Mortality risk management under the factor copula framework – with applications to insurance policy pools. (English) Zbl 1466.91263

The paper focuses on the mortality risk source and its impact on the life insurers’ business. This subject is extremely important, since the dependence structure among the uncertainty of future lifetime of each insured determines the aggregate risk of an insurance policy pool. In particular, this study proposes factor copulas to describe the dependence structure among the future lifetimes of numerous insureds. After introducing the main valuation formulas and dependence structures between a life settlement fund and a life insurance pool, the paper illustrates the data, the assumptions behind and the numerical analyses. The core of the study consists in solving for the optimal investment amount in the fund with respect to different risk measures. Numerical illustrations, supported by appropriate robustness checks, show concrete applications of the results presented in the paper.


91G05 Actuarial mathematics
Full Text: DOI


[1] Asmussen, S.; Glynn., P. W., Stochastic simulation: Algorithms and analysis (2007), New York, NY: Springer, New York, NY · Zbl 1126.65001
[2] Azzalini, A., A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171-78 (1985) · Zbl 0581.62014
[3] Azzalini, A.; Dalla Valle, A., The multivariate skew-normal distribution, Biometrika, 83, 715-26 (1996) · Zbl 0885.62062
[4] Bahna-Nolan, M.; Ritzke, C., Valuation basic table (VBT) report. Society of Actuaries (2008)
[5] Benjamin, B.; Soliman, A. S., Mortality on the move: Methods of mortality projection (1993), Oxford, UK: Institute of Actuaries, Oxford, UK
[6] Blake, D.; Burrows., W., Survivor bonds: Helping to hedge mortality risk, Journal of Risk and Insurance, 68, 339-48 (2001)
[7] Blake, D.; Cairns, A. J. G.; Dowd., K., Living with mortality: Longevity bonds and other mortality-linked securities, British Actuarial Journal, 12, 153-97 (2006)
[8] Blake, D.; Cairns, A. J. G.; Dowd., K., Longevity bonds: Financial engineering, valuation, and hedging, Journal of Risk and Insurance, 73, 647-72 (2006)
[9] Braun, A.; Gatzert, N.; Schmeiser., H., Performance and risks of open-end life settlement funds, Journal of Risk and Insurance, 79, 193-230 (2012)
[10] Burtschell, X.; Gregory, J.; Laurent., J. P., A comparative analysis of CDO pricing models under the factor copula framework, Journal of Derivatives, 16, 9-37 (2009)
[11] 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
[12] 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)
[13] Cairns, A. J. G.; Blake, D.; Dowd, K.; Coughlan, G. D.; Epstein, D.; Khalaf-Allah., M., Mortality density forecasts: An analysis of six stochastic mortality models, Insurance: Mathematics and Economics, 48, 355-67 (2011)
[14] Carriere, J. F., Bivariate survival models for coupled lives, Scandinavian Actuarial Journal, 1, 17-31 (2000) · Zbl 0959.62094
[15] Chan, J. C. C.; Kroese., D. P., Efficient estimation of large portfolio loss probabilities in t-copula models, European Journal of Operational Research, 205, 361-67 (2010) · Zbl 1188.91231
[16] Chaplin, G.; Aspinwall, J.; Venn, M., Life settlements and longevity structures: Pricing and risk management (2011), Hoboken, NJ: Wiley Finance, Hoboken, NJ
[17] Chen, H.; MacMinn, R.; Sun., T., Multi-population mortality models: A factor copula approach, Insurance: Mathematics and Economics, 63, 135-46 (2015) · Zbl 1348.91131
[18] Cherubini, U.; Luciano, E.; Vecchiato, W., Copula Methods in Finance (2004), West Sussex, UK: John Wiley & Sons, West Sussex, UK · Zbl 1163.62081
[19] Chiang, M. H.; Yueh, M. L.; Hsieh., M. H., An efficient algorithm for basket default swap valuation, Journal of Derivatives, 15, 8-19 (2007)
[20] Cousin, A.; Laurent., J. P.; Cont, R., Frontiers in quantitative finance. Credit risk and volatility modeling, An overview of factor models for pricing CDO tranches, 185-216 (2008), Chichester, UK: John Wiley & Sons, Chichester, UK
[21] Cowley, A.; Cummins., J. D., Securitization of life insurance assets and liabilities, Journal of Risk and Insurance, 72, 193-226 (2005)
[22] Cox, S. H.; Lin., Y., Natural hedging of life and annuity mortality risks, North American Actuarial Journal, 11, 1-15 (2007)
[23] Cox, S. H.; Lin, Y.; Wang., S. N., Multivariate exponential tilting and pricing implications for mortality securitization, Journal of Risk and Insurance, 73, 719-36 (2006)
[24] Denuit, M.; Dhaene, J.; Le Bailly de Tilleghem, C.; Teghem, S., Measuring the impact of dependence among insured lifelengths, Belgian Actuarial Bulletin, 1, 18-39 (2001)
[25] Denuit, M.; Devolder, P.; Goderniaux., A. C., Securitization of longevity risk: Pricing survivor bonds with Wang transform in the Lee-Carter framework, Journal of Risk and Insurance, 74, 87-113 (2007)
[26] Dowd, K., Survivor bonds: A comment on Blake and Burrows,, Journal of Risk and Insurance, 70, 339-48 (2003)
[27] Dowd, K.; Blake, D.; Cairns, A. J. G.; Dawson., P., Survivor swaps, Journal of Risk and Insurance, 73, 1-17 (2006)
[28] Eling, M.; Kiesenbauer., D., What policy features determine life insurance lapse? An analysis of the German market, Journal of Risk and Insurance, 81, 241-69 (2014)
[29] Embrechts, P.; Lindskog, F.; McNeil., A.; Rachev, S., Handbook of heavy tailed distributions in finance, Modeling dependence with copulas and applications to risk management (2003), Amsterdam, The Netherlands: Elsevier, Amsterdam, The Netherlands
[30] Frees, E. W.; Carriere, J. F.; Valdez., E. A., Annuity valuation with dependent mortality, Journal of Risk and Insurance, 63, 229-61 (1996)
[31] Frees, E. W.; Valdez., E. A., Understanding relationships using copulas, North American Actuarial Journal, 2, 1-25 (1998) · Zbl 1081.62564
[32] Gatzert, N., The secondary market for life insurance in the U.K., Germany, and the U.S.: Comparison and overview, Risk Management and Insurance Review, 13, 279-301 (2010)
[33] Giacalone, J. A., Analyzing an emerging industry: Viatical transactions and the secondary market for life insurance policies, Southern Business Review, 27, 1-7 (2001)
[34] Glasserman, P.; Kang, W.; Shahabuddin., P., Fast simulation of multifactor portfolio credit risk, Operations Research, 56, 1200-17 (2008) · Zbl 1167.91369
[35] Hougaard, P., Analysis of multivariate survival data (Statistics for biology and health) (2000), New York, NY: Springer, New York, NY
[36] Hull, J.; White., A., Valuation of a CDO and an nth to default CDs without Monte Carlo simulation, Journal of Derivatives, 12, 8-23 (2004)
[37] Ingraham, H. G., and S. S. Salani. 2004. Life settlements as a viable option. Journal of Financial Service Professionals 58: 72-6.
[38] Klugman, S.; Panjer, H.; Willmot, G., Loss models: From data to decisions (2008), Hoboken, NJ: Wiley, Hoboken, NJ · Zbl 1159.62070
[39] Kogure, A.; Kurachi., Y., A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions, Insurance: Mathematics and Economics, 46, 162-72 (2010) · Zbl 1231.91438
[40] Laurent, J. P.; Gregory., J., Basket default swaps, CDOs and factor copulas, Journal of Risk, 7, 1-20 (2005)
[41] Lee, R. D.; Carter., L. R., Modeling and forecasting U.S. mortality, Journal of the American Statistical Association, 87, 659-71 (1992) · Zbl 1351.62186
[42] Leimberg, S. R., M. D. Weinberg, B. T. Weinberg, and C. J. Callahan. 2008. Life settlements: Know when to hold and know when to fold. Journal of Financial Service Professionals 62: 61-72.
[43] Li, J. S.; Hardy., M., Measuring basis risk in longevity hedges, North American Actuarial Journal, 15, 177-200 (2011) · Zbl 1228.91042
[44] Li, J. S. H., Pricing longevity risk with the parametric bootstrap: a maximum entropy approach, Insurance: Mathematics and Economics, 47, 176-86 (2010) · Zbl 1231.91441
[45] Li, J. S. H.; Ng, A. C. Y., Canonical valuation of mortality-linked securities, Journal of Risk and Insurance, 78, 853-84 (2011)
[46] Li, N.; Lee., R., Coherent mortality forecasts for a group of population: An extension of the Lee-Carter method, Demography, 42, 575-94 (2005)
[47] Lin, Y.; Cox., S. H., Securitization of mortality risks in life annuities, Journal of Risk and Insurance, 72, 227-52 (2005)
[48] 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)
[49] Milhaud, X.; Loisel, S.; Maume-Deschamps, V., Surrender triggers in life insurance: Classification and risk predictions, Bulletin Français d’Actuariat, 22, 5-48 (2011)
[50] Oh, D. H.; Patton., A. J., Modelling dependence in high dimensions with factor copulas, Journal of Business & Economic Statistics, 35, 139-54 (2015)
[51] Oh, D. H.; Patton., A. J., Simulated method of moments estimation for copula-based multivariate models, Journal of the American Statistical Association, 108, 689-700 (2013) · Zbl 06195971
[52] Pinquet, J.; Guillen, M.; Ayuso., M., Commitment and lapse behavior in long-term insurance: A case study, Journal of Risk and Insurance, 78, 983-1002 (2011)
[53] Renshaw, A. E.; Haberman., S., Lee-Carter mortality forecasting with age specific enhancement, Insurance: Mathematics and Economics, 33, 255-72 (2003) · Zbl 1103.91371
[54] Seitel, C. L. 2006. Inside the life settlement industry: An institutional investor’s perspective. Journal of Structured Finance 12: 38-40.
[55] Shemyakin, A.; Youn., H., Copula models of joint last survivor analysis, Applied Stochastic Models in Business and Industry, 22, 211-24 (2006) · Zbl 1127.62091
[56] Smith, B. B., and S. L. Washington. 2006. Acquiring life insurance portfolios: Diversifying and minimizing risk. Journal of Structured Finance 12: 41-5.
[57] Stone, C. A.; Zissu., A., Securitization of senior life settlements: Managing extension risk, Journal of Derivatives, 13, 66-72 (2006)
[58] Stone, C. A.; Zissu., A., Using life extension-duration and life extension-convexity to value senior life settlement contracts, Journal of Alternative Investments, 11, 94-108 (2008)
[59] Stutzer, M., A simple nonparametric approach to derivative security valuation,, Journal of Finance, 51, 1633-52 (1996)
[60] Wang, J. L.; Huang, H. C.; Yang, S. S.; Tsai., J. T., An optimal product mix for hedging longevity risk in life insurance companies: The immunization theory approach, Journal of Risk and Insurance, 77, 473-97 (2010)
[61] Wang, C. W., S. W. Yang, and H. C. Huang. 2015. Modeling multi-country mortality dependence and its application in pricing survivor index swaps - A dynamic copula approach. Insurance: Mathematics and Economics 63: 30-9. · Zbl 1348.62249
[62] Youn, H.; Shemyakin, A., Statistical aspects of joint life insurance pricing, 1999 Proceedings of the Business and Statistics Section of the American Statistical Association, 34-38 (1999)
[63] Youn, H.; Shemyakin, A., Pricing practices for joint last survivor insurance, Actuarial Research Clearing House, 2001, 1 (2001)
[64] Ziser, B. 2006. Life settlements today: A secret no more. Journal of Structured Finance12 (2): 35-7.
[65] Ziser, B. 2007. An eventful year in the life settlement industry. Journal of Structured Finance13 (2): 40-3.
[66] Zhu, W.; Tan, K. S.; Wang., C. W., Modeling multi-country longevity risk with mortality dependence: A Levy subordinated hierarchical Archimedean copulas approach, Journal of Risk and Insurance, 84, 477-93 (2017)
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.