Evaluating the performance of Gompertz, Makeham and Lee-Carter mortality models for risk management with unit-linked contracts. (English) Zbl 1151.91577

Summary: The paper compares the performance of three mortality models in the context of optimal pricing and hedging of unit-linked life insurance contracts. Two of the models are the classical parametric results of Gompertz and Makeham, the third is the recently developed method of R. D. Lee and L. R. Carter [“Modeling and forecasting U.S. mortality”, J. Am. Stat. Assoc. 419, No. 87, 659–675 (1992)] for fitting mortality and forecasting it as a stochastic process. First, quantile hedging techniques of H. Föllmer and P. Leukert [Finance Stoch. 4, No. 2, 117–146 (2000; Zbl 0956.60074) and Finance Stoch. 3, No. 3, 251–273 (1999; Zbl 0977.91019)] are applied to price a unit-linked contract with payoff conditioned on the client’s survival to the contract’s maturity. Next, the paper analyzes the implications of the three mortality models on risk management possibilities for the insurance firm based on numerical illustrations with the Toronto Stock Exchange/Standard and Poor financial index and mortality data for the USA, Sweden and Japan. The strongest differences between the models are observed in Japan, where the lowest mortality for the next two decades is expected. The general mortality decline patterns, rectangularization of the survival curve and deceleration of mortality at older ages, are well pronounced in the results for all three countries.


91B30 Risk theory, insurance (MSC2010)
91G70 Statistical methods; risk measures
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Aase, K.K.; Persson, S.A, Pricing of unit-linked life insurance policies, Scandinavian actuarial journal, 1, 26-52, (1994) · Zbl 0814.62067
[2] Bacinello, A.R.; Ortu, F., Pricing of unit-linked life insurance with endogenous minimum guarantees, Insurance: mathematics and economics, 12, 245-257, (1993) · Zbl 0778.62093
[3] Biffis, E., Affine processes for dynamic mortality and actuarial valuations, Insurance: mathematics and economics, 37, 443-468, (2005) · Zbl 1129.91024
[4] Black, F.; Scholes, M., Pricing of options and corporate liabilities, Journal of political economy, 81, 3, 637-654, (1973) · Zbl 1092.91524
[5] Bolder, D.J., Johnson, G., Metzler, A., 2004. An empirical analysis of the Canadian term structure of zero-coupon interest rates. Bank of Canada: Working Paper 2004-48. Available at www.bankofcanada.ca
[6] Bongaarts, J., Population aging and the rising cost of public pensions, Population and development review, 30, 1, 1-23, (2004)
[7] Bowers, N.L.; Gerber, H.U.; Hickman, G.C.; Jones, D.A.; Nesbit, C.J., Actuarial mathematics, (1997), The Society of Actuaries Schaumburg, Illinois
[8] Boyle, P.P.; Hardy, M.R., Reserving for maturity guarantees: two approaches, Insurance: mathematics and economics, 21, 113-127, (1997) · Zbl 0894.90044
[9] Brennan, M.J.; Schwartz, E.S., The pricing of equity-linked life insurance policies with an asset value guarantee, Journal of financial economics, 3, 195-213, (1976)
[10] Brouhns, N.; Denuit, M.; Vermunt, J.K., A Poisson log-bilinear regression approach to the construction of projected life tables, Insurance: mathematics and economics, 31, 373-393, (2002) · Zbl 1074.62524
[11] Brouhns, N.; Denuit, M.; Van Keilegom, I., Bootstrapping the Poisson log-bilinear model for mortality forecasting, Scandinavian actuarial journal, 3, 212-224, (2005) · Zbl 1092.91038
[12] Cox, J.; Ingersoll, J.; Ross, S., A theory of the term structure of interest rates, Econometrica, 53, 2, 385-408, (1985) · Zbl 1274.91447
[13] Dahl, M., Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance: mathematics and economics, 35, 113-136, (2004) · Zbl 1075.62095
[14] Davis, M., 2002. Multi-asset options. Department of Mathematics, Imperial College London
[15] Ekern, S.; Persson, S.-A., Exotic unit-linked life insurance contracts, The Geneva papers on risk and insurance theory, 21, 35-63, (1996)
[16] Föllmer, H.; Leukert, P., Quantile hedging, Finance and stochastics, 3, 251-273, (1999) · Zbl 0977.91019
[17] Heligman, L.; Pollard, J.H., The age pattern of mortality, Journal institute of actuaries, 107, 49-75, (1980)
[18] Higgins, T., 2003. Mathematical models of mortality. Presented at Workshop on Mortality Modelling and Forecasting. Australian National University
[19] Horiuchi, Sh.; Wilmoth, J.R., Deceleration in the age pattern of mortality at older ages, Demography, 35, 4, 391-412, (1998)
[20] Hull, J.C., Fundamentals of futures and options markets, (2005), Pearson/Prentice Hall New Jersey, USA · Zbl 1019.91025
[21] Human mortality database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org (data downloaded between February and September, 2005)
[22] Kirch, M.; Melnikov, A.V., Efficient hedging and pricing life insurance policies in a jump-diffusion model, Stochastic analysis and applications, 23, 6, 1213-1233, (2005) · Zbl 1125.91054
[23] Koissi, M.-C.; Shapiro, A.F.; Högnäs, G., Evaluating and extending the lee – carter model for mortality forecasting: bootstrap confidence interval, Insurance: mathematics and economics, 38, 1-20, (2006) · Zbl 1098.62138
[24] Korn, G.A.; Korn, T.M., Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review, (2000), Dover Publications New York · Zbl 0535.00032
[25] Lee, R.D., The lee – carter method for forecasting mortality with various extensions and applications, North American actuarial journal, 1, 4, 80-91, (2000) · Zbl 1083.62535
[26] Lee, R.D.; Carter, L.R., Modelling and forecasting U.S. mortality, Journal of the American statistical association, 87, 14, 659-675, (1992) · Zbl 1351.62186
[27] Luciano, E., Vigna, E., 2005. Non mean-reverting affine processes for stochastic mortality. Paper presented at the 15th International AFIR Colloquium. Zurich, Switzerland
[28] Lynch, S.M.; Brown, J.S., Reconsidering mortality compression and deceleration: an alternative model of mortality rates, Demography, 38, 1, 79-95, (2001)
[29] Margrabe, W., The value of an option to exchange one asset for another, Journal of finance, 33, 177-186, (1978)
[30] Melnikov, A.V.; Romaniuk, Yu.V.; Skornyakova, V.S., Margrabe’s formula and quantile hedging of life insurance contracts, Doklady mathematics, 71, 1, 31-34, (2005), Doklady Academii Nauk 400 (2), 1-4 (in Russian)
[31] Moeller, T., Risk-minimizing hedging strategies for unit-linked life insurance contracts, ASTIN bulletin, 28, 17-47, (1998) · Zbl 1168.91417
[32] Pitacco, E., 2003. Survival models in actuarial mathematics: from Halley to longevity risk. In: Invited lecture at 7th International Congress Insurance: Mathematics and Economics, ISFA, Lyon
[33] Pollard, J.H., Mathematical models for the growth of human populations, (1973), Cambridge University Press London, Great Britain · Zbl 0295.92013
[34] Tuljapurkar, Sh.; Li, N.; Boe, C., A universal pattern of mortality decline in the G7 countries, Nature, 405, 789-792, (2000)
[35] Wilmoth, J.R., 1993. Computational methods for fitting and extrapolating the Lee-Carter model of mortality change. Technical Report. University of California, Berkeley, USA
[36] Wilmoth, J.R.; Horiuchi, Sh., Rectangularization revisited: variability of age at death within human populations, Demography, 36, 4, 475-495, (1999)
[37] Wong-Fupuy, C.; Haberman, S., Projecting mortality trends: recent developments in the united kingdom and the united states, North American actuarial journal, 8, 2, 56-83, (2004) · Zbl 1085.62517
[38] Vasicek, O., An equilibrium characterization of the term structure, Journal of financial economics, 5, 2, 177-188, (1977) · Zbl 1372.91113
[39] Yashin, A.I.; Begun, A.S.; Boiko, S.I.; Ukraintseva, S.V.; Oeppen, J., The new trends in survival improvement require a revision of traditional gerontological concepts, Experimental gerontology, 37, 157-167, (2001)
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