近年來電腦等高科技的快速成長，進而促進資訊化的過程。資料庫的蓬勃發展，使得資料大量累積，長久之下，卻造成資料過多，資訊不足的嚴重問題。因此資料庫內的知識探索議題也隨之興起，而資料採礦（Data Mining）的過程，更是其中重要的一環。 相同的，預防醫學資訊的發展，流行病學資料庫中亦累積了大量關於死亡統計資料，而這些資料中，隱藏可能存在的知識，能加強我們對疾病進展的瞭解。若將資料採礦的概念應用於流行病學領域，相信必能相輔相成。 本研究的重點在於結合統計軟體 STATISTICA，以Visual Basic 6.0語言開發一個簡易的資料採礦系統之使用者介面，並將資料採礦技術應用於死亡統計資料中。系統中的挖掘方法主要採用敘述統計、交叉分析與多變量分析中的群集分析與區別分析，根據行政院衛生署統計室所提供之民國八十三年至八十八年台灣地區人口死亡原因資料，來發現隱藏在資料中的趨勢與模式。 / For the past decade the development in computer technology has advanced so rapidly that it brings forth the enormous supply of data information. As time passes by the data information has been increasingly accumulated yet little can be inferred from the data thus resulting a loss of information which might be of significant. Bearing with the existence of such issue, this research presents the process of data mining as one of the solution. Similarly, the data base in the field of medical science may have contained a large amount of information. If one can appropriately apply the application of data mining into this huge database then we may be able to extract some valuable findings. The focus of this research is to develop a user friendly operating system using Visual Basic 6.0 and integrates the statistical software-STATISTICA into the operating system. The research applies the application of data mining on the death data provided by Statistics Office, Department of Health from 1994 to 1999. The methods used in this application are descriptive statistics, cross tabulation, cluster analysis, and discriminant analysis of multivariate analysis in an attempt to find out if there is any pattern in the cause of death.

對於商業保險公司和政府單位而言，死亡率的改善和未來死亡率的預估一直是一大重要議題。特別是對於退休金相關的社會保險、勞退或是商業年金、壽險等等，如何找尋一個準確的預估模式對未來的死亡率改善情況進行預測，並釐訂合理的保費及提列適當的準備金，是對於一個保險制度能否永續經營的重要因素。過去所使用的配適方法，大多僅以單一群體的過去資料輔助未來的預測，例如 Li and Carter (1992)所提出的 Lee-Carter Model，或是 Bell (1997)使用主成分分析法 (Principal Component Analysis, PCA)等僅針對單一群體本身變數進行分析之方式。然而綜觀全球死亡率改善趨勢，可發現國與國間、組與祖間雖有不同，但仍具備共同的趨勢。因此在考慮未來的死亡率配適方面，應加入組與組間的共同因子 (common factors) 進行考量。 Li and Lee (2005)曾提出 Augmented Lee-Carter Model，即對原本的Lee-Carter Model進行修正，加入共同因素項，並且得到更好的預測效果。 本文則採用考慮共同因子之主成分分析原理建構多重群體死亡率模型，即透過主成分分析法，同時考慮不同群體間的死亡率，並以台灣男性和女性1970年至2010年的死亡率資料，做為兩個子群體進行分析。本文使用之主成分分析法模式，和 Lee-Carter Model (Li and Carter, 1992) 和 Augmented Lee-Carter Model (Li and Lee, 2005)，以MAPE法對個別的預測能力進行分析，並得出採用PCA的模式，在預測男性短年期(5年)內的預估能力屬精確(MAPE 介於10%~20%之間)，然而在長期預估下容易失準，且所有使用的模型，在配適台灣資料時皆發生無法準確預估嬰幼兒期(0~3歲)和老年期(80歲以上)之情形。本文並以所有模型預估之死亡率計算保險公司之準備金與保費提列，並與第五回經驗生命表進行比較。 / For governments and life insurance companies, mortality rates are one of the key factors in determining premiums and reserves. Ignoring or miscalculating mortality rates might have negative influences in pricing. However, most of the mortality models do not consider the common trends between groups. In this article, we try to construct the mortality structure which considering common trends of multi-groups populations with principal component analysis (PCA) method. We choose 9 factors to set up our model and fit with the actual data in Taiwan’s gender mortality. We also compare the Lee-Carter Model (Lee and Carter, 1992) and the augmented Lee-Carter Model (Li and Hardy, 2012) with our common factors PCA model, and we find that the PCA model has the least MAPE than other model in five years forecasting in both genders. After finishing basic analysis, we use the mortality data of Taiwan (1970 to 2010) from human mortality database to construct the life expectancy model. We adopt the same criteria to choose the components we need. We also compare the level premium and reserves by different forecasting mortality rates. All of the models indicate life insurance companies to provide higher reserves and level premium than using the 5th TSO experience mortality rare. We will do following research by using company-specific data to construct unique life expectancy model.

人口統計學家在傳統上習慣將人口的種種變化視為時間的函數，皆試圖以決定型（deterministic）的函數來刻劃，例如：1825年Gompertz提出的死力法則、1838年Verhulst以羅吉斯函數描述人口成長。近年則傾向於逐項（item-by-item）分析各種可能因素，例如：1992年Lee-Carter提出的死亡率模型、目前英國實務上使用的Renshaw與Haberman(2003)提出改善Lee-Carter模型的Reduction Factor模型、加入世代（Cohort）因素的Age-Period-Cohort模型等。但台灣地區近年來生育率與死亡率皆不斷下降，且有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。 本文以另一個角度思考生育與死亡變化，將台灣人口視為一隨時間變化的動態系統，使用微分方程來刻劃，找出此動態系統的背後所隱含的規則。人口動態系統的變化，主要來源是出生、死亡與遷移，在建模的過程中，我們先各別針對其中一項，在其他條件不變的情況下，以常微分方程建模，之後再同時考慮各項變動，以偏微分方程建模，找出台灣人口變化的模型。在本文中，我們先介紹使用微分方程模型分別配適與估計出生與死亡。 由台灣地區人口統計資料顯示，不論總生育率或各年齡組的死亡率都有逐漸下降的趨勢，但是每年之間的震盪很大，因此我們提出「二次逼近法」，從出生或死亡對時間的變化率與曲度來估計生育率與死亡率，對於此種震盪幅度較大的資料，可以得到頗精確的估計。唯在連續幾年資料呈現近似線性上升或下降處，非線性的模型容易出現較大的估計誤差，針對此問題我們也提出一些可能的修正方法，以降低整體的模型誤差率。 / Conventionally the change of population is considered as a function of time and described by using deterministic functions. The well-known examples are Gompertz law of mortality (1825) and Verhulst’s logistic growth model (1838). Recently demographers favor stochastic models when analyzing factors in an item-by-item fashion. Since 1992, Lee-Carter model is a most commonly used stochastic model in demographic studies. But empirical studies indicate that the rapid declines in both fertility and mortality rates are against the assumptions of Lee-Carter model. In this study we treat Taiwan population as a dynamic system which changes over time and characterize it by differential equations. Since the changes are from birth, death and migration, we first separately build models using ordinary differential equations. Afterwards the model of Taiwan population can be built by using partial differential equations considering the three main factors simultaneously. Total fertility and age-specific mortality rates in Taiwan decline over time but with shakes between years. Consequently we propose‘parabola approximation method’and apply it to velocity and acceleration of birth or death to solve the differential equations of Taiwan fertility and mortality. Empirical study shows the method allows us to get accurate estimates of mortality and fertility when the data change a lot in a short period of time. But we found the model may over-fit the data at some time point where the function does not seem to be very continuous.

近年來臺灣高齡人口比例有明顯之增長，兩性平均餘命自1906年至今不斷的往高齡延伸，伴隨著這兩種趨勢下，臺灣高齡人口結構快速的轉型和變動，使得瞭解高齡人口死亡率模式成為估算未來臺灣人口結構發展趨勢的重要依據。然而，過去許多人口研究所依賴的資料來源是以「戶籍人口統計資料」為基礎，其資料內容雖涵蓋時間範圍甚廣，但在高齡人口的死亡率資料記載則有稍嫌簡化的問題及死亡人數紀錄不準確的限制，因此本研究擬以搭配「死因資料檔」，擷取其對於死亡人口數及死亡時間詳細紀錄的優點，來結合運用以探討臺灣高齡人口死亡率模式。 本研究以「參數式模型」、「相關模型」、「外推法」及「APC模型」四種不同估算取徑的運用，並結合現有實際的臺灣高齡人口死亡率資料，說明臺灣65歲以上高齡人口死亡率的變遷模式及發展軌跡。研究結果顯示，自1975年臺灣65歲以上高齡人口的死亡率變遷趨勢，確實往更高齡方向發展，同時，其死亡率的變遷波動越大，本身除資料紀錄上可能有所偏誤外，也可能因為90歲以上人口資料逐漸增加的情況下，變異性逐漸的明顯。此外，以各死亡率模型估算配適下，大多在90歲以上高齡人口的計算，其估算不準確的情形則越形明顯，但在65歲至85間的估算死亡率模式則有相當不錯的配適。

本論文是由三篇關於金融市場的實證研究組合而成。第一篇以權益存續期間為主題，主要是利用迴歸模型估計台灣上市產業指數的實證權益存續期間，以探討股票報酬率的利率敏感度。迴歸模型中控制了三個重要的股票風險因子─市場因子、規模因子與價值因子。但其中，我們改以正交市場因子代替市場因子，以避免因為利率變動與市場報酬間存在共線性，而造成權益存續期間有可能錯估的問題。此外，基於權益存續期間具有會隨時間改變的動態特性，本文亦對各產業指數最近一次結構性變化的發生時點進行偵測，並據以推估最近期的權益存續期間。實證結果顯示：除了鋼鐵業的權益存續期間不顯著之外，其他所有產業指數皆具有負的權益存續期間，表示其報酬率與利率變動呈現出正向關係。在程度上，則以營建類指的利率敏感度最大，汽車類指最小。 第二篇應用了Diebold and Yilmaz (2009)的外溢指標分析台灣上市產業指數間的連動性，其優點是可以瞭解到產業間相互影響的方向以及程度。實證結果顯示：台灣上市產業指數間的外溢程度頗高，並以營建業為最主要的影響者，而相反地，鋼鐵業則是主要的被影響者。外溢指標具有隨時間改變的動態特性，而且透過動態外溢指標可觀察到次貸風暴蔓延的嚴重性。 第三篇應用了Goyal, Perignon and Villa (2008)所提出的多群組因素分析法，檢測美國總人口死亡率的共同因子個數。該方法最大的優點是能夠有效地辨識出真正的共同因子，避免了一般因素分析容易將解釋能力高的群組內獨特因子誤認為共同因子的缺點。根據檢測結果顯示，美國總人口死亡率的共同因子共有兩個，而且第二個因子的重要性隨時間愈來愈明顯。 / This thesis consists of three empirical essays about financial markets. The first essay analyzes the sensitivity of stock returns to changes in interest rates by estimating empirical equity duration of 18 industrial indices in the Taiwan Stock Exchange. In the regression models, we also control for the market excess return and the Fama-French mimicking returns for size and book-to-market factors. To avoid the effects of the multicolinearity between the market excess return and the interest rate changes, we replace the market excess return by the orthogonalized market factor. In addition, considering the time-varying pattern of empirical equity duration, we further adopt the reversed ordered Cusum test proposed by Pesaran and Timmermann (2002) to identify the most recent break of the regression relationship, and then extract the post-break data to re-estimate the up-to-date empirical equity duration. The result shows that except the Steel index, all industrial indices exhibit significantly negative equity durations, indicating a positive relationship between industrial index returns and interest rate changes in Taiwan. Among them, the Construction index has the largest interest rate sensitivity, while the lowest one is for the Automobile index. The second essay focuses on the nature of financial market interdependence, both in terms of returns and returns volatilities. Being capable of identifying the direction and magnitude of linkages among financial markets, the spillover index proposed by Diebold and Yilmaz (2009) is used to measure return and volatility spillovers between the top eight industrial indices based on market value in the Taiwan Stock Exchange. We find that for both returns and volatilities, the spillover effects among industrial indices in Taiwan are substantial. In particular, the Construction index is the major transmitter of shocks to other industries, and the Steel index, in contrast, suffers the most shocks from others. The spillover index fluctuates over time and indeed detects the severity of subprime mortgage crisis. The third essay adopts the multigroup factor analysis proposed by Goyal, Perignon and Villa (2008) to estimate the number of common pervasive factors for annual age-specific mortality for the entire U.S. populations. While the standard principal component analysis easily treats any group-specific factor as pervasive one due to its high contribution to total system variance, this methodology is able to estimate the space spanned by common and group-specific pervasive factors and recognize the true common factors. Empirical result shows that there are only two common pervasive factors governing the death rates in the United States; in particular, the importance of the second factor increases over time.

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.