H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 1 Pricing Mortality-linked Securities with Dependent Lives under Threshold Life Table Hua Chen, Temple University Samuel H. Cox, University of Manitoba Jian Wen, Central University of Finance and Economics

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 2 Introduction  Extreme-age mortality modeling  Mortality improvement is a slow but persistent process  40,000 centenarians currently in the U.S.  3 million centenarians by the first decade of next century  Challenge to actuaries  since life table is usually closed earlier, say 100.  How to extrapolate extreme-age mortality and construct a reliable life table?  EVT approach Threshold life Table (Li, Hardy, Tan 2008)

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 3 Introduction  Joint survivorship of multiple lives  Independence of a pair is normally assumed  The joint survival function is simply the product of the marginal survival functions of each life.  Common risk factors for pairs of lives  Genetic factors, e.g. twins  Environmental factors, e.g., couples  Empirical evidence: broken heart syndrome  Parkes, Benjamin, and Fitzgerald (1969), Ward (1976)Jagger and Sutton (1991)  How to capture the life dependence?  Copula function

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 4 Introduction EVTCopula Multivariate Threshold Life Table

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 5 EVT and Threshold Life Table  Parametric estimator e.g. Gompertz distribution function (Frees, Carriere, Valdez 1996)  Traditional parametric methods are ill-suited to extreme probabilities  The inaccuracy and unavailability of mortality data at old ages.  Solution: EVT “estimate extreme probabilities by fitting a model to the empirical survival function of a set of data using only the extreme event data rather than all the data, thereby fitting the tail, and only the tail”(Sanders, 2005).

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 6 EVT and Threshold Life Table  For x > N where  The Pickands-Balkema-De Hann Theorem For sufficiently high threshold N, the excess distribution function may be approximated by the GPD. .

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 7 EVT and Threshold Life Table  Li, Hardy and Tan (2008): Threshold Life Table

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 8 Life Dependency and Copula  Dependency Measures  Parametric: e.g., Pearson correlation  Non-parametric: e.g., Spearman’s rho, Kendall’s tao  Copula  Copulas capture the dependence structure separately from the marginal distributions  Schweizer and Wolff (1981) For any strictly increasing functions and, and have the same copula as and

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 9 Life Dependency and Copula  Archimedean copula family where is a convex and strictly decreasing function with domain and range such that.

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 10 Life Dependency and Copula  Frank’s copula  Spearson’s rho  Kendall’s tau where and

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 11 Last-Survivor Annuity Data  Frees, Carriere and Valdez (1996)  approximately 15,000 last-survivor annuity policies  date of birth, death (if applicable), contract initiation, and sex of each annuitant.

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 12 Modeling Algorithm  Set up the initial values of parameters  Use mortality data of males  For, find the parameters that maximize the log-likelihood function;  Repeat this step for ;  The value of that gives the maximum profile log-likelihood is the optimal threshold age for male. The parameter estimates corresponding to this value are the optimal MLE estimates.  Replicate the same procedure for mortality data for females, and find the optimal estimates and  Use Gompertzian marginals and the Frank copula to find the estimate of the dependence parameter ;

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 13 Modeling Algorithm  Using the values of and obtained from step 1 as initial values, find the MLE estimates of these parameters, for any combination of and  The values of and that give the maximum value of log-likelihood function are the optimal threshold ages for males and females. The MLE estimates corresponding to this combination are our optimal MLE estimates.

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 14 Estimation Results  Spearman’s rho = 0.49 and Kendall’s tau = 0.56 A positive mortality dependence between male and female

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 15 Pricing Example: Last-Survivor Annuity  Last-survivor annuity where  Scenario analysis  Dependent lives with the threshold life table (benchmark model)  Independent lives with the threshold life table  Dependent lives without the threshold life table  Independent lives without the threshold life table

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 16 Pricing Example: Effect of Threshold Life Table  Ratio = annuity value with TLT/ that without TLT Dependent lives Independent lives  Without threshold life table, the value of the last survivor annuity is underestimated.

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 17 Pricing Example: Effect of Dependence  Ratio = annuity value assuming dependence/ that assuming independence With threshold life table Without threshold life table  Assuming independent lives, the value of the last-survivor annuity is overestimated.

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 18 Pricing Example : Overall Effect  Ratio = Dependent lives with TLT/ Independent lives without TLT  Assuming independent lives and without threshold life table, the last survivor annuity is overestimated by 5%.

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 19 Conclusion  Develop multivariate threshold life table  EVT approach  Copula approach  Apply our model to price the last-survivor annuity  Mortality-linked securities are under-priced without the threshold life table  Mortality-linked securities are over-priced assuming independent lives  Future research  How to identify an appropriate copula function?  Incorporate a stochastic process into the multivariate threshold life table.

H. Chen, S.H. Cox, and J. Wen Longevity 5 Conference September 26, 2009Multivariate Threshold Life Table 20 Thanks!