×

Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio. (English) Zbl 1152.91612

Summary: We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation with respect to an equivalent martingale measure. Via this representation, one can interpret the instantaneous Sharpe ratio as a market price of mortality risk. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. Thus, the price reflects the fact that systematic mortality risk cannot be eliminated by selling more life insurance policies. We present a numerical example to illustrate our results, along with the corresponding algorithms.

MSC:

91B30 Risk theory, insurance (MSC2010)
91B70 Stochastic models in economics
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Ballotta, L.; Haberman, S., The fair valuation problem of guaranteed annuity options: the stochastic mortality environment case, Insurance: mathematics and economics, 38, 195-214, (2006) · Zbl 1101.60045
[2] Bayraktar, E.; Young, V.R., Hedging life insurance with pure endowments, Insurance: mathematics and economics, 40, 3, 435-444, (2007) · Zbl 1183.91067
[3] Bayraktar, E., Young, V.R., 2007b. Pricing options in incomplete equity markets via the instantaneous Sharpe ratio. Working paper. Department of Mathematics, University of Michigan. Available at: http://arxiv.org/abs/math/0701650 · Zbl 1233.91256
[4] Biffis, E., Affine processes for dynamic mortality and actuarial valuation, Insurance: mathematics and economics, 37, 443-468, (2005) · Zbl 1129.91024
[5] Björk, T., Arbitrage theory in continuous time, (2004), Oxford University Press Oxford · Zbl 1140.91038
[6] Björk, T.; Slinko, I., Towards a general theory of good deal bounds, Review of finance, 10, 221-260, (2006) · Zbl 1125.91049
[7] Blanchet-Scalliet, C.; El Karoui, N.; Martellini, L., Dynamic asset pricing theory with uncertain time-horizon, Journal of economic dynamics and control, 29, 1737-1764, (2005) · Zbl 1198.91078
[8] Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; Nesbitt, C.J., Actuarial mathematics, (1997), Society of Actuaries Schaumburg, Illinois · Zbl 0634.62107
[9] Dahl, M., Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance: mathematics and economics, 35, 1, 113-136, (2004) · Zbl 1075.62095
[10] Dahl, M.; Møller, T., Valuation and hedging of life insurance liabilities with systematic mortality risk, Insurance: mathematics and economics, 39, 2, 193-217, (2006) · Zbl 1201.91089
[11] DuChateau, P.; Zachmann, D.W., Schaum’s outline of partial differential equations, (1986), McGraw-Hill New York
[12] Gavrilov, L.A.; Gavrilova, N.S., The biology of life span: A quantitative approach, (1991), Harwood Academic Publishers New York
[13] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, (1991), Springer-Verlag New York · Zbl 0734.60060
[14] Lamberton, D.; Lapeyre, B., Introduction to stochastic calculus applied to finance, (1996), Chapman & Hall/CRC Boca Raton, Florida · Zbl 0898.60002
[15] Lee, R.; Carter, L., Modeling and forecasting US mortality, Journal of the American statistical association, 87, 659-671, (1992) · Zbl 1351.62186
[16] Milevsky, M.A.; Promislow, S.D., Mortality derivatives and the option to annuitize, Insurance: mathematics and economics, 29, 3, 299-318, (2001) · Zbl 1074.62530
[17] Milevsky, M.A., Promislow, S.D., Young, V.R., 2005. Financial valuation of mortality risk via the instantaneous Sharpe ratio: Applications to pricing pure endowments, working paper. Department of Mathematics. University of Michigan. Available at: http://arxiv.org/abs/0705.1302
[18] Milevsky, M.A.; Promislow, S.D.; Young, V.R., Killing the law of large numbers: mortality risk premiums and the sharpe ratio, Journal of risk and insurance, 73, 4, 673-686, (2007)
[19] Olshansky, O.J.; Carnes, B.A.; Cassel, C., In search of methuselah: estimating the upper limits to human longevity, Science, 250, 634-640, (1990)
[20] Protter, P., Stochastic integration and differential equations, (2005), Springer Berlin
[21] Schrager, D.F., Affine stochastic mortality, Insurance: mathematics and economics, 38, 1, 81-97, (2006) · Zbl 1103.60063
[22] Schweizer, M., A guided tour through quadratic hedging approaches, (), 538-574 · Zbl 0992.91036
[23] Walter, W., Differential and integral inequalities, (1970), Springer-Verlag New York
[24] Young, V.R., 2007. Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio: Theorems and proofs. Working paper. Department of Mathematics. University of Michigan. Available at: http://arxiv.org/abs/0705.1297
[25] Zariphopoulou, T., Stochastic control methods in asset pricing, () · Zbl 1051.91512
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.