回顧二十世紀的歷程，我們可以看到人類在壽命上的一大進步，認為壽命的延長是人類的最大勝利；但是此壽命延長現象卻視為未來社會中的最主要的挑戰與風險。台灣在1993年六十五歲以上的老年人口比例已突破7%，正式步入聯合國所定義的「高齡化社會」，也正式面臨長壽風險（Longevity Risk）的問題。人口老化所帶來的衝擊，不只是提高工作人口的負擔，它同時也增加政府的養老給付和醫療保險支出，影響社會經濟安全，因此對於未來人口推估的死亡率模型所扮演角色日益重要。本研究以移動平均法和主成分分析兩種不同方式討論不同國家的死亡率變化情形，而後分析廣為人所用的Lee-Carter模型及Reduction Factor模型不足之處，並針對此兩模型不完善部分加以調整改進，建構出適合台灣死亡率的預測模型。

由於人口死亡率的改善，全球人口高齡化現象已成為各國重視的議題，此人口結構的重大改變除了增添國家經濟發展的變量，所導致的長壽風險更衝擊著政府退休基金以及提供年金商品的保險公司。本研究將探討唐提年金制度之運作，期望能以唐提年金作為政府以及保險公司解決長壽風險之工具。 引用Piggott et al. (2005)之GSA模型(Group Self-Annuitization)，並以人口資料庫(Human Mortality Database)中之台灣男性死亡率資料進行情境模擬分析。相較于Piggott et al. (2005)，本文為探討未來死亡率改善趨勢對給付之影響，使用Lee-Carter死亡率模型來預估未來死亡率；另外，不同於Piggott et al. (2005)假設固定的投資報酬率，本文考慮每期投資報酬率之波動作為給付計算的重要參數之一。 本研究發現(1).不管是哪種投資組合當中，每期的平均年金給付隨著計劃時間增加。(2).每期的平均年金給付以及給付之分配在股票部位越高的投資組合中有越高的波動性。(3).GSA模型當中， 死亡率變數對於平均年金給付的影響較投資報酬率變數為大。另外，本文亦比較唐提年金制度與確定給付制度之不同：(1).唐提年金俱有充分的基金儲備特色，基金破產機率有限。(2). 在唐提年金體制下，退休金計劃提供者無需承擔基金投資風險。 / Tontine annuity schemes are introduced as a solution for annuity providers and governments to alleviate longevity risk. Applying Taiwan male mortality data to Group Self-Annuitization (GSA) as proposed in Piggott et al. (2005), this paperuses the Lee-Carter model, which incorporates longevity risk, in a simulation study to demonstrate how benefit payments increase in elder ages underdifferent scenarios. Unlike Piggott et al. (2005), we include deviations in both mortality and rate of return from expectations to compare benefit payments amongdifferent portfolios. Moreover, this paperdescribes the two features by whichtontine annuity schemesprevail overTaiwan’s Labor Insurance Annuity Schemes (LIAS): First, tontine annuity schemes are almost always fully funded. Second, the plan sponsor of tontine annuity schemesdoes not need to bear the investment risk.

隨著醫療技術進步、環境衛生改善與人類追求健康生活的趨勢，全世界人類的死亡率不斷地下降。在死亡率不斷的改善的情形下，保險公司可能在壽險商品上獲利，但在年金部份卻會因長壽風險而有所虧損。 自然避險則是保險公司可行的避險策略之一，即透過公司整體保單的組合，來達到規避死亡率風險和利率風險。此外，不同於之前的相關研究，我們所使用的資料，是由臺灣所有的保險公司提供的經驗死亡率，而不是國民生命表。目前保險公司在定價年金和壽險商品時，使用的死亡率是國民生命表，即假設買年金商品的被保險人和買壽險商品的被保險人的死亡率是相同的。但是從經驗死亡率的資料，我們發現購買年金商品的被保險人，其死亡率會低於買壽險商品的被保險人的死亡率。上述情形，會造成保險商品定價有誤;因此，我們考慮不同性別的年金、壽險的死亡率，並研究這些死亡率之間隨機變動項的相關性，以期在未來死亡率和利率變動下，可以藉由死亡率間的相關性，而抵消總價值變動的變異數和定價差異。 根據經驗資料，我們提出一個模型，可透過調整賣出年金和壽險的比例(年齡、性別)，使得保險公司能夠針對公司整體保單組合，找到並有效地運用的自然避險策略。文中最後進行模型敏感度分析，以及提出可能採用的保險商品配置策略，可作為目前保險公司進行死亡率和利率避險的參考。 / The mortality rate of human being has decreased year by year due to the improvement of medical and hygienic techniques. With the mortality improvement over time, life insurers may gain a profit and annuity insurers may suffer losses because of longevity risk. However, natural hedging is a feasible strategy to hedge mortality risk and interest risk at the same time. In this paper, we investigate the natural hedging strategy and try to find an optimal collocation of insurance products to deal with longevity risks for the insurance companies. Different from previous literatures, we use the experienced mortality rates from life insurance companies rather than population mortality rates. This experienced mortality data set includes more than 50,000,000 policies which are collected from the incidence data of the whole Taiwan life insurance companies. In general, insurance companies use population mortality rates to price life insurance and annuity products. Nevertheless, the mortality rate of annuity purchasers is averagely lower than that of life insurance purchasers. This situation leads to mispricing problem of both life insurance and annuity products. So in this paper, we can construct four mortality tables (gender, product) and investigate the correlation of these stochastic variation terms of four mortality rates. According to the correlation relation between these four mortality rates, we can offset the variance of portfolio’s change and difference of mispricing. On the basis of the experienced mortality rates, we demonstrate that the proposed model can lead to an optimal collocation of insurance products and effectively apply the natural hedging strategy to a more general portfolio for life insurance companies.

本研究透過不同年齡與性別之保戶之壽險商品與年金商品保費淨值比例來做商品間自然避險策略，避免以往只能配適特定年齡商品間自然避險策略的限制，使保險公司能夠對於公司內不同年齡與性別之壽險商品與年金商品投資組合進行避險策略，並藉由壽險商品和年金商品的淨值保費比例來求出最適壽險與年金商品權重來配置商品組合。且使用臺灣實務資料以建構年金商品與壽險商品各自的死亡率模型進行商品間自然避險策略，並觀察在不同生命表限制下，商品間自然避險之最適權重的變化，供未來保險公司決策者在決定如何配置公司內商品間自然避險策略時，可有一量化之參考依據。

在高齡化、少子化時代的來臨，加以年金保險開放及勞退新制的實施，喚醒社會大眾對長壽風險的重視，進而檢視自身退休保障是否足夠，而壽險業即在從事管理生命風險及長期資產管理，由其提供退休保險商品，應是最為適切，退休保險商品在國外大抵係指年金保險而言，國內因保險發展之特性尚包含養老壽險及還本壽險。 保險公司於推動上述退休保險商品時，即在滿足消費者的需求及長期穩健之經營，本研究提出幾個因素應予考量，其中包括法令規範、保險商品開發、財務操作以及行銷規劃與系統建置等，本研究透過各個考量因素之解析，期能讓壽險業者在經營上有所助益。 為進一步分析與比對，本研究選取某家壽險公司與前述考量因素進行實證評估，而從驗證中發現個案公司有其特長及成功之處，也有窒礙及難為之處；前者為個案公司所固有，後者有些是整體壽險業所面臨的，有些是個案公司本身的，有些甚至是主管機關應予重視的，無論從任何角度觀察，都是退休保險商品發展，需要正視的問題。 本研究將發現之問題予以整理，並提出結論及建議，希望作為主管機關、壽業者及個案公司從事退休保險監理、經營之參考，如能獲得適當之解決，相信對產、官及消費者都是皆贏的局面。 關鍵字：高齡化、長壽風險、年金保險、養老保險、還本壽險。

在醫療與衛生技術飛快進步下，長壽風險目前已成為國際上普遍重視之議題，為因應死亡率改善所帶來之不確定性影響，壽險公司與退休基金也衍生出各種避險策略，近年來避險策略發展中當以自然避險為主軸，其中又可分為商品間避險與商品內避險法。一般市場上含有商品內避險概念的商品並不少見，如生死合險與還本型保險等，雖然商品內避險法有規避基差風險與免除因保險期間重新配置商品組合造成管理費用之優點，卻也存在無法因應實際死亡率做調整之缺點。因此本研究以英國Money-Back年金商品為例，採用存續期間配適法建構商品內避險最適組合，並配合現金流量分析自然避險策略的真實效果，提供未來壽險公司作為設計商品時之參考。 本研究發現採取商品內避險法時，壽險部分在保險期間後期會發生反轉現象，現金流量淨值波動方向變成與年金險一致，導致商品後期淨值波動過大，失去避險效果；本研究同時發現過去評估自然避險效果時普遍採用的淨值免疫指標存在缺陷，無法兼顧現金流量波動與破產機率。因此我們提出一種創新指標，同時考慮免疫理論中的三大免疫目標，研究結果顯示透過創新指標較能夠完整的評估整體自然避險效果，減少壽險公司於保險期間因現金流量波動劇烈所衍生之資金借貸成本，獲得更佳的避險效果。 / With the improvement of medical and hygienic techniques, longevity risk has become the most important issue in the world. Life insurers and the pension provider propose various kinds of hedging strategies to cope with the uncertainty due to the improvement in mortality. In recent year, the development of hedging strategies focus on natural hedging, which can classified as the hedging strategies according to different insured policies or the same insured policy with survival benefit and death benefit. Endowment is a good example for the hedging strategy from the same insured policy. Although hedging from the same insured policy can avoid basis risk and decrease the cost from rebalance in the insurance period, it couldn't adjust product portfolios by experienced mortality rates. In this paper, we attempt to analyze the natural hedging effect for the Money-Back annuity and use the immunization model to find the optimal collocation of insurance products and evaluate the effect of the natural hedging by cash flow method. We find that life insurance will happened contrary effect in the later insurance period when we try to hedging from the same insured. The changes on the liability of life insurance become the same direction with annuity and lead to more uncertain in later insurance period; We also discover that the indicator which used to evaluate the effect of natural hedging in the past has some defect, so we propose a new indicator which include three immunization goals. We find the new indicator can evaluate the natural hedging effect completely, then it may can help life insurers to avoid the cost of capital due to the unstable cash flow.

在醫療科技與衛生技術飛快地進步下，死亡率不斷改善所帶來不確定產生的長壽風險，已經成為世界各國重視的議題之一，為了因應長壽風險所帶來的衝擊，壽險公司與退休基金發展出多種避險策略，商品內自然避險為其中一種。 本文以淨值免疫和現金流免疫的方法來探討商品內自然避險的效果，發現因為長壽風險造成錯誤定價的緣故，在被保人邁向高年齡時，壽險商品因死亡率改善的效果與一般預期有明顯出入，造成商品保單期間末期自然避險效果消失，本文定義此現象為“壽險反轉效果”，本文並進一步探討其生成原因與解決方法，發現其與亡率改善以及生存曲線矩形化的現象有關，本文接著探討台灣的生存曲線矩形化現象，以釐清“壽險反轉效果”的發生原因。 / Thanks to the improvement of technology and medicine, mortality rate has been improved but also triggered the uncertainty of longevity risk, making longevity risk an important issue around the world. In order to decrease longevity risk, the insurers and pension funds has developed several hedging strategies. Natural internally hedging is one of the common hedging strategies. Some of the insurance products share the concept of Natural internally hedging, such as endowment. The advantage of Natural internally hedging is that it helps the insurer to avoid basis risks and lower the management costs and expenses. However, it fails to be adjustable by varies of the unexpected mortality rate. This thesis will discuss and analyze the trend of cash flow of life insurance and annuity, aiming at establishing principles for insurance product design, which are designated to hedge longevity risk by the offset of the value of life insurance and annuity. During the research, this thesis found that the longevity risk can’t be hedged because the impact of “The reversion of Life product”. The following parts of this thesis discussed the reason why “The reversion of Life product” happened and how to solve it.

壽險公司積極開發新商品以因應大量退休人口的需求，讓退休屋主得以所居住之房屋為抵押物，向金融機構貸款以獲得退休後之資金來源的反向房屋抵押貸款商品也應運而生。但這類的退休商品，除了讓壽險公司因人類平均壽命延長的現象而曝露在長壽風險的威脅下之外，其中所牽涉到之多樣的財務風險，也讓壽險公司在經營上面臨另外一個挑戰，但是反向房屋抵押貸款商品因其商品特性，似乎也可以提供壽險公司不同的風險分散的效果，有助於提升整體商品組合的避險效果。 本研究所提出之多因子免疫模型，可供壽險公司依照其所銷售之商品及所欲規避之風險，選擇一個最適的商品銷售數量，讓整個商品組合獲得最佳之避險效果。本研究透過多因子免疫模型進行數值分析，發現商品中加入反向房屋抵押貸款商品時，其避險效果明顯的優於未包含反向房屋抵押貸款之商品組合，顯見壽險公司發行反向房屋抵押貸款商品將有助於達到風險分散的效果，獲得更佳的避險成效。 關鍵字：長壽風險、財務風險、反向房屋抵押貸款、多因子免疫模型。 / Life insurance company try to meet the demand of the elder who has been retired by designing new products. The mortgage instruments to enable elderly homeowners to borrow by using the equity in their home as collateral, called “reverse mortgage”. With the launch this kind of product, life insurance company exposures in the threat of longevity and involves in others financial risks. However, the features of reverse mortgage may create the different effects of diversification for life insurance company to catch the better effects of hedging. We propose the Multi-Factors Immunization Approach to calculate the optimal product portfolio which attain the best hedging effects for life insurer by adjusting the number of units sold and recognizing the risks they want to hedge. We discover that the product portfolios which include reverse mortgage have the better hedging effects than these don’t include by numerical analysis. It is obviously that life insurer can acquire the effect of diversification and better hedging effects. Key words: Longevity risk, Financial risk, Reverse mortgage, Multi-factors immunization approach.

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.