The impact of longevity and investment risk on a portfolio of life insurance liabilities

Abstract

In this paper we assess the joint impact of biometric and financial risk on the market valuation of life insurance liabilities. We consider a stylized, contingent claim based model of a life insurance company issuing participating contracts and subject to default risk, as pioneered by Briys and de Varenne (Geneva Pap Risk Insur Theory 19(1):53–72, 1994, J Risk Insur 64(4):673–694, 1997), and build on their model by explicitly introducing biometric risk and its components, namely diversifiable and systematic risk. The contracts considered include pure endowments, deferred whole life annuities and guaranteed annuity options. Our results stress the predominance of systematic over diversifiable risk in determining fair participation rates. We investigate the interaction of contract design, market regimes and mortality assumptions, and show that, particularly for lifelong benefits, the choice of the participation rate must be very conservative if longevity improvements are foreseeable.

Appendix

1.1 Properties of $\tau _i$ and N

1.1.1 Law of $\tau _i$

The survival probability of a policyholder is given by

$$\begin{aligned} {}_tp_x=Q(\tau ^i>t)=E\left[ \mathrm {e}^{-\Delta \int _0^tm(v)\mathrm {d}v}\right] =\mathscr {L}_\Delta \left( \log \,_{t}p^*_{x}\right) , \end{aligned}$$

where $\mathscr {L}_\Delta$ is the moment-generating function of $\Delta$, i.e. $\mathscr {L}_\Delta (y)=E[\text {e}^{\Delta y}]$.

1.1.2 Ordering between $\tau _i$ and $\tau ^*$

Proposition 1

If $E[\Delta ]\le 1$ then $\tau _i$ is greater than $\tau ^*$ in the hazard rate order.

Proof

We need to show that the ratio $_tp_x/_tp^*_x$ is nondecreasing with t. For $t<s$, we have

$$\begin{aligned} \frac{_sp_x}{_sp^*_x}-\frac{_tp_x}{_tp^*_x} =\mathcal {M}(_sp^*_x)-\mathcal {M}(_tp^*_x)\ge 0, \end{aligned}$$

since the function $\mathcal {M}(z)=\mathcal {L}_\Delta (\log z)/z$, $0<z\le 1$, is nonincreasing when $E[\Delta ]\le 1$ as can be seen by inspecting its derivative:

$$\begin{aligned} z^2\mathcal {M}'(z)=E[z^\Delta (\Delta -1)]=\mathrm {Cov}\left( z^\Delta ,\Delta \right) +E[z^\Delta ]\,E[\Delta -1]\le 0. \end{aligned}$$

$\square$

1.1.3 Law of N

The number of survivors N has, conditionally on $\Delta$, a binomial distribution:

$$\begin{aligned} N \sim \text {Binomial} \left( N_0,\mathrm {e}^{-\Delta \int _0^T m(v)\mathrm {d}v}\right) . \end{aligned}$$

Consequently, the unconditional law of N is a mixture of binomial distributions. Denoting by $F_\Delta$ the cumulative distribution function of $\Delta$, we have, for $j=0,1,\ldots ,N_0$,

$$\begin{aligned} Q(N=j)=E\left[ \mathrm {bin}\left( j;N_0,\pi ^\Delta \right) \right] =\int _0^\infty \,\mathrm {bin}\left( j;N_0,\pi ^l\right) F_\Delta (\mathrm {d}l), \end{aligned}$$

where $\mathrm {bin}(j;M,p)=\left( {\begin{array}{c}M\\ j\end{array}}\right) p^j(1-p)^{M-j}$ is the mass function of a Binomial random variable with parameters $M\ge 1$ and $0<p<1$.

1.2 Market value of the unitary annuity

Under Assumptions 1 and 2, the market value of the unitary annuity $a_T$ is

$$\begin{aligned} a_T&=E\left[ \int _T^\infty \mathrm {e}^{-r(s-T)}1_{\{\tau ^i>s\}}\mathrm {d}s\Big |\tau ^i>T,\,\Delta \right] \\&=\int _T^\infty \mathrm {e}^{-r(s-T)}Q\left( \tau ^i>s\Big |\tau ^i>T,\,\Delta \right) \mathrm {d}s\\&=\int _T^\infty \mathrm {e}^{-r(s-T)}\mathrm {e}^{-\Delta \int _T^s m(v)\mathrm {d}v}\mathrm {d}s\\&=a(\Delta ), \end{aligned}$$

where the function a is given by:

$$\begin{aligned} a(l)=\int _T^{\infty } \mathrm {e}^{-r(s-T)}\,\left( _{s-T}p^*_{x+T}\right) ^l\mathrm {d}s. \end{aligned}$$

Note that a(l) is the value of a continuous annuity with force of mortality $l\,m$.

1.3 Valuation formulae in the finite portfolio case

We denote by C(A, r, T, K) and P(A, r, T, K) the values at time 0 of a European call, respectively put, option written on the assets of the firm, when time to maturity is T, initial assets value is A, (fixed) interest rate is r and strike is K.

Note that the individual benefit B is a function of $\Delta$, say $B=\beta (\Delta )$, where

$$\begin{aligned} \beta (l)={\left\{ \begin{array}{ll} b &\quad \text { in case (a) } \\ \rho \,a(l) &\quad \text { in case (b) } \\ b\max \{1,\rho ^\text {g}a(l)\} &\quad \text { in case (c) } \end{array}\right. }. \end{aligned}$$

1.3.1 Market value of the guaranteed amount

Conditioning on $\Delta$, it follows that

$$\begin{aligned} V_0^\text {g}= & E[\mathrm {e}^{-r T}B 1_{\{\tau ^i>T\}}]\nonumber \\= & \mathrm {e}^{-r T}E\left[ B\pi ^\Delta \right] \nonumber \\= & \mathrm {e}^{-r T}\int _0^\infty \beta (l)\pi ^l F_\Delta (\mathrm {d}l). \end{aligned}$$

(6.1)

1.3.2 Market value of the bonus option

Recalling that $N^{(i)}=1+\sum _{h\ne i}1_{\{\tau ^h>T\}}$ is independent of $\tau ^i$ conditionally on $\Delta$ and that W is independent of all biometric related factors, we have

$$\begin{aligned} V_0^\text {b}= & E\left[ \mathrm {e}^{-rT}\left[ w-\frac{B}{\alpha }\right] ^+1_{\{\tau ^i>T\}}\right] \\= & E\left[ \pi ^\Delta E\left[ \mathrm {e}^{-r T}\left[ \frac{W}{N^{(i)}}-\frac{B}{\alpha }\right] ^+\big |\,\Delta \right] \right] . \end{aligned}$$

By further conditioning on $N^{(i)}$ the inner expectation and exploiting again Assumption 4,

$$\begin{aligned} V_0^\text {b}= & E\left[ \pi ^\Delta E\left[ C\left( \frac{W_0}{N^{(i)}},r,T,\frac{B}{\alpha }\right) |\Delta \right] \right] \nonumber \\= & \int _0^\infty \pi ^l\sum _{j=1}^{N_0}C\left( \frac{W_0}{j},r,T,\frac{\beta (l)}{\alpha }\right) \mathrm {bin}\left( j-1;N_0-1,\pi ^l\right) F_\Delta (\mathrm {d}l).\nonumber \\= & \frac{1}{N_0}\int _0^\infty \sum _{j=1}^{N_0}C\left( W_0,r,T,\frac{j\beta (l)}{\alpha }\right) \mathrm {bin}\left( j;N_0,\pi ^l\right) F_\Delta (\mathrm {d}l), \end{aligned}$$

(6.2)

where the last equation is obtained after multiplying and dividing by $\frac{j}{N_0}$.

Note that Eq. (6.2) immediately highlights the valuation formula for the aggregate bonus option $N_0V_0^\text {b}$.

1.3.3 Market value of the default option

Manipulations similar to those in “Market value of the bonus option” section can be used to obtain the following expression for the default option value:

$$\begin{aligned} V_0^\text {d}= & E\left[ \mathrm {e}^{-rT}\left[ B-w\right] ^+1_{\{\tau ^i>T\}}\right] \\= & \frac{1}{N_0}\int _0^\infty \sum _{j=1}^{N_0}P\left( W_0,r,T,j\beta (l)\right) \mathrm {bin}\left( j;N_0,\pi ^l\right) F_\Delta (\mathrm {d}l). \end{aligned}$$

1.4 Valuation formulae in the large portfolio case

Recall that now $F_\Delta$ and E refer to the cumulative distribution function, respectively expectation operator, under the probability $Q= Q^\infty$.

1.4.1 Market value of the guaranteed amount

This is formally the same expression as in the case of a finite portfolio, Eq. (6.1):

$$\begin{aligned} V_0^\text {g}(\infty )= & E[\mathrm {e}^{-r T}B 1_{\{\tau ^i>T\}}]\\= & \mathrm {e}^{-r T}\int _0^\infty \beta (l)\pi ^l F_\Delta (\mathrm {d}l). \end{aligned}$$

1.4.2 Market value of the bonus option

Conditioning on $\Delta$ and exploiting the independence between financial and demographic factors, we obtain

$$\begin{aligned} V_0^\text {b}(\infty )= & E\left[ \mathrm {e}^{-rT}\left[ \frac{w_0(\infty )\mathrm {e}^R}{\pi ^\Delta }-\frac{B}{\alpha (\infty )}\right] ^+ 1_{\{\tau ^i>T\}}\right] \\= & E\left[ C\left( \frac{w_0(\infty )}{\pi ^\Delta },r,T,\frac{B}{\alpha (\infty )}\right) \pi ^\Delta \right] \\= & \int _0^{\infty }C\left( w_0(\infty ),r,T,\frac{\beta (l)\pi ^l}{\alpha (\infty )}\right) F_\Delta (\mathrm {d}l). \end{aligned}$$

1.4.3 Market value of the default option

Similarly as in “Market value of the bonus option” section, we have:

$$\begin{aligned} V_0^\text {d}(\infty )&=E\left[ \mathrm {e}^{-rT}\left[ B-\frac{w_0(\infty )\mathrm {e}^R}{\pi ^\Delta }\right] ^+1_{\{\tau ^i>T\}}\right] \\&=\int _0^{\infty }P\left( w_0(\infty ),r,T,\beta (l)\pi ^l\right) F_\Delta (\mathrm {d}l). \end{aligned}$$

1.5 Results relative to Sect. 4

1.5.1 Proof of Theorem 1

• 1. Write $N^{(N_0)}$ to stress the dependence of N on $N_0$. Note that $N^{(N_0+1)}\ge N^{(N_0)}$ almost surely and $\widetilde{Q}\left( N^{(N_0+1)}>N^{(N_0)}\right) >0$. It follows that $W_0^\epsilon$ increases with $N_0$. If the limit of $W_0^\epsilon$ as $N_0\rightarrow +\infty$ were finite, then, as $N^{(N_0)}\rightarrow +\infty$ a.s., we would have

  $$\begin{aligned} \widetilde{E}\left[ \widetilde{F}_R\left( \log \frac{NB}{W_0}\right) \right] \rightarrow 1, \end{aligned}$$

  contradicting (4.2).

• 2. Recall first that $N^{(N_0)}/N_0\rightarrow \widetilde{\pi }^\Delta >0$ and note that $B>0$. If $W_0^\epsilon /N_0\rightarrow w_0(\infty )$ then the expectation in (4.2) converges to

  $$\begin{aligned} \widetilde{E}\left[ \widetilde{F}_R\left( \log \frac{\widetilde{\pi }^\Delta B}{w_0(\infty )}\right) \right] . \end{aligned}$$

  As this limit is also equal to $\epsilon \in (0,1)$, it follows that $0<w_0(\infty )<+\infty$. Denote explicitly $W_0^\epsilon (N_0)$ the solution of (4.2) with respect to $N_0$. To prove that the limit of $W_0^\epsilon (N_0)/N_0$ exists, suppose there are two subsequences $(N_0')$ and $(N_0'')$ such that

  $$\begin{aligned} \frac{W_0^\epsilon (N_0')}{N_0'}\rightarrow w_0'(\infty ),\,\, \frac{W_0^\epsilon (N_0'')}{N_0''}\rightarrow w_0''(\infty ) \end{aligned}$$

  with $0<w_0'(\infty )<w_0''(\infty )<\infty$. Taking the limit in the expectation (4.2) under the two subsequences leads to two different limits while (4.2) states that both limits should coincide with $\epsilon$.

1.5.2 Calculation of $W_0^\epsilon$

For a finite portfolio, the expectation in (4.2) can be computed by

$$\begin{aligned} \widetilde{E}\left[ \widetilde{F}_R\left( \log \frac{NB}{W_0}\right) \right] =\int _0^\infty \sum _{j=0}^{N_0}\widetilde{F}_R\left( \log \frac{j \beta (l)}{W_0}\right) \text {bin}(j;N_0,\widetilde{\pi }^l)\widetilde{F}_\Delta (\text {d}l). \end{aligned}$$

In the infinite portfolio case, the expectation in (4.3) can be calculated by

$$\begin{aligned} \widetilde{E}\left[ \widetilde{F}_R\left( \log \frac{\widetilde{\pi }^\Delta B}{w_0(\infty )}\right) \right] =\int _0^\infty \widetilde{F}_R\left( \log \frac{\widetilde{\pi }^l \beta (l)}{w_0(\infty )}\right) \widetilde{F}_\Delta (\text {d}l). \end{aligned}$$