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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.

臺灣的男女死亡率皆呈現逐年遞減的趨勢，自1993年進入高齡化社會後，預計將會在2018年進入高齡社會；人口不斷老化的結果讓社會上不論人民或是如保險公司等年金提供者皆面臨愈來愈嚴重的長壽風險；目前現有文獻提出了許多方式以解決長壽風險，其中多數的方法皆需使用到對未來死亡率之預估。 本研究為了能夠更準確的預估未來死亡率的趨勢，參考了Lee Carter (1992)所提出之模型以及Li and Lee (2005)、Li (2013)提出之共同改善趨勢效果，提出考慮商品與商品間以及商品與整體人口間共同改善趨勢之死亡率模型；本研究利用臺灣之保險公司壽險及年金業務經驗死亡率和Human Mortality Database之臺灣人口資料對模型進行配適，並以MAE、MAPE、RMSE三項指標比較與Lee Carter模型之優劣。 最後，本研究利用所配適之模型進行預測，模擬自然避險之效果，檢視臺灣保險業進行自然避險的可能效益，並對決策者在於決定是否要進行自然避險方面給出建議。 / Taiwan became an aging society in 1993 and is expected to become an aged society in 2018. The progressive decrease in Taiwan mortality since the 20th century for both genders has made longevity risk a serious problem for both people and annuity provider in Taiwan. So far, the literature has discussed about how to deal with longevity risk and came out with several solutions which can be categorize as “industry self-insurance”, “ mortality projection improvement” and “capital market solutions” , most of them are related to the projection of mortality. In order to provide a more precise projection of future mortality trend, this article designs several models which collaborates Lee Carter Model (1992) and the common improvement trend suggested by Li and Lee (2005). Based on our models, the Taiwan insurance industry experience mortality data and the Taiwan population mortality data, we test the performance of our models and make comparison. Lastly, we use the model we find to project future mortality trend and try to make a simulation of natural hedging strategy in Taiwan. The purpose we do this is to test the performance of natural hedging method and give suggestion for the decision-maker when they are considering whether to execute a natural hedging strategy.

背景與目的： 每一項疾病都有其潛在的風險，但要有效的降低死亡率及併發症發生率，必須找出關鍵性指標加以改善或預防。我們以心臟冠狀動脈繞道手術之患者為例，希望藉由統計分析的方式，找出造成死亡以及術後併發症最相關的因素，目的不只在預測，而在於防範。 研究對象與方法： 在我們的實驗設計上，風險因子分布在手術前、手術中、以及手術後三個階段，對象是某醫學中心接受心臟冠狀動脈繞道手術的220例患者。分析採用迴歸統計建立模型，其中羅吉斯迴歸中的依變數為死亡率與罹病率，線性迴歸的依變數為加護病房留置天數以及總住院日數。ROC curve亦將被建立，以判斷模型是否能區別病患是否罹病或死亡。所得資料亦計算EuroScore及其ROC曲線面積，並與歷史資料做比較。 結果： 所建立的死亡估計模型的有兩個，預測值都在97%以上，ROC曲線面積亦都超過0.96；併發症估計模型由六個變數所構成，預測率及ROC曲線面積分別為94%和0.984。加護病房留置天數及住院天數估計模型分別由八個及十三個因子來解釋，調整後的R square分別為0.527及0.6。EuroScore對死亡與併發症的預測率，分別為93.7%和82%，ROC曲線面積分別是0.864和0.797，均高於歷史文獻記錄，未來應該廣泛應用。 結論與建議： 經由適當的風險分級和危險因子分析，我們可以找出風險高低的標準和依據，了解影響死亡率與罹病率的關鍵因子是什麼，儘可能的做事前的防範與處置，希望能夠改善結果並提高手術的存活率。 EuroScore是個值得採用的預測工具，可以廣泛應用在死亡率與併發症發生率的估計，但是必須搭配風險因子的改善，才能發揮實際的功效。我們認為，體外循環時間與再次手術是最具有空間來降低死亡率與罹病率的兩個要素，有效率的控制時間、改善造成再手術的前因後果，除了死亡率與併發症發生率的下降外，還可以及早脫離對加護病房照顧的需求並減少留置的天數。

台灣地區居民近年的死亡率下降速度加快，使得我國國民的平均壽命在公元2000年已超過美國，成為長壽的國家之一。其中我國國民死亡率的下降幅度因年齡而不同，且各個年代、世代也不相同，與APC（Age-Period-Cohort）模型採年齡、年代與世代三個因子分析死亡率頗為一致，因此本文計畫以APC模型研究台灣的死亡率。然而，由於「年代＝年齡＋世代」之線性相關，參數估計值有甄別問題（Identification Problem），使得參數估計值不唯一。 文獻中有不同方法解決APC模型的參數估計問題，近年又有Fu（2000）提出之本質估計量（Intrinsic Estimator），可直接解決參數估計及其變異數。因此本文首先以電腦模擬驗證本質估計量，以及過去其他估計方法，檢測這些方法是否可得出理論的結果。本文的第二部分則以西元1961至2005年的資料探討APC模型的實用性，分析APC與Lee-Carter模型的優劣；研究發現APC模型用於估計死亡率時，整體而言雖不如Lee-Carter模型，但可彌補Lee-Carter模型在高年齡有較大誤差的不足，唯在年輕族群則仍有改善空間，未來或可考慮APC與Lee-Carter模型的結合。 / The mortality rates in Taiwan area have been experiencing dramatic decreases in recent years. The life expectancy has surpassed that in the United States in 2000 and Taiwan has become one of the longevity countries. Besides, the falling of mortality rates varies in different age, period, and cohort groups, which corresponds to the APC (Age-Period-Cohort) model. Therefore, the goal of this paper is to study the mortality rates in Taiwan area with APC model. However, due to the linear dependency of age, period and cohort (Period = Age + Cohort), there is the identification problem, that is, the parameter estimates are not unique. A number of solutions to the identification problem in APC model have been provided in the literature. Fu (2000) introduce a new estimator, the Intrinsic Estimator (IE), which can solve parameter estimates and variance directly. In the first part of this research, computer simulation is conducted to examine the IE, compared with other methodologies. In the second part of this research, data from 1961 to 2005 are used for verifying the validity of APC model in fitting mortality rates, and we analyze the strengths and weaknesses between the APC and Lee-Carter model. The results from our study indicate that the APC model in estimating mortality rates does not show as well as the Lee-Carter model as a whole. However, the APC model performs better than the Lee-Carter model for the elderly mortality rates, but is still needed to be improved in young groups. In the future, it can be considered to combine the APC and Lee-Carter model.

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.

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.

重大傷病是我國全民健保的主要特色之一，民國105年重大傷病領證人數為95萬6626人（約4%人口），但其醫療費用超過全國四分之一，且盛行率有逐年上升的趨勢（資料來源：衛生福利部中央健康保險署）。其中，癌症又為重大傷病的首位，佔了重大傷病發證數的49%，雖然癌症發生率每年僅些微上升，但因罹癌後死亡率也逐年下降，而且癌症發生率隨年齡而增加，預期癌症盛行率將隨人口老化而快速上升，醫療利用與支出亦會愈趨上升，加重健保財務的負擔。有鑑於癌症盛行率的增加，健保署於兩年前提高癌症病患換新卡的資格，於103年停發約1萬7000張癌症領證數，但追根究底的解決之道仍在於及早發現與治療，不僅可提昇國民健康，更可有效率使用醫療資源。 本文使用全民健康保險資料庫，以探討國人罹癌前後的健康狀況為目標。透過資料庫的就醫資料，包括重大傷病證明明細檔（HV）、重大傷病門診處方及治療明細檔（HV_CD）、承保資料檔（ID）、2005年百萬人抽樣檔之門診處方及治療明細檔（CD），套用大數據的資料分析方法，探討國人罹患癌症的相關特性。首先對癌症病患進行基本資料之分析，接著探討不同準則下在判定癌症發生與罹癌死亡人數之間的估算差異，整合HV與HV_CD兩個資料庫，選擇可信度較高的方式作為估算癌症發生率與罹癌死亡率的基礎。研究發現，以退保資訊判斷癌症患者是否死亡，錯誤率優於先前根據就醫記錄。本文研究希冀可供政府擬定癌症相關的醫療策略，提高癌症病患的就醫意願及治癒率，增進國人健康，並且有效控制健保支出。

從1906到1999年間，台灣地區男性出生時平均餘命增加了43.55歲，女性增加了47.63歲，雖然台灣地區平均餘命不斷地在增加，地區間死亡風險的差異情形則未有所改善。本研究以生命表方式計算台灣地區縣市級、鄉鎮級平均餘命，尤以鄉鎮級為研究重心，配合統計地圖的使用以瞭解小地區間死亡風險的差異情形。 在縣市層級方面，男性與女性出生時平均餘命皆以七大都市較高，西部其他地區次之，東部地區的花蓮縣、台東縣最低。各縣市出生時平均餘命的差異，正指出各縣市之間的死亡風險差異甚大，特別是東部地區明顯地比西部地區更處於不利的狀態。 在鄉鎮層級方面，男性與女性各生命階段平均餘命皆呈現出西部地區高、東部地區低的明顯差異，而且平均餘命數值越低的地區其兩性平均餘命的差距也越大，此種現象尤以男性平均餘命越低的地區越為明顯。在空間模式上，男、女平均餘命最高的地區皆以大都市為中心向外擴散至其鄰近地區，平均餘命最低的地區則廣散在山地鄉之間。總的來說，中央山脈兩側的鄉鎮不僅死亡風險較其他地區高、兩性的死亡風險差異大，山地鄉之間的死亡風險結構更是呈現高度的異質性。 關鍵字：鄉鎮市區、死亡風險、平均餘命、性別差異、統計地圖、空間模式。 / Over the twentieth century (1906-1999), the life expectancy at birth in Taiwan has increased 43.55 and 47.63 years for males and females respectively. Undoubtedly, such a dramatic increase in life expectancy is resulted from the improvement of living quality in Taiwan over the past century. Nevertheless, after detailed examination of historical data, it shows that the areal variation in life expectancy over the life cycle is significant and has never diminished. This study, by applying the life table methods and statistical maps, attempts to analyze the areal variation in risks of death at small areas in Taiwan. The data sources for achieving this purpose come from the vital statistics and death-registration- profiles. The research findings show that life expectancies for both males and females are highest in the most-developed urban counties. On the contrary, the life expectancies are lower in rural counties. The gaps in life expectancies at birth between Taipei City and Taitung county reach 11 years and 7 years for males and females respectively. The spatial pattern of variation in life expectancy at the township-level is consistent with the research findings at the county-level. In general, the life expectancies for townships within the central metropolitan area are highest, and then gradually decrease for townships distant from the metropolis center. The life expectancies for both males and females are lowest in the rural-mountain townships. The gap in life expectancies between males and females is also larger in these rural-mountain areas. Finally, the life expectancies for the aboriginal townships are lowest in Taiwan. However, the age profiles for risks of death are heterogeneous in these aboriginal townships. Keywords: life expectancy, life table, statistical maps, spatial pattern

臺灣的癌症發生率及死亡率有連年增加的趨勢，研究指出原因可能與環境中的污染物質有關，檢測可能的污染源附近是否存在癌症群聚(Cluster)，將有助於未來的癌症防治。在空間統計(Spatial Statistics)有不少方法可用於檢測群聚現象，其中用來檢測某個特定位置周圍是否發生群聚的方法被稱為焦點檢定(Focused Test)，本文介紹及評估常用的焦點檢定方法，並使用較佳方法探討臺灣地區疑似污染源的地區。 首先本文使用電腦模擬，在不同情境假設下比較焦點檢定方法的檢定力(Power)，例如研究區域大小、群聚形狀等不同的情境，以判斷檢定方法之間的優劣。最後本文分析臺灣鄉鎮市(Township)層級癌症死亡資料，應用焦點檢定方法分析石門核一廠、恆春核三廠及麥寮六輕周圍的癌症死亡率，檢定結果發現核一廠及麥寮六輕附近有較高的癌症死亡率。 / The cancer incidence and mortality rate in Taiwan have been increasing over the past 30 years. Previous studies indicate that the pollution sources, especially for those creating air pollution and excess radiation, are one of the potential causes for the increment. Correctly, detecting the location of possible sources of contaminants can help for cancer prevention. In spatial statistics, focused test can be used to determine if the intensity rate are higher around a possible pollution source. We will introduce and evaluate frequently used focused tests and apply them in Taiwan. First we use computer simulation to compare the power of focused tests in different scenarios, such as study region and cluster shape. Next, we apply the focused tests to Taiwan cancer mortality data, in order to decide if the cancer mortality rates are higher around Chinshan nuclear power plant, Maanshan nuclear power plant, and Mailiao sixth naphtha cracker. The results show that the cancer mortality rates around Chinshan nuclear power plant and Mailiao sixth naphtha cracker are significantly higher.