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.

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.

Essay 1 (with Juha Alho and Edward Palmer):  All around the world, public pension schemes are moving in the direction of non-financial (NDC) and financial defined contribution (DC) schemes.  Both rely on accurate projections of life expectancy in the creation of annuities. Accurate projections are critical for system stability, individual utility and inter-generational welfare. This paper suggests a path-breaking innovation that changes the perspective from the Lee-carter (LC) family of trend models which assume a constant rate of change in mortality over time. Our approach is to project the cohort life expectancy on basis of the specific cohort rate of change in mortality. This relaxes the strong trend assumption underlying the LC model, which is the reason why LC model does not work well in the phase of accelerating or decelerating mortality. We use unisex mortality data for $8$ countries to test the performance of our approach both ex-post and ex-ante. The ex-post experiment shows that our approach generally performs better when the rate of change in mortality is accelerating and performs as well as LC model when the rate of change is time-invariant. The ex-ante experiment, on the other hand, shows that our model almost always delivers higher projection of remaining life expectancy than the LC model for the more recent cohorts, which is consistent with the ex-post experimental results. / Essay 2:  Due to the systematic underestimation of cohort life expectancy, NDC pension schemes face a financial risk that can leads to inter-generational unfairness, given the current practice. This paper proposes an alternative method of computing annuity to address this problem. The proposal is to adjust the annuity based on re-estimations of the remaining life expectancy at intervals after retirement, but only up to a ceiling age. The scheme is assessed using 208 cohort annuity pools from eight sample countries. This experiment shows that the proposed scheme succeeds in reducing the inter-generational unfairness for 60-80% of the cohort annuity pools, compared to current practice of fixing the annuity at age 65. Because the adjustment is borne by the relatively large group of younger persons, the per capita change in utility is rather small assuming risk neutrality. / Essay 3:  This paper studies how the incentive to retire in a DC (NDC) scheme is influenced by engaging private information on life expectancy. This is an important question since the decisions made under the two scenarios, optimizing using the private life expectancy or the cohort average made available by the pension provider, create different welfare and financial outcomes. The analytical framework is a standard life-cycle model, accounting for monetary gain from work and non-monetary gain from leisure. The unique feature here is that the individual life expectancy is an explicit driver of disutility of work. The theoretical result is that prevailing private information of a longer-than-average life expectancy can lead to both advancing and delaying retirement, depending on other factors determining utility. The numerical example using Swedish data proves the theoretical results and suggests a rather small average impact on the choice of retirement by engaging private information of life expectancy. / Essay 4:   Pensions in the increasingly popular Notional Defined Contribution (NDC) Pay-as-You-Go Schemes are granted based on cohort-specific life expectancy, regardless socioeconomic differences. This risks perverse intra-generational and unintended inter-generational transfers. This paper introduces an alternative with separate annuity pools for different socioeconomic classes. Using unique Swedish data and the Swedish NDC pension system as an example, the analysis shows a significant gap in life expectancy between socioeconomic classes defined by occupation. In the Swedish context, this implies a perverse transfer of 5% of the pension capital from the manual workers to the non-manual workers, which can be abolished by using the group plan. In addition, the group plan also lessens the risk of inter-generational transfers resulting from the gap in life expectancy.

This thesis analyses the COVID-19 pandemic’s effects on Swedish mortality during 2020 by investigating whether it has resulted in excess mortality. This is done using a stochastic mortality projection model from the Lee-Carter framework and by assuming the number of deaths follows a Poisson distribution. Due to the few confirmed COVID-19 deaths at younger ages, the decision is made to only include 50-to-100-year-olds in the analysis. Models in the Lee-Carter framework are fitted on historical data from 1993–2019 collected from Human Mortality Database and Statistiska Centralbyrån. After evaluating the models, inter alia using residual analysis and backtesting, we ascertain that the classical Lee-Carter model accomplishes a wanted level of fit and forecast accuracy. During the morality projection with the Lee-Carter model, three different sources of uncertainty are accounted for by constructing prediction intervals using bootstrap. The results show that the large age group 67–94-year-olds have suffered from statistically significant excess mortality during 2020. The level of excess mortality differs between ages, with the ages 70–90-year-olds having the highest number of excess deaths. Comparing the number of confirmed COVID-19 deaths to our forecasted number of excess deaths indicates the COVID-19 virus likely caused the surge in deaths.

<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>

The main aim of this thesis is to analyse different mortality models regarding the longevity risk. We focus on the well-known stochastic models (Lee-Carter model, Age-period-cohort model by Renshaw and Haberman, Cairns-Blake-Dowd two-factor model) and compare them with relatively new Taiwanese model by Yang, Yue and Huang which is based on principal component analysis. Both the theoretical and also the empirical parts are included. Empirical part evaluates all the models mentioned above on the Czech mortality data from 1970-2000 for individuals aged between 50-100 years. Final mortality predictions are made for next 30 years.

La présente thèse vise à explorer différents types de modèles log-bilinéaires dans le domaine des sciences actuarielles. Le point de départ consiste en le modèle de Lee-Carter, utilisé pour les problèmes de projection de la mortalité. Différentes variantes sont développées, et notamment le modèle de Poisson log-bilinéaire. L'introduction de variables explicatives est également analysée. Enfin, une tentative de d'exportation de ces modèles au cas des triangles IBNR est effectuée.

La présente thèse vise à explorer différents types de modèles log-bilinéaires dans le domaine des sciences actuarielles. Le point de départ consiste en le modèle de Lee-Carter, utilisé pour les problèmes de projection de la mortalité. Différentes variantes sont développées, et notamment le modèle de Poisson log-bilinéaire. L'introduction de variables explicatives est également analysée. Enfin, une tentative de d'exportation de ces modèles au cas des triangles IBNR est effectuée.

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.

[ES] En la actualidad resulta de gran importancia el análisis de los fenómenos como el crecimiento poblacional y la reducción de la mortalidad por la repercusión económica y social que dichos procesos tienen en el desarrollo de los países. En este sentido las tablas de vida constituyen una herramienta para comprender, a través de las probabilidades de muerte, la esperanza de vida y otros indicadores, la dinámica poblacional. Lee y Carter (1992), plantearon un modelo, cuyo ajuste permite a los analistas obtener una visión dinámica del comportamiento de la mortalidad durante un periodo de análisis. Esta tesis doctoral busca contribuir en la comprensión de los cambios que ha experimentado la población colombiana en cuanto a mortalidad. Para lograrlo se plantearon cuatro objetivos. El primero, construir modelos estocásticos de mortalidad como Lee-Carter para datos de Colombia y hacer un estudio comparativo de dichos modelos para evaluar su coherencia a partir de la calidad de los resultados obtenidos. El segundo, calcular y analizar algunos indicadores relacionados con la mortalidad tales como la mortalidad infantil, la esperanza de vida al nacer, la esperanza de vida a los 65 años, el índice de Gini al nacer y el índice de Gini a los 65 años. El tercero, aplicar gráficos de control para identificar los momentos en el tiempo y los intervalos de edad en los que la probabilidad de muerte observada es sustancialmente diferente de la pauta de mortalidad en el período estudiado. Para esto, los residuos de los modelos seleccionados se vigilaron mediante el gráfico de control multivariado T2 de Hotelling para detectar cambios sustanciales en la mortalidad que no fueron identificados por los modelos. El cuarto, analizar el comportamiento de la mortalidad para los departamentos de Colombia mediante técnicas de análisis multivariado como el análisis de componentes principales, el clúster jerárquico y el fuzzy clúster, para posteriormente identificar grupos de departamentos con comportamientos similares y caracterizarlos mediante los indicadores de mortalidad estudiados. La metodología descrita relacionada con los tres primeros objetivos se aplicó a datos de las tablas de vida abreviadas por sexo para Colombia para el período 1973-2005, utilizando la información disponible en The Latin America Human Mortality Database. Para el análisis de la mortalidad por departamentos se construyeron nuevas tablas de vida abreviadas por sexo con la información de los departamentos para el período 1985-2014, ajustándonos a la información disponible para los departamentos de Colombia en cuanto a defunciones y población. La metodología fue implementada a través del software estadístico libre R, lo que permite la replicabilidad y reproducibilidad de los resultados. / [CA] En l'actualitat resulta de gran importància l'anàlisi dels fenòmens com el creixement poblacional i la reducció de la mortalitat per la repercussió econòmica i social que aquests processos tenen en el desenvolupament dels països. En aquest sentit les taules de vida constitueixen una eina per a comprendre, a través de les probabilitats de mort, l'esperança de vida i altres indicadors, la dinàmica poblacional. Lee i Carter (1992), van plantejar un model, l'ajust del qual permet als analistes obtenir una visió dinàmica del comportament de la mortalitat durant un període d'anàlisi. Aquesta tesi doctoral cerca contribuir en la comprensió dels canvis que ha experimentat la població colombiana quant a mortalitat. Per a aconseguir-ho es van plantejar quatre objectius. El primer, construir models estocàstics de mortalitat com Lee-Carter per a dades de Colòmbia i fer un estudi comparatiu d'aquests models per a avaluar la seua coherència a partir de la qualitat dels resultats obtinguts. El segon, calcular i analitzar alguns indicadors relacionats amb la mortalitat tals com la mortalitat infantil, l'esperança de vida en nàixer, l'esperança de vida als 65 anys, l'índex de Gini en nàixer i l'índex de Gini als 65 anys. El tercer, aplicar gràfics de control per a identificar els moments en el temps i els intervals d'edat en els quals la probabilitat de mort observada és substancialment diferent de la pauta de mortalitat en el període estudiat. Per a això, els residus dels models seleccionats es van vigilar mitjançant el gràfic de control multivariat T2 de Hotelling per a detectar canvis substancials en la mortalitat que no van ser identificats pels models. El quart, analitzar el comportament de la mortalitat per als departaments de Colòmbia mitjançant tècniques d'anàlisi multivariada com l'anàlisi de components principals, el clúster jeràrquic i el fuzzy clúster, per a posteriorment identificar grups de departaments amb comportaments similars i caracteritzar-los mitjançant els indicadors de mortalitat estudiats. La metodologia descrita relacionada amb els tres primers objectius es va aplicar a dades de les taules de vida abreujades per sexe per a Colòmbia per al període 1973-2005, utilitzant la informació disponible en The Latin America Human Mortality Database. Per a l'anàlisi de la mortalitat per departaments es van construir noves taules de vida abreujades per sexe amb la informació dels departaments per al període 1985-2014, ajustant-nos a la informació disponible per als departaments de Colòmbia quant a defuncions i població. La metodologia va ser implementada a través del programari estadístic lliure R, la qual cosa permet la replicabilidad i reproducibilidad dels resultats. / [EN] The analysis of phenomena such as population growth and mortality reduction is currently of great importance because of the economic and social impact that these processes have on the development of countries. In this sense, life tables are a tool for understanding population dynamics through death probabilities, life expectancy and other indicators. Lee and Carter (1992) proposed a model whose adjustment allows analysts to obtain a dynamic view of the behavior of mortality during a period of analysis. This doctoral thesis seeks to contribute to the understanding of the changes experienced by the Colombian population in terms of mortality. To achieve this, four objectives were proposed. The first, to construct stochastic mortality models such as Lee-Carter for Colombian data and to make a comparative study of these models to evaluate their coherence based on the quality of the results obtained. The second is to calculate and analyze some mortality-related indicators such as infant mortality, life expectancy at birth, life expectancy at age 65, the Gini index at birth and the Gini index at age 65. The third is to apply control charts to identify moments in time and age intervals in which the observed probability of death is substantially different from the mortality pattern in the period studied. For this, the residuals of the selected models were monitored using Hotelling's T2 multivariate control chart to detect substantial changes in mortality that were not identified by the models. Fourth, to analyze the behavior of mortality for the departments of Colombia using multivariate analysis techniques such as principal component analysis, hierarchical clustering and fuzzy clustering, in order to subsequently identify groups of departments with similar behavior and characterize them by means of the mortality indicators studied. The methodology described in relation to the first three objectives was applied to data from the abbreviated life tables by sex for Colombia for the period 1973-2005, using the information available in The Latin America Human Mortality Database. For the analysis of mortality by department, new abbreviated life tables by sex were constructed with information from the departments for the period 1985-2014, adjusting to the information available for the departments of Colombia in terms of deaths and population. The methodology was implemented through the free statistical software R, which allows the replicability and reproducibility of the results. / Díaz Rojo, G. (2021). Ajuste y predicción de la mortalidad. Aplicación a Colombia [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/179995