<p>Under det gångna århundradet ökade den förväntade livslängden avsevärt såväl i Sverige som i övriga världen. 1900-talets förbättrade livslängd drevs inledningsvis av en minskad barnadödlighet medan de senare årtiondena kännetecknades av minskad dödlighet i höga åldrar.</p><p>En åldrande befolkning innebär ökade krav på sjukvård, äldreomsorg och inte minst pensionssystem. Pålitliga prognoser för vår framtida livslängd behövs för att beräkna de resurser som nämnda verksamheter kommer att ta i anspråk och utgör förutsättningen för en rättvis prissättning av försäkringsprodukter med levnadsrisk.</p><p>Lee-Carter-modellen är en av vår tids tongivande modeller för mortalitetsprognostisering. Modellen används här för att göra livslängdsprognoser utifrån svenska mortalitetsdata; prognoserna jämförs sedan med observerade utfall.</p><p>Mot bakgrund av resultatet diskuteras levnadsrisk med fokus på pensioner.</p><p>Inte oväntat presterar prognoserna ingen felfri bild av verkligheten och prognosfelet varierar i storlek mellan skattningarna; att använda dem som underlag för pensionsberäkningar hade i förlängningen varit ohållbart. Exemplet illustrerar på samma gång vår osäkerhet inför framtidens livslängdsutveckling och svårigheten i att prognostisera den.</p> / <p>During the past century, Sweden along with many other countries experienced a sharp decline in mortality rates. The increased life expectancy was initially propelled by mortality reductions among infants and subsequently by a survival improvement in advanced ages.</p><p>An ageing population has large implications for those providing services to the elderly, such as medical care and pensions, whilst also addressing the need for accurate and reliable mortality forecasts and projection methods.</p><p>The Lee-Carter model is the current gold standard for mortality forecasting and has been widely adopted in several studies. Here, the model is applied on Swedish mortality data; the projections are then compared to the observed lifespan development. Against this backdrop, a discussion on longevity risk in pensions schemes follows.</p><p>The forecasts performed in this study do not perfectly reflect the observed mortality change in the examined period; furthermore, the variation of the estimation errors limits the actuarial value of the projections. The findings illuminate the uncertainty that surrounds our future life expectancy as well as the difficulties associated with forecasting it.</p>

Under det gångna århundradet ökade den förväntade livslängden avsevärt såväl i Sverige som i övriga världen. 1900-talets förbättrade livslängd drevs inledningsvis av en minskad barnadödlighet medan de senare årtiondena kännetecknades av minskad dödlighet i höga åldrar. En åldrande befolkning innebär ökade krav på sjukvård, äldreomsorg och inte minst pensionssystem. Pålitliga prognoser för vår framtida livslängd behövs för att beräkna de resurser som nämnda verksamheter kommer att ta i anspråk och utgör förutsättningen för en rättvis prissättning av försäkringsprodukter med levnadsrisk. Lee-Carter-modellen är en av vår tids tongivande modeller för mortalitetsprognostisering. Modellen används här för att göra livslängdsprognoser utifrån svenska mortalitetsdata; prognoserna jämförs sedan med observerade utfall. Mot bakgrund av resultatet diskuteras levnadsrisk med fokus på pensioner. Inte oväntat presterar prognoserna ingen felfri bild av verkligheten och prognosfelet varierar i storlek mellan skattningarna; att använda dem som underlag för pensionsberäkningar hade i förlängningen varit ohållbart. Exemplet illustrerar på samma gång vår osäkerhet inför framtidens livslängdsutveckling och svårigheten i att prognostisera den. / During the past century, Sweden along with many other countries experienced a sharp decline in mortality rates. The increased life expectancy was initially propelled by mortality reductions among infants and subsequently by a survival improvement in advanced ages. An ageing population has large implications for those providing services to the elderly, such as medical care and pensions, whilst also addressing the need for accurate and reliable mortality forecasts and projection methods. The Lee-Carter model is the current gold standard for mortality forecasting and has been widely adopted in several studies. Here, the model is applied on Swedish mortality data; the projections are then compared to the observed lifespan development. Against this backdrop, a discussion on longevity risk in pensions schemes follows. The forecasts performed in this study do not perfectly reflect the observed mortality change in the examined period; furthermore, the variation of the estimation errors limits the actuarial value of the projections. The findings illuminate the uncertainty that surrounds our future life expectancy as well as the difficulties associated with forecasting it.

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.

In this thesis we focus on modeling stochastic mortality and parameter risk in assessing mortality. We explore two mortality stochastic models for modeling the number of deaths in portfolio which consist of one or more than one cohort. We define the term mixture of distributions and introduce Beta-Binomial and Poisson-Gamma model. We address immediate life annuities and we apply Bayesian Poisson- Gamma model to quantify longevity risk on data. The obvious increasing trend of average lifetime leads insurance companies to greater protection against longevity risk. We show how to deal with solvency rules by internal models designed consistently with the requirement in the standard formula of Solvency II. Powered by TCPDF (www.tcpdf.org)

Hoarding a too large cash reserve is often unfavourable due to lost investment opportunities. Similarly, an insufficient cash reserve can be detrimental, as one might fail to meet payment obligations. Finding the optimal balance is nothing that is done in the blink of an eye, particularly when the underlying variable is stochastic, e.g., the life span of a human being. Resscapital is a fund manager investing in the secondary and tertiary markets for life insurance policies, also known as the life settlements market. They are currently on a mission to set up a closed-end fund where one of the main challenges is balancing the invested capital and the amount of capital set aside for payment obligations. The stochastic nature of life insurance policies entails the difficulty to foresee future premium payments and face value payouts. Without a model forecasting the cash flows, decisions regarding the cash reserve are based on nothing better than a guesstimate. Thus, with the aim to help determine the minimum cash reserve required to cover the payment obligations, this thesis was initiated. By developing a methodology based on general theory, the objective of this thesis is reached and the purpose fulfilled. The proposed model uses Monte Carlo simulation to generate scenarios that eventually creates a distribution of required cash reserves. Following the inversion principle, the remaining lifetime for each and every individual is simulated from their empirical distribution of survival probabilities, respectively. After simulating the occurrences of demise, an algorithm builds up the cash flows for the entire fund term for that specific scenario based on predetermined parameters. Since cash flows stem from both assets and management, the portfolio must be revalued continuously, demanding a gradual evaluation of the cash flows during the fund term. Repeated a large number of times, the quantile corresponding to any confidence level is attained by using a Value at Risk methodology. Analysis of the results and sensitivity analysis on the parameters provides a deeper understanding of the underlying factors, revealing, among other things, that longevity risk for policies with short life expectancy is a key driver of the required cash reserve. Furthermore, validation of the model shows that the results are sufficient and serve the purpose well.

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.

This thesis addresses the absence of explicit pricing formulae and the complexity of proposed models (incomplete markets framework) in the area of mortality risk management requiring the application of advanced techniques from the realm of Financial Mathematics and Actuarial Science. In fact, this is a multi-essay dissertation contributing in the direction of designing and pricing mortality-linked derivatives and offering the state of art solutions to manage longevity risk. The first essay investigates the valuation of Catastrophic Mortality Bonds and, in particular, the case of the Swiss Re Mortality Bond 2003 as a primary example of this class of assets. This bond was the first Catastrophic Mortality Bond to be launched in the market and encapsulates the behaviour of a well-defined mortality index to generate payoffs for bondholders. Pricing this type of bond is a challenging task and no closed form solution exists in the literature. In my approach, we adapt the payoff of such a bond in terms of the payoff of an Asian put option and present a new methodology to derive model-independent bounds for catastrophic mortality bonds by exploiting the theory of comonotonicity. While managing catastrophic mortality risk is an upheaval task for insurers and re-insurers, the insurance industry is facing an even bigger challenge - the challenge of coping up with increased life expectancy. The recent years have witnessed unprecedented changes in mortality rate. As a result academicians and practitioners have started treating mortality in a stochastic manner. Moreover, the assumption of independence between mortality and interest rate has now been replaced by the observation that there is indeed a correlation between the two rates. Therefore, my second essay studies valuation of Guaranteed Annuity Options (GAOs) under the most generalized modeling framework where both interest rate and mortality risk are stochastic and correlated. Pricing these types of options in the correlated environment is an arduous task and a closed form solution is non-existent. In my approach, I employ the use of doubly stochastic stopping times to incorporate the randomness about the time of death and employ a suitable change of measure to facilitate the valuation of survival benefit, there by adapting the payoff of the GAO in terms of the payoff of a basket call option. I then derive general price bounds for GAOs by employing the theory of comonotonicity and the Rogers-Shi (Rogers and Shi, 1995) approach. Moreover, I suggest some `model-robust' tight bounds based on the moment generating function (m.g.f.) and characteristic function (c.f.) under the affine set up. The strength of these bounds is their computational speed which makes them indispensable for annuity providers who rely heavily on Monte Carlo simulations to calculate the fair market value of Guaranteed Annuity Options. In fact, sans Monte Carlo, the academic literature does not offer any solution for the pricing of the GAOs. I illustrate the performance of the bounds for a variety of affine processes governing the evolution of mortality and the interest rate by comparing them with the benchmark Monte Carlo estimates. Through my work, I have been able to express the payoffs of two well known modern mortality products in terms of payoffs of financial derivatives, there by filling the gaps in the literature and offering state of art techniques for pricing of these sophisticated instruments.

Pension schemes are facing more difficulties on matching their underlying liabilities with assets, mainly due to faster mortality improvements for their underlying populations, better environments and medical treatments and historically low interest rates. Given most of the pension schemes are relatively much smaller than the national population, modelling and forecasting the small populations' longevity risk become urgent tasks for both the industrial practitioners and academic researchers. This thesis starts with a systematic analysis on the influence of population size on the uncertainties of mortality estimates and forecasts with a stochastic mortality model, based on a parametric bootstrap methodology with England and Wales males as our benchmark population. The population size has significant effect on the uncertainty of mortality estimates and forecasts. The volatilities of small populations are over-estimated by the maximum likelihood estimators. A Bayesian model is developed to improve the estimation of the volatilities and the predictions of mortality rates for the small populations by employing the information of larger population with informative prior distributions. The new model is validated with the simulated small death scenarios. The Bayesian methodologies generate smoothed estimations for the mortality rates. Moreover, a methodology is introduced to use the information of large population for obtaining unbiased volatilities estimations given the underlying prior settings. At last, an empirical study is carried out based on the Scotland mortality dataset.

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.