回顧二十世紀的歷程，我們可以看到人類在壽命上的一大進步，認為壽命的延長是人類的最大勝利；但是此壽命延長現象卻視為未來社會中的最主要的挑戰與風險。台灣在1993年六十五歲以上的老年人口比例已突破7%，正式步入聯合國所定義的「高齡化社會」，也正式面臨長壽風險（Longevity Risk）的問題。人口老化所帶來的衝擊，不只是提高工作人口的負擔，它同時也增加政府的養老給付和醫療保險支出，影響社會經濟安全，因此對於未來人口推估的死亡率模型所扮演角色日益重要。本研究以移動平均法和主成分分析兩種不同方式討論不同國家的死亡率變化情形，而後分析廣為人所用的Lee-Carter模型及Reduction Factor模型不足之處，並針對此兩模型不完善部分加以調整改進，建構出適合台灣死亡率的預測模型。

隨著醫療技術的進步、環境衛生的改善與人類追求健康生活型態的趨勢，全世界人類死亡率不斷逐年地下降中。但死亡率的下降不僅影響政府的社會福利政策，也影響到壽險公司對於未來的不確定性。例如在年金商品定價上，如果使用不適當的死亡率預測將會導致保險公司在未來現金流量上的不穩定，進而影響到公司的財務健全度。因此用來預估死亡率的模型便扮演著相當重要的角色。本研究首先透過Reduction Factor圖形觀察台灣、日本、美國、加拿大、英國與法國的歷年死亡率變動，之後再使用廣為人使用的Lee-Carter模型與其改善方法主成分分析方法(Principal Component Analysis, PCA)預估未來死亡率，最後再比較兩種方法在預測死亡率的表現。再透過計算年金商品與壽險商品的純保費部份，了解忽略死亡率變動趨勢所可能產生的影響。最後利用上述年金商品與壽險商品對於死亡率帶來的影響，討論保險公司在上述情形之下可以採取的最佳自然避險策略。

El objetivo central de esta tesis es presentar el método de Lee Carter para la construcción de tablas dinámicas de mortalidad. Si bien es cierto que estas tablas se pueden utilizar en cualquier campo cuyo interés sea estudiar la evolución de la mortalidad en una población, en este trabajo, el desarrollo está orientado al campo actuarial. Por esta razón, en todos los capítulos, siempre que sea posible, se hace mención a conceptos actuariales. Esta tesis está organizada de la siguiente manera, en el primer capítulo se definen las funciones biométricas, las tablas de mortalidad y se realizan describen algunas aplicaciones en el sector actuarial, utilizando la notación correspondiente. Luego, en el segundo capítulo se definen las tablas dinámicas de mortalidad y se describe el método de Lee Carter. Finalmente, en el tercer capítulo, se construyen tablas de mortalidad dinámicas mediante el método de Lee Carter, se calcula la esperanza de vida al nacer y se presentan algunas aplicaciones en el análisis actuarial. Cabe indicar que, para la construcción de las tablas de mortalidad dinámicas mediante el método de Lee Carter se utiliza el paquete demography del lenguaje R.

In the presented diploma thesis we study possibilities of forecasting mortality rates and we explain the most used models to measure it. The longevity and mortality risk are characterized and options for transfer of risks to other subjects are suggested. Further we applied LifeMetrics tool to predict mortality and quantify longevity risk in our data set, also possibilities of its usage are described in more details. The aim of the work is to provide the reader with sufficient amount of theoretical information about the used stochastic models for mortality prediction. Also the work may be helpful to gain deeper knowledge about 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.

對於人口數不多的國家及地區，因為樣本數較少，死亡率的震盪較大，導致死亡率的估計值較不穩定。為解決此種問題，本研究以其他國家的死亡率資料輔助台灣，建構死亡率模型。首先，以群集分析方式選擇適合輔助台灣的國家，也就是死亡率性質相近之國家，本研究建議以死亡改善率做為主要的考量；其次，以主成分分析的方式分解多個國家死亡率，以負荷做為多個國家的共有係數，分數則是隨著資料和時間改變的變數，在研究結果中，5~6個成分個數即會有不錯的配適和預測效果，以五齡組死亡率配適模型為例，成分個數為6時，男性配適Lee-Carter模型全部國家的平均MAPE為5.40%，主成分分析則為4.13%，下降幅度將近24%，而Lee-Carter模型預測的整體MAPE為14.72%，主成分分析為12.22%，下降幅度約17%，因此主成分分析模型確實有明顯改善Lee-Carter模型。 而和台灣死亡率性質相近的國家，主要選入歐洲國家，像是奧地利、法國、愛爾蘭、挪威和西班牙，除了法國和西班牙人口數分別為六千多萬和四千多萬的國家外，其餘三個國家人口數皆不超過一千萬，這說明人口數多寡或許不是輔助小地區建構死亡率模型的唯一重點，應選取適合的國家作為輔助用途。

In Taiwan area, the mortality rates at all ages have decreased since the end of World War II, and the life expectancy of people has increased from 62 in 1950's to 75 in 2000, which is an increase of 21%. The mortality improvement of the elderly (i.e. people ages 65 and over) is especially significant, which effects in the rapid population aging in Taiwan area. For example, the proportion of the elderly has increased from 6.14%in 1990 to 8.52% in 2000. On one hand, the prolonged life span for an individual means a longer period of retirement life and thus a larger retirement fund. On the other hand, a longer life for the government is equivalent to a more thorough social system for the elderly. Therefore, a reliable mortality rates projection is essential to both personal financial and social welfare planning. 　　In this study, we have two main objectives: First, we explore some frequent used models, such as Lee-Carter, multivariate regression and principal component methods. We use the data between 1950 to 1995 as the pilot data and 1996 to 2000 as the test data to judge which method has the smallest prediction error. In addition, based on computer simulation, we also evaluate the performance of the estimation methods for the Lee-Carter method. The second part (and the other objective) of this study is to explore the effect of mortality improvement on the pure premium of annuity insurance. In particular, we calculate the pure premium of the annuity under the best model acquired from the first part, and compare those under 1989 TSO and other life tables. We found that the pure premiums under current life tables are under estimated, which may cause the insolvency of insurance companies.

The Lee-Carter mortality model provides the very first model for modeling the mortality rate with stochastic time and age mortality dynamics. The model is constructed modeling the mortality rate to incorporate both an age effect and a period effect. The Lee-Carter model provides the fundamental set up currently used in most modern mortality modeling. Various extensions of the Lee-Carter model include either adding an extra term for a cohort effect or imposing a stochastic process for mortality dynamics. Although both of these extensions can provide good estimation results for the mortality rate, applying them for the pricing of the mortality/ longevity linked derivatives is not easy. While the current stochastic mortality models are too complicated to be explained and to be implemented, transforming the cohort effect into a stochastic process for the pricing purpose is very difficult. Furthermore, the cohort effect itself sometimes may not be significant. We propose using a new modified Lee-Carter model with a Normal Inverse Gaussian (NIG) Lévy process along with the Esscher transform for the pricing of mortality/ longevity linked derivatives. The modified Lee-Carter model, which applies the Lee-Carter model on the growth rate of mortality rates rather than the level of mortality rates themselves, performs better than the current mortality rate models shown in Mitchell et al (2013). We show that the modified Lee-Carter model also retains a similar stochastic structure to the Lee-Carter model, so it is easy to demonstrate the implication of the model. We proposed the additional NIG Lévy process with Esscher transform assumption that can improve the fit and prediction results by adapting the mortality improvement rate. The resulting mortality rate matches the observed pattern that the mortality rate has been improving due to the advancing development of technology and improvements in the medical care system. The resulting mortality rate is also developed under a martingale measure so it is ready for the direct application of pricing the mortality/longevity linked derivatives, such as q-forward, longevity bond, and mortality catastrophe bond. We also apply our proposed model along with an information theoretic optimization method to construct the pricing procedures for a life settlement. While our proposed model can improve the mortality rate estimation, the application of information theory allows us to incorporate the private health information of a specific policy holder and hence customize the distribution of the death year distribution for the policy holder so as to price the life settlement. The resulting risk premium is close to the practical understanding in the life settlement market.

Pastaraisiais metais beveik visose valstybėse stebimos visuomenės senėjimo bei ilgaamžiškumo tendencijos. Ši problema yra aktuali ne tik valstybinei pensijų sistemai, bet ir gyvybės draudimo įmonėms, mokančioms ar ateityje mokėsiančioms anuitetus. Šio darbo tikslas yra modeliuoti bei prognozuoti Latvijos, Lietuvos ir Estijos mirtingumą, įvertinti Baltijos valstybių visuomenės senėjimo tendenciją, atlikti anuitetų jautrumo analizę palūkanų normos bei mirtingumo tikimybių kitimui. Mirtingumo modeliavimui ir prognozavimui buvo naudojamas R. D. Lee ir L. R. Carter modelis. Atlikus duomenų analizę ir reikalingus tarpinius skaičiavimus yra sukonstruojamos prognozuojamos mirtingumo tikimybių lentelės. Sukonstruotos lentelės naudojamos nagrinėjant aukščiau aprašytus tikslus, t.y. skaičiuojant ilgaamžiškumo faktorius: gyvenimo trukmę bei anuitetus. Gaunamos išvados: -Ilgaamžiškumo problema egzistuoja; -Stebimos mažėjančios mirtingumo tikimybės; -Anuitetai yra jautresni mirtingumo tikimybių kitimui nei palūkanų normos svyravimams. Atlikus gautų rezultatų analizę galima daryti išvadą, kad visuomenės senėjimo problema egzistuoja, bet nėra labai grėsminga. Tačiau anuitetinių išmokų mokėtojai turėtų įvertinti ilgaamžiškumo tendenciją ir konstruoti atsargius anuitetus, atsižvelgiant ne tik į mirtingumą, bet ir į palūkanų normą. / In recent years the population ageing and the longevity tendencies are observed in almost all countries. This problem is significant for the state pension system as well as for the private insurance companies, paying life annuities. The aim of this paper is to evaluate the ageing tendencies in Baltic States, perform the sensitivity for mortality and interest rate changes analysis, predict the mortality in Baltic States in the future. The mortality was modeled and forecasted by using R. D. Lee and L. R. Carter model. The mortality table is designed after analyzing data and making necessary calculations. Produced table is applied to solve the prediction problem and to calculate the longevity factors - life expectancy and annuities. Conclusions obtained: -Longevity problem exists; -Decreasing mortality probabilities are observed; -The annuity is more sensitive to mortality fluctuation than to interest rate fluctuations. The overall conclusion is obtained that population ageing problem exists, but it is not very threatening. However, the size of annuity payments should be evaluated paying careful attention to longevity tendencies. Moreover, they should also be prudent in respect of interest rates, not only the mortality.