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Hedging longevity risk under non-Gaussian state-space stochastic mortality models: A mean-variance-skewness-kurtosis approach

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  • Li, Johnny Siu-Hang
  • Liu, Yanxin
  • Chan, Wai-Sum

Abstract

Longevity risk has recently become a high profile risk among insurers and pension plan sponsors. One way to mitigate longevity risk is to build a hedge using derivatives that are linked to mortality indexes. Longevity hedging methods are often based on the normality assumption, considering only the variance but no other (higher) moments. However, strong empirical evidence suggests that mortality improvement rates are driven by asymmetric and fat-tailed distributions, so that existing longevity hedging methods should be expanded to incorporate higher moments. This paper fills the gap by adopting a mean-variance-skewness-kurtosis approach based on non-Gaussian extensions of commonly-used stochastic mortality models, formulated in a state-space setting. On the basis of a general representation of these models, the authors derive (approximate) analytical expressions for the moments of the present values of the hedging instruments and the liability being hedged. These expressions are then integrated with a polynomial goal programming model, from which the optimal hedge portfolio is identified. Finally, the paper demonstrates the theoretical results with a real mortality data set and a range of hedger preferences.

Suggested Citation

  • Li, Johnny Siu-Hang & Liu, Yanxin & Chan, Wai-Sum, 2023. "Hedging longevity risk under non-Gaussian state-space stochastic mortality models: A mean-variance-skewness-kurtosis approach," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 96-121.
  • Handle: RePEc:eee:insuma:v:113:y:2023:i:c:p:96-121
    DOI: 10.1016/j.insmatheco.2023.08.001
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    as
    1. Fung, Man Chung & Peters, Gareth W. & Shevchenko, Pavel V., 2017. "A unified approach to mortality modelling using state-space framework: characterisation, identification, estimation and forecasting," Annals of Actuarial Science, Cambridge University Press, vol. 11(2), pages 343-389, September.
    2. Eric Jondeau & Michael Rockinger, 2006. "Optimal Portfolio Allocation under Higher Moments," European Financial Management, European Financial Management Association, vol. 12(1), pages 29-55, January.
    3. Andrew Cairns & David Blake & Kevin Dowd & Guy Coughlan & David Epstein & Alen Ong & Igor Balevich, 2009. "A Quantitative Comparison of Stochastic Mortality Models Using Data From England and Wales and the United States," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(1), pages 1-35.
    4. Kim, Sungil & Kim, Heeyoung, 2016. "A new metric of absolute percentage error for intermittent demand forecasts," International Journal of Forecasting, Elsevier, vol. 32(3), pages 669-679.
    5. Gao, Quansheng & Hu, Chengjun, 2009. "Dynamic mortality factor model with conditional heteroskedasticity," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 410-423, December.
    6. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
    7. Lee, Tom K Y & Tse, Y K, 1991. "Term Structure of Interest Rates in the Singapore Asian Dollar Market," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 6(2), pages 143-152, April-Jun.
    8. Jackie Li, 2013. "A Poisson common factor model for projecting mortality and life expectancy jointly for females and males," Population Studies, Taylor & Francis Journals, vol. 67(1), pages 111-126, March.
    9. Sun, Qian & Yan, Yuxing, 2003. "Skewness persistence with optimal portfolio selection," Journal of Banking & Finance, Elsevier, vol. 27(6), pages 1111-1121, June.
    10. David Blake & Andrew Cairns & Guy Coughlan & Kevin Dowd & Richard MacMinn, 2013. "The New Life Market," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 80(3), pages 501-558, September.
    11. Zhang, Jingong & Tan, Ken Seng & Weng, Chengguo, 2017. "Optimal hedging with basis risk under mean–variance criterion," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 1-15.
    12. Wang, Zihe & Li, Johnny Siu-Hang, 2016. "A DCC-GARCH multi-population mortality model and its applications to pricing catastrophic mortality bonds," Finance Research Letters, Elsevier, vol. 16(C), pages 103-111.
    13. Gurdip Bakshi & Nikunj Kapadia & Dilip Madan, 2003. "Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options," The Review of Financial Studies, Society for Financial Studies, vol. 16(1), pages 101-143.
    14. Ling, Shiqing & McAleer, Michael, 2002. "Stationarity and the existence of moments of a family of GARCH processes," Journal of Econometrics, Elsevier, vol. 106(1), pages 109-117, January.
    15. repec:bla:jfinan:v:44:y:1989:i:4:p:889-908 is not listed on IDEAS
    16. Goliński, Adam & Zaffaroni, Paolo, 2016. "Long memory affine term structure models," Journal of Econometrics, Elsevier, vol. 191(1), pages 33-56.
    17. Campbell Harvey & John Liechty & Merrill Liechty & Peter Muller, 2010. "Portfolio selection with higher moments," Quantitative Finance, Taylor & Francis Journals, vol. 10(5), pages 469-485.
    18. Johnny Li & Mary Hardy, 2011. "Measuring Basis Risk in Longevity Hedges," North American Actuarial Journal, Taylor & Francis Journals, vol. 15(2), pages 177-200.
    19. Wai-Sum Chan & Johnny Li & Jackie Li, 2014. "The CBD Mortality Indexes: Modeling and Applications," North American Actuarial Journal, Taylor & Francis Journals, vol. 18(1), pages 38-58.
    20. Justel, Ana & Peña, Daniel & Zamar, Rubén, 1997. "A multivariate Kolmogorov-Smirnov test of goodness of fit," Statistics & Probability Letters, Elsevier, vol. 35(3), pages 251-259, October.
    21. Prakash, Arun J. & Chang, Chun-Hao & Pactwa, Therese E., 2003. "Selecting a portfolio with skewness: Recent evidence from US, European, and Latin American equity markets," Journal of Banking & Finance, Elsevier, vol. 27(7), pages 1375-1390, July.
    22. Chou-Wen Wang & Hong-Chih Huang & I-Chien Liu, 2013. "Mortality Modeling With Non-Gaussian Innovations and Applications to the Valuation of Longevity Swaps," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 80(3), pages 775-798, September.
    23. Liu, Yanxin & Li, Johnny Siu-Hang, 2018. "A strategy for hedging risks associated with period and cohort effects using q-forwards," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 267-285.
    24. Leung, Mark T. & Daouk, Hazem & Chen, An-Sing, 2001. "Using investment portfolio return to combine forecasts: A multiobjective approach," European Journal of Operational Research, Elsevier, vol. 134(1), pages 84-102, October.
    25. Yanxin Liu & Johnny Siu-Hang Li, 2021. "An Efficient Method for Mitigating Longevity Value-at-Risk," North American Actuarial Journal, Taylor & Francis Journals, vol. 25(S1), pages 309-340, February.
    26. Genton, Marc G. & He, Li & Liu, Xiangwei, 2001. "Moments of skew-normal random vectors and their quadratic forms," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 319-325, February.
    27. Sears, R Stephen & Wei, K C John, 1985. "Asset Pricing, Higher Moments, and the Market Risk Premium: A Note," Journal of Finance, American Finance Association, vol. 40(4), pages 1251-1253, September.
    28. Liu, Yanxin & Li, Johnny Siu-Hang, 2016. "It’s all in the hidden states: A longevity hedging strategy with an explicit measure of population basis risk," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 301-319.
    29. Andrew J. G. Cairns & David Blake & Kevin Dowd, 2006. "A Two‐Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 73(4), pages 687-718, December.
    30. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
    31. Andrew J.G. Cairns & Kevin Dowd & David Blake & Guy D. Coughlan, 2014. "Longevity hedge effectiveness: a decomposition," Quantitative Finance, Taylor & Francis Journals, vol. 14(2), pages 217-235, February.
    32. Shripad Tuljapurkar & Nan Li & Carl Boe, 2000. "A universal pattern of mortality decline in the G7 countries," Nature, Nature, vol. 405(6788), pages 789-792, June.
    33. Peiro, Amado, 1999. "Skewness in financial returns," Journal of Banking & Finance, Elsevier, vol. 23(6), pages 847-862, June.
    34. Plat, Richard, 2011. "One-year Value-at-Risk for longevity and mortality," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 462-470.
    35. Duan, Jin-Chuan, 2016. "Local-momentum autoregression and the modeling of interest rate term structure," Journal of Econometrics, Elsevier, vol. 194(2), pages 349-359.
    36. Chou-Wen Wang & Hong-Chih Huang & I-Chien Liu, 2011. "A Quantitative Comparison of the Lee-Carter Model under Different Types of Non-Gaussian Innovations," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 36(4), pages 675-696, October.
    37. Li, Johnny Siu-Hang & Luo, Ancheng, 2012. "Key Q-Duration: A Framework for Hedging Longevity Risk," ASTIN Bulletin, Cambridge University Press, vol. 42(2), pages 413-452, November.
    38. Li, Johnny Siu-Hang & Zhou, Rui & Hardy, Mary, 2015. "A step-by-step guide to building two-population stochastic mortality models," Insurance: Mathematics and Economics, Elsevier, vol. 63(C), pages 121-134.
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    More about this item

    Keywords

    Higher moments; State-space models; Polynomial goal programming; Longevity risk;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods

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