A Study of Elderly Mortality Models and Their Applications in Annuity Insurance

傳統上國人寄望養兒防老，但面臨少子化及壽命延長，家庭已無法獨力負擔照顧老年人的責任，必須仰賴個人（老年人自己）、國家及政府分擔人口老化造成的需求，這也是政府在過去二十年來積極投入更多資源，制訂與老年人有關的社會保險、福利及政策的原因。像是1995年開辦的全民健康保險提升了全民健康，其中老年人受惠尤多；2005年的勞工退休金條例、2008年的國民年金保險等，則是因應我國國民壽命延長的社會保險制度。對於未來費用的需求估算，需要依賴可靠的死亡率預測，但大多數預測並沒有將死亡率改善列入考量，勢必低估長壽風險的衝擊，影響個人的財務規劃、增加國家負債。 有鑑於此，本文研究常用的死亡率模型，評估哪些適合用於描述高齡死亡率的變化，且能用於計算年金商品的定價。本文考量的模型大致分成兩類：關係模型(Relational Models)及隨機模型(Stochastic Models)，第一類包括常用於高齡的Gompertz、Coale-Kisker模型，以及Discount Sequence模型，第二類則有Lee-Carter及CBD等模型。模型比較的方式以長期預測和短期預測，選用交叉驗證的方式驗證死亡率模型的預測結果與觀察值之間的差異。研究結果顯示Discount Sequence、Lee-Carter、CBD隨機模型較能準確描述台灣、日本與美國等三個國家的死亡率特性；但這三個模型在年金險保費並沒有很明顯的訂價差異。另外，若用於短期預測、長期預測比較，又以Discount Sequence的預測結果優於Lee-Carter模型的預測。Traditionally in Asia, families played the main role in caring their own elderly (i.e., parents and grand-parents), but the declining fertility rates and longer life expectancy make it difficult for the families to take care of the elderly alone. The elderly themselves and the government need to share the burden caused by the aging population. In fact, most Taiwan’s major social policies in the past 20 years are targeting the elderly, such as National Health Insurance, Labor Pension Act and National Pension Insurance. Their planning and financial solvency rely on reliable mortality models and their projections for the elderly population. However, many mortality models do not take into account the mortality improvements and thus underestimate the cost. In this study, we look for elderly mortality models which can reflect the mortality improvements in recent years and use them to price the annuity products. Two types of mortality models are of interest: relational models and stochastic models. The first group includes the Gompertz model, Coale-Kisker model and Discount Sequence; the other group includes the Lee-Carter and CBD models. We utilize these mortality models to project future mortality rates in Taiwan, Japan and U.S., along with the block bootstrap and ARIMA for projection. The model comparison is based on cross-validation, and both short-term and long-term projections are considered. The results show that the Discount Sequence, Lee-Carter model and CBD model have the best model fits for mortality rates and, for the short-term and long-term forecasts, the Discount Sequence is better than the Lee-Carter model