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.
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Notes
 1.
The probability that a portfolio be completely extinct at maturity is negligible for usual ages and maturities and reasonable portfolio sizes. For instance, with a survival probability of \(95\%\) (which may be common for a 40years old policyholder and a 20 years horizon), the probability of extinction is less than \(10^{6}\) for a group of 5 individuals. When the survival probability is only \(50\%\), the extinction probability is less than \(10^{6}\) for a group of 20 individuals.
 2.
Note that the indicator of the event \({\{N>0\}}\) can be omitted in presence of the indicator of the event \(\{\tau ^i>T\}\).
 3.
The function m is nonnegative, continuous, and satisfies \(\int _0^{+\infty } m(u)\text {d}u=+\infty \).
 4.
Formally, the random variable \(\Delta \) is measurable with respect to the \(\sigma \)algebra containing the information available to market participants at time T.
 5.
This result also holds under any probability measure equivalent to Q, in particular under the physical measure.
 6.
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Appendix
Appendix
Properties of \(\tau _i\) and N
Law of \(\tau _i\)
The survival probability of a policyholder is given by
where \(\mathscr {L}_\Delta \) is the momentgenerating function of \(\Delta \), i.e. \(\mathscr {L}_\Delta (y)=E[\text {e}^{\Delta y}]\).
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
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:
\(\square \)
Law of N
The number of survivors N has, conditionally on \(\Delta \), a binomial distribution:
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 \(j=0,1,\ldots ,N_0\),
where \(\mathrm {bin}(j;M,p)=\left( {\begin{array}{c}M\\ j\end{array}}\right) p^j(1p)^{Mj}\) is the mass function of a Binomial random variable with parameters \(M\ge 1\) and \(0<p<1\).
Market value of the unitary annuity
Under Assumptions 1 and 2, the market value of the unitary annuity \(a_T\) is
where the function a is given by:
Note that a(l) is the value of a continuous annuity with force of mortality \(l\,m\).
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
Market value of the guaranteed amount
Conditioning on \(\Delta \), it follows that
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
By further conditioning on \(N^{(i)}\) the inner expectation and exploiting again Assumption 4,
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 \(N_0V_0^\text {b}\).
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:
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 \(Q= Q^\infty \).
Market value of the guaranteed amount
This is formally the same expression as in the case of a finite portfolio, Eq. (6.1):
Market value of the bonus option
Conditioning on \(\Delta \) and exploiting the independence between financial and demographic factors, we obtain
Market value of the default option
Similarly as in “Market value of the bonus option” section, we have:
Results relative to Sect. 4
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 \(\epsilon \).
Calculation of \(W_0^\epsilon \)
For a finite portfolio, the expectation in (4.2) can be computed by
In the infinite portfolio case, the expectation in (4.3) can be calculated by
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Bacinello, A.R., Millossovich, P. & Chen, A. The impact of longevity and investment risk on a portfolio of life insurance liabilities. Eur. Actuar. J. 8, 257–290 (2018). https://doi.org/10.1007/s1338501801755
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Keywords
 Solvency
 Longevity risk
 Investment risk
 Fair valuation
 Participating life insurance