Skip to main content
Log in

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

  • Original Research Paper
  • Published:
European Actuarial Journal Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others


  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 40-years 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. See for instance [11, 12].


  1. Ballotta L (2005) A Lévy process-based framework for the fair valuation of participating life insurance contracts. Insur Math Econ 37(2):173–196

    Article  MathSciNet  Google Scholar 

  2. Ballotta L, Haberman S (2003) Valuation of guaranteed annuity conversion options. Insur Math Econ 33(1):87–108

    Article  MathSciNet  Google Scholar 

  3. Ballotta L, Haberman S (2006) The fair valuation problem of guaranteed annuity options: the stochastic mortality environment case. Insur Math Econ 38(1):195–214

    Article  MathSciNet  Google Scholar 

  4. Ballotta L, Esposito G, Haberman S (2006a) The IASB insurance project for life insurance contracts: impact on reserving methods and solvency requirements. Insur Math Econ 39(3):356–375

    Article  MathSciNet  Google Scholar 

  5. Ballotta L, Haberman S, Wang N (2006b) Guarantees in with-profit and unitized with-profit life insurance contracts: fair valuation problem in presence of the default option. J Risk Insur 73(1):97–121

    Article  Google Scholar 

  6. Barrieu P, Bensusan H, Karoui NE, Hillairet C, Loisel S, Ravanelli C, Salhi Y (2012) Understanding, modelling and managing longevity risk: key issues and main challenges. Scand Actuar J 2012(3):203–231

    Article  MathSciNet  Google Scholar 

  7. Bernard C, Le Courtois O, Quittard-Pinon F (2005) Market value of life insurance contracts under stochastic interest rates and default risk. Insur Math Econ 36(3):499–516

    Article  MathSciNet  Google Scholar 

  8. Biffis E (2005) Affine processes for dynamic mortality and actuarial valuations. Insur Math Econ 37(3):443–468

    Article  MathSciNet  Google Scholar 

  9. Biffis E, Denuit M, Devolder P (2010) Stochastic mortality under measure changes. Scand Actuar J 2010(4):284–311

    Article  MathSciNet  Google Scholar 

  10. Brémaud P (1981) Point processes and queues: martingale dynamics. Springer, New York, Heidelberg and Berlin

    Book  Google Scholar 

  11. Briys E, de Varenne F (1994) Life insurance in a contingent claim framework: pricing and regulatory implications. Geneva Pap Risk Insur Theory 19(1):53–72

    Article  Google Scholar 

  12. Briys E, de Varenne F (1997) On the risk of life insurance liabilities: debunking some common pitfalls. J Risk Insur 64(4):673–694

    Article  Google Scholar 

  13. Cairns AGC, Blake DP, Dowd K (2008) Modelling and management of mortality risk: a review. Scand Actuar J 2008(2–3):79–113

    Article  MathSciNet  Google Scholar 

  14. Chen A, Suchanecki M (2007) Default risk, bankruptcy procedures and the market value of life insurance liabilities. Insur Math Econ 40(2):231–255

    Article  MathSciNet  Google Scholar 

  15. Denuit M, Dhaene J, Goovaerts M, Kaas R (2006) Actuarial theory for dependent risks: measures, orders and models. Wiley, New York

    Google Scholar 

  16. Fung MC, Ignatieva K, Sherris M (2014) Systematic mortality risk: an analysis of guaranteed lifetime withdrawal benefits in variable annuities. Insur Math Econ 58:103–115

    Article  MathSciNet  Google Scholar 

  17. Gatzert N, Wesker H (2014) Mortality risk and its effect on shortfall and risk management in life insurance. J Risk Insur 81(1):57–90

    Article  Google Scholar 

  18. Grosen A, Jørgensen P (2002) Life insurance liabilities at market value: an analysis of insolvency risk, bonus policy, and regulatory intervention rules in a barrier option framework. J Risk Insur 69(1):63–91

    Article  Google Scholar 

  19. Haberman S, Olivieri A (2008) Risk classification/life. The encyclopedia of quantitative risk assessment and analysis. Wiley, New York, pp 1535–1540

    Google Scholar 

  20. Hari N, de Waegenaere AMB, Melenberg B, Nijman TE (2008) Longevity risk in portfolios of pension annuities. Insur Math Econ 42(2):505–519

    Article  Google Scholar 

  21. Lee R, Carter L (1992) Modeling and forecasting US mortality. J Ame Stat Assoc 87(419):659–671

    MATH  Google Scholar 

  22. Ngai A, Sherris M (2011) Longevity risk management for life and variable annuities: the effectiveness of static hedging using longevity bonds and derivatives. Insur Math Econ 49(1):100–114

    Article  MathSciNet  Google Scholar 

  23. Pitacco E, Denuit M, Haberman S, Olivieri A (2009) Modelling longevity dynamics for pensions and annuity business. Oxford University Press, Oxford

    MATH  Google Scholar 

  24. Schervish M (1995) Theory of statistics. Springer, Berlin

    Book  Google Scholar 

  25. Stevens R, de Waegenaere AMB, Melenberg B (2010) Longevity risk in pension annuities with exchange options: the effect of product design. Insur Math Econ 46(1):222–234

    Article  MathSciNet  Google Scholar 

  26. Zhu N, Bauer D (2011) Applications of forward mortality factor models in life insurance practice. Geneva Pap Risk Insur Issues Pract 36(4):567–594

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to An Chen.



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.


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(ArTK) and P(ArTK) 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

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

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,

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

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. 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. 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

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: