On the optimal product mix in life insurance companies using conditional value at risk. (English) Zbl 1231.91244

Summary: This paper proposes a Conditional Value-at-Risk Minimization (CVaRM) approach to optimize an insurer’s product mix. By incorporating the natural hedging strategy of Cox and Lin (2007) and the two-factor stochastic mortality model of A. J. G. Cairns, D. Blake and K. Dowd [Astin Bull. 36, No. 1, 79–120 (2006; Zbl 1162.91403)], we calculate an optimize product mix for insurance companies to hedge against the systematic mortality risk under parameter uncertainty. To reflect the importance of required profit, we further integrate the premium loading of systematic risk. We compare the hedging results to those using the duration match method of Wang et al. (forthcoming), and show that the proposed CVaRM approach has a narrower quantile of loss distribution after hedging – thereby effectively reducing systematic mortality risk for life insurance companies.


91B30 Risk theory, insurance (MSC2010)
91G10 Portfolio theory


Zbl 1162.91403
Full Text: DOI


[1] Artzner, P.; Delbaen, F.; Eber, J.; Heath, D., Thinking coherently, Risk, 10, 11, 68-71, (1997)
[2] Artzner, P.; Delbaen, F.; Eber, J.; Heath, D., Coherent measures of risk, Mathematical finance, 9, 3, 203-228, (1999) · Zbl 0980.91042
[3] Benjamin, B.; Soliman, A.S., Mortality on the move, (1993), Actuarial Education Service Oxford
[4] Biffis, E., Affine processes for dynamic mortality and actuarial valuations, Insurance: mathematics and economics, 37, 443-468, (2005) · Zbl 1129.91024
[5] Blake, D.; Burrows, W., Survivor bonds: helping to hedge mortality risk, Journal of risk and insurance, 68, 2, 339-348, (2001)
[6] Blake, D.; Cairns, A.J.G.; Dowd, K., Living with mortality: longevity bonds and other mortality-linked securities, British actuarial journal, 12, 153-197, (2006)
[7] Blake, D.; Cairns, A.J.G.; Dowd, K.; MacMinn, R., Longevity bonds: financial engineering, valuation, and hedging, Journal of risk and insurance, 73, 4, 647-672, (2006)
[8] 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
[9] Cairns, A.J.G., A discussion of parameter and model uncertainty in insurance, Insurance: mathematics and economics, 27, 313-330, (2000) · Zbl 0971.62063
[10] Cairns, A.J.G.; Blake, D.; Dowd, K., Pricing death: frameworks for the valuation and securitization of mortality risk, ASTIN bulletin, 36, 79-120, (2006) · Zbl 1162.91403
[11] Cairns, A.J.G.; Blake, D.; Dowd, K., A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration, Journal of risk and insurance, 73, 687-718, (2006)
[12] Cairns, A.J.G., Blake, D., Dowd, K., Goughlan, G.D., Epstein, D., Ong, A., Balevich, I., 2007. A Quantitative comparison of stochastic mortality models using data from England and Wales and the United States. Pensions Institute Discussion Paper I-0701. http://www.pensions-institute.org/workingpapers/wp0701.pdf
[13] Cowley, A.; Cummins, J.D., Securitization of life insurance assets and liabilities, Journal of risk and insurance, 72, 193-226, (2005)
[14] Cox, S.H.; Lin, Y., Natural hedging of life and annuity mortality risks, North American actuarial journal, 11, 3, 1-15, (2007)
[15] Cox, S.H.; Lin, Y.; Wang, S., Multivariate exponential tilting and pricing implications for mortality securitization, Journal of risk and insurance, 73, 719-736, (2006)
[16] 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
[17] Dahl, M.; Moller, T., Valuation and hedging of life insurance liabilities with systematic mortality risk, Insurance: mathematics and economics, 39, 193-217, (2006) · Zbl 1201.91089
[18] Denuit, M.; Devolder, P.; Goderniaux, A., Securitization of longevity risk: pricing survivor bonds with Wang transform in the Lee-Carter framework, Journal of risk and insurance, 74, 87-113, (2007)
[19] Deprez, O.; Gerber, H.U., On convex principles of premium calculation, Insurance: mathematics and economics, 4, 179-189, (1985) · Zbl 0579.62090
[20] Dowd, K., Survivor bonds: A comment on blake and Burrows, Journal of risk and insurance, 70, 2, 339-348, (2003)
[21] Dowd, K.; Blake, D., After var: the theory, estimation, and insurance applications of quantile-based risk measures, Journal of risk and insurance, 73, 2, 193-229, (2006)
[22] Dowd, K.; Blake, D.; Cairns, A.J.G.; Dawson, P., Survivor swaps, Journal of risk and insurance, 73, 1-17, (2006)
[23] Grundl, H.; Post, T.; Schulze, R.N., To hedge or not to hedge: managing demographic risk in life insurance companies, Journal of risk and insurance, 73, 1, 19-41, (2006)
[24] JPMorgan LifeMetrics, 2006. http://www.jpmorgan.com/pages/jpmorgan/investbk/solutions/lifemetrics
[25] Koissi, M.C.; Shapiro, A.F.; Hognas, G., Evaluating and extending the Lee-Carter model for mortality forecasting: bootstrap confidence interval, Insurance: mathematics and economics, 26, 1-20, (2006) · Zbl 1098.62138
[26] Lin, Y.; Cox, S.H., Securitization of mortality risks in life annuities, Journal of risk and insurance, 72, 227-252, (2005)
[27] McDonald, A.S.; Cairns, A.J.G.; Gwilt, P.L.; Miller, K.A., An international comparison of recent trends in the population mortality, British actuarial journal, 3, 3-141, (1998)
[28] Melnikov, A.; Romaniuk, Y., Evaluating the performance of Gompertz, Makeham and Lee-Carter mortality models for risk management, Insurance: mathematics and economics, 39, 310-329, (2006) · Zbl 1151.91577
[29] Milevsky, M.A.; Promislow, S.D., Mortality derivatives and the option to annuitize, Insurance: mathematics and economics, 29, 299-318, (2001) · Zbl 1074.62530
[30] 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, (2006)
[31] Renshaw, A.E.; Haberman, S., Lee-Carter mortality forecasting with age-specific enhancement, Insurance: mathematics and economics, 33, 255-272, (2003) · Zbl 1103.91371
[32] Rockafellar, R.T.; Uryasev, S., Optimization of conditional value-at-risk, Journal of risk, 2, 21-41, (2000)
[33] Schrager, D.F., Affine stochastic mortality, Insurance: mathematics and economics, 38, 81-97, (2006) · Zbl 1103.60063
[34] Stallard, E., Demographic issues in longevity risk analysis, Journal of risk and insurance, 73, 4, 575-609, (2006)
[35] Wang, J.L., Huang, H.C., Yang, S.S., Tsai, J.T., 2009. An optimal product mix for hedging longevity risk in life insurance companies: The immunization theory approach. Journal of Risk and Insurance (forthcoming)
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.