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。

由於死亡率曲線與殖利率曲線同樣可用水平(level)、斜率(slope) 、曲度(curvature)來描述，且兩者之參數皆為受到時間因素影響之動態因子，故本研究應用Nelson-Siegel(1987)系列之動態利率期間結構模型，如Diebold and Li (2006)的三因子模型，針對日本1947至2006年死亡率進行配適，再以自我相關模型檢視因子的趨勢變化進而預測；結果發現本研究所使用模型在配適死亡率曲線上效果良好，而高齡人口死亡率預測上較幼年、青少年人口精確，以日本資料而言Svensson四因子模型相較於Lee-Carter模型預測能力佳，但在年輕人口死亡率中則不然。 / The main purpose of this study is tempting to extend existing model in interest model context to mortality modeling. Since the mortality curve has resemblance of interest rate yield curve. Both of them can be describe by level, slope, and curvature terms. Also, the parameters of two curves are the function of time. We apply the Nelson and Siegel family yield rate models such like Diebold and Li (2006) model to fit and forecast the mortality term structure. By using the Japanese mortality data within 1947 to 2006, we find out that the fitting of these models are precise, especially when age dimension being truncated to age 20-103. The forecasting performances comparing with the benchmark Lee-Cater model is better in elder age but worse in younger age.

對於人口數不多的國家及地區，因為樣本數較少，死亡率的震盪較大，導致死亡率的估計值較不穩定。為解決此種問題，本研究以其他國家的死亡率資料輔助台灣，建構死亡率模型。首先，以群集分析方式選擇適合輔助台灣的國家，也就是死亡率性質相近之國家，本研究建議以死亡改善率做為主要的考量；其次，以主成分分析的方式分解多個國家死亡率，以負荷做為多個國家的共有係數，分數則是隨著資料和時間改變的變數，在研究結果中，5~6個成分個數即會有不錯的配適和預測效果，以五齡組死亡率配適模型為例，成分個數為6時，男性配適Lee-Carter模型全部國家的平均MAPE為5.40%，主成分分析則為4.13%，下降幅度將近24%，而Lee-Carter模型預測的整體MAPE為14.72%，主成分分析為12.22%，下降幅度約17%，因此主成分分析模型確實有明顯改善Lee-Carter模型。 而和台灣死亡率性質相近的國家，主要選入歐洲國家，像是奧地利、法國、愛爾蘭、挪威和西班牙，除了法國和西班牙人口數分別為六千多萬和四千多萬的國家外，其餘三個國家人口數皆不超過一千萬，這說明人口數多寡或許不是輔助小地區建構死亡率模型的唯一重點，應選取適合的國家作為輔助用途。

隨著醫療的普及與生活品質的改善，人類的死亡率持續的下降。壽命的延長是人們夢寐以求的理想，但是隨著壽命延長，人類隨之要面臨許多衍生而來的問題，諸如退休規劃、醫療照顧等問題。面臨延壽風險的問題，現行最急迫的課題莫過於探討人口死亡率預測模型。對於死亡率預測的模型，國外已有相當多的研究，近年來也看到國內有許多學者紛紛投入死亡率的研究，由於目前英國實務上所使用Reduction Factor模型，在國內尚無相關的研究，故本文以Reduction Factor模型為基礎，並透過與Lee-Carter模型的比較與各國死亡率資料的驗證，進一步加以改善並建構出適用於台灣地區死亡率預測的模型，以作為往後用來衡量延壽風險的依據。

退休基金制度由確定給付制轉換成確定提撥制，使退休基金之投資風險改由勞工承擔。而退休基金的多寡則完全視退休基金投資績效的好壞而定，如何有效地管理退休基金儼然成為一項重要議題。本文延續Yang, Huang, and Lee(2006)之研究，改採用Mitchell et al.(2013)隨機死亡率模型，除了將股票與零息債券作為退休基金之兩項投資標的，再增加一項長壽債券作為退休基金之投資標的。透過極大化勞工之預期效用函數，探究確定提撥制下退休基金之最適資產配置策略以處理長壽風險。

無 / Existing literature has shown that force of mortality has amazing resemblance of interest rate. It is then tempting to extend existing model of interest rate model context to mortality modeling. We apply the model in Diebold and Li (2006) and other models that belong to family of yield rate model originally proposed by Nelson and Siegel (1987) to forecast (force of) mortality term structure. The fitting performance of extended Nelson-Siegel model is comparable to the benchmark Lee-Carter model. While forecasting performance is no better than Lee-Carter model in younger ages, it is at the same level in elder ages. The forecasting performance increases for 5-year ahead forecast is better than 1-year ahead comparing to Lee-Carter forecast. In the end, the forecast outperforms Lee-Carter model when age dimension is trimmed to age 20-100.

隨著科技、醫療技術的進步，全世界的死亡率逐年下降，導致人口高齡化、扶老比逐年增加等問題，在這些問題下，退休老人是否有足夠的退休金來維持生活品質是每個人都很關心的議題。本論文探討反向房屋抵押貸款在台灣的應用來維持退休老人生活品質，在承做反向房屋抵押貸款得過程中，影響最大的三個因素分別為死亡率模型、房屋價值模型與利率模型。本論文中的死亡率模型採用Lee 與Carter 的死亡率模型；利率模型採用CIR-SR (Cox、Ingersoll 與Ross)模型；房價模型則是採用ARIMA 模型與布朗運動模型。最後利用台灣死亡率、利率與房價的資料進行模擬，針對各個不同的情境做分析，使用無套利的定價模型計算 反向房屋貸款在台灣的最適可貸金額。 / Progressions of technology and medical treatment has caused the dropping of death rates which raised the aging population problem. Under this circumstance, maintaining good quality of life after retirement is an issue that many of us concerned. This paper discusses how the use of reverse mortgage may help us to accomplish a quality retirement life. In addition to that, we apply the Lee-Carter model, CIR-SR model, and ARIMA model to forecast mortality, interest rate, and house prices respectively. Finally, we use the statistic from Taiwan to simulate several scenarios, and then use the no arbitrage pricing model to find the best payments of reverse mortgage.

20世紀以來，因為衛生醫療等因素的進步，各年齡死亡率均大幅下降，使得平均壽命大幅延長。壽命延長的效果近年逐漸顯現，其中的人口老化及其相關議題較受重視，因為人口老化已徹底改變國人的生活規劃，死亡率是否會繼續下降遂成為熱門的研究課題。描述死亡率變化的模型很多，近代發展的Age–Period–Cohort模型(簡稱APC模型)，同時考慮年齡、年代與世代三個解釋變數，是近年廣受青睞的模型之一。這個模型將死亡率分成年齡、年代與世代三個效應，常用於流行病學領域，探討疾病、死亡率是否與年齡、年代、世代三者有關，但一般僅作為資料的大致描述，本研究將評估APC模型分析死亡率的可能性。 APC模型最大的問題在於不可甄別(Non–identification)，即年齡、年代與世代三個變數存有共線性的問題，眾多的估計APC模型參數方法因應甄別問題而生。本研究預計比較七種較常見的APC模型估計方法，包括本質估計量(IE)、限制的廣義線性模型(cglim_age、cglim_period與cglim_cohort)、序列法ACP、序列法APC與自我迴歸模型(AR)，以確定哪一種估計方法較為穩定，評估包括電腦模擬與實證分析兩部份。 電腦模擬部份比較各估計方法，衡量何者有較小的年齡別死亡率及APC參數的估計誤差；實證分析則考慮交叉分析，尋找用於死亡率預測的最佳估計方法。另外，也將以蒙地卡羅檢驗APC的模型假設，以確定這個模型的可行性。初步研究發現，以台灣死亡資料做為實證，本研究考量的估計方法在估計年齡別死亡率大致相當，只是在年齡–年代–世代這三者有不同的詮釋，且模型假設並非很符合。交叉分析上，Lee–Cater模型及其延展模型相對於APC模型有較小的預測誤差，整體顯示Lee–Cater 模型較佳。 / Since the beginning of the 20th century, the human beings have been experiencing longer life expectancy and lower mortality rates, which can attributed to constant improvements of factors such as medical technology, economics, and environment. The prolonging life expectancy has dramatically changed the life planning and life style after the retirement. The change would be even more severe if the mortality rates have larger reduction, and thus the study of mortality become popular in recent years. Many methods were proposed to describe the change of mortality rates. Among all methods, the Age-Period-Cohort model (APC) is a popular method used in epidemiology to discuss the relation between diseases, mortality rate, age, period and cohort. Non-identification (i.e. collinearity) is a serious problem for APC model, and many methods used in the procedure included estimation of parameter. In the first part of this paper, we use simulation compare and evaluate popular estimation methods of APC model, such as Intrinsic Estimator (IE), constrained of age, period and cohort in the Generalized Linear Model (c–glim), sequential method, and Auto-regression (AR) Model. The simulation methods considered include Monte-Carlo and cross validation. In addition, the morality data in Taiwan (Data sources: Ministry of Interior), are used to demonstrate the validity and model assumption of these methods. In the second part of this paper, we also apply similar research method to the Lee-Carter model and compare it to the APC model. We found Lee–Carter model have smaller prediction errors than APC models in the cross–validation.

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.

在人口統計領域中，早期習慣將人口變化視為時間的函數，企圖以Deterministic Function來刻劃，例如：1798年Malthus提出的Malthusian Growth Model ；1825年Gompertz提出的Gompertz Model以及1838年Verhulst主張以Logistic Function描述人口成長。而近年來則是傾向於逐項分析各種因素的隨機性模型，例如：1983年Holford加入世代的APC模型；1992年Lee 和Carter提出的Lee-Carter死亡率模型以及2003年Renshaw與Haberman提出改善Lee-Carter死亡率模型的Reduction Factor模型。 人口變化主要分成自然增加與社會增加，而自然增加是為出生扣掉死亡，社會增加則為移入扣掉移出。首先，本文先不考慮遷移的部分，各別以出生與死亡人口的變化為研究對象，視其變化為一隨時間變動的動態系統，以常微分方程來刻劃。由台灣地區人口統計資料顯示，出生率或死亡率都有逐年下降的趨勢，而且隨著時間而變化加劇的傾向，使得以往使用的模型不易捕捉變化，因此我們提出「二次逼近法」，從出生、死亡人數對時間的變化率與曲度利用數值分析的方式來估計出生與死亡數，進而從中找出在此動態系統背後隱藏的規則。而後再同時考慮其他各種變項，以偏微分方程來刻劃，最後即可建立台灣地區人口變化模型。 / In early population statistics, the population changes were regarded as a function of time so that people tended to describe the variations by deterministic functions. For instance, Malthus proposed the Malthusian Growth Model in 1798; Gompertz presented Gompertz Model in 1825; Verhulst advocated using logistic function to describe an increase in population. In recent years, people tend to use the stochastic forecast method to analyse every factor term by term. For instance, the Age-Period-Cohort (APC) Model which was proposed by Holford in 1983; Lee and Carter proposed the Lee-Carter Mortality Model in 2003; and Renshaw and Haberman proposed the Reduction Factor Model in 2003 that improve the Lee-Carter Mortality Model. The population changes equal to nature and social increase, where the nature increase is the difference between birth and death population, and the social increase is the difference between immigrants and emigrants. First, we focus on natural increase rather than social increase. Moreover, we use ordinary differential equation to decribe the variation as a dynamic system over time. From the data obtained from the Ministry of Interior Taiwan, we know that the fertility and mortality has been decreasing, and the change is getting more violent year by year. Under the consideration that previous models are not able to accurately present the changes of birth and death, we proposed "second-order (or parabola) approximation method." From the variation rates and curvatures of birth and death population, we estimated the population size. Furthermore, we want to find the rule in the dynamic system. Later we will consider other factors simultaneously, and describe them by partial differential equation. Finally, the population model is constructed.