|Abstract: ||死亡率為壽險公司計算特定保單準備金水準的重要變數之一。由於以往世界各國的死亡率資料蒐集困難，使得保險公司僅能採用各國政府每隔幾年所公布的生命表或是公司內部模型來估計個別年齡層的死亡率以計算相關的準備金。考量近年來全球經濟的快速發展提升人類的生活品質，加上醫療技術進步神速等因素，人類的存活年限已見大幅改善，因此根據更新速度較慢的生命表來估計死亡率將可能造成對準備金計算的偏誤。有鑑於世界各國死亡率資料庫日趨完善，保險相關之學術界與實務界開始投入大量資料建構理論和實證模型解釋死亡率的相關成因與動態行為，如此有助於未來保險公司可以更精確地估計死亡率以降低準備金的估計偏誤，最後增進保險公司的資金運用效率。自從Lee and Carter (1992)提出解釋與預測美國死亡率的模型後，他們的模型便廣為各國政府與保險公司採用以估計未來的死亡率。Lee-Carter 模型也帶動了學術界對預測死亡率的研究興趣，各種預測模型如雨後春筍般的出現。綜觀目前的死亡率模型可知，其主要的差別在於模型中所包含的共同因子數目以及估計模型的方法。換言之，到底需要幾個共同因子以及何種型式的模型方能有效得解釋和預測死亡率的動態行為呢？到目前為止，相關文獻似乎仍莫衷一是。有鑑於此，本計畫的研究目的在於檢定解釋整體保險死亡率期限結構的共同因子個數以及針對該期限結構進行模型的配適與模型預測能力的評估。本研究計畫為三年期的研究計畫。在計畫的第一年中，我們打算透過主成分分析法（Principal Component Analysis, PCA）、因素分析法（Factor Analysis, FA）以及財務學的套利定價理論（Arbitrage Pricing Theory, APT）相關文獻中的檢定方法決定共同因子的數目。我們考慮採用的檢定法包括PCA 的碎石圖（scree plot）、FA 的概似比率檢定法（likelihood ratio test）、Connor and Korajczyk (1993, JF)的因子個數檢定法以及Bai and Ng (2002, Econometrica)的因子個數檢定法。透過這些檢定法，我們可以決定適當的因子個數，以利後續的因素模型的配適。我們擬採用 Connor and Korajczyk (1986,1988, JFE)的漸進式主成分分析法（asymptotical principal components technique）配適英國死亡率的期限結構，並利用各成分的線性迴歸探討影響各主成分的相關總體經濟變數，藉此瞭解影響死亡率動態行為的外在環境因素。在計畫的第二年中，我們擬借重財務學的利率期限結構配適法進行整體的死亡率期限結構，並研究該期限結構的動態行為，以期能預測死亡率期限結構的未來動態。我們配適的利率期限模型包括Diebold, Piazzesi, and Rudebusch (2005, AER)的二因子模型、Nelson and Siegel (1987, JB)的三因子模型以及Svensson (1994, NBER) 的四因子模型，並比較這些模型的樣本內配適度和樣本外的預測能力。最後在我們挑選出最適模型後，我們將參考Diebold and Li (2006, J. Econometrics)的方法建構該模型的動態模型，並評估此動態模型的效率性。在計畫的第三年中，我們將上述的研究方法應用到其他國家的死亡率期限結構上，例如台灣、美國、日本以及一些歐洲國家。換言之，我們將估計這些國家之死亡率期限結構的共同因子，接著採用Connor and Korajczyk 的漸進式主成分分析法（asymptotical principal components technique）配適這些國家的死亡率期限結構，並同時探討影響死亡率動態行為的背後成因。我們也將由上述的許多不同結構的 Nelson-Siegel 模型中選擇各國的最適死亡率期限結構模型，然後預測這些國家之未來死亡率的可能變化。|
Mortality rate is a key element in the calculation of reserve for an insurance firm. Due to the difficulty of collecting reliable data of global mortality rates, insurance firms usually calculate the reserves of insurance contracts based on either the official mortality table or on their own internal models. However, the fact that the improving life quality and medical treatment techniques in last decades of human society have greatly reduce the mortality rates of policyholders suggests that the mortality rate itself has been becoming more a dynamic process rather than a static one. In other words, the reserves calculated based on the official mortality table or the internal models of insurance companies may be subject to potential biases. Hence, mortality models that are able to capture the dynamic behavior of mortality rate have become imperative for insurance companies to calculate more accurate reserves. Recently, the research team in University of California, Berkeley and Max Planck Institute for Demographic Research has constructed the human mortality database in which mortality rates for different ages in 37 countries were collected with good quality and were regularly updated. This database has accelerated both the academic research and the practical study in explaining and predicting the dynamic behavior of global mortality rates. Such studies will definitely facilitate better estimation of necessary reserves for insurance firms and enhance their capital efficiency. Since Lee and Carter (1992) proposed a single-factor mortality model in their seminal paper, a large number of various mortality models have been suggested and tested in the literature. Overall, these models differ in the number of common factors used to explain the dynamic behavior of mortality rates and the method of model estimation. In other words, exactly how many common factors are needed to adequately fit the term structure of mortality rate and which method is the best one for estimating the mortality model are still hot debatable issues. In view of this, this research project contributes to the literature by first testing the most probable number of common factors for the term structure of mortality rates and then adopting the curve-fitting models in the literature of fitting yield curve to fit the term structure of mortality rates. This project is a three-year research project. In the first year of the project, we will employ several commonly used tests in the literatures of principal component analysis (PCA), factor analysis (FA), and arbitrage pricing theory (APT) to determine the most probable number of common factors for the term structure of mortality rate. In particular, the tests include the scree plot from PCA, the likelihood ratio test from FA, and the tests of Connor and Korajczyk (1993, JF) and Bai and Ng (2002, Econometrica) from APT. After deciding the number of common factors, we will use the asymptotical principal components technique of Connor and Korajczyk (1986, 1988, JFE) to fit the term structure of mortality rate in England. Finally, we would like to investigate the relevant macroeconomic variables that influence the mortality rates by estimating linear regression models of the common factors on the selected variables. In the second year of the project, in order to further understand the dynamic behavior of mortality rate, we plan to fit the mortality rate by the curve-fitting method in the literature of interest rate term structure. The models we utilize to fit the term structure of mortality rate and study its dynamics include the two-factor model of Diebold, Piazzesi, and Rudebusch (2005, AER), the three-factor model of Nelson and Siegel (1987, J. Business) and the four-factor model of Svensson (1994, NBER). We will evaluate the in-sample fitting and out-of-sample forecasting performance of these models in order to determine the best one. After that, we will follow the method of Diebold and Li (2006, J. Econometrics) to construct a dynamic version of the best model and specifically study the performance this dynamic term structure model of mortality rate. In the third year of the project, we would like to extend our study to other counties, e.g., Taiwan, the U.S.A., Japan, and some European countries. In particular, we will estimate the proper number of common factors in the mortality term structures of these countries, construct the factor model by the asymptotical principal components analysis and examine the underlying causes of the dynamics of mortality term structure, and finally search for the optimal model from the Nelson-Siegel class of term structure models.