## Longevity Greeks: what do insurers and capital market investors need to know?(English)Zbl 1465.91099

The authors derive three longevity Greeks – delta, gamma, and vega – on the basis of an extended version of the Lee-Carter model that incorporates stochastic volatility. The properties of each longevity Greek is studied and the levels of effectiveness that different longevity Greek hedges can possibly achieve are estimated.
The results reveal several facts. For example, in a delta-vega hedge formed by $$q$$-forwards, the choice of reference ages does not materially affect hedge effectiveness, but the choice of times to maturity does. These facts may aid insurers to better formulate their hedge portfolios and issuers of mortality-linked securities to determine what security structures are more likely to attract liquidity.

### MSC:

 91G05 Actuarial mathematics 62P05 Applications of statistics to actuarial sciences and financial mathematics

### Keywords:

longevity Greeks; $$q$$-forward; Greek hedging strategies

### Software:

LifeMetrics; FinTS
Full Text:

### References:

 [1] Brouhns, N., Bootstrapping the Poisson log-bilinear model for mortality forecasting, Scandinavian Actuarial Journal, 2005, 3, 212-24 (2005) · Zbl 1092.91038 [2] Brouhns, N., A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance: Mathematics and Economics, 31, 3, 373-93 (2002) · Zbl 1074.62524 [3] Cairns, A. J. G., Modelling and management of longevity risk: Approximations to survivor functions and dynamic hedging, Insurance: Mathematics and Economics, 49, 3, 438-53 (2011) · Zbl 1230.91068 [4] Cairns, A. J. G., Robust hedging of longevity risk, Journal of Risk and Insurance, 80, 3, 621-48 (2013) [5] Cairns, A. J. G., A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration, Journal of Risk and Insurance, 73, 4, 687-718 (2006) [6] Cairns, A. J. G., Longevity hedge effectiveness: A decomposition, Quantitative Finance, 14, 2, 217-35 (2014) · Zbl 1294.91072 [7] Chai, C. M. H., A double-exponential GARCH model for stochastic mortality, European Actuarial Journal, 3, 2, 385-406 (2013) [8] Chen, H., Multi-population mortality models: A factor copula approach, Insurance: Mathematics and Economics, 63, 135-46 (2015) · Zbl 1348.91131 [9] Coughlan, G. D., The handbook of insurance-linked securities, Longevity risk transfer: Indices and capital market solutions, 261-81 (2009) [10] Coughlan, G. D., LifeMetrics: A toolkit for measuring and managing longevity and mortality risks (2007) [11] Coughlan, G. D., Longevity hedging 101, North American Actuarial Journal, 15, 2, 150-76 (2011) [12] Crépey, S., Delta-hedging vega risk?, Quantitative Finance, 4, 5, 559-79 (2004) · Zbl 1405.91604 [13] Dahl, M., Mixed dynamic and static risk-minimization with an application to survivor swaps, European Actuarial Journal, 1, Suppl.2, 233-60 (2011) [14] Dahl, M., On systematic mortality risk and risk-minimization with survivor swaps, Scandinavian Actuarial Journal, 2-3, 114-46 (2008) · Zbl 1224.91054 [15] Dahl, M., Valuation and hedging of life insurance liabilities with systematic mortality risk, Insurance: Mathematics and Economics, 39, 2, 193-217 (2006) · Zbl 1201.91089 [16] Engle, R. F., GARCH gamma, Journal of Derivatives, 2, 4, 47-59 (1995) [17] Engle, R. F., Testing the volatility term structure using option hedging criteria, Journal of Derivarives, 8, 1, 10-28 (2000) [18] Gao, Q., Dynamic mortality factor model with conditional heteroskedasticity, Insurance: Mathematics and Economics, 45, 3, 410-23 (2009) · Zbl 1231.91187 [19] Giacometti, R., A comparison of the Lee-Carter model and AR-ARCH model for forecasting mortality rates, Insurance: Mathematics and Economics, 50, 1, 85-93 (2012) · Zbl 1235.91089 [20] Graziani, G., Longevity risk—A fine balance, Institutional Investor, 1, 35-37 (2014) [21] Javaheri, A., GARCH and volatility swaps, Quantitative Finance, 4, 5, 589-95 (2004) · Zbl 1405.91578 [22] Koissi, M. C., Evaluating and extending the Lee-Carter model for mortality forecasting: Bootstrap confidence interval, Insurance: Mathematics and Economics, 38, 1, 1-20 (2006) · Zbl 1098.62138 [23] Lee, R. D., Modeling and forecasting U.S. mortality, Journal of the American Statistical Association, 87, 419, 659-71 (1992) · Zbl 1351.62186 [24] Lee, R., Evaluating the performance of the Lee-Carter method for forecasting mortality, Demography, 38, 4, 537-49 (2001) [25] Lehar, A., GARCH vs. stochastic volatility: Option pricing and risk management, Journal of Banking & Finance, 26, 323-45 (2002) [26] Li, J., A quantitative comparison of simulation strategies for mortality projection, Annals of Actuarial Science, 8, 2, 281-97 (2014) [27] Li, J. S.-H., Measuring basis risk in longevity hedges, North American Actuarial Journal, 15, 2, 177-200 (2011) · Zbl 1228.91042 [28] Li, J. S.-H., Key Q-duration: A framework for hedging longveity risk, ASTIN Bulletin, 42, 2, 413-52 (2012) [29] Li, J. S.-H., Canonical valuation of mortality-linked securities, Journal of Risk and Insurance, 78, 4, 853-84 (2011) [30] Li, N., Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method, Demography, 42, 3, 575-94 (2005) [31] Lin, T., On the mortality/longevity risk hedging with mortality immunization, Insurance: Mathematics and Economics, 53, 3, 580-96 (2013) · Zbl 1290.91093 [32] Lin, T., Applications of mortality durations and convexities in natural hedges, North American Actuarial Journal, 18, 3, 417-42 (2014) · Zbl 1414.91215 [33] Liu, X., Investigating mortality uncertainty using the block bootstrap, Journal of Probability and Statistics, 2010, 813583 (2010) [34] Luciano, E., Efficient versus inefficient hedging strategies in the presence of financial and longevity (value at) risk, Insurance: Mathematics and Economics, 55, 1, 68-77 (2014) · Zbl 1296.91164 [35] Luciano, E., Delta-gamma hedging of mortality and interest rate risk, Insurance: Mathematics and Economics, 50, 3, 402-12 (2012) · Zbl 1237.91134 [36] Luciano, E., Single- and cross-generation natural hedging of longevity and financial risk, Journal of Risk and Insurance, 84, 3, 961-86 (2017) [37] McDonald, R. L., Derivatives markets (2012), Boston: Pearson, Boston [38] Michaelson, A., Strategy for increasing the global capacity for longevity risk transfer: Developing transactions that attract capital markets investors, The Journal of Alternative Investments, 17, 1, 18-27 (2014) [39] Ngai, A., Longevity risk management for life and variable annuities: The effectiveness of static hedging using longevity bonds and derivatives, Insurance: Mathematics and Economics, 49, 1, 100-114 (2011) [40] Plat, R., One-year Value-at-Risk for longevity and mortality, Insurance: Mathematics and Economics, 49, 3, 462-70 (2011) [41] Renshaw, A. E., On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling, Insurance: Mathematics and Economics, 42, 2, 797-816 (2008) · Zbl 1152.91598 [42] Rosa, C. D., Basis risk in static versus dynamic longevity-risk hedging, Scandinavian Actuarial Journal, 2017, 4, 343-65 (2017) · Zbl 1401.91129 [43] Tsai, C. C. L., Actuarial applications of the linear hazard transform in mortality immunization, Insurance: Mathematics and Economics, 53, 1, 48-63 (2013) · Zbl 1284.91272 [44] Tsai, C. C.-L., Actuarial applications of the linear hazard transform in life contingencies, Insurance: Mathematics and Economics, 49, 1, 70-80 (2011) · Zbl 1218.91073 [45] Tsai, J. T., On the optimal product mix in life insurance companies using conditional value at risk, Insurance: Mathematics and Economics, 46, 1, 235-41 (2010) · Zbl 1231.91244 [46] Tsay, R. S., Analysis of financial time series (2005), Hoboken, NJ: John Wiley & Sons · Zbl 1086.91054 [47] Wang, J. L., An optimal product mix for hedging longevity risk in life insurance companies: The immunization theory approach, Journal of Risk and Insurance, 77, 2, 473-97 (2010) [48] Wang, Z., A DCC-GARCH multi-population mortality model and its applications to pricing catastrophic mortality bonds, Finance Research Letters, 16, 103-11 (2016) [49] Wong, T. W., Time-consistent mean-variance hedging of longevity risk: Effect of cointegration, Insurance: Mathematics and Economics, 56, 56-67 (2014) · Zbl 1304.91136 [50] Yang, B., Using bootstrapping to incorporate model error for risk-neutral pricing of longevity risk, Insurance: Mathematics and Economics, 62, 16-27 (2015) · Zbl 1318.91126 [51] Zhou, K. Q., Dynamic longevity hedging in the presence of population basis risk: A feasibility analysis from technical and economic perspectives, Journal of Risk and Insurance, 84, Suppl. 1, 417-37 (2017)
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