Optimal investment problem for a hybrid pension with intergenerational risk-sharing and longevity trend under model uncertainty

The financial market consists of a risk-free asset (bond) and a risky asset (stocks). The price of the risk-free asset and risky asset at time $t\geq 0$ is described by

where $r>0$ denotes the risk-free interest rate. $\mu$ is the appreciation rate of the stock and $\sigma$ is the volatility rate, both $\mu$ and $\sigma$ being positive constants, and $B(t)$ is a standerd Brownian motion. To exclude arbitrage opportunities, we assume that $\mu>r$.

For population, We will develop a continuous deterministic framework considering that age and time are continous variables. Plan members are assumed to enter the labor market at age $x_{0}$ and to keep active up to a retirement age $x_{r}$. As above mentioned, it is reasonable that the force of mortality decrease with time $t$ and increase with age $x$ because of the decreasing fertility and increasing life expectancy. Here, we consider an age and time-dependent mortality $\mu(x,t)$ followed by modified Makeham’s Law, which may capture longevity trend dynamically as following assumption.

where $\omega$ is a positive parameter called longevity parameter which shows longevity trend that is every $\omega$ year, the average life expectancy increases by one year. More precisely, for a $x$-year-old cohort at time 0, the force of mortality is $\mu(x,0)$. $\omega$ years later, the force of mortality at time $\omega$ is $\mu(x+\omega,\omega)=\mathcal{A}+\mathcal{B}\theta^{x+\omega-1}$. And theoretically, if there is no longevity parameter, the force of mortality at time $\omega$ could be like $\mathcal{A}+\mathcal{B}\theta^{x+\omega}$. Therefore the model we considered shows that people have relatively low force of mortality and is equivalent to be one-year “younger”. On the whole, the average life expectancy would increases by one year.

This assumption shows that the maximum age increases in a linear fashion and is in line with the empirical literature. Knell (2018) and Oeppen and Vaupel (2002) e.g., analyze ‘record female life expectancy’ (i.e., the highest value for female life expectancy reported in any country for which data are available) from 1840 to 2000 and they show that it follows an almost perfect linear development. Actually, Strulik and Vollmer (2013) have shown that from the second half of the 20th century onwards improvements in life expectancy have been driven to a large part by expansions of the maximum age.

Denote $p(x,t)$ the survival function that the cohort who enter the system at time $t-(x-x_{0})$ survives to age $x$. It holds that $p(x_{0},t)=1,p(m(t),t)=0$ and that survivorship declines with age, i.e., $\textrm{d}p(x,t)/\textrm{d}x\leq 0$ for $x\in[x_{0},m(t)]$. The mortality hazard rate of cohort $t$ at age $x$ is given by

therefore, $p(x,t)$ and $\mu(x,t)$ are not only age-specific but also include a cohort-specific time index. It holds that:

Thus the age and time-dependent survival function $p(x,t)$ can be expressed as

We denote by $n(t)$ the entry rate which describes the density of sizes of new cohort entering in working life at age $x_{0}$ at time $t$. $n(t)$ follows Malthusian demographic model $n(t)=n_{0}e^{\kappa t}$, where $n_{0}$ is the density of the new $x_{0}$-year-old entrants at time $0$, $\kappa$ is the growth rate of the new entrants. Inspired by He et al. (2020) and He et al. (2021), $\kappa<0$ represents the demographic model with serious aging problem, which is caused by low fertility and low mortality. Then the number of those attaining age $x$ at time $t$ is

where $t-(x-x_{0})$ is the time at which this cohort joined the pension system. Note that with $t\geq 0$, the value of $t-(x-x_{0})$ could be negative, meaning that a member whose age is $x$ at time $t\geq 0$ joined the pension system $(x-x_{0})$ years ago. The total number of active member who make contributions at time $t$ is given by

and the total number of retired members who recieve the benefits at time $t$ is

Here, we assume that during the working period, each cohort receives his labor income, which is normalized to 1. Therefore the total contribution income $TC=\{TC(t):t\geq 0\}$ is

and the total benefit payment $TB=\{TB(t):t\geq 0\}$ is

where $c(t)$ and $b(t)$ is the amount of the contribution and benefit at time t, respectively. The specific forms about $c(t)$ and $b(t)$ will be given by the next section.

The surplus (deficit) between the total contribution income and the total benefit payment will increase (decrease) the accumulation of the pension fund. Meanwhile, the accumulation is dynamically allocated to a risk-free asset and a risky asset.

The pension fund trustee is responsible for the asset-liability management. The amount of money invested in the risky asset at time $t$ is denoted by $\pi=\{\pi(t),t\geq 0\}$ and then the amount invested in risk-free asset is $A(t)-\pi(t)$. The dynamics of the asset process $A(t)$ can be described by

where $a_{0}$ is the initial asset at time 0. Using (2.1) and (2.2), we can easily rewrite (2.7) as

Consider the target contribution $c$ and target benefit $b$ at time $0$ and a finite time interval $[0,T]$ with $T$ being the so-called terminal time. For each age cohort, denoted by $x$, the target liability equals the difference between the present value of risk-free benefits and the present value of the yet-to-be-paid risk-free contributions. This is in line with the one in the literature of Cui et al. (2011).

where $\tau$ is the target instantaneous growth rate of the target contribution and the target benefit.

At the aggregate level, the target liabilities of the fund can be calculated simply as the sum of the liabilities for each age cohort.

Given the sationary age composition of the fund and the fixed target benefit level, the target liability $L$ is time invariant. Actually, if the fund invests fully in the risk-free asset, then the actual liability follows exactly as the target liability $L$. If the fund accepts mismatch risk, for example by investing in stocks, there may be funding surpluses or deficits with respect to the target liability.

The fund surplus is defined as the diference between assets and the target liability level

Notice that the collective pension schemes may leave surpluses or deficits which lead to intergenerational transfers and hence to intergenerational risk sharing.

Besides the adjustment of the plan members’ traget liabilities as described in (2.10) by linking to the fund assets as decribed in (2.8), for the collective hybrid pension model, we also dynamically and simultaneously adjust both the current individual contribution paid by the active members and individual benefit payments payable to the retirees, implicitly influenced by the surplus level of the pension fund in order to make intergenerational risk sharing.

To achieve our goal mentioned above, we specify a simple cohort contribution $c(t)$ and benefit $b(t)$ policy, where the contribution per cohort is a function of the target contribution rate level $c\rm{e}^{\tau t}$ and the funding residual per active cohort $\frac{SP(t)}{NC(t)}$, and the benefit per cohort is a function of the target benefit rate level $b\rm{e}^{\tau t}$ and the funding residual per retire cohort $\frac{SP(t)}{NB(t)}$.

where $\lambda_{1}(t)=\alpha\frac{A(t)-L}{NC(t)}$, $\lambda_{2}(t)=\beta\frac{A(t)-L}{NB(t)}$ which represent the adjustments to the pension contributions and benefit payments, respectively. The slope coefficient $\alpha$ and $\beta$ change over time according to the adjustment of the hybrid pension system thus reflect the speed of absorbing the funding imbanlances, being the so-called spread parameters for contribution income and benefit payment, respectively. $\alpha=1$ or $\beta=1$ implies that a funding imbalance is immediately and fully abdorbed. More specifically, a fraction $\alpha$ of the funding residual is shared among employees and a fraction $\beta$ of the funding residual is shared among retirees. A lower vaule of $\alpha$ and $\beta$ imply that part of the funding surplus is shifted to the future, and shared across generations i.e. the lower vaule of $\alpha$ and $\beta$, the higher the degree of intergenerational risk sharing.

The two adjusted processes $\{c(t);t\geq 0\}$ and $\{b(t);t\geq 0\}$ describe the pension system over the time which consist of decision on contribution and benefit payment for each cohort at time $t$. When the underlying fund value at time t has surplus ($SP(t)>0$), indicating an out performance in assets investment, we expect both positive policy adjustment of $\lambda_{1}(t)$ and $\lambda_{2}(t)$ so that at time t the benefit payments would be increased while the contributions required being decreased. When the underlying fund incurs a deficit due to investment losses ($SP(t)<0$), we in turn expect both negative policy adjustment of $\lambda_{1}(t)$ and $\lambda_{2}(t)$. Here, increasing the contributions and decreasing the benefits can re-balanced the budget. All in all, high and low asset returns of the fund are spread over active members and retirees, which implies an intergenerational risk transfer or sharing. Such hybrid pension model has a similar mathematical structure as the hybrid model studied in Cui et al. (2011), Khorasanee (2012) and Wang and Lu (2019).

In the following section we present, for the pension model described above, we set a continuous-time robust stochastic optimal control problem for the hybrid pension plan.

In the traditional framework of pension plan or investment problem, the member is assumed to be ambiguity-neutral, which implies that the manager has complete confidence in the above model provided by probability measure $\mathbb{P}$. However, the fact is that in many cases the member cannot know exactly the true model, and thus any particular probability measure used to describe the model would lead to potential model misspecification. The parameters of financial markets are difficult to estimate with precision in practice and numerous experimental studies demonstrate that the investors are averse not only to risk (the known probability distribution) but also to ambiguity (the unknown probability distribution) (refer to Peter et al. (2010)). Therefore, we assume that the manager is ambiguity-averse and aims to protect himself against the worst-case scenario. The manager recognizes that the models under the reference probability $\mathbb{P}$ only approximate the true models and takes into account some alternative models, which can be defined via a family of probability measures equivalent to $\mathbb{P}$ as follows

Based on this assumption, we can use Girsanov’s theorem to change the probability measures, so that the models defined by the alternative measures only differ in terms of drift function. By Girsanov’s theorem, for each $\mathbb{Q}\in\mathscr{Q}$, there is a progressively measurable process $\phi(t),t\in[0,T]$, which can be referred as the probability distortion process, such that

where

Moreover, if $\phi(t),t\in[0,T]$ satisfies Novikov’s condition

then the Radon-Nikodym derivative process $\Lambda^{\phi}(t)$ is a $\mathbb{P}$-martingale.

According to Girsanov’s theorem, under the alternative measure $\mathbb{Q}$, the stochastic process $B^{\mathbb{Q}}(t)$ is one-dimensional standard Brownian motion, and we have

Hence, for the given measure $\mathbb{P}$, choosing an alternative measure $\mathbb{Q}$ is equivalent to determining the process $\phi$. Furthermore, the price of the stock and dynamic asset process under alternative measure $\mathbb{Q}$ become

Recall that the hybrid pension plan considered consists of both active members who make contributions to the fund and retired members who receive pension benefits. Since the contribution and benefit are time variant, the plan trustees must choose an asset allocation, contribution and benefit strategies which strike a reasonable compromise between the interests of these two groups. Ideally, the fund should make contributions and benefits in a stable and sustainable manner to each cohort as close to the target as possible. As the pension plan we considered is a collective one with different generations involved, taking the all generations into consideration, the goal of the pension trustees is that the pension plan can be run stably and sustainably and the investment risks are shared intergenerationally.

Mathematically, the objective of this continuous-time hybrid pension system is to seek a robust optimal strategy $\boldsymbol{\pi}$ to minimize the expected discounted cost of unstable contribution risk, unstable benefit risk and discontinuity risk over the time interval $[t,T]$. The model setting is in line with He et al. (2020) and Wang et al. (2018). In particular, the contribution and benefit should be stable and comparable with respect to target contibutionn and target benefit, and the final accumulation should be neither too large nor too small, comparing with the initial accumulation. As such, we propose a cost function that consists of three parts. The first part is the deviation between the actual contribution and the target contribution. The target contribution $ce^{\tau t}$ is a reference level, At any time $t\in[0,T]$, when $c(t)$ is ralatively larger (smaller) than $ce^{\tau t}$, the current working cohorts (cohorts at previous times) naturally feel unstatisfied because of bearing higher contribution burden, comparing to the cohorts at previous times (current working cohorts). In this cricumastance, there is welfare cost due to the unstable contribution risk. The second part is the deviation between the actual benefit and the target benefit. As same as the first part, $b(t)$ should aproach the $be^{\tau t}$ as much as possible in order to make sure that is fair to all the cohorts. If not, in this cricumastance, there is also welfare cost due to the unstable benefit risk. The third part is the initial accumulation to depict the cost of the two risks. When the final accumulation is smaller (larger) than the initial accumulation, there exist intergenerational transfer from the current working cohorts (cohorts at other times) to the cohorts at other times (current working cohorts), In this circumstance, there is welfare cost due to the discontinuous accumulation risk. Hence, the objective is to minimize the deviations within the finite time horizon $[t,T]$. In the literature Josa-Fombellida et al. (2001), Ngwira and Gerrard (2007) and Wang et al. (2018), both the finite and infinite time horizon are studied, and the finite time horizon vary from years to decades. In this paper, we assumed a reasonable finite time horizon because that the pension manager pays more attention to the interests of the working cohorts in their term of mangement, which means that they do not take the interests of the unborn into consideration. Further reasons about finite time horizon can be found in He et al. (2020). Employing the quadratic deviations as the cost function, We formulate the robust optimal investment problem as follows:

where $\Pi$ is a set of all the admissible strageies of $(\pi(t),\lambda_{1}(t),\lambda_{2}(t))$. $\gamma_{1}$,$\gamma_{2}$ and $\gamma_{3}$ are nonnegative constants, interpreted as the weight parameter which measures the importance of the cost of unstable contribution risk, the cost of the unstable benefit risk and the cost of the discontinuous risk in the overall cost function, respectively. The expectation is computed under the alternative measure $\mathbb{Q}$ defined by $\phi$. The last part in (3.4) is the penalty term depending on the relative entropy which measures the discrepancy between the alternative measure and the reference measure, and the increase in the relative entropy from $t$ to $t+\textrm{d}t$ equals $\frac{1}{2}\phi^{2}\textrm{d}t$. This integral term is scaled by $\Psi(t)$ which is nonnegative and stand for the preference parameters for ambiguity aversion, measuring the degree of confidence to the reference probability $\mathbb{P}$. The larger $\Psi(t)$ is, the more ambiguity-averse the insurer is, the smaller the penalty for deviating from the reference measure is, and the less faith in the reference measure the manager has. Finally, when $\Psi=0$, the manager is completely convinced that the true measure is the reference measure $\mathbb{P}$. In this case problem (3.4) degenerates to the classical the quadratic deviations minization in the absence of ambiguity. When $\Psi=\infty$, the penalty term vanishes, which implies that the member is extremely ambiguous (see Uppal and Wang (2003); Yi et al. (2014);Wang and Li (2018)). And for analytical tractability, we assume that $\Psi(t)=k/V$, where $k$ is positive constants and stand for the ambiguity aversion parameter, meaning manager’s the different levels of ambiguity aversion about the stock’s price.

Thus, the continuous-time robust stochastic optimal control problem for the hybrid pension plan is established. By choosing strategies, the manager seeks to minimize the penalized objective function in the worst-case scenario. In the section below we aim at finding the optimal policy which solves the optimal control problem.

In this section, we use standard methods to solve the optimal control problem (3.4) and derive closed-form expressions of the optimal policy, denoted by $(\phi^{*},\pi^{*},\lambda_{1}^{*},\lambda_{2}^{*})$, within all the admissible policies.

First, we define a variational operator $\mathscr{A}^{\boldsymbol{\pi},\phi}$

where $V_{t}$, $V_{a}$, $V_{aa}$ are partial derivatives of $V(t,a)$.

According to the principle of stochastic dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation can be derived as(see Uppal and Wang (2003); Yi et al. (2014)):

Based on the above setting, we derive a solution to the HJB equation (3.6) with $\Psi(t)=k/V$ . We state our findings on the optimal asset allocation, contribution and benefit adjustment policies for robust optimal control problem (3.4) in the following theorem. For notation simplicity, we let $\varphi=(\mu-r)/\sigma$, which is the Sharpe ratio of the risky asset.

Substituting (3.20) into (3.18), the optimal policies (control variables) can be expressed in terms of the functions $P(t)$, $Q(t)$.We have

The coefficients of $a^{2}$, $a$ are all zeros, which leads to the following system of differential equations

Differential equations (3.22)with boundary conditions in (3.19) can be easily solved. Denote $g_{3}=\frac{\varphi^{2}}{2k-1}$, we have

where

Above, we have derived closed-form expressions for the optimal investment strategy and optimal risk-sharing arrangemnts taking into account the three competing objectives of contibution rate stabilty and benefit stability and continous intergenerational risk-sharing.

Then, we will interpret the optimal controls presented in (3.8), (3.9) and (3.10), providing economic intuition for some of their more surprising behavior.

First, $Q(t)$ in (3.22) can be rewrite in actuarial notation as

where $\overline{a}_{\overline{T-t|}r}$ is the present value of an annuity certain payable continuously for the remainder of the horizon, valued at a force of interest $r$. Besides, $g_{2}$ in (3.12) represent the difference between the target total contribution and target total benefit. Therefore the first term in $Q(t)$ is the present value at time $t$ of difference between the target total contribution and target total benefit during the remaining horizon, and the second is the accumulated asset at time $t$ for the initial fund. Now, we can have next remark.

Next, moving our focus to the optimal risk-sharing adjustment policy. Like $Q(t)$, $P(t)$ can also be rewrite in actuarial notation as

where $\gamma=2r+g_{3}$ and $\overline{a}_{\overline{T-t|}\gamma}$ is the present value of an annuity certain payable continuously for the remainder of the horizon, valued at a force of interest $\gamma$. Therefore

where the denominator represent the present value of a continuous investment of $(1+\frac{\gamma_{1}}{\gamma_{2}}(\frac{NB(t)}{NC(t)})^{2})$ at time $t$ for $T-t$ years, valued at the adjusted rate $\gamma$, while leaving a lump sum at the end of $T-t$ years equal to $\frac{\gamma_{1}}{\gamma_{3}NC(t)^{2}}$, valued at the adjusted rate $\gamma-r$. And $\frac{\gamma_{3}}{\gamma_{2}}NB(t)^{2}P(t)$ can be showed as same way.