This study examines the effect of pensions on the timing of retirement, focusing on the differences between defined benefit (DB) plans and defined contribution (DC) plans. I find that DC plans have different effects on the accumulation of retirement wealth, the incentives for retirement and the risk of retirement benefits than DB plans. Thereby, DC plans have different effects from DB plans on the decision to retire. This paper is the first empirical study to investigate the effect of longevity risk in pension plans on retirement. It is an important addition to the literature on retirement behavior since longevity risk will become more important as individuals have longer life expectancies and bear more longevity risk due to increasing likelihood of coverage by DC plans or Social Security personal accounts. Previous research has found that DB plans have an age-incentive effect on retirement. That is, the structure of DB plans may induce individuals to retire at a specific age. By contrast, the structure of DC plans does not have age-incentive effects. Thereby, individuals with DC plans may retire either earlier or later on average than individuals with DB plans because of the absence of age-related incentives in DC plans. To shed further light on these issues, this study introduces risk factors, and particularly longevity risk, to an option value model of the retirement decision. Longevity risk is important to DC participants since DC plans usually offer a lump-sum benefit at retirement. Since payouts are not guaranteed over life expectancy, retirees with DC plans bear a greater risk of outliving their resources, i.e., longevity risk. The additional risks in DC plans may make workers save more, and retire later. This paper extends a standard intertemporal model of consumption and retirement by incorporating risk factors for different pension types into the retirement decision problem. Comparative statics from the optimal solution show that increases in risk factors (i.e. longevity risk) during retirement induce workers with DC plans to retire later than workers with defined benefit (DB) plans. This study then test the predictions of this model empirically, using the data from the Health and Retirement Study (HRS). Empirical results confirm the predictions of the theoretical model. First, workers with DC plans expect to retire later than workers with DB plans. Next, increase in pension option value, measured as the difference between the maximum pension value and the pension value of 1992, decreases the probability of retirement, thereby increasing the expected retirement wage. By contrast, greater pension wealth increases the probability of retirement, reducing the expected retirement age. Considering that pension wealth in DC plans is about half of pension wealth in DB plans, it is reasonable to conclude that workers with DC plans retire later than workers with DB plans. Finally, longevity risk, as measured by the Annuity Equivalent Wealth (AEW), decreases probability of retirement, increasing the expected retirement age.

In product pricing and reserving, actuaries are often required to make predictions of future death rates. In the past, this has been performed by using deterministic improvement scales that give only a single mortality trajectory. However, there is enormous likelihood that future death rates will turn out to be different from the projected ones, and so a better assessment of longevity risk would be one that consists of both a mean estimate and a measure of uncertainty. Such assessment can be performed using a stochastic mortality model, which is the core of this thesis. The Lee-Carter model is one of the most popular stochastic mortality models. While it does an excellent job in mean forecasting, it has been criticized for providing overly narrow prediction intervals that may have underestimated uncertainty. This thesis mitigates this problem by relaxing the assumption on the distribution of death counts. We found that the generalization from Poisson to negative binomial is equivalent to allowing gamma heterogeneity within each age-period cells. The proposed extension gives not only a better fit, but also a more conservative prediction interval that may reflect better the uncertainty entailed. The proposed extension is then applied to the construction of mortality improvement scales for Canadian insured lives. Given that the insured lives data series are too short for a direct Lee-Carter projection, we build an extra relational model that could borrow strengths from the Canadian population data, which covers a far longer period. The resultant scales consist of explicit measures of uncertainty. The prediction of the tail of a survival distribution requires a special treatment due to the lack of high quality old-age mortality data. We utilize the asymptotic results in modern extreme value theory to extrapolate death probabilities to the advanced ages, and to statistically determine the age at which the life table should be closed. Such technique is further integrated with the Lee-Carter model to produce a stochastic analysis of old-age mortality, and a prediction of the highest attained age for various cohorts. The mortality models we considered are further applied to the valuation of mortality-related financial products. In particular we investigate the no-negative-equity-guarantee that is offered in most fixed-repayment lifetime mortgages in Britain. The valuation of such guarantee requires a simultaneous consideration of both longevity and house price inflation risk. We found that house price returns can be well described by an ARMA-EGARCH time-series process. Under an ARMA-EGARCH process, however, the Black-Scholes formula no longer applies. We derive our own pricing formula based on the conditional Esscher transformation. Finally, we propose some possible hedging and capital reserving strategies for managing the risks associated with the guarantee.

In product pricing and reserving, actuaries are often required to make predictions of future death rates. In the past, this has been performed by using deterministic improvement scales that give only a single mortality trajectory. However, there is enormous likelihood that future death rates will turn out to be different from the projected ones, and so a better assessment of longevity risk would be one that consists of both a mean estimate and a measure of uncertainty. Such assessment can be performed using a stochastic mortality model, which is the core of this thesis. The Lee-Carter model is one of the most popular stochastic mortality models. While it does an excellent job in mean forecasting, it has been criticized for providing overly narrow prediction intervals that may have underestimated uncertainty. This thesis mitigates this problem by relaxing the assumption on the distribution of death counts. We found that the generalization from Poisson to negative binomial is equivalent to allowing gamma heterogeneity within each age-period cells. The proposed extension gives not only a better fit, but also a more conservative prediction interval that may reflect better the uncertainty entailed. The proposed extension is then applied to the construction of mortality improvement scales for Canadian insured lives. Given that the insured lives data series are too short for a direct Lee-Carter projection, we build an extra relational model that could borrow strengths from the Canadian population data, which covers a far longer period. The resultant scales consist of explicit measures of uncertainty. The prediction of the tail of a survival distribution requires a special treatment due to the lack of high quality old-age mortality data. We utilize the asymptotic results in modern extreme value theory to extrapolate death probabilities to the advanced ages, and to statistically determine the age at which the life table should be closed. Such technique is further integrated with the Lee-Carter model to produce a stochastic analysis of old-age mortality, and a prediction of the highest attained age for various cohorts. The mortality models we considered are further applied to the valuation of mortality-related financial products. In particular we investigate the no-negative-equity-guarantee that is offered in most fixed-repayment lifetime mortgages in Britain. The valuation of such guarantee requires a simultaneous consideration of both longevity and house price inflation risk. We found that house price returns can be well described by an ARMA-EGARCH time-series process. Under an ARMA-EGARCH process, however, the Black-Scholes formula no longer applies. We derive our own pricing formula based on the conditional Esscher transformation. Finally, we propose some possible hedging and capital reserving strategies for managing the risks associated with the guarantee.

This dissertation studies the adverse financial implications of "longevity risk" and "mortality risk", which have attracted the growing attention of insurance companies, annuity providers, pension funds, public policy decision-makers, and investment banks. Securitization of longevity/mortality risk provides insurers and pension funds an effective, low-cost approach to transferring the longevity/mortality risk from their balance sheets to capital markets. The modeling and forecasting of the mortality rate is the key point in pricing mortality-linked securities that facilitates the emergence of liquid markets. First, this dissertation introduces the discrete models proposed in previous literature. The models include: the Lee-Carter Model, the Renshaw Haberman Model, The Currie Model, the Cairns-Blake-Dowd (CBD) Model, the Cox-Lin-Wang (CLW) Model and the Chen-Cox Model. The different models have captured different features of the historical mortality time series and each one has their own advantages. Second, this dissertation introduces a stochastic diffusion model with a double exponential jump diffusion (DEJD) process for mortality time-series and is the first to capture both asymmetric jump features and cohort effect as the underlying reasons for the mortality trends. The DEJD model has the advantage of easy calibration and mathematical tractability. The form of the DEJD model is neat, concise and practical. The DEJD model fits the actual data better than previous stochastic models with or without jumps. To apply the model, the implied risk premium is calculated based on the Swiss Re mortality bond price. The DEJD model is the first to provide a closed-form solution to price the q-forward, which is the standard financial derivative product contingent on the LifeMetrics index for hedging longevity or mortality risk. Finally, the DEJD model is applied in modeling and pricing of life settlement products. A life settlement is a financial transaction in which the owner of a life insurance policy sells an unneeded policy to a third party for more than its cash value and less than its face value. The value of the life settlement product is the expected discounted value of the benefit discounted from the time of death. Since the discount function is convex, it follows by Jensen's Inequality that the expected value of the function of the discounted benefit till random time of death is always greater than the benefit discounted by the expected time of death. So, the pricing method based on only the life expectancy has the negative bias for pricing the life settlement products. I apply the DEJD mortality model using the Whole Life Time Distribution Dynamic Pricing (WLTDDP) method. The WLTDDP method generates a complete life table with the whole distribution of life times instead of using only the expected life time (life expectancy). When a life settlement underwriter's gives an expected life time for the insured, information theory can be used to adjust the DEJD mortality table to obtain a distribution that is consistent with the underwriter projected life expectancy that is as close as possible to the DEJD mortality model. The WLTDDP method, incorporating the underwriter information, provides a more accurate projection and evaluation for the life settlement products. Another advantage of WLTDDP is that it incorporates the effect of dynamic longevity risk changes by using an original life table generated from the DEJD mortality model table. / text

In this thesis we deal with the longevity risk originating from the uncertain future evolution of mortality at adult-old ages. It may emerge in particular because of an unanticipated reduction in mortality rates. That risk is significant for annuity and pension providers. We consider a model portfolio represented by one cohort of recipients of immediate life annuities. We introduce possibilities for assessing the risk of such portfolio. A comparison of the impact of longevity risk is made with random deviations in mortality rates. We also deal with the question of solvency of the insurer by investigating the solvency capital requirement for longevity risk.

A evolução do aumento da expectativa de vida registrada nas últimas décadas foi uma conquista significativa para a sociedade e trouxe novos desafios em diversas áreas do conhecimento humano. Dentre os impactos do aumento da longevidade, destaca-se sua influência no equilíbrio técnico dos planos previdenciários. Nas entidades de previdência complementar, a identificação oportuna de possíveis desvios da premissa da mortalidade à realidade subjacente visa garantir a solvência e a manutenção dos benefícios de longo prazo. Assim, o presente estudo tem por objetivo estimar os fatores de improvement (fator redutor de mortalidade) para a população coberta por planos privados de aposentadoria, com base no método Lee-Carter e na abordagem CMI (Continuous Mortality Investigation), bem como analisar o impacto da incorporação da estimativa do aumento da expectativa de vida no fluxo de caixa atuarial em uma carteira de benefício definido. Em virtude da carência de informações históricas de tábuas de mortalidade para o Brasil, fez-se uso da técnica de pareamento (propensity score), o qual consiste na identificação do país que mais se assemelha ao Brasil no que se refere às variáveis socioeconômicas relevantes para prever a evolução da expectativa de vida. Essa técnica foi aplicada para uma amostra de 21 países da OCDE. As variáveis socioeconômicas consideradas no estudo foram: Fertilidade, PIB per capita, Crescimento anual do PIB, Saúde, Desemprego, Gini, Analfabetismo e Escolaridade. Diante dos testes efetuados, Portugal foi escolhido para servir de base para as projeções da mortalidade e obtenção dos fatores de improvement, em decorrência da técnica de pareamento e do teste de aderência realizado. Comparando-se as médias dos fluxos de caixa da AT-2000 com e sem improvement e levando-se em consideração os cenários de taxas de juros de 3%, 4%, 5% e 6% ao ano, observou-se que, não considerar o improvement, gera uma elevação do fluxo atuarial entre 7,15% a 10,51% para a carteira simulada. A projeção pelo método CMI forneceu resultado semelhante, sendo que o impacto variou entre 7,05% a 10,32%. Embora os métodos de improvement sejam bem diferentes, é importante destacar que os resultados foram bem semelhantes. Um ponto que merece preocupação é a questão da taxa de juros, pois com a tendência de queda, no longo prazo, maior será a sensibilidade do impacto da projeção do risco de longevidade. Adicionalmente, compararam-se os resultados obtidos com a Tábua Geracional RP-2000 e a Tábua SUSEP BR-EMS. Assim, os resultados anteriores mostram que não considerar a tendência de aumento da expectativa de vida na constituição das provisões técnicas pode expor as entidades de previdência a riscos pouco suportáveis no longo prazo. / The evolution of increased life expectancy recorded in recent decades has been a significant achievement for the society and brought new challenges in various areas of human knowledge. Among those, living longer has impacted the technical balance of the pension plans. In the private pension entities, the timely identification of possible deviations from the assumption of mortality to the underlying reality is to ensure the solvency and the maintenance of long-term benefits. Thus, based on Lee-Carter method and approach CMI (Continuous Mortality Investigation Bureau), this study aims to estimate the factors of improvement (reduction factor of mortality) for the population covered by pension plans as well as analyze the impact of incorporating an estimated longer life expectancy on actuarial cash flow into a portfolio of defined benefits. Due to a lack of historical information about mortality tables of Brazil, the matching technique (propensity score) was used to identify the country which is the most similar to Brazil concerning relevant socioeconomic variables, in order to predict the evolution of life expectancy. This technique was applied on 21 OECD sample countries. Socioeconomic variables considered were: Fertility, GDP per capita, annual growth of GDP, Health, Unemployment, Gini, Illiteracy and Schooling. According to test results, Portugal was chosen as the basis for projections of mortality and acquisition of factors of improvement, due to the matching technique and the adherence test performed. Comparing the averages of the cash flows of the AT-2000 with and without improvement and taking into account the scenarios of interest rates of 3%, 4%, 5% and 6% a year, it was observed that not considering the improvement generates an increased actuarial flow between 7.15% and 10.51% for the simulated portfolio. The CMI method provided similar projection, and the impact varied from 7.05% to 10.32%. Even though the methods of improvement are quite different, it is important to emphasize that the results were much the same. One point that deserves concern is the issue of interest rate since, due to the declining trend in the long run more sensitive will be the impact of the projection of longevity risk. Additionally, those results were compared with the table Generational RP-2000 and BRTable SUSEP EMS. Thus, previous results show that not considering the trend of increasing life expectancy in the establishment of technical provisions can expose the private pension entities to a little bearable risk in the long term.

A evolução do aumento da expectativa de vida registrada nas últimas décadas foi uma conquista significativa para a sociedade e trouxe novos desafios em diversas áreas do conhecimento humano. Dentre os impactos do aumento da longevidade, destaca-se sua influência no equilíbrio técnico dos planos previdenciários. Nas entidades de previdência complementar, a identificação oportuna de possíveis desvios da premissa da mortalidade à realidade subjacente visa garantir a solvência e a manutenção dos benefícios de longo prazo. Assim, o presente estudo tem por objetivo estimar os fatores de improvement (fator redutor de mortalidade) para a população coberta por planos privados de aposentadoria, com base no método Lee-Carter e na abordagem CMI (Continuous Mortality Investigation), bem como analisar o impacto da incorporação da estimativa do aumento da expectativa de vida no fluxo de caixa atuarial em uma carteira de benefício definido. Em virtude da carência de informações históricas de tábuas de mortalidade para o Brasil, fez-se uso da técnica de pareamento (propensity score), o qual consiste na identificação do país que mais se assemelha ao Brasil no que se refere às variáveis socioeconômicas relevantes para prever a evolução da expectativa de vida. Essa técnica foi aplicada para uma amostra de 21 países da OCDE. As variáveis socioeconômicas consideradas no estudo foram: Fertilidade, PIB per capita, Crescimento anual do PIB, Saúde, Desemprego, Gini, Analfabetismo e Escolaridade. Diante dos testes efetuados, Portugal foi escolhido para servir de base para as projeções da mortalidade e obtenção dos fatores de improvement, em decorrência da técnica de pareamento e do teste de aderência realizado. Comparando-se as médias dos fluxos de caixa da AT-2000 com e sem improvement e levando-se em consideração os cenários de taxas de juros de 3%, 4%, 5% e 6% ao ano, observou-se que, não considerar o improvement, gera uma elevação do fluxo atuarial entre 7,15% a 10,51% para a carteira simulada. A projeção pelo método CMI forneceu resultado semelhante, sendo que o impacto variou entre 7,05% a 10,32%. Embora os métodos de improvement sejam bem diferentes, é importante destacar que os resultados foram bem semelhantes. Um ponto que merece preocupação é a questão da taxa de juros, pois com a tendência de queda, no longo prazo, maior será a sensibilidade do impacto da projeção do risco de longevidade. Adicionalmente, compararam-se os resultados obtidos com a Tábua Geracional RP-2000 e a Tábua SUSEP BR-EMS. Assim, os resultados anteriores mostram que não considerar a tendência de aumento da expectativa de vida na constituição das provisões técnicas pode expor as entidades de previdência a riscos pouco suportáveis no longo prazo. / The evolution of increased life expectancy recorded in recent decades has been a significant achievement for the society and brought new challenges in various areas of human knowledge. Among those, living longer has impacted the technical balance of the pension plans. In the private pension entities, the timely identification of possible deviations from the assumption of mortality to the underlying reality is to ensure the solvency and the maintenance of long-term benefits. Thus, based on Lee-Carter method and approach CMI (Continuous Mortality Investigation Bureau), this study aims to estimate the factors of improvement (reduction factor of mortality) for the population covered by pension plans as well as analyze the impact of incorporating an estimated longer life expectancy on actuarial cash flow into a portfolio of defined benefits. Due to a lack of historical information about mortality tables of Brazil, the matching technique (propensity score) was used to identify the country which is the most similar to Brazil concerning relevant socioeconomic variables, in order to predict the evolution of life expectancy. This technique was applied on 21 OECD sample countries. Socioeconomic variables considered were: Fertility, GDP per capita, annual growth of GDP, Health, Unemployment, Gini, Illiteracy and Schooling. According to test results, Portugal was chosen as the basis for projections of mortality and acquisition of factors of improvement, due to the matching technique and the adherence test performed. Comparing the averages of the cash flows of the AT-2000 with and without improvement and taking into account the scenarios of interest rates of 3%, 4%, 5% and 6% a year, it was observed that not considering the improvement generates an increased actuarial flow between 7.15% and 10.51% for the simulated portfolio. The CMI method provided similar projection, and the impact varied from 7.05% to 10.32%. Even though the methods of improvement are quite different, it is important to emphasize that the results were much the same. One point that deserves concern is the issue of interest rate since, due to the declining trend in the long run more sensitive will be the impact of the projection of longevity risk. Additionally, those results were compared with the table Generational RP-2000 and BRTable SUSEP EMS. Thus, previous results show that not considering the trend of increasing life expectancy in the establishment of technical provisions can expose the private pension entities to a little bearable risk in the long term.

隨著醫療技術進步、環境衛生改善與人類追求健康生活的趨勢，全世界人類的死亡率不斷地下降。在死亡率不斷的改善的情形下，保險公司可能在壽險商品上獲利，但在年金部份卻會因長壽風險而有所虧損。 自然避險則是保險公司可行的避險策略之一，即透過公司整體保單的組合，來達到規避死亡率風險和利率風險。此外，不同於之前的相關研究，我們所使用的資料，是由臺灣所有的保險公司提供的經驗死亡率，而不是國民生命表。目前保險公司在定價年金和壽險商品時，使用的死亡率是國民生命表，即假設買年金商品的被保險人和買壽險商品的被保險人的死亡率是相同的。但是從經驗死亡率的資料，我們發現購買年金商品的被保險人，其死亡率會低於買壽險商品的被保險人的死亡率。上述情形，會造成保險商品定價有誤;因此，我們考慮不同性別的年金、壽險的死亡率，並研究這些死亡率之間隨機變動項的相關性，以期在未來死亡率和利率變動下，可以藉由死亡率間的相關性，而抵消總價值變動的變異數和定價差異。 根據經驗資料，我們提出一個模型，可透過調整賣出年金和壽險的比例(年齡、性別)，使得保險公司能夠針對公司整體保單組合，找到並有效地運用的自然避險策略。文中最後進行模型敏感度分析，以及提出可能採用的保險商品配置策略，可作為目前保險公司進行死亡率和利率避險的參考。 / The mortality rate of human being has decreased year by year due to the improvement of medical and hygienic techniques. With the mortality improvement over time, life insurers may gain a profit and annuity insurers may suffer losses because of longevity risk. However, natural hedging is a feasible strategy to hedge mortality risk and interest risk at the same time. In this paper, we investigate the natural hedging strategy and try to find an optimal collocation of insurance products to deal with longevity risks for the insurance companies. Different from previous literatures, we use the experienced mortality rates from life insurance companies rather than population mortality rates. This experienced mortality data set includes more than 50,000,000 policies which are collected from the incidence data of the whole Taiwan life insurance companies. In general, insurance companies use population mortality rates to price life insurance and annuity products. Nevertheless, the mortality rate of annuity purchasers is averagely lower than that of life insurance purchasers. This situation leads to mispricing problem of both life insurance and annuity products. So in this paper, we can construct four mortality tables (gender, product) and investigate the correlation of these stochastic variation terms of four mortality rates. According to the correlation relation between these four mortality rates, we can offset the variance of portfolio’s change and difference of mispricing. On the basis of the experienced mortality rates, we demonstrate that the proposed model can lead to an optimal collocation of insurance products and effectively apply the natural hedging strategy to a more general portfolio for life insurance companies.

One of the classical problems in finance is that of an economic unit who aims at maximizing his expected life-time utility from consumption and/or terminal wealth by an effective asset-liability management. The purpose of this thesis is to determine the optimal investment strategies , from the point of view of their economic utility, for individual and institutional investors such pension funds.

This thesis is constituted of three chapters. he first part of my Ph.D. dissertation develops a Bayesian stochastic model for computing the reserves of a non-life insurance company. The first chapter is the product of my research experience as an intern at the Risk Management Department of Fondiaria-Sai S.p.A.. I present a short review of the deterministic and stochastic claims reserving methods currently applied in practice and I develop a (standard) Over-Dispersed Poisson (ODP) Bayesian model for the estimation of the Outstanding Loss Liabilities (OLLs) of a line of business (LoB). I present the model, I illustrate the theoretical foundations of the MCMC (Markov Chain Monte Carlo) method and the Metropolis-Hastings algorithm used in order to generate the non-standard posterior distributions. I apply the model to the Motor Third Party Liabil- ity LoB of Fondiaria-Sai S.p.A.. Moreover, I explore the problem of computing the prudential reserve level of a multi-line non-life insurance company. In the second chapter, then, I present a full Bayesian model for assessing the reserve requirement of multiline Non-Life insurance companies. The model combines the Bayesian approach for the estimation of marginal distribution for the single Lines of Business and a Bayesian copula procedure for their aggregation. First, I consider standard copula aggregation for different copula choices. Second, I present the Bayesian copula technique. Up to my knowledge, this approach is totally new to stochastic claims reserving. The model allows to "mix" own-assessments of dependence between LoBs at a company level and market wide estimates. I present an application to an Italian multi-line insurance company and compare the results obtained aggregating using standard copulas and a Bayesian Gaussian copula. In the second part of my Dissertation I propose a theoretical model that studies optimal capital and organizational structure choices of financial groups which incorporate two or more business units. The group faces a VaR-type regulatory capital requirement. Financial conglomerates incorporate activities in different sectors either into a unique integrated entity, into legally separated divisions or in ownership-linked holding company/subsidiary structures. I model these different arrangements in a structural framework through different coinsurance links between units in the form of conditional guarantees issued by equityholders of a firm towards the debtholders of a unit of the same group. I study the effects of the use of such guarantees on optimal capital structural and organizational form choices. I calibrate model parameters to observed financial institutions' characteristics. I study how the capital is optimally held, the costs and benefits of limiting undercapitalization in some units and I address the issues of diversification at the holding's level and regulatory capital arbitrage. The last part of my Ph.D. Dissertation studies the hedging problem of life insurance policies, when the mortality rate is stochastic. The field developed recently, adapting well-established techniques widely used in finance to describe the evolution of rates of mortality. The chapter is joint work with my supervisor, prof. Elisa Luciano and Elena Vigna. It studies the hedging problem of life insurance policies, when the mortality and interest rates are stochastic. We focus primarily on stochastic mortality. We represent death arrival as the first jump time of a doubly stochastic process, i.e. a jump process with stochastic intensity. We propose a Delta-Gamma Hedging technique for mortality risk in this context. The risk factor against which to hedge is the difference between the actual mortality intensity in the future and its "forecast" today, the instantaneous forward intensity. We specialize the hedging technique first to the case in which survival intensities are affine, then to Ornstein-Uhlenbeck and Feller processes, providing actuarial justifications for this restriction. We show that, without imposing no arbitrage, we can get equivalent probability measures under which the HJM condition for no arbitrage is satisfied. Last, we extend our results to the presence of both interest rate and mortality risk, when the forward interest rate follows a constant-parameter Hull and White process. We provide a UK calibrated example of Delta and Gamma Hedging of both mortality and interest rate risk.