A strategy for hedging risks associated with period and cohort effects using q-forwards. (English) Zbl 1398.91344

Summary: The stochastic nature of future mortality arises from both period (time-related) and cohort (year-of-birth-related) effects. Existing index-based longevity hedging strategies mitigate the risk associated with period effects, but often overlook cohort effects. The negligence of cohort effects may lead to sub-optimal hedge effectiveness, if the liability being hedged is a deferred pension or annuity which involves cohorts that are not covered by the data sample. In this paper, we propose a new hedging strategy that incorporates both period and cohort effects. The resulting longevity hedge is a value hedge, reducing the uncertainty surrounding the \(\tau\)-year ahead value of the liability being hedged. The proposed method is illustrated with data from the male population of England and Wales. It is found that the benefit of incorporating cohort effects into a longevity hedging strategy depends heavily on the persistence of cohort effects and the choice of q-forwards.


91B30 Risk theory, insurance (MSC2010)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Blake, D.; Cairns, A.; Coughlan, G.; Dowd, K.; MacMinn, R., The new life market, J. Risk Insurance, 80, 501-557, (2013)
[2] Blake, D.; MacMinn, R.; Li, J. S.-H.; Hardy, M., Longevity risk and capital markets: the 2012-2013 update, N. Am. Actuar. J., 18, 501-557, (2014)
[3] Box, G. E.P.; Jenkins, G. M., Time Series Analysis: Forecasting and Control, (1976), Holden-Day San Francisco · Zbl 0363.62069
[4] Cairns, A. J.G., Modelling and management of longevity risk: approximations to survival functions and dynamic hedging, Insurance Math. Econom., 49, 438-453, (2011) · Zbl 1230.91068
[5] Cairns, A. J.G., Robust hedging of longevity risk, J. Risk Insurance, 80, 621-648, (2013)
[6] Cairns, A. J.G.; Blake, D.; Dowd, K., A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration, J. Risk Insurance, 73, 687-718, (2006)
[7] Cairns, A. J.G.; Blake, D.; Dowd, K.; Coughlan, G. D., Longevity hedge effectiveness: A decomposition, Quant. Finance, 14, 217-235, (2014) · Zbl 1294.91072
[8] 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 Math. Econom., 48, 355-367, (2011)
[9] Cairns, A. J.G.; Blake, D.; Dowd, K.; Coughlan, G. D.; Epstein, D.; Ong, A.; Balevich, I., A quantitative comparison of stochastic mortality models using data from england and wales and the united states, N. Am. Actuar. J., 13, 1-35, (2009)
[10] Cairns, A. J.G.; Blake, D.; Dowd, K.; Kessler, A. R., Phantoms never die: living with unreliable population data, J. Roy. Statist. Soc. Ser. A, (2016)
[11] Cairns, A.; Blake, D.; Dowd, K.; MacMinn, R., Longevity bonds: financial engineering, valuation, and hedging, J. Risk Insurance, 73, 647-672, (2006)
[12] Chuang, S.-L.; Brockett, P. L., Modeling and pricing longevity derivatives using stochastic mortality rates and the esscher transform, N. Am. Actuar. J., 18, 22-37, (2014) · Zbl 1412.91040
[13] Continuous Mortality Investigation Bureau 2002. An Interim Basis for Adjusting the “92” Series Mortality Projections for cohort Effects. CMI Working Paper 1. London: Institute of Actuaries and Faculty of Actuaries.
[14] Coughlan, G.; Blake, D.; MacMinn, R.; Cairns, A. J.G.; Dowd, K., Longevity risk and hedging solutions, (Handbook of Insurance, (2013), Springer New York), 997-1035
[15] Coughlan, G. D.; Khalaf-Allah, M.; Ye, Y.; Kumar, S.; Cairns, A. J.G.; Blake, D.; Dowd, K., Longevity hedging 101: A framework for longevity basis risk analysis and hedge effectiveness, N. Am. Actuar. J., 15, 150-176, (2011)
[16] Dahl, M., Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance Math. Econom., 35, 113-136, (2004) · Zbl 1075.62095
[17] Dahl, M.; Melchior, M.; Møller, T., On systematic mortality risk and risk minimization with mortality swaps, Scand. Actuar. J., 108, 114-146, (2008) · Zbl 1224.91054
[18] Dahl, M.; Møller, T., Valuation and hedging of life insurance liabilities with systematic mortality risk, Insurance Math. Econom., 39, 193-217, (2006) · Zbl 1201.91089
[19] Dowd, K.; Cairns, A. J.G.; Blake, D.; Coughlan, G. D.; Epstein, D.; Khalaf-Allah, M., Evaluating the goodness of fit of stochastic mortality models, Insurance Math. Econom., 47, 255-265, (2010) · Zbl 1231.91179
[20] Dowd, K.; Cairns, A. J.G.; Blake, D.; Coughlan, G. D.; Epstein, D.; Khalaf-Allah, M., Backtesting stochastic mortality models: an ex-post evaluation of multi-period-ahead density forecasts, N. Am. Actuar. J., 14, 281-298, (2010)
[21] Gong, G.; Webb, A., Evaluating the advanced life deferred annuity-an annuity people might actually buy, Insurance Math. Econom., 46, 210-221, (2010) · Zbl 1231.91189
[22] Haberman, S., Kaishev, V., Millossovich, P., Villegas, A., Baxter, S., Gaches, A., Gunnlaugsson, S., Sison, M., 2014. Longevity basis risk: A methodology for assessing basis risk. Research investigation and report by Cass Business School and Hymans Robertson LLP for the Institute and Faculty of Actuaries and the Life and Longevity Markets Association. Available at http://www.actuaries.org.uk/research-and-resources/documents/sessional-research-event-longevity-basis-risk-methodology.
[23] Holmes, E. C., Enhanced: 1918 and all that, Science, 303, 1787-1788, (2004)
[24] Horneff, W.; Maurer, R.; Rogalla, R., Dynamic portfolio choice with deferred annuities, J. Bank. Finance, 34, 2652-2664, (2010)
[25] Human Mortality Database, 2015. University of California Berkeley (USA) and Max Planck Institute of Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de (data downloaded on 01.04.15).
[26] Hunt, A., Blake, D., 2015. forward mortality rates in discrete time I: calibration and securities pricing. Pensions Institute Discussion Paper PI-1511.
[27] Hunt, A., Blake, D., 2016. Forward mortality rates in discrete time II: longevity risk and hedging strategies. Pensions Institute Discussion Paper PI-1602.
[28] Li, J. S.-H.; Chan, W.-S.; Zhou, R., Semicoherent multipopulation mortality modeling: the impact on longevity risk securitization, J. Risk and Insurance, (2016)
[29] Li, J. S.-H.; Hardy, M. R., Measuring basis risk in longevity hedges, N. Am. Actuar. J., 15, 177-200, (2011) · Zbl 1228.91042
[30] Li, J. S.-H.; Hardy, M. R.; Tan, K. S., Uncertainty in mortality forecasting: an extension to the classical Lee-Carter approach, ASTIN Bull., 39, 137-164, (2009) · Zbl 1203.91113
[31] Li, N.; Lee, R., Coherent mortality forecasts for a group of populations: an extension of the Lee-Carter method, Demography, 42, 575-594, (2005)
[32] Li, J. S.-H.; Luo, A., Key q-duration: A framework for hedging longevity risk, ASTIN Bull., 42, 413-452, (2012) · Zbl 1277.91089
[33] Liu, Y., Li, J.S.-H., 2014. The locally-linear cairns-blake-dowd model: A note on delta-nuga hedging of longevity risk. Paper presented at the 10th International Longevity Risk and Capital Markets Solutions Conference, Santiago, Chile.
[34] Luciano, E.; Regis, L.; Vigna, E., Delta-gamma hedging of mortality and interest rate risk, Insurance Math. Econom., 50, 402-412, (2012) · Zbl 1237.91134
[35] Milevsky, M., Real longevity insurance with a deductible: introduction to advanced-life delayed annuities, N. Am. Actuar. J., 9, 109-122, (2005) · Zbl 1215.91036
[36] Ngai, A.; Sherris, M., Longevity risk management for life and variable annuities: the effectiveness of static hedging using longevity bonds and derivatives, Insurance Math. Econom., 49, 100-114, (2011)
[37] Richards, S. J., Detecting year-of-birth mortality patterns with limited data, J. Roy. Statist. Soc. Ser. A, 171, 279-298, (2008)
[38] Villegas, A.M., Millossovich, P., Kaishev, V.K., 2016. StMoMo: An R Package for Stochastic Mortality Modelling. R package version 0.3.1. URL http://CRAN.R-project.org/package=StMoMo.
[39] Zhou, K.Q., Li, J.S.-H., 2014. Dynamic Longevity hedging in the presence of population basis risk: A feasibility analysis from technical and economic perspectives. Paper presented at the 10th International Longevity Risk and Capital Markets Solutions Conference, Santiago, Chile.
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