Skip to main content
Log in

A Model-Point Approach to Indifference Pricing of Life Insurance Portfolios with Dependent Lives

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

In this paper, we study the pricing of life insurance portfolios in the presence of dependent lives. We assume that an insurer with an initial exposure to n mortality-contingent contracts wanted to acquire a second portfolio constituted of m contracts. The policyholders’ lifetimes in these portfolios are correlated with a Farlie-Gumbel-Morgenstern (FGM) copula, which induces a dependency between the two portfolios. In this setting, we compute the indifference price charged by the insurer endowed with an exponential utility. The indifference price is characterized as a solution to a backward stochastic differential equation (BSDE), which can be decomposed into (n − 1) n! auxiliary BSDEs. In this general case, the derivation of the indifference price is computationally infeasible. Therefore, while focusing on the example of death benefit contracts, we develop a model-point based approach in order to ease the computation of the price. It consists on replacing each portfolio with a single policyholder that replicates some risk metrics of interest. Also, the two representative contracts should adequately reproduce the observed dependency between the initial portfolios. We implement the proposed procedure and compare the computed prices to classical valuation approach.

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.

Institutional subscriptions

Similar content being viewed by others

References

  • Barrieu P, Bensusan H, El Karoui N, 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  MATH  Google Scholar 

  • Bayraktar E, Milevsky MA, Promislow SD, Young VR (2009) Valuation of mortality risk via the instantaneous sharpe ratio: applications to life annuities. J Econ Dyn Control 33(3):676–691

    Article  MathSciNet  MATH  Google Scholar 

  • Becherer D (2003) Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insur: Math Econ 33(1):1–28

    MathSciNet  MATH  Google Scholar 

  • Cairns AJG, Blake D, Dowd K (2004) Pricing frameworks for securitization of mortality risk. In: Proceedings of the 14th AFIR Colloquium, vol 1, pp 509–540

  • Chevalier E, Lim T, Roméro RR (2014) Indifference fee rate for variable annuities

  • Cox S, Lin Y (2007) Natural hedging of life and annuity mortality risks. North Amer. Actuar J 11(3):1–15

    Article  MathSciNet  Google Scholar 

  • Delong Ł (2012) No-good-deal, local mean-variance and ambiguity risk pricing and hedging for an insurance payment process. Astin Bullet 42(01):203–232

    MathSciNet  MATH  Google Scholar 

  • El Karoui N, Jeanblanc M, Jiao Y (2013) Density approach in modelling multi-defaults. preprint hal-00870492

  • Genest C, Neṡlehová J, Ben Ghorbal N (2011) Estimators based on kendall’s tau in multivariate copula models. Aust Z J Stat 53(2):157–177

    Article  MathSciNet  MATH  Google Scholar 

  • Gobet E, Lemor J-P, Warin X (2005) A regression-based monte carlo method to solve backward stochastic differential equations. Ann Appl Probab 15(3):2172–2202

    Article  MathSciNet  MATH  Google Scholar 

  • Goffard P-Q, Guerrault X (2015) Is it optimal to group policyholders by age, gender, and seniority for bel computations based on model points? Eur Actuar J 1–16

  • Hodges SD, Neuberger A (1989) Optimal replication of contingent claims under transaction costs. Rev Futur Mark 8(2):222–239

    Google Scholar 

  • Hu Y, Imkeller P, Muller M (2005) Utility maximization in incomplete markets. Ann Probab 15(3):1691–1712

    Article  MathSciNet  MATH  Google Scholar 

  • Jaworski P, Durante E, Hardle WK, Rychlik T (2010) Copula theory and its applications. Springer, Berlin

    Book  MATH  Google Scholar 

  • Jeulin T (1980) Semi-martingales et grossissement d’une filtration. Lecture Notes in Mathematics, 833 Springer-Verlag Berlin Heidelberg

  • Kchia Y (2011) Semimartingales and contemporary issues in quantitative finance. PhD thesis, Ecole Polytechnique X

  • Kharroubi I, Lim T (2011) A decomposition approach for the discrete-time approximation of fbsdes with a jump i: the lipschitz case. arXiv:1103.3029

  • Kharroubi I, Lim T (2012) A decomposition approach for the discrete-time approximation of bsdes with a jump ii: the quadratic case. arXiv:1211.6231

  • Kharroubi I, Lim T, Ngoupeyou A (2013) Mean-variance hedging on uncertain time horizon in a market with a jump. Appl Math Optim 68(3):413–444

    Article  MathSciNet  MATH  Google Scholar 

  • Kharroubi I, Lim T (2014) Progressive enlargement of filtrations and backward stochastic differential equations with jumps. J Theor Probab 27(3):683–724

    Article  MathSciNet  MATH  Google Scholar 

  • Lee RD, Carter LR (1992) Modeling and forecasting us mortality. J Amer Stat Assoc 87(419):659–671

    MATH  Google Scholar 

  • Ludkovski M, Young VR (2008) Indifference pricing of pure endowments and life annuities under stochastic hazard and interest rates. Insur: Math Econ 42(1):14–30

    MathSciNet  MATH  Google Scholar 

  • Marceau E, Gaillardetz P (1999) On life insurance reserves in a stochastic mortality and interest rates environment. Insur: Math Econ 25(3):261–280

    MATH  Google Scholar 

  • Milevsky MA, Promislow SD (2001) Mortality derivatives and the option to annuitise. Insur: Math Econ 29(3):299–318

    MathSciNet  MATH  Google Scholar 

  • Remillard B (2013) Statistical Methods for Financial Engineering. CRC Press

  • Young VR (2003) Equity-indexed life insurance: pricing and reserving using the principle of equivalent utility. North Amer Actuar J 7(1):68–86

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work is funded by ANR Research Project “LoLitA” (ANR-13-BS01-0011). Y. Salhi’s work has been supported by the BNP Paribas Cardif Chair ”Data Analytics and Models in Insurance”. The views expressed in this document are the authors owns and do not necessarily reflect those endorsed by BNP Paribas Cardif. The authors would like to thank the anonymous referee for his/her careful reading and his/her remarks which improved the presentation of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yahia Salhi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Blanchet-Scalliet, C., Dorobantu, D. & Salhi, Y. A Model-Point Approach to Indifference Pricing of Life Insurance Portfolios with Dependent Lives. Methodol Comput Appl Probab 21, 423–448 (2019). https://doi.org/10.1007/s11009-017-9611-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-017-9611-2

Keywords

Mathematics Subject Classification (2010)

Navigation