drivers is still preserved under Q. We refer to Wang (2007)for the technical details.
39
Risk driver Distribution under P Distribution under Q
Trend change occurrence Bernoulli(p(i)[R]
)Bernoulli
(Φ(
Φ−1(p(i)[R]
)+ λ
(i)O
))Trend change sign P
(S
(i)[R]t = −1
)= 0.5 Q
(S
(i)[R]t = −1
)= Φ
(Φ−1 (0.5) + λ
(i)S
)Trend change magnitude LN
(µ
(i)[R]M , σ
(i)[R]M
2)LN
(µ
(i)[R]M + λ
(i)M σ
(i)[R]M , σ
(i)[R]M
2)Noise around AMTa N
(0,Σ[R]
)N(−D
12
Σ[R]Σ[R]ρ Λε,Σ
[R])
Mortality differentialsb N(0,Σ[Sub]
)N(−D
12
Σ[Sub]Σ[Sub]ρ ΛSub,Σ
[Sub])
Table 4: Objective and risk-adjusted model dynamics.
aHere, DΣ[R] denotes a diagonal matrix with entries Σ[R]i,i , i = 1, 2, Λε := (λ
(1)ε , λ
(2)ε )
′denotes the market
price of risk vector for random noise around the AMT, and Σ[R]ρ denotes its correlation matrix.
bFor Σ[Sub] ∈ {Σ[RWD],Σ[V AR]} , DΣ[Sub] denotes a diagonal matrix with entries Σ[Sub]i,i , i = 1, . . . , 2NSub,
ΛSub := 1NSub
(λ(1)Sub, λ
(2)Sub, . . . , λ
(1)Sub, λ
(2)Sub)
′∈ R2NSub denotes the market price of risk vector for socioeconomic
mortality differentials, and Σ[Sub]ρ denotes the correlation matrix. Note that we included a division by the
number of subpopulations in the market price of risk vector to ensure that the pricing is not excessivelyskewed by the chosen number of subpopulations. As pointed out by Freimann (2019), this might be an issuewhen applying the same distortion parameter to all risk drivers instead of fitting them to actual market prices.
pendence among random variables, the whole model structure is preserved under Q allowingto directly simulate the risk-adjusted distribution. Second, as argued by Wang (2007), nor-malized exponential tilting ensures a consistent interpretation of the tilting parameter amongdifferent risks. Finally, the pricing operator is directly linked to the structure of the under-lying simulation model and additive which ensures applicability to all hedging instrumentsunder consideration.
Regarding the rolling portfolios of call spread options, the pathwise pricing is performedas follows: Given the observable mortality information up to time t, the hedge premium isdetermined as p[IP ](t) := (1 + r)−1EQ (h[IP ](t+ 1)|Ft
), where mortality from time t to t+ 1
is simulated under the risk-adjusted AMT simulation model. Here, unobservable informationabout the prevailing AMT may not be included to avoid the misestimation of rolling risk.In practice, the risk taker would conduct a careful analysis of parameter uncertainty in thecurrent AMT (see for instance Borger and Schupp (2018)) and incorporate the prevailinglevel of uncertainty into the pricing. To avoid over-complexity, we refrain from modelingparameter uncertainty beyond time zero and simply follow the current EMT (instead of theunobservable AMT) over the year for the pathwise pricing of one-year hedge contracts.28
B Stochastic mortality model calibration
In this appendix, we calibrate the whole stochastic mortality model to the historical mortalityexperience of English and Welsh males.
28A slight distributional inconsistency in the sense that E(H [IP ](0)) = 0 does not necessarily hold for λ = 0is accepted for rolling hedge programs since it does not systematically distort our results.
Table 5: Empirical distributions for the AMT starting values.
Data
For the reference population, we use data of the male population of England and Wales for theyears 1841 to 2016 over the age range of 60 to 109, which is available in the Human MortalityDatabase (2018). Regarding the subpopulations, we rely on mortality data for English malessorted by quintiles of the Index of Multiple Deprivation (IMD) (the official measure of relativedeprivation for small areas in England) for the years 2001 to 2016 over the age range of 60to 89, which we obtained from the Office for National Statistics (ONS). The subpopulationsare ordered from the most to the least deprived areas. For details on this data set, we referto Li et al. (2019), who also use it for their study on basis risk in longevity hedging.
The whole multi-population CBD model structure is fitted via a standard two-stage maxi-mum likelihood estimation (MLE) approach based on the assumption of binomially distributeddeaths as implemented in the R package ’StMoMo’, see Villegas et al. (2018). Note that theapplied relative model setup can deal with a shorter mortality history and a different agerange for the subpopulations than for the reference population.
Stochastic trend process
For the calibration of the AMT process, we apply an iterative pseudo MLE approach asproposed by Schupp (2019) and refer to this paper for technical details. As discussed byBorger et al. (2019a), the calibration of stochastic mortality trend processes typically involvesa considerable amount of uncertainty since data on historical mortality trend changes is sparse.For this reason, we explicitly account for parameter uncertainty in the starting values of thesimulation as well as in the trend change parameters.
Figure 10 shows the historical trend processes κ(i)[R]t , i = 1, 2 and the best possible realiza-
tions for the underlying trend processes κ(i)[R]t , i = 1, 2 for all relevant numbers of historical
trend changes which are identified by the calibration algorithm. The corresponding parameterestimates and weights, which the calibration algorithm assigns to each possible set of start-ing values, are provided in Table 5. In our numerical application, we account for parameteruncertainty by sampling the starting values from these empirical distributions.
The trend change parameters are estimated as(p(1)[R], µ
(1)[R]M , σ
(1)[R]M
)= (0.0223,−4.5453, 0.4105) ,(
p(2)[R], µ(2)[R]M , σ
(2)[R]M
)= (0.0246,−7.4134, 0.2027)
41
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Figure 10: Historical trend processes for English and Welsh males (dotted), best possible realizationsfor the actual trend processes given different numbers of trend changes k (colored dashed lines), and90% prediction intervals (black solid lines).
with corresponding covariance matrices of standard errors of
SE(1) =
1.353 · 10−4 3.535 · 10−5 −1.754 · 10−5
3.535 · 10−5 4.616 · 10−2 3.331 · 10−4
−1.754 · 10−5 3.331 · 10−4 2.322 · 10−2
,
SE(2) =
1.860 · 10−4 −1.211 · 10−3 8.060 · 10−4
−1.211 · 10−3 4.285 · 10−2 −2.153 · 10−2
8.060 · 10−4 −2.153 · 10−2 2.124 · 10−2
.
To account for parameter uncertainty in the trend change parameters for κ(i)[R]t , i = 1, 2 we
follow Borger et al. (2019a): At the start of each simulation path, we generate a multivari-ate normal random vector with mean equal to the estimated trend change parameters andcovariance matrix SE(i). Subsequently, the first component of the vector is transformed toa Beta distribution with same mean and variance to obtain reasonable trend change prob-abilities between zero and one. Similarly, the third component is transformed to a Gammadistribution with unchanged mean and variance to ensure positivity.
Finally, the covariance structure of the random noise around the AMT is estimated as
Σ[R] =
(3.865 · 10−4 1.720 · 10−5
1.720 · 10−5 2.036 · 10−6
),
for which we neglect parameter uncertainty since it is typically not material.
42
EMT weights
Given the calibrated AMT simulation model, we follow Borger et al. (2019b) and determinethe optimal EMT weights numerically by minimizing the mean squared errors between theunobservable AMT and the derived EMT in a Monte Carlo simulation and obtain(
ψ(1), ψ(2))
= (2.225, 2.752) .
Time processes for socioeconomic mortality differentials
Given the fitted CBD model structure, the stochastic processes for socioeconomic mortal-ity differentials are calibrated via standard statistical methods. For the RWD, we apply astandard MLE approach and the VAR(1) process is calibrated via ordinary least squares asimplemented in the function ’VAR’ from the R package ’vars’. For simplicity, we refrain frommodeling parameter uncertainty for the subpopulations.
