此研究使用一個考慮世代、年齡、期間、學習效果的 CAPL 模型描繪多群體死 亡率，與過去文獻上 Lee-Carter 等死亡率模型不同之處在於，我們的主要觀察 對象為各個世代，並將年齡、期間、世代以及學習效果納入我們的實證模型。 此模型能夠直觀地解釋多變量死亡率變動率，並且在預測未來死亡率以及其變 動率上亦有很好的表現。以美國、英國、法國、日本、加拿大以及澳洲等六國 資料為例，在十年期樣本外測試上，我們的方法在大部分的國家都具有較小的 RMSE，優於 Lee-Carter(1992)、Renshaw and Haberman (2003, 2006)以及 Cairns, Blake, and Dowd (2006)等模型。

This article proposes a cohort-age-period-learning (CAPL) model to characterize multi-population mortality processes using cohort, age, period, and learning variables. Distinct to the factor-based decomposition mortality model (e.g., the Lee-Carter, 1992), this approach is empirically based and include the age, period, cohort, and learning variables into the equation system. The model not only provides a fruitful intuition for explaining multivariate mortality change rates but also has a better performance on forecasting future patterns. Using the six countries mortality data and ten-year out-of-sample tests, our approach shows smaller mean square errors in most of the countries, comparing to the models of Lee and Carter (1992), Renshaw and Haberman (2003, 2006), and Cairns, et al. (2006).

1 INTRODUCTION ..................................................................................................................5
2 LITERATURE .......................................................................................................................7
2.1 LEE-CARTER MODEL............................................................................................................. 7
2.2 CBD MODEL ......................................................................................................................... 8
2.3 RENSHAW-HABERMAN MODEL ............................................................................................. 8
3 METHODOLOGY .................................................................................................................9
3.1 CONSTRUCTING THE COHORT DATA..................................................................................... 9
3.2 CAPL MODEL ...................................................................................................................... 10
3.3 TIME-VARYING SUR (TV-SUR) MODEL............................................................................... 13
4 EMPIRICAL RESULTS ......................................................................................................14
4.1 DATA................................................................................................................................... 14
4.2 MODEL SELECTION .............................................................................................................. 16
4.3 FORECASTING ...................................................................................................................... 18
4.4 PRICING THE KORTIS LONGEVITY BOND ............................................................................ 21
5 CONCLUSION ....................................................................................................................22
REFERENCE. ............................................................................................................................... 23
APPENDIX A. STATIONARITY TESTS..................................................................................26
APPENDIX B. AUTOCORRELATION ....................................................................................28
APPENDIX C. THE SUR COEFFICIENTS OF THE US, FRANCE, JAPAN, CANADA, AND
AUSTRALIA. ................................................................................................................................29

1. A.E. Renshaw, S. Haberman*, 2006. A cohort-based extension to the Lee- Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38, 556-570.
2. A. Hunt and A. M. Villegas, 2015. Robustness and convergence in the Lee– Carter model with cohort effects. Insurance: Mathematics and Economics, 64, 186–202.
3. Arnold Zellner, 1962. An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias. Journal of the American Statistical Association, 57(298), 348-368
4. Cairns, A.J.G., Blake, D., Dowd, K., 2006. A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance, 73, 687–718.
5. Cairns, A., Blake, D., Dowd, K., Coughlan, G., Epstein, D., Ong, A., and Balevich, I., 2009. A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal, 13(1), 1–35.
6. Cairns, A. J. G., D. Blake, K. Dowd, G. D. Coughlan, and M. Khalaf-Allah, 2011. Bayesian Stochastic Mortality Modelling for Two Populations, ASTIN Bulletin, 41, 29-59.
7. Casas I, Ferreira E, Orbe S, 2017. “Time-varying coefficient estimation in SURE models. Application to portfolio management.” Working paper, CREATES, Aarhus University.
8. Chen, H., R. D. MacMinn, and T. Sun, 2015. Multi-Population Mortality Models: A Factor Copula Approach, Insurance: Mathematics and Economics, 63, 135-146.
23
9. Chen, H., MacMinn, R.D., Sun, T, 2017. Mortality dependence and longevity bond pricing: A dynamic factor copula mortality model with the gas structure. Journal of Risk and Insurance, 84(S1), 393–415.
10. Choi, I., 2001. Unit root tests for panel data. Journal of International Money and Finance, 20(2), 249-272.
11. Dowd, K., A. J. G. Cairns, D. Blake, G. D. Coughlan, and M. Khalaf-Allah, 2011. A Gravity Model of Mortality Rates for Two Related Populations, North American Actuarial Journal, 15(2), 334-356.
12. Giacometti, R., M. Bertocchi, S. T. Rachev, and F. J. Fabozzi., 2012. A Comparison of the Lee-Carter Model and AR-ARCH Model for Forecasting Mortality Rates, Insurance: Mathematics and Economics, 50(1), 85-93.
13. Hadri, K., 2000. Testing for stationarity in heterogeneous panel data. The Econometrics Journal, 3(2), 148-161.
14. Henderson DJ, Kumbhakar SC, Li Q, Parmeter CF., 2015. “Smooth coefficient estimation of a seemingly unrelated regression.” Journal of Econometrics, 189, 148–162.
15. Im, K. S., Pesaran, M. H., & Shin, Y., 2003. Testing for unit roots in heterogeneous panels. Journal of Econometrics, 115(1), 53-74.
16. Lee, R.D., Carter, L., 1992. Modeling and Forecasting the Time Series of U.S. Mortality. Journal of the American Statistical Association, 87(419), 659–675.
17. Levin, A., Lin, C. F., & Chu, C. S. J., 2002. Unit root tests in panel data: asymptotic and finite- sample properties. Journal of Econometrics, 108(1), 1- 24.
18. Li, H., and Lu, Y., 2017. Coherent forecasting of mortality rates: A sparse vector autoregression approach. ASTIN Bull.: J. IAA 47(2), 563–600. http://dx.doi.org/10.1017/asb.2016.37.
19. Li, J. and Hardy, M., 2011. Measuring basis risk in longevity hedges. North American Actuarial Journal, 15(2), 177–200.
20. Maddala, G. S., & Wu, S., 1999. A comparative study of unit root tests with panel data and a new simple test. Oxford Bulletin of Economics and Statistics, 61(S1), 631-652.
21. Orbe S, Ferreira E, Rodriguez-Poo J, 2005. “Nonparametric estimation of time varying parameters under shape restrictions.” Journal of Econometrics, 126, 53–77.
22. Plat, R., 2009. On stochastic mortality modeling. Insurance: Mathematics and Economics, 45(3), 393-404.
24

23. Quentin Guibert, Olivier Lopez, and Pierrick Piette, 2019. Forecasting mortality rate improvements with a high dimensional VAR. Insurance: Mathematics and Economics, 88, 255-272.
24. Renshaw, A. E., & Haberman, S., 2003. Lee–Carter mortality forecasting with age-specific enhancement. Insurance: Mathematics and Economics, 33(2), 255-272.
25. Wang, C.-W., H.-C Huang, and I.-C Liu, 2013. Mortality Modeling with Non- Gaussian Innovations and Applications to the Valuation of Longevity Swaps, Journal of Risk and Insurance, 80(3), 775-798.
26. Yang, S. S., and C.-W Wang, 2013. Pricing and Securitization of Multi- Country Longevity Risk with Mortality Dependence, Insurance: Mathematics and Economics, 52(2), 157-169.
27. Zhou, R., J. S.-H. Li, and K. S. Tan, 2013. Pricing Standardized Mortality Securitizations: A Two-Population Model with Transitory Jump Effects, Journal of Risk and Insurance, 80(3), 733-774.
28. Zhou, R., Y. Wang, K Kaufhold, J. S.-H. Li, and K. S. Tan, 2014. Modeling Period Effects in Multi-Population Mortality Models: Applications to Solvency II, North American Actuarial Journal, 18(1), 150-167.