改良動態因子關聯結構於死亡率的估計

Modified Dynamic Factor Copula Model for Mortality Estimation

賴冠維(Kuan-Wei Lai)

指導教授 ： 蔡子晧

國立清華大學/科技管理學院/計量財務金融系/碩士(2021年)

摘要

近年來隨著科技快速發展、醫療技術進步，人們面臨非預期的生存風險，長壽風險， 近來台灣的壽險轉型推動「保障型保單」，包含了高齡化商品、年金險等，死亡率的精確估計對於保險公司來說越發重要。
本研究使用資料為年齡介於60至99歲的女性，時間為1940年至2017年的多國資料，共澳大利亞、加拿大、丹麥、芬蘭、法國、冰島、義大利、荷蘭、挪威、匈牙利、西班牙、瑞士、瑞典、美國、英國，總計十五國的死亡率資料，以估計美國女性高齡死亡率為目標，建立兩階段因子關聯結構模型。
本研究第一階段，按年齡從60歲至99歲共分40個子集，每個子集建立縮減SUR模型，以AIC選取SUR模型最佳落後項，並加入機器學習(Boosting)估計值作為外生變數，最後以個別t檢定篩選掉不顯著的變數，以建立各特定年齡的邊際分配。
接續將縮減SUR模型的估計殘差標準化，得到標準化估計殘差，以此建立各年齡間，關聯結構下的殘差分配，參考Chen, MacMinn, and Sun (2017)，先以單因子模型縮減參數，再根據模型建立潛在變數，模擬樣本，最後建立出各年齡殘差間的聯合機率分配，在此也參考Oh, Patton (2013)所使用的GAS結構，讓因子關聯結構能有效地動態調整。
經過上述步驟模型建立完成後，本研究於樣本內、樣本外跟原模型進行比較，若以AIC衡量模型表現，發現本研究模型表現最佳，並且能有效改善原模型殘差間存在正相關性的問題。
本研究推論，因為在第一階段本研究使用了多國資料，亦採用演算法基礎、非線性的機器學習方式協助估計，理應比原模型捕捉更多的資訊，讓各年齡間的殘差相關性有效降低。但仍然可以發現，年齡在85歲以上，甚至是95至99歲的超高年齡族群間，仍存在明顯相關，如何處理超高齡死亡率間的相依性，是未來研究的一大方向。

關鍵字

長壽風險 ； 關聯結構 ； 稀疏向量自迴歸 ； 機器學習 ； 集成學習

並列摘要

In this study, we used the multiple countries female mortality data from 1940 to 2017, and aged 60 to 99. With the goal of estimating the mortality rate of the US females at an advanced age, we established a two-stage factor copula model in this study.
In the first stage of this study, divided data into 40 subsets according to age from 60 to 99 years old, and a reduced SUR model was established for each subset. And the Boosting estimated value was added as the exogenous variables. Also, individual t-tests are used to filter out insignificant variables to establish the marginal distribution of each specific age.
Then standardize the estimated residuals to establish the distribution of residuals under the correlation structure between ages. Refer to Chen, MacMinn, and Sun (2017), We estimated a single factor model and establish the latent variables according to the model. Finally, we established the joint probability distribution among the residuals. We used the GAS structure referred by Oh, Patton (2013), so that the factor structure can be effectively adjusted dynamically.
Through the above steps, we compared our model with the original one in and out of the sample. We measured performances of the model by AIC, it’s found that the model in this study performs the best and can effectively improve the positive correlation between the residuals of the original.
Because, in the first stage, we used data from multiple countries and also used algorithm-based and non-linear machine learning methods to assist in the estimation, it should capture more information than the original model. However, it can still be found that there is still a significant correlation between the ages of 85 and over, even between the ages of 95 and 99. How to deal with the interdependence of the mortality rate of the very elderly is a major direction of future research.

並列關鍵字

Longevity Risk ； Copula ； Sparse VAR ； Machine Learning ； Gradient Boosting

參考文獻

1. H. Chen, R. D. MacMinn, and T. Sun, “Mortality Dependence and Longevity Bond Pricing:A Dynamic Factor Copula Mortality Model with the GAS Structure,” The Journal of Riskand Insurance, vol. 84, no. S1R, pp. 393–415, 2017.

2. D. H. Oh and A. J. Patton, “Time-Varying Systemic Risk:Evidence From a Dynamic CopulaModel of CDS Spreads,” Journal of Business and Economic Statisticss, vol. 36, no. 2,pp. 181–195, 2018.

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4. R. D. Lee and L. R. Carter, “Modeling and forecasting U.S. mortality. ,” Journal of the American Statistical Association, vol. 87, no. 419, pp. 659–671, 1992.

5. H. Chen, R. D. MacMinn, and T. Sun, “Multi-population mortality models:A factor copula approach,” Insurance: Mathematics and Economics, no. 63, pp. 135–146, 2015.