2. Model formulation

In this section, we establish the longevity risk transfer problem for an insurer, who constructs a life annuity plan. Let $(\Omega, \mathcal{F}, \mathbb{P})\,:\!=\,(\Omega_{\tau}\times\Omega_{\Psi},\mathcal{F}_{\tau}\otimes\mathcal{F}_{\Psi},\mathbb{P}_{\tau}\otimes\mathbb{P}_{\Psi})$ be a complete, filtered probability space on which all random variables in this paper are well defined. The space $(\Omega_{\tau},\mathcal{F}_{\tau},\mathbb{P}_{\tau})$ is the probability space constructed from the random ages at death of countably many individuals, and the space $(\Omega_{\Psi},\mathcal{F}_{\Psi},\mathbb{P}_{\Psi})$ is the probability space constructed from the random future longevity improvement. The random elements $w_{\tau}\in\Omega_{\tau}$ and $w_{\Psi}\in\Omega_{\Psi}$ are independent under $\mathbb{P}$ , that is,

\begin{equation*}\nonumber \mathbb{P}((w_{\tau},w_{\Psi})\in A\times B)=\mathbb{P}_{\tau}(A) \, \mathbb{P}_{\Psi}(B),\quad \forall A\in\mathcal{F}_{\tau},\forall B\in\mathcal{F}_{\Psi}.\end{equation*}

Throughout the paper, we denote $\mathbb{R}=(\!-\infty,\infty)$ as the set of real numbers and $\mathbb{R}^{+}=[0,\infty)$ as the set of nonnegative real numbers. For any event A, the notation $\mathbb{I}_{A}$ denotes the indicator function of A, which equals 1 if A occurs and 0 otherwise. We write $\mathbb{E}[X]$ for the expectation of the random variable X. For any function $f(\!\cdot\!)$ , the notations $f'(\!\cdot\!)$ , $f''(\!\cdot\!)$ and $f^{-1}(\!\cdot\!)$ denote its first derivative, second derivative and inverse function, respectively, when they exist. The symbols $\buildrel \mathrm{a.s.} \over =$ and $\buildrel \mathrm{a.e.} \over =$ denote ‘almost surely equal’ and ‘almost everywhere equal’, respectively. We use boldfaced $\boldsymbol{c}=(c_1,\ldots,c_n)$ to denote a n-dimensional vector and write $||\cdot||$ for the Euclidean norm $||\boldsymbol{c}||=\sqrt{c_1^2+\ldots+c_n^2}$ .

The insurer constructs a life annuity pool and then arranges a reinsurance contract for the annuity plan with a reinsurer to mitigate longevity risk. On the one hand, some retired individuals (policyholders) aged R participate in the life annuity constructed by the insurer at time 0. Let n be the initial number of homogeneous policyholders in the pool. Each policyholder annuitizes wealth x either as a single lump sum or through installment payment, whose total present value equals x. Then, the insurer has an obligation to continuously pay benefit to each policyholder until death, where the benefit payment function is denoted by $b=\{b(t)\}_{t\, \geqslant \,0}$ . For a given $\psi\in\mathbb{R}$ , let $\tau_i(\psi)\,:\!=\,\tau_i(w_\tau,\psi)$ denote the age at death for policyholder i and bounded above by W, that is, the support of $\tau_i(\psi)$ is [R,W]. Then, $\tau_i(\psi)$ , $ i\in\{1,\ldots,n\}$ , are independent and identically distributed. The parameter $\psi$ represents the longevity parameter, and a larger value of $\psi$ implies greater longevity risk. For convenience, we denote $\tau(\psi)\,:\!=\,\tau(w_{\tau},\psi)$ , $w_{\tau}\in\Omega_{\tau}$ as the random age at death for a representative policyholder.

The insurer and reinsurer obtain the longevity parameter $\psi$ from national mortality tables at time 0. However, the future longevity parameter in practice may differ from $\psi$ . Since systematic longevity risk exists, the difference between the future longevity parameter and $\psi$ , that is, the longevity improvement, is unknown to both parties, and this uncertainty cannot be diversified by increasing the number of policyholders. To model the systematic longevity risk, the insurer and reinsurer treat longevity improvement as a random variable $\Delta\Psi$ defined on the probability space $(\Omega_{\Psi},\mathcal{F}_{\Psi},\mathbb{P}_{\Psi})$ . The random longevity improvement $\Delta\Psi$ is independent of $\{\tau_i(\psi)\}_{i=1,\ldots,n}$ , and its density is denoted as $f_{\Delta\Psi}(\!\cdot\!)$ . Then, the ages at death of all policyholders are formulated as $\{\tau_i(\psi+\Delta\Psi)\}_{i=1,\ldots,n}$ . The true longevity improvement is denoted as $\Delta\psi$ , which is a realization of $\Delta\Psi$ . If the realization $\Delta\psi\gt 0$ , there is an upward longevity trend. If the realization $\Delta\psi\lt 0$ , there is a downward longevity trend. The case $\Delta\psi=0$ implies that the future longevity parameter is consistent with the initial one, which means that there is no longevity trend. A larger value of the realization $\Delta\psi$ indicates greater underestimation of lifespan, and therefore greater longevity risk for the insurer.

We denote the mortality rate of the policyholder at age t by $q(t; \, \psi+\Delta\Psi)$ . For the given $\Delta\Psi =\Delta\psi$ , the conditional mortality rate is $q(t; \, \psi+\Delta\psi)$ , which is decreasing in $\Delta\psi$ . It means that larger $\Delta\psi$ corresponds to longer lifespan, and therefore greater longevity risk for the insurer. We define the conditional and unconditional survival functions of an a-year-old policyholder, denoted by $S_R(a; \, \psi+\Delta\psi)$ and $s_R(a)$ , as follows:

\begin{align*} S_R(a; \, \psi+\Delta\psi)\,:\!=\,&\mathbb{P}(\tau(\psi+\Delta\Psi)\, \geqslant \, a|\Delta\Psi=\Delta\psi)=\exp\bigg(-\int_R^a q(s; \, \psi+\Delta\psi)\mathrm{d} s\bigg),\quad &a\, \geqslant \, R,\\ s_R(a)\,:\!=\,&\mathbb{E}\left[\exp\bigg(-\int_R^a q(s;\, \psi+\Delta\Psi)\mathrm{d} s\bigg)\right],\quad &a\, \geqslant \, R,\end{align*}

where $S_R(a; \, \psi+\Delta\psi)$ is the conditional probability that a policyholder who retires at age R survives to age a given $\Delta\Psi=\Delta\psi$ , and $s_R(a)$ is the expectation of $S_R(a; \, \psi+\Delta\psi)$ with respect to $\Delta\Psi$ . Then, the conditional survival function $S_R(a; \, \psi+\Delta\psi)$ is an increasing function with respect to $\Delta\psi$ , which indicates that the larger value of $\Delta\psi$ leads to a longer expected lifespan for the policyholder. The insurer’s present value of aggregate benefit payment to all policyholders is given by

\begin{equation*} B=\sum_{i=1}^n\int_0^{\tau_i(\psi+\Delta\Psi)-R} \mathrm{e}^{-\xi t}b(t)\mathrm{d} t,\end{equation*}

where $\xi\gt 0$ represents the constant discount rate for the time preference.

On the other hand, the insurer is exposed to longevity risk due to the uncertainty of each policyholder’s remaining lifetime. In order to mitigate the exposure to longevity risk, the insurer purchases reinsurance from the reinsurer. Specifically, the insurer pays a fixed reinsurance premium in exchange for continuously transferring part of benefit payment to the reinsurer via an indemnity strategy. Since all policyholders are homogeneous, the insurer adopts the same longevity risk transfer policy to each policyholder. Let $g = \{g(t, b)\}_{t,b \, \geqslant \, 0}$ denote the rate of benefit payment transferred to the reinsurer at time t given the benefit payment function b. We define g as longevity risk transfer policy. Thus, the reinsurer covers the benefit payment at the rate of g(t, b), and the insurer is only liable for paying the rate of $b(t) - g(t, b(t))$ to the policyholder. Under this payment structure, the benefit paid to each policyholder is divided into two parts, as follows:

\begin{equation*} b(t)=\underbrace{g(t,b)}_{\text{reinsurer's cover}}+\underbrace{b(t)-g(t,b)}_{\text{insurer's liability}}.\end{equation*}

Therefore, the present values of the total benefit paid by the insurer and covered benefit paid by the reinsurer at time 0 are formulated as

\begin{align*} &\text{Liable benefit paid by the insurer:}\quad\sum_{i=1}^n\int_0^{\tau_i(\psi+\Delta\Psi)-R} \mathrm{e}^{-\xi t}(b(t)-g(t,b))\mathrm{d} t.\\ &\text{Covered benefit paid by the reinsurer:}\quad\sum_{i=1}^n\int_0^{\tau_i(\psi+\Delta\Psi)-R} \mathrm{e}^{-\xi t}g(t,b)\mathrm{d} t.\end{align*}

In insurance economics, the reinsurance strategy is typically assumed to satisfy the principle of indemnity to reduce the ex post moral hazard. Thus, we give the following definition of the longevity risk transfer policy for life annuity formally.

Condition (i) implies that the longevity risk transfer policy is independent of each policyholder’s age at death. This condition is realistic, since the transfer policy is predetermined by the insurer at time 0, and is not affected by the policyholder’s state. Condition (ii) in Definition 2.1 is referred to as the principle of indemnity, which means that the longevity risk transfer policy is non-negative and smaller than the benefit payment rate. For any time interval $[t,t+\Delta t]\subset[0,W-R]$ , we say a longevity risk transfer policy is a full-transfer policy on $[t,t+\Delta t]$ if $g(\nu,b)\buildrel \mathrm{a.e.} \over = b(\nu)$ for all $\nu\in[t,t+\Delta t]$ and a no-transfer policy on $[t,t+\Delta t]$ if $g(\nu,b)\buildrel \mathrm{a.e.} \over = 0$ for all $\nu\in[t,t+\Delta t]$ . We call the longevity risk transfer policies in $\mathcal{G}$ moral hazard free and will only consider them in later analysis. As will become clear, the optimal longevity risk transfer policy for the life annuity obtained in this paper is naturally moral hazard free. In Definition 2.1, we define the admissible longevity risk transfer policy for the life annuity. In general, there are two commonly used reinsurance strategies in insurance economics, namely proportional reinsurance and excess of loss reinsurance. Motivated by these strategies, we provide following remark to illustrate two specific forms of longevity risk transfer policies.

The proportional transfer policy in Equation (2.1) can be viewed as transferring a portion of the benefit payment rate to the reinsurer, where m represents the transfer proportion. As a result, both the insurer and the reinsurer continuously pay benefit until the policyholder’s death. However, the excess of age transfer policy in Equation (2.2) differs significantly. In this strategy, the insurer is only liable for paying benefit up to the policyholder’s age $R+T$ , while any benefit payment after age $R+T$ is covered by the reinsurer, where T represents the deductible level. In later analysis, we will show that the optimal longevity risk transfer policy for life annuity takes the form of excess of age policy, as presented in Theorem 3.2.

Suppose that the insurer pays reinsurance premium to the reinsurer at time 0. We use the expected premium principle to calculate the reinsurance premium, and it is determined under measure $\mathbb{P}$ . Let $\delta(g)$ denote the reinsurance premium defined under the longevity risk transfer policy g, we obtain

(2.3) \begin{equation} \delta(g)=(1+\theta)\mathbb{E}\bigg[\sum_{i=1}^n\int_0^{\tau_i(\psi+\Delta\Psi)-R} \mathrm{e}^{-\xi t} g(t,b)\mathrm{d} t\bigg],\end{equation}

where $\theta\, \geqslant \,0$ is the safety loading of the reinsurer. Note that the reinsurance premium in Equation (2.3) reduces to the pure premium principle when $\theta=0$ . To present the transfer policy more intuitively, Figure 1 shows the payment structure among each policyholder, insurer and reinsurer. Note that the contribution and reinsurance premium are paid at time 0, whereas the covered benefit rate is paid at each time t until the policyholder’s death.

Figure 1.

The relationship between the policyholder, insurer and reinsurer in the longevity risk transfer policy of life annuity.

Since the insurer models longevity improvement via $\Delta\Psi$ , the insurer’s terminal average wealth per policyholder, evaluated at the time when all policyholders are dead, is formulated as

(2.4) \begin{equation} F(g)=x-\frac{1}{n}\sum_{i=1}^n\int_0^{\tau_i(\psi+\Delta\Psi)-R} \mathrm{e}^{-\xi t}(b(t)-g(t,b))\mathrm{d} t-(1+\theta)\int_0^\infty \mathrm{e}^{-\xi t}s_R(t+R) g(t,b)\mathrm{d} t.\end{equation}

The expression F(g) represents the average profit obtained from each policyholder and consists of three components: (i) the contribution x paid by each policyholder; (ii) the present value of average benefit payment paid by the insurer and (iii) the present value of the average reinsurance premium paid at time 0. Based on the expected utility framework in Chi and Wei (Reference Chi and Wei2020) and Boonen et al. (Reference Boonen, Jiang, Yong and Zhang2025), we assume the insurer possesses a utility function $U(\!\cdot\!)$ satisfying $U{'}(\!\cdot\!)\gt 0$ and $U{''}(\!\cdot\!) \lt 0$ . The insurer aims to determine an optimal longevity risk transfer policy $g^*(t,b)$ to maximize the expected utility of the terminal average wealth. Then, the optimal longevity risk transfer problem is formulated as

(2.5) \begin{equation} \sup_{g\in\mathcal{G}}\mathbb{E}[U(F(g))].\end{equation}

3. Main results

In this section, we first discuss existence of the optimal longevity risk transfer policy. Then, we provide a necessary and sufficient condition for the optimal longevity risk transfer policy $g^*(t,b)$ via the calculus of variations. After obtaining $g^*(t,b)$ , we discuss its uniqueness.

Based on the existence of the optimal longevity risk transfer policy, the optimal transfer policy is characterized by the following theorem.

Theorem 3.1 The strategy $g^*(t,b)$ is the optimal longevity risk transfer policy to Problem (2.5) if and only if it satisfies

(3.1) \begin{equation} g^*(t,b)=\left\{ \begin{aligned} &b(t),\quad &H(t;\,g^*) \geqslant 1+\theta;\\ &0,\quad &H(t;\,g^*)\lt 1+\theta, \end{aligned} \right. \end{equation}

where $H(t; \, g)$ is defined as

(3.2) \begin{equation} H(t; \, g)=\frac{\mathbb{E}\left[U'(F(g))|\tau_i(\psi+\Delta\Psi)\, \geqslant \, t+R\right]}{\mathbb{E}\left[U'(F(g))\right]}. \end{equation}

Remark 3.1 In Equation (3.2), we use the notation $H(t; \,g)$ instead of $H_i(t; \, g)$ since $\{\tau_i(\psi+\Delta\Psi)\}_{i=1,\ldots,n}$ are conditionally independent and identically distributed. It implies that

\begin{equation*} \frac{\mathbb{E}\left[U'(F(g))|\tau_i(\psi+\Delta\Psi)\, \geqslant \, t+R\right]}{\mathbb{E}\left[U'(F(g))\right]}=\frac{\mathbb{E}\left[U'(F(g))|\tau_j(\psi+\Delta\Psi)\, \geqslant \, t+R\right]}{\mathbb{E}\left[U'(F(g))\right]},\quad \forall i, j\in\{1,\ldots,n\}. \end{equation*}

Therefore, we denote the right-hand side of Equation (3.2) by $H(t; \, g)$ for sake of convenience, and the similar notations are used in Theorem 3.2 and Corollary 3.2.

Theorem 3.1 provides a necessary and sufficient condition for characterizing the optimal longevity risk transfer policy $g^*(t,b)$ . The function $H(t; \,g^*)$ is defined as the ratio between the expected marginal utility given that the policyholder survives to time t and the unconditional expected marginal utility. This ratio describes the insurer’s relative weight on marginal utility associated with the overpayment occurring after age $t+R$ . When $H(t; \, g^*)\, \geqslant \,1+\theta$ , the relative weight for transferring overpayment exceeds the reinsurance safety loading, then the insurer effectively adopts full-transfer policy. Conversely, when $H(t; \, g^*)\lt 1+\theta$ , the reinsurance safety loading exceeds the relative weight for transferring overpayment, then no-transfer policy is optimal. The solutions to $H(t; \, g^*)\lt 1+\theta$ and $H(t; \, g^*)\, \geqslant \,1+\theta$ are still determined by $g^*(t,b)$ , resulting in an implicit form for $g^*(t,b)$ . However, it gives an insight into the form that the optimal longevity risk transfer policy should take. Therefore, before presenting an explicit optimal policy, we first present a corollary and a lemma to develop the characteristics of the optimal longevity risk transfer policy.

According to Lemma 3.1, we deduce that $H(t;\,g^*)$ is also bounded and increasing in t. If $H(t;\,g^*)$ up-crosses $1+\theta$ , then there exists a point $T^*$ such that $g^*(t,b)=0$ is optimal for any $t\in[0,T^*]$ , and $g^*(t,b)=b(t)$ is optimal for $t\in[T^*,W-R]$ . This implies that there exists a deductible age $R+T^*$ such that for each policyholder aged less than $R+T^*$ , the insurer pays the full benefit payment. In contrast, for each policyholder aged greater than $R+T^*$ , the whole benefit payment is covered by the reinsurer. For later convenience, we refer to $R+T^*$ as the optimal deductible age and $T^*$ as the optimal deductible level. Therefore, the optimal strategy takes the form of the excess of age transfer policy. It remains challenging to explicitly derive the inverse function of $H(t;\,g^*)$ in that its formulation is implicit. However, we transform the original Problem (2.5) into obtaining the optimal deductible level $T^*$ for the excess of age transfer policy. We now present the following theorem to obtain $T^*$ .

Theorem 3.2 Denote that

\begin{equation*} \mathcal{H}(T)\,:\!=\,\frac{\mathbb{E}\left[U'\left(x-\frac{1}{n}\sum_{j=1}^n\hat{F}(\tau_j(\psi+\Delta\Psi);\,T)\right)|\tau_i(\psi+\Delta\Psi)\, \geqslant \, R+T\right]}{\mathbb{E}\left[U'\left(x-\frac{1}{n}\sum_{j=1}^n\hat{F}(\tau_j(\psi+\Delta\Psi);\,T)\right)\right]}, \end{equation*}

where $\hat{F}(s;\,T)=\int_0^{s-R} \mathrm{e}^{-\xi t}b(t)\mathbb{I}_{\{t \lt T\}}\mathrm{d} t+(1+\theta)\int_0^\infty \mathrm{e}^{-\xi t}s_R(t+R)b(t)\mathbb{I}_{\{t\, \geqslant \, T\}}\mathrm{d} t$ . There exists $T^*$ such that the optimal longevity risk transfer policy defined in Theorem 3.1 can be expressed as

(3.3) \begin{equation} g^*(t,b)=\left\{ \begin{aligned} &b(t),\quad &t \, \geqslant \, T^*;\\ &0,\quad &t\, \lt \,T^*, \end{aligned} \right. \end{equation}

where $T^*$ is given by

(3.4) \begin{equation} T^*=\inf\{T\in[T_\theta,W-R] \, : \, \mathcal{H}(T)\, \geqslant \,1+\theta\}, \end{equation}

and $T_\theta$ is defined in Corollary 3.1.

Theorem 3.2 provides an intuitive result. The insurer is required to pay the benefit up to a predetermined age $R+T^*$ . Specifically, the insurer is only liable for paying the benefit to any policyholders aged in $[R,R+T^*]$ , and the present value of the average benefit payment is given by $\frac{1}{n}\sum_{i=1}^n\int_0^{\tau_i(\psi+\Delta\Psi)-R} \mathrm{e}^{-\xi t}b(t)\mathbb{I}_{\{t\lt T^*\}}\mathrm{d} t$ . Any benefits payable after the age of $R+T^*$ are covered by the reinsurer, and the present value of total benefit payment for the reinsurer is

$$\sum_{i=1}^n\left(\int_0^{\tau_i(\psi+\Delta\Psi)-R} \mathrm{e}^{-\xi t}b(t)\mathrm{d} t-\int_0^{\tau_i(\psi+\Delta\Psi)-R} \mathrm{e}^{-\xi t}b(t)\mathbb{I}_{\{t\lt T^*\}}\mathrm{d} t\right)=\sum_{i=1}^n\int_0^{\tau_i(\psi+\Delta\Psi)-R} \mathrm{e}^{-\xi t}b(t)\mathbb{I}_{\{t\, \geqslant \, T^*\}}\mathrm{d} t.$$

Therefore, the optimal longevity risk transfer strategy $g^*(t,b)$ defined in Equation (3.3) effectively acts as an excess of age transfer policy over any policyholder age phase, as illustrated in Equation (2.2). This policy can be considered as a defense against longevity risk, since overpayment caused by the policyholder with a longer lifespan (policyholder’s age exceeds $R+T^*$ ) is covered by the reinsurer. The insurer’s liability of the total benefit payment is bounded by $n\int_0^{T^*}\mathrm{e}^{-\xi t}b(t)\mathrm{d} t$ . Moreover, the optimal excess of age transfer policy truncates the right tail risk of benefit payment arising from the uncertainty in the random longevity improvement $\Delta\Psi$ after the age $R+T^*$ , then eliminating the insurer’s exposure to systematic longevity risk beyond $R+T^*$ . Compared with transferring all benefit payment to the reinsurer, the excess of age transfer policy requires the insurer to pay the benefit to all policyholders until a deductible age $R+T^*$ , and the reinsurer is only liable for the benefit payment after the policyholder reaches the age $R+T^*$ . Thus, it reduces the reinsurance premium cost paid by the insurer and lowers the reinsurer’s benefit payment burden. As a result, it improves the solvency for both parties.

Since the optimal longevity risk transfer policy satisfies the excess of age transfer policy, proving uniqueness of $g^*(t,b)$ is equivalent to proving uniqueness of the optimal deductible level $T^*$ . The following proposition states this.

Before ending this part, we present some corollaries that state the effect of parameters on the optimal deductible level $T^*$ .

Corollary 3.3 Let $m(\Delta\psi)=\mathbb{E}[\hat{F}(\tau(\psi+\Delta\Psi);\,T)|\Delta\Psi=\Delta\psi]$ . When n converges to $\infty$ , the optimal deductible time $T^*$ tends to $\tilde{T}$ , where $\tilde{T}$ is given by

\begin{equation*} \tilde{T}=\inf\left\{T\in[T_\theta,W-R]\,:\,\frac{\mathbb{E}[U'(x-m(\Delta\Psi))|\tau(\psi+\Delta\Psi)\, \geqslant \, R+T]}{\mathbb{E}[U'(x-m(\Delta\Psi))]}\, \geqslant \,1+\theta\right\}. \end{equation*}

Proof. Since $\{\tau_i(\psi+\Delta\Psi)\}_{i=1,\ldots,n}$ are conditionally independent and identically distributed under given $\Delta\Psi=\Delta\psi$ , the conditional strong law of large numbers implies

\begin{equation*} \frac{1}{n}\sum_{i=1}^n\hat{F}(\tau_i(\psi+\Delta\psi);\,T)\xrightarrow{n\rightarrow\infty}m(\Delta\psi).\end{equation*}

According to dominated convergence theorem and $U''(\!\cdot\!)\lt 0$ , we have

\begin{equation*} \lim_{n\rightarrow\infty}\mathcal{H}(T)=\frac{\mathbb{E}[U'(x-m(\Delta\Psi))|\tau(\psi+\Delta\Psi)\, \geqslant \, T+R]}{\mathbb{E}[U'(x-m(\Delta\Psi))]}.\end{equation*}

Then, we obtain $T^*\rightarrow\tilde{T}$ , as $n\rightarrow\infty$ .

4. Impact of longevity risk transfer on insurer’s terminal wealth

In this section, we examine the effect of the optimal longevity risk transfer policy on insurer’s true terminal average wealth. In Sections 2 and 3, the insurer and the reinsurer calibrate the survival function using the random longevity improvement parameter $\Delta\Psi$ and obtain $s_R(t)$ . There exists a true longevity improvement denoted as $\Delta\psi$ , which is the realization of $\Delta\Psi$ . Under the optimal longevity risk transfer policy $g^*(t,b)$ , the optimal deductible age $R+T^*$ is determined by the random longevity improvement $\Delta\Psi$ and is invariant to changes. Any overpayment beyond $R+T^*$ that results from longevity improvement is transferred to the reinsurer. Since the true longevity parameter is $\psi+\Delta\psi$ , the insurer’s true terminal average wealth under the optimal excess of age transfer policy is given by

\begin{equation*} \tilde{F}(g^*)=x-\frac{1}{n}\underbrace{\sum_{i=1}^n\int_0^{\tau_i(\psi+\Delta\psi)-R} \mathrm{e}^{-\xi t}b(t)\mathbb{I}_{\{t\lt T^*\}}\mathrm{d} t}_{\text{true average benefit payment}}-(1+\theta)\int_0^\infty \mathrm{e}^{-\xi t}s_R(t+R)b(t)\mathbb{I}_{\{t\, \geqslant \, T^*\}}\mathrm{d} t.\end{equation*}

To evaluate the effect of longevity risk transfer, we first consider the insurer’s true average terminal wealth under no-transfer policy, given by

\begin{equation*} \tilde{F}=x-\frac{1}{n}\sum_{i=1}^n\int_0^{\tau_i(\psi+\Delta\psi)-R} \mathrm{e}^{-\xi t}b(t)\mathrm{d} t,\end{equation*}

which represents the difference between the contribution paid by the policyholder and the true average benefit payment made by the insurer under no-transfer. We define the relative gains from adopting the optimal excess of age transfer policy as the difference between $\mathbb{E}[\tilde{F}(g^*)]$ and $\mathbb{E}[\tilde{F}]$ . Mathematically,

(4.1) \begin{equation} \Delta F(\Delta\psi)\,:\!=\,\mathbb{E}[\tilde{F}(g^*)]-\mathbb{E}[\tilde{F}]=\int_0^\infty (S_R(t+R;\,\psi+\Delta\psi)-(1+\theta)s_R(t+R))\mathrm{e}^{-\xi t} b(t)\mathbb{I}_{\{t\, \geqslant \, T^*\}}\mathrm{d} t.\end{equation}

The relative gains $\Delta F(\Delta\psi)$ describe the improvement of insurer’s true average terminal wealth when the optimal excess of age transfer policy is adopted, which depends on the true longevity improvement $\Delta\psi$ . The larger value of $\Delta F(\Delta\psi)$ indicates greater gains to the insurer from the transfer policy.

We state the following proposition to illustrate the effect of the optimal longevity risk transfer.

Proposition 4.1 A necessary condition for the insurer to have nonnegative relative gains under the optimal excess of age transfer policy, that is, the relative gains $\Delta F(\Delta\psi)\, \geqslant \,0$ , is

(4.2) \begin{equation} S_R(t+R;\,\psi+\Delta\psi)\, \geqslant \,(1+\theta)s_R(t+R),\quad\forall t\in[0,W-R]. \end{equation}

Proposition 4.1 shows that the insurer attains non-negative expected gains, that is, $\Delta F(\Delta\psi)\, \geqslant \,0$ , when the true survival function $S_R(t+R;\,\psi+\Delta\psi)$ dominates the unconditional survival function $s_R(t+R)$ scaled by the reinsurance safety loading. Conversely, if longevity improvement is insufficient, that is, $S_R(t+R;\,\psi+\Delta\psi)\lt (1+\theta)s_R(t+R)$ , the reinsurance premium outweighs the gains of transferring longevity risk and the insurer’s relative gains are negative.

5. The analysis of reinsurer

In this section, we investigate the conditions under which the reinsurer adopts the optimal longevity risk transfer policy proposed by the insurer. In the previous section, we observed that the optimal longevity risk transfer policy takes the form of an excess of age policy. Under this policy, the reinsurer receives the premium at time 0 and begins to cover the benefit payment after time $T^*$ . Therefore, the reinsurer has sufficient time to invest in the financial market and accumulate additional wealth, which then be used to offset potential loss from bearing longevity risk.

We assume that the reinsurer invests in a financial asset, with the average return rate denoted by r. The price process of the financial asset is described by the following differential equation.

\begin{equation*} \mathrm{d} S_0(t)=rS_0(t)\mathrm{d} t,\quad S_0(0)=1.\end{equation*}

We assume that the reinsurer invests the total premium in the financial asset until time $T^*$ , where $T^*$ is defined in Equation (3.4). The constant discount rate for the reinsurer is $\xi$ , which is the same as that for the insurer. Therefore, the present values of reinsurer’s total wealth at time 0 and time $T^*$ are given by

\begin{align*} &\text{The reinsurer's wealth at time 0:}\quad (1+\theta)n\int_0^\infty \mathrm{e}^{-\xi t}s_R(t+R)b(t)\mathbb{I}_{\{t\, \geqslant \, T^*\}}\mathrm{d} t,\\ &\text{The reinsurer's wealth at time $T^*$:}\quad(1+\theta) \mathrm{e}^{(r-\xi)T^*}n\int_0^\infty \mathrm{e}^{-\xi t}s_R(t+R)b(t)\mathbb{I}_{\{t\, \geqslant \, T^*\}}\mathrm{d} t.\end{align*}

After time $T^*$ , the reinsurer begins to cover the benefit payment. We define the reinsurer’s terminal wealth under any longevity parameter $\phi$ as

(5.1) \begin{equation} W_R(r;\,\phi)\,:\!=\,(1+\theta) \mathrm{e}^{(r-\xi)T^*}n\int_0^\infty \mathrm{e}^{-\xi t}s_R(t+R)b(t)\mathbb{I}_{\{t\, \geqslant \, T^*\}}\mathrm{d} t-\sum_{i=1}^n\int_0^{\tau_i(\phi)-R} \mathrm{e}^{-\xi t}b(t)\mathbb{I}_{\{t\, \geqslant \, T^*\}}\mathrm{d} t.\end{equation}

Since the initial longevity parameter is $\psi$ and the reinsurer uses the random longevity improvement $\Delta\Psi$ to calibrate the survival function, the reinsurer’s estimated terminal wealth is given by $W_R(r;\,\psi+\Delta\Psi)$ , while the true terminal wealth is $W_R(r;\,\psi+\Delta\psi)$ . The difference between $W_R(r;\,\psi+\Delta\Psi)$ and $W_R(r;\,\psi+\Delta\psi)$ evaluates the shortfall arising from underestimating longevity improvement.

We investigate the condition under which the reinsurer adopts the insurer’s proposed longevity risk transfer policy under the risk-adjusted liability framework and the impact of longevity improvement on reinsurer’s decision. We first define the reinsurer’s objective function under any longevity parameter $\phi$ as

\begin{equation*} J_R(r;\,\phi)\,:\!=\,\mathbb{E}[-W_R(r;\,\phi)]+\delta\varrho(-W_R(r;\,\phi)-\mathbb{E}[-W_R(r;\,\phi)]), \end{equation*}

where $\delta\in(0,1)$ , and $\varrho(\!\cdot\!)$ is a law-invariant coherent risk measure. Then, $J_R(r;\,\psi+\Delta\Psi)$ and $J_R(r;\,\psi+\Delta\psi)$ denote the reinsurer’s objective functions under the random and true longevity parameter, respectively. If the reinsurer rejects the insurer’s proposed transfer policy, that is, when the reinsurance contract for the life annuity does not take effect, then the reinsurer’s terminal wealth is zero, then $J_R(r;\,\phi)=0$ holds for any $\phi$ . Since the reinsurer calibrates the survival function using the random longevity improvement $\Delta\Psi$ , the reinsurer adopts the transfer policy if and only if its objective after accepting $g^*(t,b)$ is no larger than the objective when not accepting it. Mathematically,

(5.2) \begin{equation} J_R(r;\,\psi+\Delta\Psi)\leqslant0. \end{equation}

We then present the following proposition to illustrate under which condition the reinsurer adopts the excess of age transfer policy defined in Equation (3.3).

6. Numerical examples

In the previous section, we explicitly derived the optimal longevity risk transfer policy $g{^*}(t,b)$ for the life annuity under the random longevity improvement $\Delta\Psi$ . In this section, we present numerical examples to analyse the theoretical results under the Gompertz–Makeham law. Specifically, we calibrate parameters to mortality data for the United States from the Human Mortality Database (HMD), due to its large population, high-quality data and sizable insurance market. The parameters calibrated for 2001 serve as the initial longevity parameters used to obtain the optimal deductible age $R+T^*$ , and we hold $R+T^*$ fixed for 2001–2019. For any year i after 2001, longevity improvement is defined as the difference between the parameter calibrated for year i and those for 2001.

6.1. Parameter assumptions

In this subsection, we make several numerical assumptions. We assume that the mortality rate satisfies the Gompertz–Makeham law. Corresponding to the model formulation in Section 2, since the insurer and the reinsurer use the random longevity improvement $\Delta\Psi$ to represent the systematic longevity risk, then the Gompertz–Makeham law is formulated as

\begin{equation*} q(t;\,\psi+\Delta\Psi)=\lambda+\frac{1}{\beta} \mathrm{e}^{\frac{t-(\psi+\Delta\Psi)}{\beta}},\quad t\in[R,W],\end{equation*}

where $\lambda\, \geqslant \, 0$ is constant representing non-age-dependent accidental death, $\beta$ represents the rate of increase in the hazard with age and $\psi+\Delta\Psi$ represents the age at which the exponential term begins to affect the mortality rate. It is observed that $q(t;\,\psi)$ is decreasing in $\psi$ . This implies that larger value of $\psi$ results in lower mortality rate, leading to a longer lifespan for the policyholder. Then, the logarithm of conditional survival function $S_R(t;\,\psi+\Delta\psi)$ given $\Delta\Psi=\Delta\psi$ is formulated as

\begin{equation*}\ln S_R(t;\,\psi+\Delta\psi)=-\lambda(t-R)- \mathrm{e}^{\frac{R-(\psi+\Delta\psi)}{\beta}} \big( \mathrm{e}^{\frac{t-R}{\beta}}-1\big),\quad t\in[R,W].\end{equation*}

Unless specified otherwise, we use the following assumptions for our numerical example.

• • The retirement age is $R=66$ , which is the full retirement age for people born before 1960 in the United States. The maximum age at death is $W=122$ . The number of policyholder at time 0 is $n=5000$ .

• • Using the method described in Milevsky (Reference Milevsky2020), we estimate the parameters of the Gompertz–Makeham law from historical United States data for ages 0–110 from 2001 to 2019. The initial longevity parameters are $\lambda=4.28\times10^{-6}$ , $\psi=82.372$ and $\beta=12.606$ .

• • The random longevity improvement $\Delta\Psi$ satisfies the normal distribution, with $\mathbb{E}[\Delta\Psi]=0$ , $\mathrm{Var}[\Delta\Psi]=2$ , that is, $\Delta\Psi\sim N(0,2)$ .

• • The insurer’s utility function satisfies exponential utility, that is, $U(x)=-\frac{1}{\gamma} \mathrm{e}^{-\gamma x}$ , with $\gamma=0.1$ . We assume that the reinsurer’s objective function is

  \begin{equation*} \mathbb{E}[-W_R(r;\,\psi+\Delta\Psi)]+\delta\mathrm{CVaR}_{\alpha}(-W_R(r;\,\psi+\Delta\Psi)-\mathbb{E}[-W_R(r;\,\psi+\Delta\Psi)]),\end{equation*}
  where $\delta=0.2$ and $\alpha=0.95$ .

• • The benefit function for each policyholder at time t is given by $b(t)=b_0\mathrm{e}^{\eta t}$ , where $b_0\in\{0.9,1,1.1\}$ is the initial benefit payment, and $\eta=0.0269$ is the average inflation rate in the United States from 1983 to 2021.

• • The reinsurance safety loading is $\theta=0.1$ .

• • The discount rate is $\xi=0.015$ .

Figure 2.

The effects of $\theta$ on $T^*$ .

6.3. The influence of Gompertz–Makeham parameters

To assess the robustness of the optimal deductible age $R+T^*$ to the changes in the Gompertz parameter $\psi$ , we analyze the sensitivity of $R+T^*$ to $\psi$ . Figure 3 shows $R+T^*$ as $\psi$ varies over [80, 85]. In both figures, all other parameters are fixed at their previously assumption. In Subsection 6.1, we conclude that larger $\psi$ implies longer expected lifespan for the policyholder. Figures 3 shows that the optimal deductible age $R+T^*$ increases with $\psi$ . Intuitively, the insurer is more inclined to retain the benefit payment rather than transfer the benefit payment to the reinsurer, as the policyholder generally has a longer lifespan (larger value of $\psi$ ). This is because transferring at earlier ages increases the duration of reinsurance exposure, and thus raises reinsurance premium, since the safety loading $\theta\, \geqslant \,0$ .

Figure 3.

The effects of $\psi$ on $T^*$ .

6.4. The influence of risk-aversion parameter

The risk-aversion coefficient $\gamma$ represents the insurer’s level of risk aversion. A higher value of $\gamma$ indicates greater risk aversion, while a lower value suggests a higher tolerance for risk. Figure 4 illustrates the impact of the risk-aversion parameter $\gamma$ on the optimal deductible age $R+T^*$ . As $\gamma$ increases, the optimal deductible age decreases. This means that a more risk-averse insurer chooses to transfer the benefit payment to the reinsurer earlier in the policyholders’ lifespan compared to a less risk-averse insurer. This is because a more risk-averse insurer is reluctant to pay excessive benefit to the policyholder.

Figure 4.

The effects of $\gamma$ on $R+T^*$ .

6.5. The influence of parameters on benchmark return rate $\bar{r}$

In the previous discussion, we used $\Delta\psi$ to denote the difference between the future and initial longevity parameters, which is the realized value of $\Delta\Psi$ . Figure 5 shows the effect of the true longevity improvement $\Delta\psi$ on $\bar{r}$ . According to Proposition 5.1, the intersections of the curve $J_R(r;\,\psi+\Delta\psi)$ with the slanted lines in Figure 5 determine the benchmark return rate $\bar{r}$ . The blue line in the panel represents the case in which the future longevity parameter equals to the initial one, that is, the realization $\Delta\psi=0$ . The line for $\Delta\psi\lt 0$ corresponds to an overestimation of longevity improvement, whereas the lines for $\Delta\psi\gt 0$ correspond to an underestimation. We find that $\bar{r}$ increases as $\Delta\psi$ increases. Indeed, larger value of $\Delta\psi$ corresponds to greater underestimation of longevity improvement, increasing the reinsurer’s longevity risk. Consequently, the reinsurer will accept the optimal longevity risk transfer policy only if the investment yields a sufficiently high average return to sustain its profit.

Figure 5.

The effects of $\Delta\psi$ on $R+T^*$ .