Sharing of longevity basis risk in pension schemes with income-drawdown guarantees
SSharing of longevity basis risk in pension schemes withincome-drawdown guarantees
Ankush Agarwal (cid:3)
Christian-Oliver Ewald y Yongjie Wang z Adam Smith Business School, University of Glasgow, G12 8QQ Glasgow, United KingdomFebruary 14, 2020
Abstract
This work studies a stochastic optimal control problem for a pension scheme which provides an income-drawdown policy to its members after their retirement. To manage the scheme efficiently, the manager andmembers agree to share the investment risk based on a pre-decided risk-sharing rule. The objective is tomaximise both sides’ utilities by controlling the manager’s investment in risky assets and members’ benefitwithdrawals. We use stochastic affine class models to describe the force of mortality of the members’ populationand consider a longevity bond whose coupon payment is linked to a survival index. In our framework, wealso investigate the longevity basis risk, which arises when the members’ and the longevity bond’s referencepopulations show different mortality behaviours. By applying the dynamic programming principle to solvethe corresponding HJB equations, we derive optimal solutions for the single- and sub-population cases. Ournumerical results show that by sharing the risk, both manager and members increase their utility. Moreover,even in the presence of longevity basis risk, we demonstrate that the longevity bond acts as an effective hedginginstrument.
Keywords:
Pension scheme, longevity basis risk, mortality-linked instrument, stochastic control, dynamicprogramming principle
Pension schemes face a wide range of risks such as investment risk, interest rate risk, inflation risk and longevityrisk. Among all the risks, the longevity risk is becoming a non-negligible challenge to pension schemes. Thelongevity risk is the risk that the actual life expectancy may be higher than anticipated. It is a positive trendfor society that people’s average life expectancy is increasing over the last decades. However, the pension schemesare suffering from the loss caused by the unexpected increasing benefit outgo accompanied by the longevity trend.Cocco and Gomes [2012] shows that the average life expectancy of 65-year-old US (UK) males increases by 1.2(1.5) years per decade. As a consequence, a defined benefit scheme (DB scheme) for those populations would haveneeded 29% more wealth in 2007 than in 1970.Instead of a classical DB or DC scheme, this work considers a pension scheme which provides an income-drawdown option to its members in the decumulation phase. The decumulation phase refers to the period afterthe retirement of pension scheme members. The drawdown option gives the members the right to withdraw moneyperiodically from the scheme until death time, while keeps the remaining amount in the pension scheme. Themanagement of the scheme starts from the retirement time of the members and ends until the last member passesaway. Thus, the investment period can last for decades. Due to the long investment horizon, the scheme may facesignificant interest rate risk and inflation risk. However, the main focus of this work is on hedging the longevity (cid:3) e-mail : [email protected] y e-mail : [email protected] z e-mail : [email protected] a r X i v : . [ q -f i n . R M ] F e b isk. We assume the risk-free interest rate to be constant and do not consider the inflation risk. In other words,we only consider the investment risk and longevity risk. The scheme manager invests in a representative stock togain investment returns, and he is exposed to the investment risk. As the members tend to live longer, the numberof surviving members each year is more than expected, resulting in an increase of benefit withdrawals. Also, thelength of decumulation period is increasing due to the members’ growing life expectancy. One way to reduce thepension scheme’s unexpected loss caused by its exposure to the longevity risk is to gain a better understanding ofthe future mortality evolution. Another way is to seek some financial instruments to transfer the longevity risk tothe financial market.The force of mortality, which represents the instantaneous rate of mortality at a certain time, is useful whenstudying mortality behaviour. In a continuous-time framework, Luciano and Vigna [2005] modelled the force ofmortality by affine processes and calibrated the models to the observed and projected UK mortality tables. Theyclaimed that the affine process with a deterministic part that increases exponentially could describe the force ofmortality properly. This paper applies Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) process to modelthe force of mortality of certain populations. To hedge the longevity risk using financial instruments, Blake andBurrows [2001] introduced a survivor bond which provides coupon payment based on the number of survivorsin a chosen reference population. Under a stochastic force of mortality framework, Menoncin [2008] studied theoptimal consumption and investment strategy for an individual investor using the longevity bond to hedge theinvestor’s longevity risk. He argued that the longevity bond plays a crucial role in the individual’s longevity riskmanagement. However, the longevity basis risk arises when the force of mortality of the longevity bond referencepopulation correlates imperfectly with the pension members’ force of mortality. To deal with the longevity basisrisk, Wong et al. [2014] applied the cointegration technique to study the mean-variance hedging of longevity riskfor an insurance company with a longevity bond. They suggested that cointegration is vital in longevity riskmanagement. In this paper, we introduce a rolling zero-coupon longevity bond to hedge the scheme’s longevityexposure. We show that the longevity bond provides an efficient way to hedge the longevity risk both in the caseswith and without longevity basis risk.Under a continuous-time framework, this work aims to determine the optimal investment and benefit with-drawal strategy for a pension scheme that provides an income-drawdown policy in the decumulation phase. Afterretirement, the members continuously withdraw money from the pension scheme until death. Upon death, a deter-ministic proportion of the passed away members’ pension account balance is given to the manager as compensation.The proposed optimal control problem considers both benefit of the scheme members and profit of the managerby incorporating a risk-sharing rule parameter. By applying the dynamic programming principle, we first solvethe optimisation problem under a general case. Then, we drive explicit solutions for single- and sub-populationcases. Numerical studies on single-population and sub-population OU models are conducted to investigate theoptimal portfolio strategy and optimal benefit withdrawal rate dynamically. Comparison studies are provided toanalyse the performance of longevity risk hedge. We also implement sensitivity analyses to look into the impactof the market price of risk and risk-sharing rule parameter. Our results show that with the absence of longevitybasis risk, the longevity bond provides efficient longevity risk hedge when the members show a longevity trend.When the members are a sub-population of the longevity bond reference population, the presence of the longevitybasis risk may weaken the longevity bond’s hedging performance. However, it still provides a way to hedge thelongevity risk and offers a risk premium. We also establish that an equal-risk-sharing rule is beneficial to boththe members and the manager in the long run. This paper contributes to the literature by studying a stochasticoptimal control problem for pension schemes in the presence of longevity risk and longevity basis risk. Our optimalcontrol problem also incorporates a risk-sharing rule parameter which decides the agreement on how to share therisk between the members and the manager such that both parties benefit in the long run.The rest of this paper is organised as follows. Section 2 introduces the mathematical framework of the problemand derives the optimal solution in the general case. Explicit solutions under single-population and sub-populationmodels are given in Section 3. In Section 4, numerical simulations are carried out to investigate the optimalportfolio strategy and benefit withdrawal. Comparison and sensitivity analyses are also conducted to discuss therole of longevity bond and impact of model parameters in the optimal solutions. Section 5 concludes the paper.2 Mathematical Framework
The literature on optimal control problems for pension schemes with deterministic mortality behaviour of thepopulations is rich. However, given the fluctuations in the mortality behaviour over time, it is more practical touse stochastic mortality rates. In this work, we use affine class models for the force of mortalities of members’population and the reference population for longevity bond. When, the two populations are different, we use n = 2populations in our framework and when they are the same, we use n = 1 population. We consider an infinite time horizon T = [0, ∞ ) where time 0 represents the retirement time of all the populations.Let {W( t ) | t ∈ T } = {(W ( t ), ..., W n ( t ), W S ( t )) | t ∈ T } denote an ( n + 1)-dimensional standard Brownianmotion on a complete filtered probability space ( Ω , F , { F ( t )} t ≥ , P ). Here, n denotes the number of populationsand P denotes the physical measure where we observe the longevity behaviours of the populations, and the financialmarket.For i ≤ n , let p i ( t ) and λ i ( t ) denote the survival probability and the force of mortality, respectively, of the i thpopulation at time t . The two are related through the following relation:d p i ( t ) p i ( t ) = – λ i ( t )d t , p i (0) = 1.For any s ≥ t ∈ T , p i ( s ) p i ( t ) = e – R st λ i ( u )d u is the survival probability of the i th population between time t and s . Fornotational simplicity, we denote by { λ ( t ) | t ∈ T } = {( λ ( t ), ..., λ n ( t )) | t ∈ T }, the vector of force of mortalitiesand assume that it evolves as d λ ( t ) = A ( t , λ )d t + Σ ( t , λ )dW( t ), (1)where A ( t , λ ) = a ( t , λ )... a n ( t , λ ) , Σ ( t , λ ) = σ ( t , λ ) · · · σ n ( t , λ n ) 0... . . . ... ... σ n ,1 ( t , λ ) · · · σ n , n ( t , λ n ) 0 .For any i , j = 1, . . . , n , a i ( t , λ ) and σ ij ( t , λ j ) are assumed to be continuous functions. In the following, we willuse different affine class models for λ ( t ). We consider a frictionless financial market consisting of a stock and a rolling zero-coupon longevity bond . Themoney market account is denoted by R( t ),dR( t )R( t ) = r d t , R(0) = 1,where r denotes the constant risk-free interest rate. In this work we consider a constant risk-free rate of interestas our focus is to understand the impact of longevity and investment risk in a pensions scheme. Our analysis canalso be performed in the presence of a stochastic interest rate.Under a risk-neutral probability measure, the price of a financial derivative is the discounted expected valueof its future payoff. We thus introduce an equivalent risk-neutral probability measure Q by the following Radon-Nikodym derivative d Q d P = Z(T) = exp – Z T0 θ ( t , λ )dW( t ) – 12 Z T0 k θ ( t , λ ) k d t ! ,3here { θ ( t , λ ) | t ∈ T } = {( θ ( t , λ ), · · · , θ n ( t , λ n ), θ S ) | t ∈ T } is R n -valued, F -adapted process such that Z( t )is a martingale and E [Z] = 1. θ ( t , λ ) is called the vector of market prices of risks and measures the additionalamount of investment return when risk increases by one unit. By Girsanov’s theorem, {W Q ( t ) | t ∈ T } is an( n + 1)-dimensional standard Brownian motion under Q such thatW Q ( t ) = W( t ) + Z t θ ( s , λ )d s . (2)The stock price process {S( t ) | t ∈ T } is given asdS( t )S( t ) = r d t + σ S [ θ S d t + dW S ( t )] , S(0) = S ,where σ s denotes the constant stock price volatility. The risk premium of the stock is θ S σ S .In the literature, several types of mortality-linked securities are proposed to hedge the longevity risk. Thevalues of these securities depend on the mortality index for some given populations: higher the survival rate,more valuable the securities. The definition below provides a short description of one such security, a zero-couponlongevity bond . Definition 1.
A zero-coupon longevity bond is a contract paying a face amount which is equal to the survivalprobability of the reference population from time 0 until a fixed maturity time T.There may be multiple longevity bonds based on different reference populations in the market. However, as anillustration of the use of longevity bond, we only consider one longevity bond in this work. Let the i th populationbe the reference population of the zero-coupon longevity bond which pays p i (T) p i (0) at maturity. The arbitrage-freeprice of the longevity bond at time t is given asL( t , T) = E Q t (cid:20) R(T)R( t ) p i (T) p i (0) (cid:21) = E Q t (cid:20) e – r (T– t )– R T0 λ i ( u )d u (cid:21) = e – R t λ i ( u )d u – r (T– t ) E Q t (cid:20) e – R T t λ i ( u )d u (cid:21) .Applying Itô’s formula and using (2) givesdL( t , T)L( t , T) = r d t + n X i , j =1 σ ij L ( t , T) h θ j ( t , λ j )d t + dW j ( t ) i ,where σ ij L ( t , T) = ∂ L( t ,T) ∂λ j ( t ) σ ij ( t , λ j )L( t ,T) .For any t ∈ [0, T], the time to maturity of the zero-coupon longevity bond is T – t . In practice, it is impossiblefor an investor to find all zero-coupon longevity bonds in the market with any time to maturity T – t . By takinginspiration from the arguments in Boulier et al. [2001] on rolling zero-coupon bonds, we introduce a rolling zero-coupon longevity bond L( t ) (with a little abuse of notation) with constant time to maturity T. The use of a rollingzero-coupon longevity bond in our set-up simplifies the calculations. The dynamics of L( t ) under P is given asdL( t )L( t ) = r d t + n X i , j =1 σ ij L ( t ) (cid:2) θ j ( t , λ j ) + dW j ( t ) (cid:3) .From the above, we can see that L( t ) provides a longevity risk premium of P ni , j =1 σ ij L ( t ) θ j ( t , λ j ). Any zero-couponlongevity bond L( t , T) can be replicated by using the rolling longevity bond L( t ) and cash. The following equationshows the relationship between L( t , T) and L( t )dL( t , s )L( t , s ) = P ni , j =1 σ ij L ( t , s ) P ni , j =1 σ ij L ( t ) ! dR( t )R( t ) + P n –1 i , j =1 σ ij L ( t , s ) P ni , j =1 σ ij L ( t ) dL( t )L( t ) .4 .3 The optimisation problem We study the optimal benefit withdrawal rate and investment strategy in the decumulation phase for a pensionscheme that provides an income-drawdown option. That is, the members are allowed to withdraw money periodi-cally from the scheme until death. Upon death, π ∈ [0, 1] fraction of the passed away members’ pension balance isdelivered to the scheme manager as compensation. While the remaining 1 – π fraction of the balance stays in thescheme’s fund pool. Let α S ( t ), α L ( t ) and α ( t ) denote the investments in stock, rolling longevity bond and moneymarket, respectively. In addition, let β ( t ) denote the amount of benefit withdrawal and Y( t ) the scheme’s wealth.We assume that the force of mortality of pension members is described by λ j ( t ). In other words, population j represents the pension members.To study the dynamics of the scheme’s wealth, we employ a similar method used in He and Liang [2013b] byfirst looking at the discrete-time changes in scheme’s wealth. For any t ∈ T and a small positive number ∆ , thereturns on stock, longevity bond and money market account are given byS( t + ∆ ) – S( t )S( t ) α S ( t ), L( t + ∆ ) – L( t )L( t ) α L ( t ), R( t + ∆ ) – R( t )R( t ) α ( t ).Let µ ( t , t + ∆ ) denote the rate of investment return. Then, we have µ ( t , t + ∆ )Y( t ) = S( t + ∆ ) – S( t )S( t ) α S ( t ) + L( t + ∆ ) – L( t )L( t ) α L ( t ) + R( t + ∆ ) – R( t )R( t ) α ( t ).Let q ( t , t + ∆ ) denote the proportion of the members who pass away in time interval ( t , t + ∆ ). The total amountof passed away members’ pension equals to q ( t , t + ∆ )Y( t ). During this time period, the scheme manager receives π q ( t , t + ∆ )Y( t ) as compensation. In the meantime, the members withdraw β ( t ) ∆ as benefits. Thus, the scheme’scashflow in this period is given by µ ( t , t + ∆ )Y( t ) – π q ( t , t + ∆ )Y( t ) – β ( t ) ∆ .At time t + ∆ , 1 – q ( t , t + ∆ ) fraction of the members survive, and the scheme wealth is equally distributed intoeach surviving member’s pension account. Therefore, we haveY( t + ∆ ) = (cid:16) Y( t ) + µ ( t , t + ∆ )Y( t ) – q ( t , t + ∆ ) π Y( t ) – β ( t ) ∆ (cid:17)
11 – q ( t , t + ∆ ) .Since λ j ( t ) is the force of mortality of members, we have 1 – q ( t , t + ∆ ) = e – R t + ∆ t λ j ( u )d u . The Taylor seriesapproximation gives 11 – q ( t , t + ∆ ) = e R t + ∆ t λ j ( u )d u = 1 + λ j ( t ) ∆ + o ( ∆ ).Thereafter, we haveY( t + ∆ ) = (cid:16) Y( t ) + µ ( t , t + ∆ )Y( t ) – q ( t , t + ∆ ) π Y( t ) – β ( t ) ∆ (cid:17)(cid:16) λ j ( t ) ∆ + o ( ∆ ) (cid:17) .Since µ ( t , t + ∆ ) λ j ( t ) ∆ = o ( ∆ ), q ( t , t + ∆ ) λ j ( t ) ∆ = o ( ∆ ),we get Y( t + ∆ ) = Y( t ) + µ ( t , t + ∆ )Y( t ) – q ( t , t + ∆ ) π Y( t ) – β ( t ) ∆ + λ j ( t ) ∆ Y( t ) + o ( ∆ ).Furthermore, Y( t + ∆ ) – Y( t ) ∆ = µ ( t , t + ∆ ) ∆ Y( t ) – π Y( t ) q ( t , t + ∆ ) ∆ – β ( t ) + λ j ( t )Y( t ) + o ( ∆ ).5he following equations hold: lim ∆ → q ( t , t + ∆ ) ∆ = λ j ( t ), lim ∆ → µ ( t , t + ∆ )Y( t ) = dS( t )S( t ) α S ( t ) + dL( t )L( t ) α L ( t ) + dR( t )R( t ) α ( t ).Thus, taking ∆ → t ) = h r Y( t ) + (1 – π ) λ j ( t )Y( t ) + α S ( t ) σ S θ S + α L ( t ) n –1 X i , j =1 σ ij L ( t ) θ j ( t ) – β ( t ) i d t + α S ( t ) σ S dW S ( t ) + α L ( t ) n –1 X i , j =1 σ ij L ( t )dW j ( t ).Here, we use the fact that α ( t ) = Y( t ) – α S ( t ) – α L ( t ). It is worth noticing that the wealth process here is nota self-financing wealth process since there are continuous benefits withdrawal from the scheme. Also, at any time t ∈ T , the manager receives πλ j ( t )Y( t )d t amount of money out from the scheme as compensation. Meanwhile, thescheme continuously receives (1 – π ) λ j ( t )Y( t ) from the passed away members which is called the mortality creditin He and Liang [2013a].At any time, the pension scheme manager decides the benefits withdrawal rate and the investment strategy.The manager works not only for his own benefit but also for the benefit of scheme members. This is also known asfirst-best principal-agent problem where the agent is paid a fraction of the scheme’s wealth at the stochastic deathtime. More specifically, we consider an optimisation problem which combines the manager’s and the members’utilities. Denote by τ the stochastic death time of the pension members, the objective function is given asJ(Y, λ ; α S , α L , β ) = E h Z ∞ e – rs U P ( β ( s )) { τ ≥ s } d s i + φ E h e – r τ U A ( π Y( τ )) i = E h Z ∞ e – rs p j ( s ) (cid:16) U P ( β ( s )) + φλ j ( s )U A ( π Y( s )) (cid:17) d s i where U P ( · ) and U A ( · ) denote the utility functions of the principal (members) and the agent (manager). Thenon-negative constant φ can be viewed as a parameter that determines the risk-sharing rule between the principaland the agent. The case φ = 0 corresponds to the situation when the manager works for only the sake of members.In this case, the objective is to maximise the members’ running utility from benefit withdrawals while the managerpays no attention to his own utility. The case 0 < φ < 1 prioritizes the members’ utility. When φ = 1, the objectivefunction puts equal importance on members’ and manager’s utility.To specify the optimisation problem, we set U P ( · ) and U A ( · ) as log utility functions:U A ( x ) = ln x , U P ( x ) = ln x , ∀ x > 0.Besides, upon death, we assume the total amount of passed away members’ remaining pension is paid to themanager (i.e. π = 1). The wealth process is nowdY( t ) = h r Y( t ) + α S ( t ) σ S θ S + α L ( t ) n –1 X i , j =1 σ ij L ( t ) θ j ( t ) – β ( t ) i d t + α S ( t ) σ S dW S ( t ) + α L ( t ) n –1 X i , j =1 σ ij L ( t )dW j ( t ). (3)It should be noticed that, at any time t ∈ T , the manager receives c ( t ) = λ i ( t )Y( t ) as his compensation.6he optimisation problem is thus defined as sup α S , α L , β E h R ∞ e – rs p j ( s ) (cid:16) ln ( β ( s )) + φλ j ( s ) ln (Y( s )) (cid:17) d s i s.t. (1) and (3) hold. (4)This optimal control problem can be solved by applying the dynamic programming principle, and the solution isgiven in the following proposition. Proposition 1.
The solution to optimisation problem (4) is β ∗ ( t )Y( t ) = 1G( t , λ ) , α ∗ S ( t )Y( t ) = θ S σ S , α ∗ L ( t )Y( t ) = P n –1 i , j =1 σ ij L ( t ) θ j ( t , λ j ) P n –1 j =1 (cid:16)P n –1 i =1 σ ij L ( t ) (cid:17) + P n –1 i , j =1 σ ij L ( t ) Σ G λ ( t , λ ) P n –1 j =1 (cid:16)P n –1 i =1 σ ij L ( t ) (cid:17) G( t , λ ) , where G( t , λ ) = E t (cid:20)Z ∞ t ( φλ j ( s ) + 1) e – R st ( r + λ j ( u ))d u d s (cid:21) . Proof.
The proof is given in Appendix A.
This section gives the explicit solutions to the optimal control problem proposed before in single-population andsub-population cases. In the literature, e.g. Luciano and Vigna [2005] and Wong et al. [2014], several continuous-time stochastic models for force of mortality are studied. For example, OU process, CIR process and Feller process.In this section, we use OU and CIR processes to model the stochastic force of mortality.
Let n = 1, that is, we assume that the reference population of the longevity bond happens to be the schememembers’ population. We expect that the investment in the longevity bond can hedge the scheme’s longevity riskeffectively, since the uncertainty in the longevity bond value correlates the members’ longevity risk perfectly. Now,(1) takes the following form: d λ ( t ) = ( a ( t ) – b λ ( t )) d t + σ ( t , λ )dW ( t ),where b is a constant. According to Menoncin and Regis [2017], a ( t ) is defined as a ( t ) = b (cid:18) ν + 1 ∆ (cid:18) b ∆ (cid:19) e t – m ∆ (cid:19) ,where m , ν and ∆ are constants. Particularly, m equals the modal value of life expectancy of the members.The initial value of the force of mortality is calculated according to the Gompertz-Makeham function λ (0) = ν + 1 ∆ e – m ∆ .The continuous function σ ( t , λ ) takes different forms for different processes. For instance:7 OU process: σ ( t , λ ) = σ , and d λ ( t ) = ( a ( t ) – b λ ( t )) d t + σ dW ( t ). (5) (cid:15) CIR process: σ ( t , λ ) = σ p λ ( t ), andd λ ( t ) = ( a ( t ) – b λ ( t )) d t + σ p λ dW ( t ). (6)In what follows, we apply dynamic programming principle to solve the single-population optimisation problem.Proposition 2 and 3 give the explicit solutions under OU process and CIR process settings, respectively. Proposition 2.
Suppose that λ ( t ) follows the OU process (5) . Then, we have E t (cid:20) e – R st λ ( u ) du (cid:21) = e A ( t , s )–A ( t , s ) λ ( t ) , where A ( t , s ) = 1 – e – b ( s – t ) b , A ( t , s ) = – Z st (cid:18) a ( u )A ( u , s ) – 12 σ A ( u , s ) (cid:19) d u . Suppose θ ( t , λ ) := θ . Then, the solution to the single-population optimisation problem (4) is given as β ∗ ( t )Y( t ) = 1G( t , λ ) , α ∗ S ( t )Y( t ) = θ S σ S , α ∗ L ( t )Y( t ) = θ σ L ( t ) + σ σ L ( t ) G λ ( t , λ )G( t , λ ) , where σ L ( t ) = – A ( t , t + T) σ ,G( t , λ ) = φλ ( t ) Z ∞ t e –( b + r )( s – t )+A ( t , s )–A ( t , s ) λ ( t ) d s + φ Z ∞ t Z st h a ( u ) – σ A ( u , s ) i d u e – r ( s – t )+A ( t , s )–A ( t , s ) λ ( t ) d s + Z ∞ t e – r ( s – t )+A ( t , s )–A ( t , s ) λ ( t ) d s .To derive the optimal solution, we define a new equivalent probability measure e P . The complete proof is givenin Appendix B. Proposition 3.
Suppose that λ ( t ) follows the CIR process (6) . Then, we have E t (cid:20) e – R st λ ( u ) du (cid:21) = e A ( t , s )–A ( t , s ) λ ( t ) , where A ( t , s ) = 2( e η ( s – t ) – 1)( b + η )( e η ( s – t ) – 1) + 2 η , η = q b + 2 σ ,A ( t , s ) = – Z st a ( u )A ( u , s )d u . Suppose θ ( t , λ ) := θ p λ ( t ). Then, the solution to the single-population optimisation problem (4) is givenas β ∗ ( t )Y( t ) = 1G( t , λ ) , α ∗ S ( t )Y( t ) = θ S σ S , α ∗ L ( t )Y( t ) = θ p λ ( t ) σ L ( t ) + σ p λ ( t ) σ L ( t ) G λ ( t , λ )G( t , λ ) ,8 here σ L ( t ) = – A ( t , t + T) σ p λ ( t ),G( t , λ ) = φλ ( t ) Z ∞ t e –( b + r )( s – t )+A ( t , s )–A ( t , s ) λ ( t )– σ R st A ( u , s )d u d s + φ Z ∞ t Z st a ( u ) e – b ( s – u )– σ R su A ( v , s )d v d u e – r ( s – t )+A ( t , s )–A ( t , s ) λ ( t ) d s + Z ∞ t e – r ( s – t )+A ( t , s )–A ( t , s ) λ ( t ) d s .As the proof of above result is similar to the proof of Proposition 2, we omit it here to avoid repetition.We learn from Proposition 2 and 3 that the optimal portfolio weight on stock under OU and CIR processesare the same and keep constant over time. However, it is difficult to see from the solution how the longevity bondinvestment changes over time, because α ∗ L ( t )Y( t ) depends on G( t , λ ) and G λ ( t , λ ). Later in Section 4, we performnumerical simulations to investigate the optimal investment strategy and benefit withdrawal rate dynamically. Based on different mortality indices and maturity times, there may be different longevity bonds in the market. Itmay be interesting to study our problem in a market setting with multiple longevity bonds. However, we onlyconsider one longevity bond in our framework as our main focus is on longevity risk hedging. In this section, westudy the case where the longevity bond’s reference population is different from the pension members’ population.More specifically, we set n = 2 and denote the index population of the longevity bond by Population 1, and usePopulation 2 to denote the pension members’ population. In practice, the reference population of longevity bondtends to be large and could be much larger than pension schemes’ reference population. Therefore, we assume thatthe pension members are a sub-population of the index population. Under this sub-population assumption (see,Wong et al. [2014]), vectors A ( t , λ ) and Σ ( t , λ ) in (1) take the following form: A ( t , λ ) = (cid:20) a ( t ) + b λ ( t ) a ( t ) + b λ ( t ) + b λ ( t ) (cid:21) , Σ ( t , λ ) = (cid:20) σ ( t , λ ) 0 0 σ ( t , λ ) σ ( t , λ ) 0 (cid:21) , (7)where b , b and b are constant numbers. a i ( t ) = b i (cid:18) ν i + 1 ∆ i (cid:18) b i ∆ i (cid:19) e t – mi ∆ i (cid:19) , for i = 1, 2,where ν i are ∆ i constants. The constants m and m are modal values of life expectancy of Population 1 andPopulation 2, respectively.Under OU and CIR processes, the continuous functions σ ( t , λ ), σ ( t , λ ) and σ ( t , λ ) in vector Σ ( t , λ )are given as: (cid:15) OU process: Σ ( t , λ ) = (cid:20) σ σ σ (cid:21) . (cid:15) CIR process: Σ ( t , λ ) = (cid:20) σ p λ ( t ) 0 0 σ p λ ( t ) σ p λ ( t ) 0 (cid:21) .9n this sub-population model, there are two state variables. Moreover, the state variable λ ( t ) correlates with thestate variable λ ( t ). This increases the difficulty of solving the optimisation problem. In this case, an analyticalsolution may not always be available. In the following, we provide Proposition 4 and 5 for the solutions to thesub-population optimisation problem under OU and CIR settings, respectively. Proposition 4.
Suppose Population 2 is a sub-population of Population 1, and the force of mortalities followOU processes: the dynamics of λ ( t ) and λ ( t ) are described as d λ ( t ) = (cid:16) a ( t ) + b λ ( t ) (cid:17) d t + σ dW ( t ),d λ ( t ) = (cid:16) a ( t ) + b λ ( t ) + b λ ( t ) (cid:17) d t + σ dW ( t ) + σ dW ( t ). Then, we have E t (cid:20) e – R st λ ( u ) du (cid:21) = e A ( t , s )–A ( t , s ) λ ( t ) , E t (cid:20) e – R st λ ( u ) du (cid:21) = e C ( t , s )–C ( t , s ) λ ( t )–C ( t , s ) λ ( t ) , where functions A ( t , s ) and A ( t , s ) are the same as in Proposition 2. The functions C ( t , s ) , C ( t , s ) and C ( t , s ) are C ( t , s ) = 1 b (cid:16) e – b ( s – t ) (cid:17) ,C ( t , s ) = b b ( b – b ) (cid:16) e – b ( s – t ) (cid:17) + b b ( b – b ) ,C ( t , s ) = – Z st h a ( u )C ( u , s ) + a ( u )C ( u , s ) – 12 σ C ( u , s )– 12 ( σ + σ ) C ( u , s ) – σ σ C ( u , s )C ( u , s ) i d u . Suppose θ ( t , λ ) := θ . Then, the solution to the sub-population optimisation problem is given as β ∗ ( t )Y( t ) = 1G( t , λ , λ ) , α ∗ S ( t )Y( t ) = θ S σ S , α ∗ L ( t )Y( t ) = θ σ L ( t ) + σ σ L ( t ) G λ ( t , λ , λ )G( t , λ , λ ) + σ σ L ( t ) G λ ( t , λ , λ )G( t , λ , λ ) , where σ L ( t ) = – A ( t , t + T) σ ,G( t , λ , λ ) = φ b b – b λ ( t ) Z ∞ t (cid:16) e – b ( s – t ) – e – b ( s – t ) (cid:17) f ( t , s )d s + φλ ( t ) Z ∞ t e – b ( s – t ) f ( t , s )d s – φ Z ∞ t Γ ( t , s ) f ( t , s )d s + φ b b – b Z ∞ t Γ ( t , s ) f ( t , s )d s + Z ∞ t f ( t , s )d s , f ( t , s ) = e – r ( s – t )+C ( t , s )–C ( t , s ) λ ( t )–C ( t , s ) λ ( t ) , Γ ( t , s ) = Z st e – b ( s – u ) h b b – b a ( u ) – a ( u ) + σ σ C ( u , s )10 (cid:18) σ + σ – b σ σ b – b (cid:19) C ( u , s ) i d u , Γ ( t , s ) = Z st e – b ( s – u ) h a ( u ) – σ C ( u , s ) – σ σ C ( u , s ) i d u . Proof.
The proof is given in Appendix C
Proposition 5.
Suppose Population 2 is a sub-population of Population 1, and the force of mortalities followCIR processes: the dynamics of λ ( t ) and λ ( t ) are described as d λ ( t ) = (cid:16) a ( t ) + b λ ( t ) (cid:17) d t + σ p λ ( t )dW ( t ),d λ ( t ) = (cid:16) a ( t ) + b λ ( t ) + b λ ( t ) (cid:17) d t + σ p λ ( t )dW ( t ) + σ p λ ( t )dW ( t ). Then, we have E t (cid:20) e – R st λ ( u ) du (cid:21) = e A ( t , s )–A ( t , s ) λ ( t ) , E t (cid:20) e – R st λ ( u ) du (cid:21) = e C ( t , s )–C ( t , s ) λ ( t )–C ( t , s ) λ ( t ) , where functions A ( t , s ) and A ( t , s ) are the same as in Proposition 3. The functions C ( t , s ) , C ( t , s ) and C ( t , s ) are solutions to the following ODE system ∂ t + a C + a C ,0 = – ∂ C ∂ t + b C + b C + 12 σ C + 12 σ C + σ σ C e A ,0 = – ∂ C ∂ t + b C + 12 σ C – 1, (8) with terminal conditions C ( s , s ) = 0 , C ( s , s ) = 0 and C ( s , s ) = 0 .Suppose θ ( t , λ ) := θ p λ ( t ). Then, the solution to the sub-population optimisation problem is given as β ∗ ( t )Y( t ) = 1G( t , λ , λ ) , α ∗ S ( t )Y( t ) = θ S σ S , α ∗ L ( t )Y( t ) = θ p λ ( t ) σ L ( t ) + σ p λ ( t ) σ L ( t ) G λ ( t , λ , λ )G( t , λ , λ ) + σ p λ ( t ) σ L ( t ) G λ ( t , λ , λ )G( t , λ , λ ) , where σ L ( t ) = – A ( t , t + T) σ p λ ( t ).G( t , λ , λ ) satisfies G( t , λ , λ ) = Z ∞ t e – r ( s – t )+C ( t , s )–C ( t , s ) λ ( t )–C ( t , s ) λ ( t ) d s + φ Z ∞ t e – r ( s – t ) e E t [ λ ( s )] d s . e E [ · ] denotes the expectation under an equivalent measure e P which is defined as d e P d P = e Z( s ) = exp (cid:18) – Z s ˜ θ ( u , s )dW( u ) – 12 Z s (cid:13)(cid:13) ˜ θ ( u , s ) (cid:13)(cid:13) d u (cid:19) ,11 here ˜ θ ( u , s ) = σ p λ ( u )C ( u , s ) + σ p λ ( u )C ( u , s ) σ p λ ( u )C ( u , s )0 . Proof.
The Picard-LindelÃűf theorem ensures the existence of unique solutions to ODEs in (8). The rest of theproof is similar to the proof of Proposition 4 and is omitted here.From Proposition 4 and 5, we see that the optimal portfolio weight on stock stays the same as in the single-population case. The OU case has an analytical solution; however, the CIR case does not. In Section 4, we conductnumerical studies to observe the hedging effect of the longevity bond in the sub-population OU process case.
This section provides numerical simulations in the single- and sub-population cases using the results from Propo-sition 2 and 4. A numerical study involving CIR models is not presented here as the results are similar to the OUcase. We observe the dynamics of the survival probability and look into the impact of the mortality behaviour onthe optimal strategy. We investigate the hedging performance of the longevity bond in the pension scheme’s riskmanagement and provide a sensitivity analysis on the market price of longevity risk. We are also interested in theeffect of the risk-sharing rule between the members and the manager.Table 1 shows the values of parameters in our numerical examples. The time interval ∆ = 1/10 means thatwe observe the mortality rates 10 times a year. Most of the values of the mortality model parameters are asconsidered in other works (e.g. [Menoncin and Regis, 2017] and Milevsky [2001]). The values of other financialmarket parameters are meant to be representative.Table 1: Values of parameters for optimization problem. Population 1 Population 2 Market Others ν = 0.0009944 ν = 0.0009944 r = 0.04 T = 35 ∆ = 11.4000 ∆ = 12.9374 θ = –0.0005 Y = 100 m = 86.4515 m = 89.18 θ S = 0.05 ∆ = 1/10 b = 0.5610 b = 0.0028 σ S = 0.15 φ = 0.8 σ = 0.0035 b = 0.6500 T L = 20 σ = 0.0040 σ = 0.0050 In the single-population case, we assume that the scheme members happen to be the reference population ofa longevity bond. The manager invests in this longevity bond to hedge the scheme’s longevity exposure. Weassume that the scheme members have similar mortality behaviour, and their retirement age is 65. The pensionscheme’s management starts from the retirement time and ends until the last member passes away. According tothe Gompertz-Makeham law of mortality, less than 5% of the population could survive until 100 years old giventhat they are alive at 65 years old. Thus, we conduct numerical simulations with a 35-year time horizon (i.e.T = 35).Figure 1 shows three simulation paths to illustrate the dynamic mortality behaviour of the members. We seethat the survival probability decreases with time. In the bottom-right plot, we observe that for all three paths,12ess than 5% of the members survive until 100 years old indicating that it is reasonable to set T = 35. Suppose theGompertz-Makeham mortality law describes the average trend of the members’ mortality. We learn from the plotsin Figure 1 that the survival probability of the simulated path 1 is always higher than average. On the contrary, thesurvival probability of path 2 is lower than expected. Path 3 does not show any particular trend. Figure 1 showsthe cumulative distribution function f ( · ) of the stochastic death time τ . We see from the plot that f ( τ ) peaks at τ = 86.5 approximately. This is consistent with our model settings and choice of parameter values: m = 86.4515in our model is the modal value of life span of the members. Generally, our observations imply that: (cid:15) path 1 shows a prominent longevity trend; (cid:15) path 2 on average has a lower survival rate; (cid:15) path 3 does not show any particular trend; (cid:15) For all three paths, most of the members’ deaths happen before the age of 86.5. t p (t) Path 1Path 2Path 3Mortality law 12 14 16 18 20 22 24 t p (t) Path 1Path 2Path 3Mortality law24 26 28 30 32 34 t p (t) Path 1Path 2Path 3Mortality law 65 70 75 80 85 90 95 100 t p (t) Path 1Path 2Path 3Mortality law
Figure 1: τ In the base scenario, we investigate the optimal benefit withdrawal rate and investment strategy. Besides, weobserve the scheme’s wealth level and manager’s compensation dynamically. Figure 2 shows the average investmentstrategy over 100 simulation paths. We observe that the portfolio weight in the longevity bond drops over time.As members get older, the scheme’s exposure to the longevity risk reduces, and the need for longevity protectiondecreases. Accordingly, the manager reduces the portfolio weight in the longevity bond. The flat line which showsthe investment proportion in stock is a direct result from the optimal solutions in Proposition 2: the portfolio weightin stock keeps constant and equals to θ S σ S . This constant investment strategy coincides with the result in the classicalMerton’s portfolio problem. The interpretation is that the constant market price of risk causes no change in themanager’s investment behaviour. The proportion of the portfolio in the money market is α ( t )Y( t ) = 1 – α L ( t )Y( t ) – α S ( t )Y( t ) .We see that the portfolio weight in the money market is negative at first and then increases over time. The negative13osition in the initial years indicates that the manager borrows money from the money market, and invests in therisky assets to gain risk premiums and increase the scheme’s wealth level. As α S ( t )Y( t ) keeps constant, and α L ( t )Y( t ) decreases over time, the manager puts more weight in the money market. With the passage of time, the managerbecomes more conservative to avoid unexpected losses. Overall, the longevity bond dominates the investmentportfolio throughout the investment horizon. Even when members reach the age of 100, the manager puts around50% of the portfolio in the longevity bond, indicating that the longevity bond could provide not only longevityprotection but also considerable risk premium. t -0.4-0.200.20.40.60.81 p r opo r t i on longevity bondstockmoney market Figure 2: average optimal investment strategies over 100 paths
To observe general results, we show the average of 100 simulation paths of the optimal benefit withdrawalproportion and rate, the scheme’s wealth and the manager’s compensation. From the top-left plot in Figure 3, wesee the optimal proportion of the scheme wealth β ∗ Y( t ) withdrawn by members increases over time. Meanwhile, inthe top-right plot, we observe that the optimal benefit withdrawal rate β ∗ ( t ) reduces over time. This phenomenonis explained by the scheme’s declining wealth process shown in the bottom-right plot. Although the managerinvests in the financial market, the average scheme level is decreasing throughout the time horizon due to thecontinuous benefit withdrawals and compensation payments. The decreasing number of survival members, as wellas the declining scheme wealth, result in the drop in benefit withdrawal rate. The average compensation receivedby the manager shows an interesting trend - it increases at first, peaks at around the 19th year and then dropsrapidly. The reason is that most members pass away before the age of 84. The compensation increases before themode of the members’ life span even though the scheme’s wealth level keeps decreasing. After reaching the modalvalue of life expectancy, as most of the members have already passed away, the manger receives less compensation. To test the hedging performance of longevity bond, we give the results in the case when the manager does notinclude the longevity bond in the investment portfolio. Without the investment in longevity bond, the optimalbenefit withdrawal proportion β ∗ ( t )Y( t ) = t , λ ) , and the optimal portfolio weight on stock α ∗ s ( t )Y( t ) = θ S σ S , are the sameas given in Proposition 2. The portfolio weight in money market equals to 1 – α ∗ S ( t )Y( t ) and keeps constant over time.Let β ( t ) ( c ( t )) and β ( t ) ( c ( t )) denote the benefit withdrawal (compensation) without and with investment inlongevity bond, respectively. Figure 4 shows the benefit withdrawal and compensation improvement by investing inlongevity bond. It shows that, for path 1, investing in longevity bond always results in higher benefit withdrawalsand compensations. For path 3, investing in longevity bond in general increases both members’ benefit withdrawalsand the manager’s compensation. Although, during some short period, longevity bond investment decreases thewithdrawals and compensations. However, for path 2, investment in longevity bond seems to cut down bothbenefit withdrawals and compensations. As discussed earlier (see Figure 1), the survival probability on path 2 isoverall lower than the Gompertz-Makeham survival probability. It implies that the pension members tend to live14 t Y ( t ) t t ( t ) t c ( t ) Figure 3: average benefit withdrawal proportion, withdrawal rate, wealth, and compensation over 100 paths shorter than expected. Likewise, Figure 1 suggests that for path 2, the random death age τ is more likely to bethe younger ages compared to the other paths - path 1 or path 3. As a result, the pension scheme does not facethe longevity risk and investing in the longevity bond actually loses money rather than making gains. As it isa global trend that people’s average life expectancy is increasing, we argue that the pension schemes will benefitfrom longevity bond investment as mirrored by the situation in path 1 and 3. To support our claim, we show theaverage improvements of benefit withdrawal and compensation over 100 simulation paths in Figure 5. As shown,there are low improvements in the first few years, but overall the improvements are significant over most of theinvestment period. It indicates that investing in longevity bond increases both members’ benefit withdrawal rateand manager’s compensation, and that it is beneficial to both members and manager. t -0.3-0.2-0.100.10.20.30.4 (t)- (t) Path 1Path 2Path 3 0 5 10 15 20 25 30 35 t -0.1-0.0500.050.10.150.2 c (t)- c (t) Path 1Path 2Path 3
Figure 4: improvement of the benefit withdrawal and compensation t -0.00500.0050.010.015 0 5 10 15 20 25 30 35 t -0.00500.0050.010.0150 5 10 15 20 25 30 35 t -0.0500.050.1 0 5 10 15 20 25 30 35 t -0.0200.020.04 Figure 5: average improvements of benefit withdrawal and compensation over 100 paths
We are interested in the impact of the market price of risk on pension scheme’s risk management. It is difficult todecide the market price of longevity risk (in other words θ ) due to the absence of longevity bonds in the market.But we examine a few values of θ to give an illustration of the effects of the market price of longevity risk onthe optimal strategy, the benefit withdrawal and the manager’s compensation. In our base scenario, θ is set as–5 × –4 , and the longevity risk premium offered by the longevity bond is 4.4563 × –6 . Compared with the stock’srisk premium (7.5 × –3 ), the longevity risk premium is minimal. We show the optimal investment strategies inthe cases where θ = 0, –1.5 × –3 and –3 × –3 . We know that a large absolute value of θ indicates highrisk premium offered by the longevity bond. Besides, large premiums should lead to more investment in longevitybond. Figure 6 shows the investment strategies with different values of θ . We can see that when the managerdoes not add longevity bond to his portfolio, the optimal investment proportions in stock and money market keepconstant over time. The investment in the longevity bond does not affect the portfolio weight in stock while affectsthe investment in the money market. The top-right plot shows that even in the case when the longevity bondoffers no risk premium (i.e. θ = 0), the proportion invested in the longevity bond is always higher than 40%. Itillustrates that the longevity bond provides a good way to hedge the scheme’s longevity risk. As shown in Figure6, as the longevity risk premium decreases (i.e. lower θ ), more portfolio weight is put on the longevity bond. Inthe bottom-right plot, θ = –0.0030 and the longevity risk premium equals to 2.6738 × –5 which is again far lessthan the stock’s risk premium. It implies that the manager continuously borrows money from the money marketto invest in risky assets throughout the whole time horizon. The intuition is that the longevity bond not onlyprovides longevity risk hedge, but also provides an attractive risk premium.Figure 7 shows the improvements of benefit withdrawal and compensation when investing in longevity bond. Asshown for path 1, high market prices of longevity risk lead to high improvements in both manager’s compensationand members’ benefit withdrawal. As discussed in Section 4.1.2, investing in longevity bond sometimes decreasesthe benefit withdrawal and compensation and thus causes a ‘loss’. Here by ‘loss’, we mean loss in the membersbenefit and manager compensation as the improvements by investing in longevity bond are negative. Nevertheless,we observe from the plots in Figure 7 that a smaller θ reduces this loss.We now investigate the impact of the risk-sharing parameter φ on the benefit withdrawal and manager com-pensation. As stated before, φ decides the agreement between the manager and members on how to share the risk.16 t P r opo r t i on Inevtment strategy without longevity bond stockcash 0 5 10 15 20 25 30 35 t -0.200.20.40.60.81 P r opo r t i on With longevity bond when = 0 stockcashlongevity bond0 5 10 15 20 25 30 35 t -0.500.51 P r opo r t i on With longevity bond when = -0.0015 stockcashlongevity bond 0 5 10 15 20 25 30 35 t -0.500.511.5 P r opo r t i on With longevity bond when = -0.0030 stockcashlongevity bond Figure 6: impact of θ on the optimal investment strategy t -0.2-0.100.10.20.30.4 =0 Path 1Path 2Path 3 0 5 10 15 20 25 30 35 t -0.4-0.200.20.40.6 = -0.0015 Path 1Path 2Path 3 0 5 10 15 20 25 30 35 t -0.4-0.200.20.40.6 = -0.0030 Path 1Path 2Path 30 5 10 15 20 25 30 35 t -0.1-0.0500.050.10.150.2 = 0 Path 1Path 2Path 3 0 5 10 15 20 25 30 35 t -0.1-0.0500.050.10.150.2 = -0.0015 Path 1Path 2Path 3 0 5 10 15 20 25 30 35 t -0.2-0.100.10.20.3 = -0.0030 Path 1Path 2Path 3
Figure 7: impact of θ on the benefit withdrawal and compensation improvements
17 high value of φ implies that more weight is put on the manager’s utility. When φ = 0, the manager works onbehalf of the members and only cares about the members’ benefit. This case corresponds to no risk-sharing. When0 < φ < 1, more attention is put on the members’ benefit. In the case where φ = 1, the manager treats his ownprofit and members’ benefit equally which corresponds to the case of equal risk-sharing. We test the cases when φ takes values 0, 0.5 and 1. The case with φ = 0 is chosen as the reference case. Figure 8 shows the improve-ment rates on benefit withdrawal and compensation (i.e. β φ ( t )– β φ =0 ( t ) β φ =0 ( t ) and c φ ( t )– c φ =0 ( t ) c φ =0 ( t ) ). As shown in the rightplot, higher the φ , higher the improvement in compensation. Compared to the case φ = 0, the equal risk-sharingrule agreement improves the manager’s compensation by more than 20% at the end of the time horizon. For thebenefit withdrawal, a higher φ leads to higher withdrawals in the last 10 years, but reduces the withdrawals inthe early periods. To illustrate the impact of φ , we calculate the average discounted values of benefit withdrawalsand compensations over 100 simulation paths. We find that compared to the case φ = 0, φ = 1 increases thediscounted benefit withdrawal by 4.71% and the discounted compensation by 12.82%. Thus, both manager andmembers benefit from sharing the risk equally. t b e n e f i t i m p r o ve m e n t =1=0.8=0.50 5 10 15 20 25 30 35 t -0.0200.020.040.060.080.1 c o m p e n sa t i on i m p r o ve m e n t =1=0.8=0.5 Figure 8: impact of φ on the benefit withdrawal and compensation The previous discussions can be summarised as follows. The longevity bond offers an efficient way to hedgethe pension scheme’s longevity risk as it always dominates the investment portfolio. Moreover, both members andmanager benefit from investment in the longevity bond. Higher the longevity risk premium, more portfolio weightis put on the longevity bond, and more the manager and members benefit from investing in it. Finally, an equalrisk-sharing rule is the most beneficial to both members and scheme’s manager.
On the one hand, a pension scheme faces the longevity risk caused by its members’ more extended life span. Onthe other hand, a pension scheme faces the longevity basis risk if the mortality behaviour of the scheme members isimperfectly correlated with the longevity bond mortality index. In practice, pension schemes face longevity basisrisk as it is difficult to find a longevity bond in the market that is based exactly on the scheme members. Therefore,the sub-population model may be more practical compared to the single-population model. In this section, weassume that the longevity bond is based on a large population and the scheme members are a sub-population ofthis large population. Furthermore, we assume that the life expectancy of the scheme members is higher than thelongevity bond reference population. If it is lower, the pension scheme may not face the longevity risk and thereis no need to invest in the longevity bond. We are interested in the optimal strategy and the hedging performanceof longevity bond in the presence of longevity basis risk.18 .2.1 The base scenario
The values of parameters used in this section are as given in Table 1. For i = 1, 2, m i in (7) is the mode of lifeexpectancy of the i th population. m is set to be greater than m implying that the possible death age of thepension members of Population 2 is higher than the longevity bond reference population in Population 1. Again,we present three simulation paths in this section to observe the two populations’ mortality behaviour. Figure 9shows the survival probabilities for Population 1 and Population 2. As expected, the average survival probabilityof Population 1 is lower than Population 2. The plots below illustrate the following: (cid:15) Population 1 – path 1 has no particular trend; – path 2 has a higher survival probability than expected; – path 3 shows shorter life expectancy. (cid:15) Population 2: – path 1 displays a lower survival probability in the first half of the time horizon; – path 2 & 3 on average show longevity trend. t p (t) Path 1Path 2Path 3Mortality law 18 20 22 24 26 28 30 32 34 t p (t) Path 1Path 2Path 3Mortality law0 2 4 6 8 10 12 14 16 t p (t) Path 1Path 2Path 3Mortality law 18 20 22 24 26 28 30 32 34 t p (t) Path 1Path 2Path 3Mortality law
Survival Probability for Population 1Survival Probability for Population 2
Figure 9: survival probabilities for Population 1 and 2
Figure 10 shows the optimal investment strategy and benefit withdrawal in the sub-population case. Theoptimal strategy is similar to the single-population case: (cid:15) the longevity bond dominates the portfolio; (cid:15) the portfolio weight on the longevity bond decreases over time while the holding in the money marketincreases; (cid:15) the optimal investment proportion in the stock keeps constant; (cid:15) the percentage of the funds withdrawn by the members increases with time.We learn from it that the longevity bond plays an essential role in the pension scheme’s risk management.19 t p r opo r t i on benefitstockcashlongevity bond Figure 10: average optimal portfolio strategy and benefit withdrawal in sub-population case
In Section 4.1, we show that in the single-population case, both the manager and members benefit from investing inthe longevity bond. Now, we conduct a comparison study in the sub-population case to see whether the longevitybond still brings the advantage. For the simulation path 1, Figure 11 shows that investing in the longevity bondin general increases the members’ benefit withdrawal and manager’s compensation. For path 2, the members andmanager benefit from investing in the longevity bond in the late 20-year time period, although the scheme suffersfrom some loss in early years. While the simulation path 3 shows that neither the manager nor members takeadvantage of longevity bond investment. To find out the reason for these phenomena, we check the simulatedsurvival probabilities. We learn from the bottom plots in Figure 9 that on path 3, scheme members in Population2 live much longer than anticipated. Whereas, the survival probability of longevity bond reference population(Population 1) on path 3 is lower. In this situation, the scheme suffers from severe longevity risk. , In this case, thelongevity bond can not provide an efficient longevity risk hedge because the reference population displays shorterlife expectancy. Since we assume the members are a sub-population of the longevity bond reference population,the members should have similar mortality trend with the reference population, although with slightly differentbehaviour. We believe that investing in the longevity bond can still be beneficial to both sides - manager andscheme members - in the two population case. t -0.3-0.2-0.100.10.2 ( t )- ( t ) Path 1Path 2Path 3 0 5 10 15 20 25 30 35 t -0.1-0.0500.050.1 c ( t )- c ( t ) Path 1Path 2Path 3
Figure 11: benefit and compensation improvement in sub-population case
Overall, we claim that the longevity bond provides a way to hedge the longevity risk in the sub-populationcase. However, due to the presence of longevity basis risk, the hedging performance may be less effective than in20he single-population case. The situation with multiple longevity bonds in the market may show different resultsand is worthy of an independent future study. The sensitivity analyses on the market price of longevity risk andthe risk-sharing rule in the sub-population case deliver results which are similar to the single-population case.
This work studies the optimal portfolio strategy and benefit withdrawal rate for a pension scheme with an income-drawdown policy in the decumulation phase. The optimal solutions under single- and sub-population cases areobtained by applying the dynamic programming principle. The numerical study shows that the longevity bond canbe used to hedge the longevity risk, and both members and manager benefit from the longevity bond investment.We believe that the development of a longevity market is required to provide a solution to capital markets forlongevity risk hedging. Moreover, both members and manager benefit from an agreement on the risk-sharing rulein the long run. The problem with multiple longevity bonds issued on different populations is an interesting topicwhich we will explore in our future study.
A Proof of Proposition 1
Proof.
The corresponding value function v (Y, λ ) of the optimisation problem (4) is given by v (Y, λ ) = sup α S , α L , β E h Z ∞ e – R s ( r + λ j ( u ))d u (cid:16) ln ( β ( s )) + φλ j ( s ) ln (Y( s )) (cid:17) d s i .Applying Dynamic Programming Principle (DPP) to the value function gives v ( y , z ) ≥ E (cid:20)Z t e – R s ( r + λ j ( u ))d u h ln ( β ( s )) + φλ j ( s ) ln (Y( s )) i d s (cid:21) + E (cid:20) e – R t ( r + λ i ( u ))d u v (Y yt , λ λ t ) (cid:21) . (9)Applying Itô’s formula to e – R t ( r + λ i ( u ))d u v (Y yt , λ λ t ) and substituting it into the DPP (9) leads to0 ≥ E (cid:20)Z t e – R s ( r + λ j ( u ))d u h ln ( β ( s )) + φλ j ( s ) ln (Y( s )) i d s (cid:21) – E (cid:20)Z t ( r + λ j ( s )) e – R s ( r + λ j ( u ))d u v (Y ys , λ λ s )d s (cid:21) + E (cid:20)Z t e – R s ( r + λ j ( u ))d u L α S , α L , β v (Y ys , Z zs )d s (cid:21) .Dividing by t and taking t → ≥ φλ j ln y – ( r + λ j ) v ( y , λ ) + ln ( β ) + L α S , α L , β v ( y , λ ),and we obtain the HJB equation0 = φλ j ln y – ( r + λ j ) v ( y , λ ) + sup α S , α L , β h ln ( β ) + L α S , α L , β v ( y , λ ) i ,where L α , b v ( y , λ ) =V y ry + α S σ S θ S + α L n –1 X i , j =1 σ ij L θ j – β + A ( t , λ )V λ + 12 tr (cid:0) Σ Σ V λλ (cid:1) + 12 α σ + α n –1 X j =1 n –1 X i =1 σ ij L ! V yy + α L n –1 X k =1 n –1 X j =1 n –1 X i =1 σ ij L ! σ kj V y λ k .21olving the first order conditions on β ( t ), α S ( t ) and α L ( t ) gives β ∗ ( t ) = 1V y , α ∗ S ( t ) = – θ S V y σ S V yy , α ∗ L ( t ) = – P n –1 i , j =1 σ ij L θ j V y + P n –1 i , j =1 σ ij L Σ V y λ P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) V yy . (10)Substituting (10) into the HJB equation leads to0 = φλ j ln y – ( r + λ j )V – ln (cid:0) V y (cid:1) + ry V y – 1 + A V λ + 12 tr (cid:0) Σ Σ V λλ (cid:1) – 12 V y V yy θ – (cid:16)P n –1 i , j =1 σ ij L θ j (cid:17) P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) – 12 h P n –1 k =1 (cid:16)P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) σ kj (cid:17) V y λ k i V yy P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) – V y V yy (cid:16)P n –1 i , j =1 σ ij L θ j (cid:17) (cid:16)P n –1 k =1 (cid:16)P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) σ kj (cid:17) V y λ k (cid:17)P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) . (11)We guess the solution to the PDE (11) is of the following formV( y , λ ) = G( λ ) ln y + H( λ ),with boundary conditions lim t →∞ v (Y yt , Z zt ) = 0, lim t →∞ G(Z zt ) = 0, lim t →∞ H(Z zt ) = 0.The PDE (11) now becomes0 = φλ j ln y – ( r + λ j )(G ln y + H) – ln G + ln y + r G – 1 + A (G λ ln y + H λ ) + 12 tr (cid:0) Σ Σ (G λλ ln y + H λλ ) (cid:1) + 12 θ – (cid:16)P n –1 i , j =1 σ ij L θ j (cid:17) P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) G + 12 h P n –1 k =1 (cid:16)P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) σ kj (cid:17) G λ k i G P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) + (cid:16)P n –1 i , j =1 σ ij L θ j (cid:17) (cid:16)P n –1 k =1 (cid:16)P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) σ kj (cid:17) G λ k (cid:17)P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) .Separating the ln y terms and we get two ODEs0 = φλ j – ( r + λ j )G + 1 + A G λ + 12 tr (cid:0) Σ Σ G λλ (cid:1) ,0 = – ( r + λ j )H – ln G + r G – 1 + A H λ + 12 tr (cid:0) Σ Σ H λλ (cid:1) + 12 θ – (cid:16)P n –1 i , j =1 σ ij L θ j (cid:17) P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) G+ (cid:16)P n –1 i , j =1 σ ij L θ j (cid:17) (cid:16)P n –1 k =1 (cid:16)P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) σ kj (cid:17) G λ k (cid:17)P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) + 12 h P n –1 k =1 (cid:16)P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) σ kj (cid:17) G λ k i G P n –1 j =1 (cid:16)P n –1 i =1 σ ij L (cid:17) .22e only need G( λ ) to get the optimal solutions so we solve the first ODE for G( λ ) and getG( λ ) = E (cid:20)Z ∞ ( φλ j ( s ) + 1) e – R s ( r + λ j ( u ))d u d s (cid:21) .G( λ ) has a dynamic version as G( t , λ ) = E t (cid:20)Z ∞ t ( φλ j ( s ) + 1) e – R st ( r + λ j ( u ))d u d s (cid:21) .Substituting G( t , λ ) into (10) gives the optimal solutions in Proposition 1. B Proof of Proposition 2
Proof.
We first give the calculations for A ( t , s ) and A ( t , s ). Under OU setting (5), denote by h ( t , s ) = E t (cid:20) e – R st λ ( u )d u (cid:21) and applying Itô’s formula to e – R t λ ( u )d u h ( t , s ) and letting d t term equals to 0, we obtain0 = – λ h + h t + h λ ( a – b λ ) + 12 σ h λλ As the λ ( t ) follows affine class model, we guess h ( t , s ) has the following form h ( t , s ) = e A ( t , s )–A ( t , s ) λ ( t ) ,with terminal conditions h ( s , s ) = 1, A ( s , s ) = 0 and A ( s , s ) = 0.Differentiating h ( t , s ) and plugging into (B) leads to two ODEs0 = ∂ A ∂ t – a A + 12 σ A ,0 = – ∂ A ∂ t + b A – 1.Solving these ODEs gives A ( t , s ) and A ( t , s ) in Proposition 2.Similarly, denote by h Q ( t , s ) = E Q t (cid:20) e – R st λ ( u )d u (cid:21) and assume h Q ( t , s ) = e A Q ( t , s )–A Q ( t , s ) λ ( t ) , we obtain thatA Q ( t , s ) = A ( t , s ). Moreover, we have σ L ( t ) = –A Q ( t , t + T) σ = –A ( t , t + T) σ .Next, we show the derivation of G( t , λ ). According to Proposition 1, we haveG( t , λ ) = φ Z ∞ t E t (cid:20) λ ( s ) e – R st λ ( u )d u (cid:21) e – r ( s – t ) d s + Z ∞ t E t (cid:20) e – R st λ ( u )d u (cid:21) e – r ( s – t ) d s .To solve E t (cid:20) λ ( s ) e – R st λ ( u )d u (cid:21) , we denote by e Z( t ) = e E t [ e – R s λ ( u )d u ] e E [ e – R s λ ( u )d u ] , m ( t , s , λ ) = e A ( t , s )–A ( t , s ) λ ( t ) and D( t ) = e – R t λ ( u )d u and have E t (cid:20) λ ( s ) e – R st λ ( u )d u (cid:21) = E t λ ( s ) e – R st λ ( u )d u h ( t , s ) h ( t , s ) = E t " λ ( s ) e Z( s ) e Z( t ) h ( t , s ).Let E t (cid:20) λ ( s ) e Z( s ) e Z( t ) (cid:21) = e E t [ λ ( s )], where e E [ · ] denotes the expectation under measure e P which is equivalent to P andwill be specify later. Applying Itô’s formula to e Z( t ) givesd e Z( t ) =dD( t ) m ( t , s , λ ) m (0, s , λ ) + D( t ) m (0, s , λ ) d m ( t , s , λ )23 – λ ( t ) e Z( t )d t + D( t ) m (0, s , λ ) (cid:18) m t + m λ ( a – b λ ) + 12 σ m λλ (cid:19) d t + D( t ) m (0, s , λ ) m λ σ dW ( t )= σ m λ ( t , s , λ ) m ( t , s , λ ) e Z( t )dW ( t )= – σ A ( t , s ) e Z( t )dW ( t )Now, we can define e P through e Z( s )d e P d P = e Z( s ) = exp (cid:18) – Z s σ A ( u , s )dW ( u ) – 12 Z s σ A ( u , s )d u (cid:19) ,and we have d f W ( t ) = dW ( t ) + σ A ( t , s )d t .Thus, we have d λ ( t ) = ( a ( t ) – b λ ( t )) d t + σ (cid:16) d f W ( t ) – σ A ( t , s )d t (cid:17) = (cid:16) a ( t ) – b λ ( t ) – σ A ( t , s ) (cid:17) d t + σ d f W ( t ),taking expectation under e P leads tod e E t [ λ ( u )]d u = a ( u ) – σ A ( u , s ) – b e E t [ λ ( u )].The solution of the above ODE is then obtained e E t [ λ ( s )] = λ ( t ) e – b ( s – t ) + Z st (cid:16) a ( u ) – σ A ( u , s ) (cid:17) e – b ( s – u ) d u .Substituting e E t [ λ ( s )] gives the function G( t , λ ) in Proposition 2. C Proof of Proposition 4
Proof.