Title: Stochastic mortality modelling for multiple populations Abstract: This thesis deals with the possibilities of modelling and forecasting of age-specific mortality rates. The introductory part summarizes the basic terms from demo- graphy, which are related to mortality, and specifies elementary approaches to the mortality modelling. Subsequently there are in detail described the three most commonly used stochastic mortality models - Lee-Carter, Renshaw-Haberman and Cairns-Blake-Dowd. The fundamental part of this thesis deals with the possi- bilities of using these models for mortality modelling simultaneously in correlated populations. These theoretical bases are in the final part of this thesis numerically illustrated on the mortality models for populations of Czech and Slovak Republic. 1

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Yes / We outline the valuation process for a No-Negative Equity Guarantee in an Equity Release Mortgage loan and for an Equity Release Mortgage that has such a guarantee. Illustrative valuations are provided based on the Black ’76 put pricing formula and mortality projections based on the M5, M6 and M7 mortality versions of the Cairns–Blake–Dowd (CBD) family of mortality models. Results indicate that the valuations of No-Negative Equity Guarantees are high relative to loan amounts and subject to considerable model risk but that the valuations of Equity Release Mortgage loans are robust to the choice of mortality model. Results have significant ramifications for industry practice and prudential regulation.

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由於誤差與人口數成反比，資料多寡影響統計分析的穩定性及可靠性，因此常用於推估大區域人口的方法，往往無法直接套用至縣市及其以下層級，尤其當小區域內部地理、社會或經濟的異質性偏高時，人口推估將更為棘手。本文以兩個面向對臺灣小區域人口進行探討：其一、臺灣人口結構漸趨老化，勢必牽動政府政策與資源分配，且臺灣各縣市的人口老化速度不一，有必要針對各地特性發展適當的小區域人口推估方法；其二、因為壽命延長，全球皆面臨長壽風險(Longevity Risk)的挑戰，包括政府退休金制度規劃、壽險保費釐定等，由於臺灣各地死亡率變化不盡相同，發展小區域死亡率模型也是迫切課題。 小區域推估面臨的問題大致可歸納為四個方向：「資料品質」、「地區人數」、「資料年數」與「推估年數」，資料品質有賴資料庫與制度的建立，關於後三個問題，本文引進修勻(Smoothing, Graduation)等方法來提高小區域推估及小區域死亡模型的穩定性。人口推估方面結合修勻與區塊拔靴法(Block Bootstrap)，死亡率模型的建構則將修勻加入Lee-Carter與Age-Period-Cohort模型。由於小區域人口數較少，本文透過標準死亡比(Standard Mortality Ratio)及大區域與小區域間的連貫(Coherence)，將大區域的訊息加入小區域，降低因為地區人數較少引起的震盪。 小區域推估通常可用的資料時間較短，未來推估結果的震盪也較大，本文針對需要過去幾年資料，以及未來可推估年數等因素進行研究，希冀結果可提供臺灣各地方政府的推估參考。研究發現，參考大區域訊息有穩定推估的效果，修勻有助於降低推估誤差；另外，在小區域推估中，如有過去十五年資料可獲得較可靠的推估結果，而未來推估年數盡量不超過二十年，若地區人數過少則建議合併其他區域增加資料量後再行推估；先經過修勻而得出的死亡率模型，其效果和較為複雜的連貫模型修正相當。 / The population size plays a very important role in statistical estimation, and it is difficult to derive a reliable estimation for small areas. The estimation is even more difficult if the geographic and social attributes within the small areas vary widely. However, although the population aging and longevity risk are common phenomenon in the world, the problem is not the same for different countries. The aim of this study is to explore the population projection and mortality models for small areas, with the consideration of the small area’s distinguishing characteristic. The difficulties for small area population projection can be attributed into four directions: data quality, population size, number of base years, and projection horizon. The data quality is beyond the discussion of this study and the main focus shall be laid on the other three issues. The smoothing methods and coherent models will be applied to improve the stability and accuracy of small area estimation. In the study, the block bootstrap and the smoothing methods are combined to project the population to the small areas in Taiwan. Besides, the Lee-Cater and the age-period-cohort model are extended by the smoothing and coherent methods. We found that the smoothing methods can reduce the fluctuation of estimation and projection in general, and the improvement is especially noticeable for areas with smaller population sizes. To obtain a reliable population projection for small areas, we suggest using at least fifteen-year of historical data for projection and a projection horizon not more than twenty years. Also, for developing mortality models for small areas, we found that the smoothing methods have similar effects than those methods using more complicated models, such as the coherent models.

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.

本篇論文將厚尾分配（Heavy-Tailed Distribution）應用在財務及保險精算上。本研究主要有三個部分：第一部份是用厚尾分配來重新建構Lee-Carter模型（1992），發現改良後的Lee-Carter模型其配適與預測效果都較準確。第二部分是將厚尾分配建構於具有世代因子（Cohort Factor）的Renshaw and Haberman模型（2006）中，其配適及預測效果皆有顯著改善，此外，針對英格蘭及威爾斯（England and Wales）訂價長壽交換（Longevity Swaps），結果顯示此模型可以支付較少的長壽交換之保費以及避免低估損失準備金。第三部分是財務上的應用，利用Schmidt等人（2006）提出的多元仿射廣義雙曲線分配（Multivariate Affine Generalized Hyperbolic Distributions; MAGH）於Boyle等人（2003）提出的低偏差網狀法（Low Discrepancy Mesh; LDM）來定價多維度的百慕達選擇權。理論上，LDM法的數值會高於Longstaff and Schwartz（2001）提出的最小平方法（Least Square Method; LSM）的數值，而數值分析結果皆一致顯示此性質，藉由此特性，我們可知道多維度之百慕達選擇權的真值落於此範圍之間。 / The thesis focus on the application of heavy-tailed distributions in finance and actuarial science. We provide three applications in this thesis. The first application is that we refine the Lee-Carter model (1992) with heavy-tailed distributions. The results show that the Lee-Carter model with heavy-tailed distributions provide better fitting and prediction. The second application is that we also model the error term of Renshaw and Haberman model (2006) using heavy-tailed distributions and provide an iterative fitting algorithm to generate maximum likelihood estimates under the Cox regression model. Using the RH model with non-Gaussian innovations can pay lower premiums of longevity swaps and avoid the underestimation of loss reserves for England and Wales. The third application is that we use multivariate affine generalized hyperbolic (MAGH) distributions introduced by Schmidt et al. (2006) and low discrepancy mesh (LDM) method introduced by Boyle et al. (2003), to show how to price multidimensional Bermudan derivatives. In addition, the LDM estimates are higher than the corresponding estimates from the Least Square Method (LSM) of Longstaff and Schwartz (2001). This is consistent with the property that the LDM estimate is high bias while the LSM estimate is low bias. This property also ensures that the true option value will lie between these two bounds.