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.

近年隨著油價、物價上漲所導致的通貨膨脹風險，壽險業者以增額型終身壽險來吸引潛在消費者。另外，由於醫療技術的進步，使得死亡率逐年改善，因此將造成保單在設計時可能將遭受到長壽風險的影響。本篇文章的主要目的即探討長壽風險對於保單責任準備金的影響，並以增額型終身壽險作為本文主要分析標的。首先建構死亡率模型(Lee-Carter模型)，用來配適並模擬死亡率，接著探討增額型終身壽險在各保單年度下之現金流量以及責任準備金的提存，進一步再引進不同的死亡率來探討其現金流量分佈情形與責任準備金之提存。本文研究結果發現，在保險公司未採用遞迴方式計算保費時，當繳費期間愈短、複利利率愈高以及投保年齡愈低時，保險公司所面臨之長壽風險愈大，其後在帶入各種不同死亡率模型，發現死亡改善率愈高，保險公司所面臨之長壽風險愈大，而保險公司在提存責任準備金時，並未考慮到死亡改善率的部分，此對保險公司的財務健全將造成隱憂，本文於此部分建議監理機關透過法規(RBC)的制訂，調整準備金提存的係數，以降低長壽風險對保險公司財務之衝擊。 關鍵字：長壽風險、死亡率模型、增額型終身壽險、保單責任準備金、增額準備金、Lee-Carter Model以及RBC制度。 / With the improvement of medical technology, the life expectancy around the world is increasing year by year during the past decade. Therefore, the increasing whole life insurance policy is popular during these years because its benefits are escalating with time and policyholders think they could gain more benefits when they live longer. Like annuity policies, the increasing whole life insurance could also suffer from the longevity risk, which may have enormous impact on the financial statements of insurers. The purpose of this paper is to discuss the impact of longevity risk on reserves, based on increasing whole life insurance policy. First, we construct Lee-Carter model to fit and simulate mortality rate and assume different mortality improvements from the 2002 Taiwan Standard Ordinary Experience Mortality Table (2002TSO) for further comparisons. And then, we construct a simple model to analyze the cash flows of the increasing whole life policies based on the mortality rates we observed. By constructing a simple model and simulation, we find that if the insurance company does not correctly estimate longevity risk, the insurance company will lose money on the increasing whole life policies. In order to mitigate the insufficiency of life insurers for the increasing whole life policies, we try to provide some supervision suggestion from the view of the risk-based capital (RBC) requirements. We calculate the factor of insurance risk (C2) of RBC requirements because this factor represents the surplus needed to provide for excess claims over expected, both from random fluctuations and from inaccurate pricing for future levels of claims. Keywords: longevity risk, increasing whole life insurance policy, Lee-Carter model, risk-based capital (RBC).

本篇論文主要討論在死亡率改善不確定性之下的避險策略。當保險公司負債面的人壽保單是比年金商品來得多的時候，公司會處於死亡率的風險之下。我們假設死亡率和利率都是隨機的情況，部分的死亡率風險可以經由自然避險而消除，而剩下的死亡率風險和利率風險則由零息債券和保單貼現商品來達到最適避險效果。我們考慮mean variance、VaR和CTE當成目標函數時的避險策略，其中在mean variance的最適避險策略可以導出公式解。由數值結果我們可以得知保單貼現的確是死亡率風險的有效避險工具。 / This paper proposes hedging strategies to deal with the uncertainty of mortality improvement. When insurance company has more life insurance contracts than annuities in the liability, it will be under the exposure of mortality risk. We assume both mortality and interest rate risk are stochastic. Part of mortality risk is eliminated by natural hedging and the remaining mortality risk and interest rate risk will be optimally hedged by zero coupon bond and life settlement contract. We consider the hedging strategies with objective functions of mean variance, value at risk and conditional tail expectation. The closed-form optimal hedging formula for mean variance assumption is derived, and the numerical result show the life settlement is indeed a effective hedging instrument against mortality risk.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Mortality information is very important for national planning and health of a country. Mortality rate forecasting is a basic contribution for the projection of financial improvement of pension plans, well-being and social strategy planning. In the first part of the thesis, we fit the selected mortality rate models, namely the Power-exponential function based model, the ModifiedPerks model and the Heligman and Pollard (HP4) model to the data obtained from the HumanMortality Database [22] for the male population ages 1–70 of the USA, Japan and Australia. We observe that the Heligman and Pollard (HP4) model performs well and better fit the data as compared to the Power-exponential function based model and the Modified Perks model. The second part is to systematically compare the quality of the mortality rate forecasting using the second order Lee–Carter method with the selected mortality rate models. The results indicate that Power-exponential function based model and the Heligman and Pollard (HP4) model gives a more reliable forecast depending on individual countries.

由於台灣地區的生育率變化較大，之前的研究發現其他各國的生育率模型不見得適用，亟需建立可反映我國國情的生育率模型。本文引用台灣、日本、荷蘭、美國（亞洲、歐洲、美洲）等經濟發達國家的出生資料，配適包括Gamma、Lee-Carter、主成份分析、年齡組個別估計法、擴散模型等較為常用的模型，比較這些國家配適結果的異同。分析發現，如果要預測總生育率，台灣、日本、美國都是以年齡組生育率個別配適擴散模型的結果最佳，荷蘭則是年齡組個別估計法；在年齡組生育率的預測方面，台灣、日本、荷蘭都是以年齡組生育率個別配適擴散模型最好，美國則是以年齡組個別估計最好。此外再從相對穩定性或相對效率的角度來評判，一樣是以年齡組生育率個別配適擴散模型或年齡組個別估計的總生育率預測結果最佳。最後還觀察到台灣地區和各縣市的有偶婦女比例和生育率呈正向關係，平均生育年齡和生育率呈反向關係，各縣市在有偶婦女比例、生育率、平均生育年齡的變化並不一致，各年齡組有偶婦女比例和生育率的改變也不盡相同。

Selected Methodical Approaches to Regional Population Forecast: A case study in the South Bohemian Region Abstract The aim of this thesis is to introduce selected methodological approaches to population forecasts, focusing on the regional level and considering different lengths of time series. Specific procedures are applied to create a population forecast for the South Bohemian Region. In the theoretical part of this thesis, the stages of population forecasts processing are determined. The Cohort Component method with migration, which can be used to create population forecast, is characterized. Another part describes selected analytical models and functions for partial mortality, fertility and migration forecasts, including Indirect estimation of net migration. To extrapolate parameters, selected trending functions and the Box-Jenkins methodology are characterized in the part of the time series analysis. The analytical part of this thesis focuses on the creation of the South Bohemian Region forecast from short initial time series and long initial time series. From short initial time series, the partial forecast of mortality is analyzed by the Heligman-Pollard model, the partial forecast of fertility is analyzed by the Beta function and the partial forecast of migration is analyzed by 25%, 50% and 75%...

Dissertation thesis creates a complex and modern scheme for stochastic modeling of demographic processes, which is universally applicable to any population in the world. All calculations are described in detail on the data of the Czech Republic. Throughout the work the attention is drawn to the issues, that every analyst must necessarily take into account in order to obtain correct results. Data comes mostly from the Czech Statistical Office database. However, some data matrices had to be calculated for the purposes of the thesis. Particular demographic processes (mortality, fertility and migration) are modeled using selected modern approaches (ARIMA models, Lee-Carter method) and based on the constructed models these processes are forecasted to the future. Using partially projected results a comprehensive demographic projection of the population of the Czech Republic is created up to the year 2050. However, not on the basis of the current state and expert expectations of the future development, but based on sophistically projected demographic events, which are explained using the trends and main components of their previous development. This demographic projection is created in three scenarios (marked SC1, SC2 and SC3), which are made from selected optimal models, presented in particular sections of the work. One part of the thesis is also the backward retropolation of age-specific number of net migrants by sex in the Czech Republic since 1948. On its basis the analysis and prediction of the migration can be done. The thesis is a synthesis of the projections of demographic processess of mortality, fertility, and migration. Final results are confronted with three scenarios of population projections of the Czech Republic created by the Czech Statistical Office and five scenarios of population projections by Eurostat. The purely statistical approach of demographic forecasting in comparison with deterministic models and expert expectations has its positives and negatives. Therefore, the different results due to various methodological approaches are discussed and compared in the conclusion of the thesis.