2.1. Longevity risk

The aging population is driven by two demographic factors: declining fertility rates and increasing life expectancy. We set n(h) to be the instantaneous density of the newborn cohort at time h, which drops over time to represent declining fertility rates.

The longevity risk can be classified into systematic and nonsystematic (Boon et al., Reference Boon, Brière and Werker2022). Here we focus on the systematic longevity risk. To describe the systematic longevity risk, we introduce the assumed survival function $\bar{S}(x,h)$ to be the assumed probability that a member of cohort h survives to age x. We denote the random death time of cohort h as k(h), and then we have $\mathbb{P}(k(h)>x)=\bar{S}(x,h)$ . The values of $\bar{S}(x,h)$ come from an initial life table. When signing a pension annuity contract, the insurer determines the value of the annuity according to $\bar{S}(x,h)$ .

The real survival function is denoted by S(x, h) and is nonnegative. The real survival function often exceeds the insurer’s expectation due to the systematic longevity risk, which will bring losses to the insurer. The difference between the real survival function S(x, h) and the hypothetical survival function $\bar{S}(x,h)$ constitutes the systematic longevity risk. It holds that $S(0,h)=1$ and $S(\omega,h)=0$ , where $\omega$ is a sufficient large constant to be an upper bound lifespan for all cohorts h considered in this model. For fixed h, S(x, h) is decreasing in x, and for fixed x, S(x, h) is increasing in h. It follows that the life expectancy of cohort h is nondecreasing which can be calculated by integrating the survival function with respect to age x from 0 to $\omega$ .

Assume all the members join the TBP at age a. Let the retirees’ minimum age (also the active members’ maximum age) at time t be r(t). Then the density of those who reach age x at time t is $n(t-x)S(x,t-x)$ . Then the expected total number of active members at time t is given by

\begin{equation*}A(t)=\int_{a}^{r(t)} n(t-x)S(x,t-x){\rm d}x.\end{equation*}

Similarly, the expected total number of retired members at time t is

\begin{equation*}R(t)=\int_{r(t)}^{\omega} n(t-x)S(x,t-x){\rm d}x.\end{equation*}

The old-age dependency ratio which describes the average number of elderly people to support per working population is thus

\begin{equation*}D(t)=\frac{R(t)}{A(t)}.\end{equation*}

Generally speaking, the aging population is accompanied by the increase of the dependency ratio.

2.2. Postponing the retirement age

By assumption, the TBP fund manager decides to implement a postponing retirement policy. Before time $t=0$ , active members retire at the age of $r_0$ . Then at time $t=0$ , a postponing retirement policy is carried out that the statutory retirement age is set to $r_1$ , which means active members need to work until they are $r_1$ years old. Those who have retired before $t=0$ can continue to enjoy their retirement life, even if part of them do not reach the new retirement age $r_1$ . After $t=0$ , active members are supposed to work until they reach the new retirement age $r_1$ . For example, if we set the initial retirement age $r_0$ to 55 and the new retirement age $r_1$ to 60, then at time 0, the minimum age of retirees is still 55, while at time $t=1$ , the retirees’ minimum age becomes 56. More specifically, at time $t=1$ , except those who died, the retired members are the same as a year ago, because the increase of the statutory retirement age causes that there are no new retirees this year. However, when $t\ge5$ , new retirees begin to appear, then the retirees’ minimum age will be fixed at age 60. To conclude, when $t\le r_1-r_0$ , the retirees’ minimum age is $r_0+t$ and the active members’ ages range from a to $r_0+t$ . For $t>r_1-r_0$ , the retirees’ minimum age (also the active members’ maximum age) will always be $r_1$ . We now introduce r(t) to be the retirees’ minimum age at time t, which is subtly different from the statutory retirement age. It holds that

\begin{equation*}r(t)=\left\{\begin{aligned}&r_0,&t\le 0,\\[5pt] &r_0+t,&0<t\le r_1-r_0,\\[5pt] &r_1,&t>r_1-r_0.\end{aligned}\right.\end{equation*}

2.3. Plan provisions

Consider a plan consisting of active and retirement members. While the active members make contributions to the pension fund, the retired members receive benefits paid from the pension fund.

Denote by Y(x, h) the annual salary rate of an active member aged x of cohort h. Then each active member aged x contributes $cY(x,t-x)$ to the pension fund at time t, where c is the contribution ratio related to the salary. Thus, the aggregate instantaneous contribution rate of all active members at time t is given by

\begin{equation*}C(t)=\int_a^{r(t)} n(t-x)S(r(t),t-x)cY(x,t-x){\rm d}x.\end{equation*}

The reason why we use $S(r(t),t-x)$ rather than $S(x,t-x)$ here is that for those who pass away before retirement, their contributions are refunded. When a member joins the pension fund, she will continue to make contributions to her account. At the moment, her account is part of the pension fund, and the manager will invest it. If she dies before retirement, the pension fund will return all the accumulated contributions in her account. We do not care how the manager invests the funds in her account, because no matter what the investment is, when she dies before retirement, the funds in her account will be withdrawn from the pension fund. Therefore, we only consider the contribution of people who live to retirement age.

When the retirees reach their retirement age $r_0$ or $r_1$ , they begin to receive pension benefits. Let b(t, h) be the actual benefit at time t for retirees of cohort h, which is given by

(2.1) \begin{equation}b(t,h)=\bar{b}(h)+\theta(t),\end{equation}

where $\bar{b}(h)$ is the target annuity for the cohort h, and $\theta(t)$ reflects the risk sharing at time t, including investment risks and longevity risks. Their expressions are discussed in the next two subsections.

2.4. Financial market

Let $(\Omega,\mathscr{F},\mathbb{P})$ be a complete probability space, where $\mathbb{P}$ is the real-world probability measure on $\Omega$ and $\mathscr{F}=\{\mathscr{F}_t\}_{t\ge 0}$ is a complete and right continuous filtration generated by a standard Brownian motion.

We assume that there are two underlying assets available to the pension trustees: one risk-free asset (a bank account) and one risky asset (a stock). The evolution of the value of the risk-free asset $S_1(t)$ is given by

(2.2) \begin{equation}{\rm d}S_1(t)=m(t)S_1(t){\rm d}t, \ \ \ \ \ t\ge 0,\end{equation}

where m(t) represents the risk-free interest rate.

The price process of the risky asset is denoted by $S_2(t)$ , and $S_2(t)$ is supposed to follow a stochastic differential equation of the form

(2.3) \begin{equation}{\rm d}S_2(t)=S_2(t)(\mu(t) {\rm d}t+\sigma(t) {\rm d}W(t)),\ \ \ \ \ t\ge 0,\end{equation}

where $\mu(t)$ is the appreciation rate of the stock and $\sigma(t)$ is the volatility rate, and W(t) is a standard Brownian motion. To exclude arbitrage opportunities, we assume that $\mu(t)>m(t)$ .

2.5. Target benefit: intergenerational equity

The pension plan provides members with a life annuity payable continuously commencing at retirement age $r_0$ or $r_1$ and has a pre-set retirement benefit target $\bar{b}(h)$ for retired members of cohort h (which means they are born at time h). The target benefit is assumed as a fixed annuity, which means retirees of the cohort h get a fixed annuity $\bar{b}(h)$ until they die. The value of $\bar{b}(h)$ is based on the principle of actuarial equity, that is, the expected future value of accumulated contributions at the retirement date is equal to the present value at that time of the expected total benefits to be received in retirement. Since we only consider people who can live to retirement age, we introduce the conditional probability that an individual aged z at time $h+z$ can live to x years old here, which is denoted by

\begin{equation*}\bar{S}(x,z,h)=\mathbb{P}(k(h)>x|k(h)>z)=\frac{\bar{S}(x,h)}{\bar{S}(z,h)}.\end{equation*}

When the pension contract is generated, the basis of the insurer is the assumed conditional survival function $\bar{S}(x,z,h)$ rather than the real survival function S(x, h).

For the cohort h with $h\ge -r_0$ (thus their retirement age is deferred), they encounter a delay in retirement and the value of $\bar{b}(h)$ should conform to actuarial equity, that is,

\begin{equation*}\int_a^{r_1}e^{\int_x^{r_1}m(s){\rm d}s}cY(x,h){\rm d}x=\int_{r_1}^{\omega}e^{\int_{r_1}^x-m(s){\rm d}s}\bar{b}(h)\bar{S}(x,r_1,h){\rm d}x.\end{equation*}

The left of the equal sign is the value of accumulated contributions at retirement $r_1$ , and the right is the discounted present value of the expected overall annuity income at $r_1$ .

So the target fixed annuity is

(2.4) \begin{equation}\bar{b}(h)=\frac{\int_a^{r_1}e^{\int_x^{r_1}m(s){\rm d}s}cY(x,h){\rm d}x}{\int_{r_1}^{\omega}e^{\int_{r_1}^x-m(s){\rm d}s}\bar{S}(x,r_1,h){\rm d}x},\end{equation}

when $h\ge -r_0$ .

Similarly, for $h< -r_0$ , we have

\begin{equation*}\bar{b}(h)=\frac{\int_a^{r_0}e^{\int_x^{r_0}m(s){\rm d}s}cY(x,h){\rm d}x}{\int_{r_0}^{\omega}e^{\int_{r_0}^x-m(s){\rm d}s}\bar{S}(x,r_0,h){\rm d}x}.\end{equation*}

In this way, even if the retirement is delayed, the interests of the policyholders can be guaranteed. In addition, this form of $\bar{b}(h)$ can also correspond well to the change of longevity trends ( $\bar{S}(x,r_0,h)$ ) and the external environment (Y(x, h)).

From the perspective of the insurer, the target aggregate payment rate $\bar{B}(t)$ for all retirees at time t is given by

\begin{equation*}\bar{B}(t)=\int_{r(t)}^{\omega} n(t-x)S(x,t-x)\bar{b}(t-x){\rm d}x.\end{equation*}

Since the target benefit $\bar{b}(h)$ of each cohort h is based on actuarial equity, we say that this target aggregate payment $\bar{B}(t)$ reflects intergenerational equity.

2.6. Actual benefit: intergenerational risk sharing

The actual aggregate payment rate is B(t) for all retirees at time t, which is a strategy determined by the plan trustees and thus is affected by market fluctuations. Since B(t) is affected by the return from the financial market, it contains investment risks. The difference between B(t) and $\bar{B}(t)$ is $B(t)-\bar{B}(t)$ which actually reflects risks (including longevity risks and investment risks). The adjustment is denoted by $\theta(t)$ , which shares the risks among all retirees. We have

(2.5) \begin{equation}\theta(t)=\frac{B(t)-\bar{B}(t)}{R(t)}.\end{equation}

The value of $\bar{B}(t)$ and R(t) is affected by longevity risks. We aim to achieve this effect: when $ B(t)-\bar{B}(t)< 0$ , the trend of longevity makes more people share the loss; when $ B(t)-\bar{B}(t)> 0$ , the trend of longevity makes more people share the benefits. Therefore, $\theta$ shares the longevity risk and the investment risk.

Since $\theta(t)$ is independent of cohort h, the value of risk sharing is the same for all cohorts, and it reflects that the risk is shared among all generations.

2.7. Terminal wealth: future security

Let $\{F(t), t \in [0, T]\}$ be the wealth process of the pension fund, where $T > 0$ is the terminal time point. The target terminal wealth is a value that F(T) is supposed to get close and is denoted as M, which contains two components: $M=M_1+M_2$ . To keep the fund size from shrinking, $M_1$ is set to $F(0)e^{\int_0^T m(s){\rm d}s}$ , which is the future value at time T of the initial wealth F(0). If F(T) is close to $M_1$ , it may imply that the pension fund operates stably and is prepared for the next cycle.

Let $M_2$ be the capital reserve which can ensure that the pension is supported for at least $\tau$ years. It is the accumulation of the present discounted value of the difference between income and target expenditure in the next $\tau$ years at time T:

\begin{equation*}M_2=\int_T^{T+\tau}e^{-\int_T^s m(u){\rm d}u}(\bar{B}(s)-C(s)){\rm d}s.\end{equation*}

If the terminal wealth F(T) reaches $M_2$ , the pension fund will last for at least $\tau$ years with risk-free investments only. This provides a guarantee for the pension payment in the next $\tau$ years.

Therefore M is used to achieve the following two goals: to maintain the fund pool size and to leave enough funds to run for $\tau$ years.

3.1. Objective function

Assume that the plan trustees can invest in both the risk-free and risky assets and use the fund to pay retirement benefits. Let $f_0$ denote the initial wealth of this fund, $\pi(t)$ denote the amount that the plan manager invests in the risky asset at time t, and F(t) denote the wealth of the pension fund at time t after adopting the investment strategy $\pi(t)$ . Then the fund’s value follows the dynamics:

(3.1) \begin{equation}\left\{\begin{aligned}&{\rm d}F(t)=\pi(t)\frac{{\rm d}S_2(t)}{S_2(t)}+(F(t)-\pi(t))\frac{{\rm d}S_1(t)}{S_1(t)}+[C(t)-B(t)] {\rm d}t,\\[5pt] &F(0)=f_0.\end{aligned}\right.\end{equation}

Using (2.2) and (2.3), we can easily rewrite (3.1) as

(3.2) \begin{equation}\left\{\begin{aligned}&{\rm d}F(t)=\pi(t)(\mu(t){\rm d}t+\sigma(t){\rm d}W(t))+(F(t)-\pi(t))(m(t){\rm d}t)+[C(t)-B(t)]{\rm d}t,\\[5pt] &F(0)=f_0.\end{aligned}\right.\end{equation}

We now define $\eta=\{(\pi(s),B(s))\}_{s\in[t,T]}$ to be a strategy adopted by the plan trustees between time t and T, where $T>0$ is the terminal time point. This strategy includes the investment strategy $\pi(s)$ which is the amount to be invested in the risky asset at time s and the pension benefit B(s) which is the amount of annuity for the retirees at time s, for $s\in [t,T]$ . The definition for a strategy being an $Admissible\ Strategy$ with respect to SDE (3.2) is provided below.

We now turn to the objective function of the problem. The composition of the objective function mainly considers the following three aspects. First, in order to reduce the general deviation between the actual benefit and the target benefit, we integrate the squared deviation between B(t) and $\bar{B}(t)$ with respect to time t, and if this term is smaller, B(t) gets closer to $\bar{B}(t)$ . Next, under the precondition that the actual and target benefit are close, the deviations are expected to be positive, that is, it will be better if the actual benefits are higher than target benefits. Finally, the actual terminal value of the fund F(T) is supposed to approach M to ensure the security of the pension fund. Combined with the three factors above, then the objective function J(t, f) at time t with the fund value f is defined as

(3.3) \begin{align}J(t,f)={\rm E}_{\pi,B}\bigg\{\int_t^T&[(B(s)-\bar{B}(s))^2-\lambda_1(B(s)-\bar{B}(s))]{\rm d}s\\[5pt] \nonumber&+\lambda_2(F(T)-M)^2|F(t)=f\bigg\},\end{align}

where both $\lambda_1$ and $\lambda_2$ are nonnegative constants, interpreted as penalty weights given to the negative integration of deviation between B(s) and $\bar{B}(s)$ and the achievement of the terminal fund target, respectively. The expectations in (3.3) are conditional on the state of f at time t.

The value function is defined as

(3.4) \begin{equation}V(t,f)\;:\!=\;\min_{\eta \in \Pi} J(t,f),\ \ \ \ \ t>0,\end{equation}

where $\Pi$ is a set of all the admissible strategies of $\eta$ .

3.2. Optimal retirement age

Under the certain condition of the aging population, we expect to obtain different values of $V(0,f_0)$ with different deferred retirement ages $r_1$ . In Section 3.1, we find that the smaller value of $V(0,f_0)$ means the benefit B(t) and the terminal wealth F(T) are more close to their targets $\bar{B}(t)$ and M, and the overpayment $B(t)-\bar{B}(t)$ is larger. Thus, our purpose is to determine an optimal deferred retirement age to minimize the value function $V(0,f_0)$ .

If the new retirement age $r_1$ can minimize $V(0,f_0)$ , we denote it as the optimal retirement age $r^*$ :

(3.5) \begin{equation}r^*=\inf\{\underset{r_1\in[r_0,\omega]}{\arg \inf}\; V(0,f_0)\}.\end{equation}

The deferred retirement ages $r_1$ which make $V(0,f_0)$ reach its minimum value may not be unique, so we select the smallest possible $r_1$ to be $r^*$ .

5.1. Assumptions

Now we make two assumptions that allow for numerical solutions with respect to the changing demographic structure. First, we assume that the number of people entering the plan per unit time evolves in an exponential manner, that is:

\begin{equation*}n(h)=N\exp\!(\!-\!w(h-t_0)\chi_{\{h\ge t_0\}}),\end{equation*}

where w is the decreasing rate, $t_0=-80$ is the time point when the fertility decline begins, and N is the initial pension entry rate at time $t=t_0$ . This setting is similar to Knell (Reference Knell2018). Note that the fertility rate begins to decrease at time $t_0$ rather than 0. We begin to observe the behavior of the pension fund at $t=0$ , while the demographic structure may have changed for a long time.

The second assumption is that the survival function S(x, h) satisfies the Gompertz-Makeham law, which goes back to Gompertz (Reference Gompertz1825). Milevsky (Reference Milevsky2020) summarizes a Gompertz-Makeham calculation model with the compensation law from the life data of various countries. This enlightens us that we can use the mortality rate of low life span countries evolving to that of high life span countries with generation to describe the longevity trend. Now we introduce the mortality rate q(x, h):

(5.1) \begin{equation}q(x,h)=\left\{\begin{aligned}&\rho+\frac{1}{\beta(h)}e^{\frac{x-\alpha(h)}{\beta(h)}},& x\in [0,x^*],\\[5pt] &\rho+e^{L},& x\in (x^*,\omega],\end{aligned}\right.\end{equation}

where $\rho$ is a nonage-dependent accidental death rate (Makeham constant), and $x^*$ , L, $\alpha$ , and $\beta$ are Gompertz parameters. In Milevsky (Reference Milevsky2020), it is observed that $\rho+e^{L}$ is an upper limit of the mortality rate, which is reached by all countries near the age of $x^*$ , and $x^*$ and L are suggested to be set to 100 and −1, respectively. This setting is followed in this paper. Note that q(x, h) has an argument h, which means the mortality rate changes with generation. We assume $\alpha$ and $\beta$ involves with the cohort h to simulate the longevity trend, thus they are denoted as $\alpha(h)$ and $\beta(h)$ . If we adopt an equivalent form of q(x), that is, $q(x)=\rho+\kappa e^{\varsigma x}$ , $x\in [0,x^*]$ , where $\kappa=(1/\beta)e^{-\frac{\alpha}{\beta}}$ and $\varsigma=1/\beta$ , then the compensation law means a linear relationship between $\kappa$ and $\varsigma$ , that is, $ln[\kappa]=L-x^*\varsigma$ . We can understand the compensation law as countries with a relatively low initial natural mortality rate $\kappa$ tend to have a higher mortality growth rate $\varsigma$ and vice versa. This consequently forces a relationship between $\alpha(h)$ and $\beta(h)$ :

\begin{equation*}\alpha(h)=x^*-\beta(h)(L+ln(\beta(h))).\end{equation*}

Next, we assume that

\begin{equation*}\beta(h)=\beta_0-d(h-t_0)\chi_{\{h\ge t_0\}},\end{equation*}

where d is the parameter that controls the speed of the longevity trend, and $t_0=-80$ is the time point when the longevity trend begins. Milevsky (Reference Milevsky2020) provides a table of Gompertz-Makeham parameters around the world, which implies the smaller the value of $\beta$ , the longer the lifespan of people. The initial value $\beta_0$ of cohort $t_0$ is set to 14, which is similar with that of Belarus ( $\beta=14.5$ ). If $d=0.05$ (the maximum value of d in the following numerical analysis), after 80 years from $h=t_0$ , we have $\beta(0)=10$ (the value of $\beta$ is 9.73 for Japan), which means the lifespan increases with the cohort h. Thus, we establish a way to reflect the change of the mortality rate with cohorts.

The survival function S(x, h) has the following form:

\begin{equation*}S(x,h)=\exp\!\left(-\int_0^{x}q(s,h){\rm d}s\right).\end{equation*}

And the equation can be calculated as follows by (5.1):

\begin{equation*}S(x,h)=\left\{\begin{aligned}&\rho x+e^{-\frac{\alpha(h)}{\beta(h)}}(e^{\frac{x}{\beta(h)}}-1),& x\in [0,x^*],\\[5pt] &\rho x^*+e^{-\frac{\alpha(h)}{\beta(h)}}(e^{\frac{x^*}{\beta(h)}}-1)+(\rho+e^{L})(x-x^*),& x\in (x^*,\omega].\\[5pt] \end{aligned}\right.\end{equation*}

The assumed survival function $\bar{S}(x,h)$ has the same expression with S(x, h) and the longevity speed is relatively $\bar{d}$ .

We also make the following assumptions for our numerical illustration.

• • The time horizon is 20 years, that is, $T=20$ .

• • Members enter the TBP at age 25 and retire at age 55 initially, that is, $a=25,\ r_0=55$ .

• • The maximum age for all cohorts considered in this model is 130, that is, $\omega=130$ .

• • The initial pension entry rate N at time $t=t_0$ is 10.

• • The Makeham constant $\rho$ is set to $2.66\times 10^{-4}$ , which is an average constant from Milevsky (Reference Milevsky2020).

• • The parameters in the two asset models (2.2) and (2.3) are $m(t)=m=0.01$ , $\mu(t)=\mu=0.05$ , $\sigma(t)=\sigma=0.15$ (thus the Sharpe ratio of the risky asset is about 0.27).

• • The salary rate Y(x, h) increases in accordance with the risk-free interest in order to account for inflation, that is, $Y(x,h)=e^{0.01(x+h)}$ .

• • The penalty weight $\lambda_1$ given to the benefit shortfall risk is chosen to be 8, and the weight $\lambda_2$ given to the risk at the terminal time is chosen to be 0.1. The estimation is given in Section 5.6.

• • The initial fund wealth F(0) is set to 100, and the contribution rate c of each active member is set to 0.1.

• • The reserved preparation time $\tau$ is 5 years.

5.2. Demographic transition

The dependency ratio D(t) is the quotient of the number of retirees R(t) and the number of workers A(t), which can reflect the degree of the social pension burden. Figure 1(a) shows that the longevity trend continuously increases D(t) with time t and D(t) grows faster with a larger longevity speed d. Figure 1(b) shows that the fertility decline also increases D(t) with time t and when the decline speed w gets larger, D(t) will rise.

Figure 1. Factors affecting D(t).

5.3. Intergenerational equity

Figure 2 shows the impact of the assumed longevity speed $\bar{d}$ and the deferred retirement age $r_1$ on the individual target annuity $\bar{b}(h)$ . In Figure 2(a), before $t_0=-80$ (the time point when the longevity trend begins), $\bar{b}(h)$ increases with Y(x, h) and stays the same for all $\bar{d}$ because the longevity trend has not started yet. After $t_0=-80$ , people’s life expectancy has increased and thus $\bar{b}(h)$ decreases with h according to actuarial equity. Besides, if we delay the retirement age, then people will work longer years and their remaining life expectancy will also be reduced, so according to (2.4), their annuity $\bar{b}(h)$ will increase. In Figure 2(b), it can be found that if we delay retirement for a longer time, the target annuity $\bar{b}(h)$ will also increase.

Figure 2. Factors affecting $\bar{b}(h)$ .

Figure 3 shows the impact of the longevity speed d, the deferred retirement age $r_1$ , and the fertility decline speed w on aggregate target benefit $\bar{B}(t)$ . In Figure 3(a), we assume $\bar{d}=0$ , $r_1=r_0$ , and $w=0$ and find that as the longevity trend becomes serious, $\bar{B}(t)$ increases, which is consistent with common sense. Figure 3(b) focuses on the change of new retirement age $r_1$ with $d=0.03$ and $w=0$ . We find that delaying retirement can significantly reduce $\bar{B}(t)$ within $r_1-r_0$ years. This is because if we set the new retirement age be $r_1$ at time $t=0$ , then there will be no new retirees in the next $r_1-r_0$ years. Meanwhile the number of original retirees will decrease with age. So the number of retirees R(t) decreases which lowers $\bar{B}(t)$ . In addition, we also find that the later the delayed retirement, the faster the growth rate of $\bar{B}(t)$ after $r_1-r_0$ . This is because the delay in retirement increases the personal annuity $\bar{b}(h)$ (for $h\ge -r_0$ ), and the cohorts affected by delaying retirement will begin to receive the pension at time $r_1-r_0$ . Figure 3(c) shows that the decline in fertility will reduce the target payment $\bar{B}(t)$ because fewer people enter the pension plan.

Figure 3. Factors affecting $\bar{B}(t)$ .

5.4. Intergenerational risk sharing

In Figure 4, we study the value of risk sharing $\theta(t)$ at time T with different levels of overpayment $B(T)-\bar{B}(T)$ . We assume that $\bar{d}=d$ , $w=0$ , $r_1=r_0$ . When $B-\bar{B}<0$ , which means the actual payment does not reach the target payment, we have $\theta<0$ according to (2.5) and the loss will be borne by all retirees. As d increases, $\theta$ increases, which implies the loss shared by everyone decreases with the trend of longevity. When $B-\bar{B}>0$ , which means the adjustment $\theta$ is positive and in addition to the target annuity $\bar{b}$ , each retiree also receives the income from the insurer’s investment. Under this condition, $\theta$ decreases as d increases, implying that the longevity trend also thins everyone’s extra benefit. The phenomena reflected in these two figures verify the effectiveness of the risk-sharing mechanism.

Figure 4. The value of risk sharing $\theta$ for different longevity speeds d.

We next study the behaviors of the optimal investment and benefit strategies $\pi^*(t)$ and $B^*(t)$ by Monte Carlo methods. Figure 5 shows three investment paths $\pi^*(t)$ and three benefit paths $B^*(t)$ with $r_1=60$ , $w=0.003$ , $\bar{d}=0.02$ , and $d=0.05$ . The interval between adjacent time points is 0.1. The paths are the 25th, 50th, and 75th percentiles of the investment and the benefit payment at each date in time (calculated from the 10,000 simulations). Note that $\pi^*(t)$ shows a downward trend on the whole. This is because in the early investment, the discounted future income is much higher than the current pension fund which implies a high leverage strategy. Besides, it can be found that the actual payment $B^*(t)$ fluctuates up and down around $\bar{B}(t)+\frac{\lambda_1}{2}$ , and it can be explained by (4.14).

Figure 5. The 25th, 50th, and 75th percentiles of investment and benefit strategies.

5.5. Effects of postponing retirement

In Figure 6, we compare $V(0,f_0)$ under different $r_1$ and d with $\bar{d}=0$ and $w=0.006$ . By definition, a smaller V implies that the pension fund gets closer to the targets and can afford more benefit payments. We find that all functions of V experience a decline and an increase w.r.t. $r_1$ and the minimum point is the optimal deferred retirement age $r^*$ . The decline is because the pension burden is reduced for at least $r_1-r_0$ years, and this period gives an opportunity to try more risky and profitable investments. The increase is because the loss of the pension fund in the trend of longevity is caused by the underestimation of the survival function, while due to actuarial fairness, deferred retirement does not really reduce the amount of benefits, but delays the time of benefits. Therefore, appropriately delaying the retirement age is good for the pension fund, but too much delay will bring losses. Note that the value of V is smaller with a larger d. This does not mean that a larger d will be better, because when changing d, the scale of the pension fund has changed, and more benefits $B^*(t)$ are brought by more retirees, which will not be better for individuals. (Note that the adjustment $\theta(t)$ will be diluted by the number of retirees R(t).) For example, even if $\theta(t)$ is the same, the total value V of the pension scheme with more retirees will be larger than that with fewer retirees, which does not mean that the former is better than the latter. Besides, we find that the distribution of the optimal retirement age $r^*$ with respect to the speed of longevity d is relatively concentrated ( $r^*=61$ or 62).

Figure 6. The total value V for different retirement ages $r_1$ and longevity speeds d.

Figure 7 shows the values of $V(0,f_0)$ under different $r_1$ and w with $\bar{d}=0$ and $d=0.05$ . We find that the optimal deferred retirement age $r^*$ (the minimum point of each curve) increases with the fertility decline rate w. Note that fertility is essentially the rate of expansion of the pension plan. This may explain the importance of scale for the pension plan. A pension plan that expands faster will have less demand for delaying retirement.

Figure 7. The total value V for different retirement ages $r_1$ and fertility decline rates w.

5.6. The sensitivity of $\lambda_1$ and $\lambda_2$

We first estimate the values of $\lambda_1$ and $\lambda_2$ . Basically, we have $\lambda_1>0$ and $\lambda_2>0$ by definition. Besides, we assume that the overpayment $B^*(t)-\bar{B}(t)$ is bounded, that is, $B^*(t)-\bar{B}(t)\in [\mathscr{I}_{\min}, \mathscr{I}_{\max}]$ , where $\mathscr{I}_{\min}$ and $\mathscr{I}_{\max}$ are lower and upper bounds determined by the plan trustees. Note that $P(t)\in[0,\lambda_2]$ . To avoid borrowing and short selling, the risky investment $\pi^*(t)$ should be larger than 0 and smaller than F(t), although this is not considered in the solution (see Remark 3.1). Since we set the terminal wealth target F(T) to M (abbreviated as M), $\pi^*(t)$ can be assumed to be smaller than M. By (4.14) and assuming that $\pi^*(t)\in[0,M]$ , we can roughly estimate the range of $B^*(t)-\bar{B}(t)$ as $[\frac{\lambda_1}{2}+\frac{\sigma^2}{m-\mu}M\lambda_2,\frac{\lambda_1}{2}]$ . Therefore, we rearrange the restrictions of $\lambda_1$ and $\lambda_2$ as the following linear programming problem:

\begin{equation*}\left\{\begin{aligned}&\lambda_1>0,\\[5pt] &\lambda_1<2\mathscr{I}_{\max}\\[5pt] &\lambda_2>0,\\[5pt] &\frac{\lambda_1}{2}+\frac{\sigma^2}{m-\mu}M\lambda_2>\mathscr{I}_{\min}.\end{aligned}\right.\end{equation*}

The values of $\mathscr{I}_{\min}$ and $\mathscr{I}_{\max}$ are set to $-6$ and 6, respectively. Then we take $(\lambda_1,\lambda_2)\in [8,12]\times [0.06,0.1]$ satisfying above restrictions for the numerical analysis.

Figure 8 shows the values of the optimal investment $\pi^*(t)$ and the optimal benefit $B^*(t)$ at time $t=10$ with different $\lambda_1$ and $\lambda_2$ , where $F(10)=100$ , $w=0.003$ , $\bar{d}=0.02$ , and $d=0.05$ . We find that when $\lambda_1$ and $\lambda_2$ increase, the optimal investment $\pi^*(10)$ also increases. This is because the rising $\lambda_1$ and $\lambda_2$ bring more pressure on the pension fund to pay and to accumulate terminal wealth, and the investment strategy tends to be more risky. It can be found in Figure 8(b) that the optimal benefit $B^*(10)$ increases with $\lambda_1$ and decreases with $\lambda_2$ . This is because the pension fund tends to overpay with a large $\lambda_1$ , and a large $\lambda_2$ means the pension fund reduces benefits to ensure sufficient terminal wealth.

Figure 8. Strategies response to $\lambda_1$ and $\lambda_2$ at $t=10$ .