## It’s all in the hidden states: a longevity hedging strategy with an explicit measure of population basis risk.(English)Zbl 1371.91103

Summary: In this paper, we propose the generalized state-space hedging method for use when the populations associated with the hedging instruments and the liability being hedged are different. In this method, the hedging strategy is derived by first reformulating the assumed multi-population stochastic mortality model in a state-space representation, and then considering the sensitivities of the hedge portfolio and the liability being hedged to all relevant hidden states. Inter alia, this method allows us to decompose the underlying longevity risk into components arising solely from the hidden states that are shared by all populations and components stemming exclusively from the hidden states that are population-specific. The latter components collectively represent an explicit measure of the population basis risk involved. Through this measure, we can infer that a portion of population basis risk depends on how the longevity hedge is constructed while another portion exists no matter what the notional amounts of the hedging instruments are. We present the proposed hedging method in both static and dynamic settings.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 62P05 Applications of statistics to actuarial sciences and financial mathematics 91D20 Mathematical geography and demography

Human Mortality
Full Text:

### References:

 [1] Ahmadi, S.; Li, J. S.-H., Coherent mortality forecasting with generalized linear models: A modified time-transformation approach, Insurance Math. Econom., 59, 194-221, (2014) · Zbl 1306.91067 [2] Blake, D.; Cairns, A.; Coughlan, G.; Dowd, K.; MacMinn, R., The new life market, J. Risk Insur., 80, 501-557, (2013) [3] Blake, D.; Cairns, A.; Dowd, K.; MacMinn, R., Longevity bonds: financial engineering, valuation, and hedging, Journal of Risk and Insurance, 73, 647-672, (2006) [4] Börger, M., Deterministic shock vs. stochastic value-at-risk—-an analysis of the solvency II standard model approach to longevity risk, Blätter der DGVFM, 31, 2, 225-259, (2010) · Zbl 1232.91341 [5] Brouhns, N.; Denuit, M.; Van Keilegom, I., Bootstrapping the Poisson log-bilinear model for mortality forecasting, Scand. Actuar. J., 2005, 3, 212-224, (2005) · Zbl 1092.91038 [6] 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 [7] Cairns, A. J.G., Robust hedging of longevity risk, Journal of Risk and Insurance, 80, 621-648, (2013) [8] 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) [9] 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 [10] Cairns, A. J.G.; Blake, D.; Dowd, K.; Coughlan, G. D.; Khalaf-Allah, M., Bayesian stochastic mortality modelling for two populations, Astin Bull., 41, 29-59, (2011) [11] Canadian Institute of Actuaries, 2014. Final Report on Canadian Pensioners Mortality. Available at http://www.cia-ica.ca/docs/default-source/2014/214013e.pdf. [12] Coale, A.; Kisker, E., Defects in data on old-age mortality in the united states: new procedures for calculating mortality schedules and life tables at the highest ages, Asian Pac. Popul. Forum, 4, 131, (1990) [13] Continuous Mortality Investigation Bureau, 2009a. Working Papers 38, A Prototype Mortality Projections Model, Part One—An Outline of the Proposed Approach. Available at http://www.actuaries.org.uk/research-and-resources/documents/cmi-working-paper-38-prototype-mortality-projections-model-part-one. [14] Continuous Mortality Investigation Bureau, 2009b. Working Papers 39, A Prototype Mortality Projections Model, Part Two—Detailed Analysis. Available at http://www.actuaries.org.uk/research-and-resources/documents/cmi-working-paper-39-prototype-mortality-projections-model-part-two. [15] Coughlan, G., Longevity risk transfer: indices and capital market solutions, (Barrieu, P. M.; Albertini, L., The Handbook of Insurance Linked Securities, (2009), Wiley London) [16] 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) [17] Dahl, M., Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance Math. Econom., 35, 113-136, (2004) · Zbl 1075.62095 [18] 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 [19] 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 [20] D’Amato, V.; Haberman, S.; Piscopo, G.; Russolillo, M.; Trapani, L., Detecting common longevity trends by a multiple population approach, N. Am. Actuar. J., 18, 139-149, (2014) [21] Dowd, K., Blake, D., Cairns, A.J.G., Coughlan, G.D., 2011b. Hedging pension risks with the age-period-cohort two-population gravity model. In: Seventh International Longevity Risk and Capital Markets Solutions Conference, Frankfurt, September 2011. [22] Dowd, K.; Cairns, A. J.G.; Blake, D.; Coughlan, G. D.; Epstein, D.; Khalaf-Allah, M., A gravity model of mortality rates for two related populations, N. Am. Actuar. J., 15, 331-356, (2011) · Zbl 1228.91032 [23] Graziani, G., Longevity risk—A fine balance, Institutional Investor Journals: Special Issue on Pension and Longevity Risk Transfer for Institutional Investors, 2014, (2014), 35-27 [24] Hatzopoulos, P.; Haberman, S., Common mortality modeling and coherent forecasts. an empirical analysis of worldwide mortality data, Insurance Math. Econom., 52, 320-337, (2013) · Zbl 1284.91238 [25] Human Mortality Database. 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 1 January 2015). [26] Hunt, A.; Blake, D., Modelling longevity bonds: analysing the swiss re kortis bond, Insurance Math. Econom., 63, 12-29, (2015) · Zbl 1348.91150 [27] Jarner, S. F.; Kryger, E. M., Modelling adult mortality in small populations: the SAINT model, ASTIN Bulletin, 41, 377-418, (2011) · Zbl 1239.91128 [28] Lee, R. D.; Carter, L. R., Modeling and forecasting U.S. mortality, J. Amer. Statist. Assoc., 87, 659-671, (1992) · Zbl 1351.62186 [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, N.; Lee, R., Coherent mortality forecasts for a group of populations: an extension of the Lee-Carter method, Demography, 42, 575-594, (2005) [31] Li, J. S.-H.; Luo, A., Key q-duration: A framework for hedging longevity risk, ASTIN Bulletin, 42, 413-452, (2012) · Zbl 1277.91089 [32] Liu, Y., Li, J.S.-H., 2014. The locally-linear Cairns-Blake-Dowd model: A note on delta-nuga hedging of longevity risk. In: Tenth International Longevity Risk and Capital Markets Solutions Conference, Santiago, Chile, September 2014. · Zbl 1390.91198 [33] 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 [34] Michaelson, A.; Mulholland, J., Strategy for increasing the global capacity for longevity risk transfer: developing transactions that attract capital markets investors, J. Altern. Investment., 17, 18-27, (2014) [35] Ngai, A.; Sherris, M., Longevity risk management for life and variable annuities: the effectiveness of static hedging using longevity bonds and derivatives, Insur. Math. Econ., 49, 100-114, (2011) [36] Olivieri, A.; Pitacco, E., Stochastic mortality: the impact on target capital, ASTIN Bulletin, 39, 541-563, (2009) · Zbl 1179.91108 [37] Plat, R., One-year value-at-risk for longevity and mortality, Insurance Math. Econom., 49, 462-470, (2011) [38] Richards, S. J.; Currie, I. D.; Ritchie, G. P., A value-at-risk framework for longevity trend risk, Br. Actuar. J., 19, 116-139, (2014) [39] Society of Actuaries, 2014. Mortality Improvement Scale MP-2014 Report. Available at https://www.soa.org/Files/Research/Exp-Study/research-2014-mp-report.pdf. [40] Tan, K. S.; Blake, D.; MacMinn, R., Longevity risk and capital markets: the 2013-14 update, Insurance Math. Econom., 63, 1-11, (2015) · Zbl 1321.00138 [41] Yang, B.; Li, J.; Balasooriya, U., Using bootstrapping to incorporate model error for risk-neutral pricing of longevity risk, Insurance Math. Econom., 62, 16-27, (2015) · Zbl 1318.91126 [42] Yang, S. S.; Wang, C. W., Pricing and securitization of multi-country longevity risk with mortality dependence, Insurance Math. Econom., 52, 157-169, (2013) · Zbl 1284.91556 [43] Zhou, K. Q.; Li, J. S.-H., Dynamic longevity hedging in the presence of population basis risk: A feasibility analysis from technical and economic perspectives, Journal of Risk and Insurance, (2016) [44] Zhou, R.; Li, J. S.-H.; Tan, K. S., Pricing mortality risk: A two-population model with transitory jump effects, Journal of Risk and Insurance, 80, 733-774, (2013) [45] Zhou, R.; Wang, Y.; Kaufhold, K.; Li, J. S.-H.; Tan, K. S., Modeling period effects in multi-population mortality models: applications to solvency II, N. Am. Actuar. J., 18, 150-167, (2014)
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