Quadratic stochastic intensity and prospective mortality tables. (English) Zbl 1140.91418

Gourieroux, C.; Monfort, A.

Insur. Math. Econ. 43, No. 1, 174-184 (2008).

Summary: We consider a quadratic stochastic intensity model with a Gaussian autoregressive factor, derive explicit formulas for predictive mortality tables and recursive updating formulas are also provided. We also explain how to use appropriately the Kalman filter to estimate the parameters of the model and to approximate the values of the underlying factor. This methodology is applied to French human mortality tables.

MSC:

91B30	Risk theory, insurance (MSC2010)
91B82	Statistical methods; economic indices and measures

Keywords:

mortality table; life insurance; longevity risk; mortality linked securities (MLS); affine model; quadratic model; Kalman filter

References:

[1]	Ahn, D.; Dittmar, R.; Gallant, A., Quadratic term structure models: theory and evidence, Review of financial studies, 15, 243-255, (2002)
[2]	Anderson, T.; Goodman, L., Statistical inference about Markov chains, Journal of the American mathematical society, 28, 89, (1957) · Zbl 0087.14905
[3]	Biffis, E., Affine processes for dynamic mortality and actuarial valuations, Insurance: mathematics and economics, 37, 3, 443-468, (2005) · Zbl 1129.91024
[4]	Blake, D.; Burrows, W., Survivor bonds: helping to hedge mortality risk, Journal of risk and insurance, 68, 339-348, (2001)
[5]	Brouhns, N.; Denuit, M.; Vermunt, J., A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance: mathematics and economics, 31, 373-393, (2002) · Zbl 1074.62524
[6]	Bravo, J., Pereira da Silva, C., 2008. Pricing longevity bonds using affine-jump diffusion models. DP University of Evora
[7]	Cairns, A., Blake, D., Dowd, K., 2004. Pricing death: Frameworks for the valuation and securitization of mortality risk. DP Actuarial Mathematics and Statistics, Heriot-Watt University · Zbl 1162.91403
[8]	Cairns, A.; Blake, D.; Dowd, K., A two factor model for stochastic mortality with parameter uncertainty: theory and calibration, Journal of risk and insurance, (2006)
[9]	Cairns, A., Blake, D., Dowd, K., Coughlan, G., Epstein, D., Ong, A., Balevich, I., 2007. A quantitative comparison of stochastic mortality models using data from England & Wales and the United States, Heriot-Watt University DP
[10]	Chen, H., Cox, S., 2007. Modeling Mortality with Jumps: Transitory Effects and Pricing Implication to Mortality Securitization, DP Georgia State University
[11]	Chen, L.; Filipovic, D.; Poor, V., Quadratic term structure models for riskfree and defaultable rates, Mathematical finance, 14, 515-536, (2004) · Zbl 1098.91048
[12]	Constantinides, G., A theory of the nominal term structure of interest rates, Review of financial studies, 5, 531-552, (1992)
[13]	Currie, I., 2007. Smoothing and forecasting mortality rates with p-splines, DP Heriot Watt University
[14]	Currie, I.; Kirkby, J.; Richard, S., The importance of year of birth in two-dimensional mortality rate, British actuarial journal, 12, 5-61, (2006)
[15]	Dahl, M., Stochastic mortality in life insurance: market reserves and mortality linked insurance contracts, Insurance: mathematics and economics, 35, 113-136, (2004) · Zbl 1075.62095
[16]	Dahl, M.; Moller, T., Valuation and hedging of life insurance liabilities with systematic mortality risk, Insurance: mathematics and economics, 39, 193-217, (2006) · Zbl 1201.91089
[17]	Darolles, S.; Gourieroux, C.; Jasiak, J., Structural Laplace transform and compound autoregressive models, Journal of time series analysis, 27, 477-503, (2006) · Zbl 1112.62090
[18]	De Jong, F.; Tickle, L., Extending lee – carter mortality forecasting, Mathematical population studies, 13, 1-18, (2006) · Zbl 1151.91742
[19]	Delwarde, A., 2004. Modèle Log-bilinéaire pour l’élaboration de tables de mortalité prospectives, Ph.D. Thesis, Louvain University
[20]	Denuit, M.; Robert, C., Actuariat des assurances des personnes, Economica, (2007)
[21]	Duffie, D.; Filipovic, D.; Schachermayer, W., Affine processes and application in finance, Annals of applied probability, 13, 984-1053, (2003) · Zbl 1048.60059
[22]	Duffie, D.; Kan, R., A yield factor model of interest rates, Mathematical finance, 6, 379-406, (1996) · Zbl 0915.90014
[23]	Feng, D.; Gourieroux, C.; Jasiak, J., The ordered qualitative model for credit rating transitions, Journal of empirical finance, 15, 111-130, (2008)
[24]	Gagliardini, P.; Gourieroux, C., Stochastic migration models, Journal of financial econometrics, 3, 188-226, (2005)
[25]	Gagliardini, P., Gourieroux, C., Monfort, A., 2007. Error in Factor Model, CREST DP
[26]	Gourieroux, C., Continuous time Wishart process for stochastic risk, Econometric reviews, 25, 177-217, (2006) · Zbl 1105.62104
[27]	Gourieroux, C., 2007. Car and Affine Processes, Lecture Notes in Time Series, Springer (in press) · Zbl 1178.91223
[28]	Gourieroux, C.; Jasiak, J., The econometrics of individual risk: credit, insurance and marketing, (2007), Princeton University Press · Zbl 1112.91038
[29]	Gourieroux, C., Jasiak, J., Sufana, R., 2007. The wishart autoregressive process for stochastic volatility. Journal of Econometrics (in press) · Zbl 1429.62397
[30]	Gourieroux, C.; Monfort, A.; Polimenis, V., Affine term structure models for credit risk, Journal of financial econometrics, 4, 494-530, (2006)
[31]	Gourieroux, C.; Sufana, R., Classification of affine term structure models, Journal of financial econometrics, 4, 31-52, (2006)
[32]	Lando, D., On Cox processes and credit risky securities, Review of derivative research, 2, 99-120, (1998) · Zbl 1274.91459
[33]	Lazar, D., 2004. On forecasting mortality using lee-carter Method, Babes-Bolyai University DP
[34]	Lee, R., The lee – carter method of forecasting mortality, with various extensions and applications, North-American actuarial journal, 4, 80-93, (2000) · Zbl 1083.62535
[35]	Lee, R.; Carter, L., Modeling and forecasting the time series of US mortality, Journal of the American statistical association, 87, 659-671, (1992)
[36]	Lee, R.; Miller, T., Evaluating the performance of the lee – carter method for forecasting mortality, Demography, 38, 537-549, (2001)
[37]	Leippold, M.; Wu, L., Asset pricing under the quadratic class, Journal of financial and quantitative analysis, 37, (2002), 374-295
[38]	Luciano, E., Vigna, E., 2005. A Note on Stochastic Survival Probabilities and Their Calibration, DP University of Turin
[39]	Maghsoodi, Y., Solution of the extended CIR term structure and bond option valuation, Mathematical finance, 6, 89-109, (1996) · Zbl 0915.90026
[40]	Menoncin, F., The role of longevity bonds in optimal portfolios, Insurance: mathematics and economics, 42, 343-358, (2008) · Zbl 1141.91537
[41]	Menoncin, F., 2008b. Death bonds with stochastic force of mortality. DP Univ. of Brescia · Zbl 1141.91537
[42]	Makeham, W., On the law of mortality and the construction of mortality tables, Journal of the institute of actuaries, 8, 301-310, (1860)
[43]	Renshaw, A.; Haberman, S., Lee – carter mortality forecasting with age specific enhancement, Insurance: mathematics and economics, 33, 255-272, (2003) · Zbl 1103.91371
[44]	Renshaw, A.; Haberman, S., On the forecasting of mortality reduction factors, Insurance: mathematics and economics, 32, 379-401, (2003) · Zbl 1025.62041
[45]	Schrager, D., Affine stochastic mortality, Insurance: mathematics and economics, 38, 81-97, (2006) · Zbl 1103.60063
[46]	Sunamoto, N., 2005. Cohort effect structure in the lee-carter residual term, DP Fukoku Mutual Life Insurance Company
[47]	White, K., Longevity advance in high income countries, 1955-96, Population and development review, 28, 59-76, (2002)
[48]	Yang, S., 2001. Reserving, Pricing and Hedging for Guaranteed Annuity Options, Ph.D. Thesis. Heriot-Watt University, Edinburgh