×

Pricing and securitization of multi-country longevity risk with mortality dependence. (English) Zbl 1284.91556

Summary: To deal with multi-country longevity risk, this article investigates the long-run equilibrium of mortality rates and introduces mortality correlations across countries as a means for pricing a multi-country longevity bond. The examination of the long-run equilibrium of the mortality rate relies on co-integration analysis, and a vector error correction model (VECM) is proposed for mortality forecasts. Mortality correlations among different countries under a VECM model are then derived. We take into account the mortality correlations across countries and utilize the multivariate Wang transform to derive the valuation formula for pricing the longevity bonds, with payoffs based on a combined weighted mortality index. This study illustrates the pattern of mortality correlations for men and women in the US and the UK, according to the Human Mortality Database. Our results show that mortality correlations across countries have a significant impact on pricing longevity bonds.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
91D20 Mathematical geography and demography
91B30 Risk theory, insurance (MSC2010)

Software:

LifeMetrics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Biffis, E., Blake, D., 2009. Mortality-linked securities and derivatives. Discussion Paper PI-0901. The Pensions Institute.
[2] Biffis, E.; Denuit, M.; Devolder, P., Stochastic mortality under measure changes, Scandinavian Actuarial Journal, 4, 284-311, (2010) · Zbl 1226.91022
[3] Blake, D., Cairns, A.J.G., Coughlan, G.D., Dowd, K., MacMinn, R., 2012. The new life market. Working Paper.
[4] Blake, D.; Cairns, A. J.G.; Dowd, K., Living with mortality: longevity bonds and other mortality-linked securities, British Actuarial Journal, 12, 153-197, (2006)
[5] Blake, D.; Dawson, P.; Dowd, K.; Cairns, A. J.G., Survivor derivatives: a consistent pricing framework, Journal of Risk and Insurance, 77, 579-596, (2010)
[6] Blake, D.; Dowd, K.; Cairns, A. J.G., Longevity risk and the grim reaper’s toxic tail, Insurance: Mathematics and Economics, 42, 1062-1066, (2008) · Zbl 1141.91485
[7] Brouhns, N.; Denuit, M.; Vermunt, J. K., A Poisson log-bilinear regression approach to the construction of projected life tables, Insurance: Mathematics and Economics, 31, 373-393, (2002) · Zbl 1074.62524
[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.; Epstein, D.; Ong, A.; Balevich, I., A quantitative comparison of stochastic mortality models using data from england and wales and the united states, North American Actuarial Journal, 13, 1-35, (2009)
[10] Chen, H.; Cox, S. H.; Wang, S. S., Is the home equity conversion mortgage in the united states sustainable? evidence from pricing mortgage insurance premiums and non-recourse provisions using the conditional esscher transform, Insurance: Mathematics and Economics, 46, 2, 371-384, (2010) · Zbl 1231.91154
[11] Cox, S. H.; Lin, Y.; Wang, S., Multivariate exponential tilting and pricing implications for mortality securitization, Journal of Risk and Insurance, 73, 719-736, (2006)
[12] Darkiewicz, G., Hoedemakers, T., 2004. How the Co-integration Analysis can Help in Mortality Forecasting. Actuarial Science Research Group, Catholic Univ. of Leuven.
[13] 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)
[14] Dickey, D. A.; Fuller, W. A., Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, 49, 1057-1072, (1981) · Zbl 0471.62090
[15] Dowd, K.; Blake, D.; Cairns, A. J.G.; Dawson, P., Survivor swaps, Journal of Risk and Insurance, 73, 1-17, (2006)
[16] Engle, R. F.; Granger, C. W.J., Cointegration and error-correction: representation, estimation and testing, Econometrica, 55, 251-276, (1987) · Zbl 0613.62140
[17] Gaille, S., Sherris, M., 2010. Improving longevity and mortality risk models with common stochastic long-run trends. Working Paper.
[18] Gerber, H. U.; Shiu, E. S.W., Option pricing by esscher transforms, Transactions of the Society of Actuaries, 46, 99-191, (1994)
[19] Hainaut, D., Multidimensional Lee-Carter model with switching mortality processes, Insurance: Mathematics and Economics, 50, 236-246, (2012) · Zbl 1235.91091
[20] Heligman, L.; Pollard, J., The age pattern of mortality, Journal of the Institute of Actuaries, 107, 49-80, (1980)
[21] Hull, J. C., Options futures and other derivatives, (1997), Prentice-Hall Englewood Cliffs, NJ · Zbl 1087.91025
[22] Huynh, C. B., Back to baskets, Risk, 5, 59-61, (1994)
[23] Kijima, M., A multivariate extension of equilibrium pricing transforms: the multivariate esscher and Wang transforms for pricing financial and insurance risks, ASTIN Bulletin, 36, 269-283, (2006) · Zbl 1162.91418
[24] Lazar, D., 2004. On forecasting mortality using the Lee-Carter model. In: 3rd Conference in Actuarial Science & Finance in Samos.
[25] Lee, R. D.; Carter, L., Modelling and forecasting the time series of US mortality, Journal of the American Statistical Association, 87, 659-671, (1992)
[26] Lee, Y. T.; Wang, C. W.; Huang, H. C., On the valuation of reverse mortgages with regular tenure payments, Insurance: Mathematics and Economics, 51, 2, 430-441, (2012) · Zbl 1284.91550
[27] Li, J. S.H.; Chan, W. S., The Lee-Carter model for forecasting mortality, revisited, North American Actuarial Journal, 11, 68-89, (2007)
[28] Li, J. S.H.; Hardy, M. R., Measuring basis risk in longevity hedges, North American Actuarial Journal, 15, 177-200, (2011) · Zbl 1228.91042
[29] Li, J. S.H.; Hardy, M. R.; Tan, K. S., On pricing and hedging the no-negative-equity guarantee in equity release mechanisms, Journal of Risk and Insurance, 77, 2, 499-522, (2010)
[30] Liao, H.H., Yang, S.S., Huang, I.H., 2007. The design of securitization for longevity risk: pricing under stochastic mortality model with tranche technique. Presented at the Third International Longevity Risk and Capital Market Solutions Symposium, Taipei.
[31] Lin, Y.; Cox, S. H., Securitization of mortality risks in life annuities, Journal of Risk and Insurance, 72, 227-252, (2005)
[32] Lin, Y.; Cox, S. H., Securitization of catastrophe mortality risks, Insurance: Mathematics and Economics, 42, 628-637, (2008) · Zbl 1152.91593
[33] Lutkepohl, H., New introduction to multiple time series analysis, (2005), Springer New York · Zbl 1072.62075
[34] Milevsky, M. A.; Posner, S. E., Asian options, the sum of lognormals, and the reciprocal gamma distributions, Journal of Financial and Quantitative Analysis, 33, 409-422, (1998)
[35] Milevsky, M. A.; Posner, S. E., A closed-form approximation for valuing basket options, Journal of Derivatives, 5, 54-61, (1998)
[36] Njenga, N.C., Sherris, M., 2009. Longevity risk and the econometric analysis of mortality trends and volatility. Working Paper.
[37] Phillips, C. B.; Perron, P., Testing for a unit root in time series regression, Biometrika, 75, 335-346, (1988) · Zbl 0644.62094
[38] Plat, R., On stochastic mortality modelling, Insurance: Mathematics and Economics, 45, 393-404, (2009) · Zbl 1231.91227
[39] Renshaw, A. E.; Haberman, S., Lee-Carter mortality forecasting with age-specific enhancement, Insurance: Mathematics and Economics, 33, 255-272, (2003) · Zbl 1103.91371
[40] Renshaw, A. E.; Haberman, S., A cohort-based extension to the Lee-Carter model for mortality reduction factors, Insurance: Mathematics and Economics, 38, 556-570, (2006) · Zbl 1168.91418
[41] Siu, T. K.; Tong, H.; Yang, H., On pricing derivatives under GARCH models: a dynamic gerber-shiu approach, North American Actuarial Journal, 8, 17-31, (2004) · Zbl 1085.91531
[42] Wang, S. S., A class of distortion operators for pricing financial and insurance risks, Journal of Risk and Insurance, 67, 15-36, (2000)
[43] Wang, S. S., A universal framework for pricing financial and insurance risks, ASTIN Bulletin, 32, 213-234, (2002) · Zbl 1090.91555
[44] Wang, S. S., Normalized exponential tilting: pricing and measuring multivariate risks, North American Actuarial Journal, 11, 3, 89-99, (2007)
[45] Wang, C.W., Huang, H.C., Liu, I.C., 2011. A quantitative comparison of the Lee-Carter model under different types of non-Gaussian innovations. In: Geneva Papers on Risk and Insurance—Issues and Practice, vol. 36, pp. 675-696.
[46] Wilson, C., On the scale of global demographic convergence 1950-2000, Population and Development Review, 27, 155-172, (2001)
[47] Yang, S.S., 2011. Securitization and tranching longevity and house price risk for reverse mortgage products. In: Geneva Papers on Risk and Insurance-Issues and Practice, vol. 36, pp. 648-674.
[48] Yang, S.S., Yue, J.C., Yeh, Y.Y., 2011. Coherent mortality modeling for a group of populations. In: Living to 100 Symposium.
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.