×

Modeling multi-country mortality dependence and its application in pricing survivor index swaps – a dynamic copula approach. (English) Zbl 1348.62249

Summary: This paper introduces mortality dependence in multi-country mortality modeling using a dynamic copula approach. Specifically, we use time-varying copula models to capture the mortality dependence structure across countries, examining both symmetric and asymmetric dependence structures. In addition, to capture the phenomenon of a heavy tail for the multi-country mortality index, we consider not only the setting of Gaussian innovations but also non-Gaussian innovations under the Lee-Carter framework model. As tests of the goodness of fit of different dynamic copula models, the pattern of mortality dependence, and the distribution of the innovations, we used empirical mortality data from Finland, France, the Netherlands, and Sweden. To understand the effect of mortality dependence on longevity derivatives, we also built a valuation framework for pricing a survivor index swap, then investigated the fair swap rates of a survivor swap numerically. We demonstrate that failing to consider the dynamic copula mortality model and non-Gaussian innovations would lead to serious underestimations of the swap rates and loss reserves.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
91D20 Mathematical geography and demography
91G20 Derivative securities (option pricing, hedging, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Aas, K.; Haff, I. H., The generalized hyperbolic skew student’s \(t\)-distribution, J. Financ. Econ., 4, 275-309, (2006)
[2] Akaike, H., A new look at the statistical model identification, IEEE Trans. Automat. Control, 19, 716-723, (1974) · Zbl 0314.62039
[3] Alexandra, D.; Paul, E., Modeling exchange rate dependence dynamics at different time horizons, J. Int. Money Finance, 29, 1687-1705, (2010)
[4] Ang, A.; Bekaert, G., International asset allocation with regime shifts, Rev. Financ. Stud., 15, 1137-1187, (2002)
[5] Ang, A.; Chen, J., Asymmetric correlations of equity portfolios, J. Financ. Econ., 63, 3, 443-494, (2002)
[6] Barndorff-Nielsen, O. E., Exponentially decreasing distributions for the logarithm of particle size, Proc. R. Soc. Lond., 353, 409-419, (1977)
[7] Barndorff-Nielsen, O. E., Normal inverse Gaussian processes and the modeling of stock returns. technical report 300, (1995), Department of Theoretical Statistics, Institute of Mathematics
[8] Barndorff-Nielsen, O. E.; Blæsild, P., Hyperbolic distributions and ramifications: contributions to theory and application, (Taillie, C.; Patil, G.; Baldessari, B., Statistical Distributions in Scientific Work. Vol. 4, (1981), Reidel Dordrecht), 19-44 · Zbl 0489.62020
[9] Biffis, E., Blake, D., 2009. Mortality-linked securities and derivatives. Discussion Paper PI-0901, The Pensions Institute.
[10] Biffis, E., Blake, D., Pitotti, L., Sun, A., 2011. The cost of counterparty risk and collateralization in longevity swaps, Pensions Institute Discussion Paper PI-1107, June.
[11] Blake, D., Reply to ‘survivor bonds: a comment on blake and burrows’, J. Risk Insurance, 70, 349-351, (2003)
[12] Blake, D., Cairns, A.J.G., Coughlan, G.D., Dowd, K., MacMinn, R., 2012. The new life market. Working Paper.
[13] Blake, D.; Cairns, A. J.; Dowd, K., Living with mortality: longevity bonds and other mortality-linked securities, Br. Actuar. J., 12, 153-197, (2006)
[14] Blake, D.; Dawson, P.; Dowd, K.; Cairns, A. J., Survivor derivatives: a consistent pricing framework, J. Risk Insurance, 77, 3, 579-596, (2010)
[15] Bu, R.; Giet, L.; Hadri, K.; Lubrano, M., Modeling multivariate interest rates using time-varying copulas and reducible nonlinear stochastic differential equations, J. Financ. Econ., 9, 1, 198-236, (2011)
[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, 2, 150-176, (2011)
[17] Cox, S. H.; Lin, Y.; Wang, S., Multivariate exponential tilting and pricing implications for mortality securitization, J. Risk Insurance, 73, 719-736, (2006)
[18] Dawson, P., 2002. Mortality swaps. Mimeo, Cass Business School.
[19] Dawson, P.; Blake, D.; Cairns, A. J.G.; Dowd, K., Survivor derivatives: A consistent pricing framework, J. Risk Insurance, 77, 579-596, (2010)
[20] Demarta, S.; McNeil, A. J., The \(t\) copula and related copulas, Int. Statist. Rev., 73, 1, 111-129, (2005) · Zbl 1104.62060
[21] Denuit, M.; Devolder, P.; Goderniaux, A. C., Securitization of longevity risk: pricing survivor bonds with Wang transform in the Lee-Carter framework, J. Risk Insurance, 74, 87-113, (2007)
[22] Dias, A., Embrechts, P., 2004. Dynamic copula models for multivariate high-frequency data in finance. Working Paper.
[23] Dowd, K., Survivor bonds: a comment on blake and Burrows, J. Risk Insurance, 70, 339-348, (2003)
[24] Dowd, K.; Blake, D.; Cairns, A. J.G.; Dawson, P., Survivor swaps, J. Risk Insurance, 73, 1-17, (2006)
[25] Engle, R. F., Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models, J. Bus. Econom. Statist., 20, 339-350, (2002)
[26] Erb, C. B.; Harvey, C. R.; Viskanta, T. E., Forecasting international equity correlations, Financ. Anal. J., 50, 32-45, (1994)
[27] Hoesli, M.; Reka, K., Volatility spillovers, comovements and contagion in securitized real estate markets, J. Real Estate Finance Econ., 47, 1, 1-35, (2013)
[28] Huang, H. C.; Wang, C. C.; Miao, Y. C., Securitization of crossover risk in reverse mortgages, Geneva Pap. Risk Insur. - Issues Pract., 36, 4, 622-647, (2011)
[29] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517
[30] Joe, H., Xu, J.J., 1996. The estimation method of inference functions for margins for multivariate models, Department of Statistics, University of British Columbia, Technical Report, 166.
[31] Jondeau, E.; Rockinger, M., The copula-GARCH model of conditional dependencies: an international stock market application, J. Int. Money Finance, 25, 827-853, (2006)
[32] Kumar, M. S.; Okimoto, T., Dynamics of international integration of government securities’ markets, J. Bank. Finance, 35, 1, 142-154, (2011)
[33] Lee, R. D., The Lee-Carter method for forecasting mortality, with warious extensions and applications, N. Am. Actuar. J., 4, 1, 80-91, (2000)
[34] Lee, R. D.; Carter, L., Modelling and forecasting the time series of US mortality, J. Amer. Statist. Assoc., 87, 419, 659-671, (1992)
[35] Li, J. S.H.; Hardy, M. R., Measuring basis risk in longevity hedges, N. Am. Actuar. J., 15, 177-200, (2011) · Zbl 1228.91042
[36] Lin, Y.; Cox, S. H., Securitization of mortality risks in life annuities, J. Risk Insurance, 72, 2, 227-252, (2005)
[37] Longin, F.; Solnik, B., Extreme correlation of international equity markets, J. Finance, 56, 649-676, (2001)
[38] Madan, D.; Seneta, E., The variance gamma (VG) model for share market returns, J. Bus., 63, 511-524, (1987)
[39] Madan, D. B.; Seneta, E., The variance gamma (VG) model for share market returns, J. Bus., 63, 511-524, (1990)
[40] Manner, H., Reznikova, O., 2010. Survey on time-varying copulas: specification, simulations and application. Working Paper.
[41] Okimoto, T., New evidence of asymmetric dependence structures in international equity markets, J. Finan. Quant. Anal., 43, 787-815, (2008)
[42] Patton, A. J., On the out-of-sample importance of skewness and asymmetric dependence for asset allocation, J. Financ. Econ., 2, 130-168, (2004)
[43] Patton, A. J., Modeling asymmetric exchange rate dependence, Internat. Econom. Rev., 47, 527-556, (2006)
[44] Pelletier, D., Regime-switching for dynamic correlation, J. Econometrics, 131, 445-473, (2006) · Zbl 1337.62277
[45] Prause, K., The generalized hyperbolic models: estimation, financial derivatives and risk measurement, (1999), Mathematics Faculty, University of Freiburg, (Ph.D. thesis) · Zbl 0944.91026
[46] Ramchand, L.; Susmel, R., Volatility and cross correlation across major stock markets, J. Empir. Finance, 17, 581-610, (1998)
[47] Reboredo, J. C., How do crude oil prices co-move? A copula approach, Energy Econ., 33, 5, 948-955, (2011)
[48] Schwarz, G., Estimating the dimension of a model, Ann. Statist., 6, 461-464, (1978) · Zbl 0379.62005
[49] Serban, M.; Brockwell, A.; Lehoczky, J.; Srivastava, S., Modelling the dynamic dependence structure in multivariate financial time series, J. Time Ser. Anal., 28, 763-782, (2007) · Zbl 1150.62064
[50] Sklar, A., Fonctions de répartition á n dimensionsetleursmarges, Publ. Inst. Statist. Univ. Paris, 8, 229-231, (1959)
[51] Vogiatzoglou, M., 2010. Dynamic Copula Toolbox. University of Macedonoa, Egnatias 156, Thessaloniki, Greece.
[52] Wang, S. S., A class of distortion operators for pricing financial and insurance risks, J. Risk Insurance, 67, 1, 15-36, (2000)
[53] Wang, C. W.; Huang, H. C.; Liu, I. C., A quantitative comparison of the Lee-Carter model under different types of non-Gaussian innovations, Geneva Pap. Risk Insur. - Issues Pract., 36, 675-696, (2011)
[54] Wang, C. C.; Yang, S. S., Pricing survivor derivatives with cohort mortality dependence under the Lee-Carter framework, J. Risk Insurance, 80, 4, 1027-1056, (2013)
[55] Yang, S. S.; Wang, C. W., Pricing and securitization of multi-country longevity risk with mortality dependence, Insurance Math. Econom., 52, 2, 157-169, (2013) · Zbl 1284.91556
[56] Yang, S.S., Yue, J.C., Yeh, Y.Y., 2011. Coherent mortality modeling for a group of populations. In: Living to 100 Symposium.
[57] Zhou, J.; Gao, Y., Tail dependence in international real estate securities markets, J. Real Estate Finance Econ., 45, 1, 128-151, (2012)
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.