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.

A general methodology is described in which policyholder behaviour is decoupled from the pricing of a variable annuity based on the cost of hedging it, yielding two sequences of weakly coupled systems of partial differential equations (PDEs): the pricing and utility systems. The utility systems are used to generate policyholder withdrawal behaviour, which is in turn fed into the pricing systems as a means to determine the cost of hedging the contract. This approach allows us to incorporate the effects of utility-based pricing and factors such as taxation. As a case study, we consider the Guaranteed Lifelong Withdrawal and Death Benefits (GLWDB) contract. The pricing and utility systems for the GLWDB are derived under the assumption that the underlying asset follows a Markov regime-switching process. An implicit PDE method is used to solve both systems in tandem. We show that for a large class of utility functions, the two systems preserve homogeneity, allowing us to decrease the dimensionality of solutions. We also show that the associated control for the GLWDB is bang-bang, under which the work required to compute the optimal strategy is significantly reduced. We extend this result to provide the reader with sufficient conditions for a bang-bang control for a general variable annuity with a countable number of events (e.g. discontinuous withdrawals). Homogeneity and bang-bangness yield significant reductions in complexity and allow us to rapidly generate numerical solutions. Results are presented which demonstrate the sensitivity of the hedging expense to various parameters. The costly nature of the death benefit is documented. It is also shown that for a typical contract, the fee required to fund the cost of hedging calculated under the assumption that the policyholder withdraws at the contract rate is an appropriate approximation to the fee calculated assuming optimal consumption.

Taiwan has an ageing population, with more people in the concern of not having sufficient income stream during retired life. This study investigates the ageing issue through socioeconomic perspectives. It is recommended that apart from low interest-bearing term deposits, Interest-Sensitive Annuity is the most suitable solution for countering longevity risk. Through case studies, it has been found that 1, due to Annuity Puzzle sentiment, term depositors will continue to invest in Interest-Sensitive Annuities, regardless of the low interest rate environment. 2, Interest-Sensitive Annuity investors are as risk-averse as term depositors, implying that they do not necessarily choose the surrender option upon expiry. 3, due to customer sentiment, the Interest-Sensitive Annuity policy fees charged are inversely correlated to customers¡¦ willingness to invest. 4, by selling low-commission products, namely one- and two-year Interest-Sensitive Annuities through bancassurance channel, insurance companies enjoy the benefit of low cost capital and are able to reduce interest spread risk through efficient investments. Moreover, customers have their retirement needs covered while insurance salespeople of different channels are able to meet respective sales targets. It is therefore shown that Interest-Sensitive Annuities have the following benefits. For investors, it is the product type that best meets their needs. For insurance salespeople, they enjoy a diverse and complete product portfolio and for insurance companies, it maximizes operation efficiency. Unfortunately, after the termination of one- and two-year Interest-Sensitive Annuities on the market, insurance company capital costs have been negatively impacted, with retirement and longevity risks unsatisfied and insurance salespeople having less products to choose from. It is suggested that the regulator considers re-introducing one- and two-year Interest-Sensitive Annuities, using Risk-Based Capital as a complement in monitoring insurance companies.

A general methodology is described in which policyholder behaviour is decoupled from the pricing of a variable annuity based on the cost of hedging it, yielding two sequences of weakly coupled systems of partial differential equations (PDEs): the pricing and utility systems. The utility systems are used to generate policyholder withdrawal behaviour, which is in turn fed into the pricing systems as a means to determine the cost of hedging the contract. This approach allows us to incorporate the effects of utility-based pricing and factors such as taxation. As a case study, we consider the Guaranteed Lifelong Withdrawal and Death Benefits (GLWDB) contract. The pricing and utility systems for the GLWDB are derived under the assumption that the underlying asset follows a Markov regime-switching process. An implicit PDE method is used to solve both systems in tandem. We show that for a large class of utility functions, the two systems preserve homogeneity, allowing us to decrease the dimensionality of solutions. We also show that the associated control for the GLWDB is bang-bang, under which the work required to compute the optimal strategy is significantly reduced. We extend this result to provide the reader with sufficient conditions for a bang-bang control for a general variable annuity with a countable number of events (e.g. discontinuous withdrawals). Homogeneity and bang-bangness yield significant reductions in complexity and allow us to rapidly generate numerical solutions. Results are presented which demonstrate the sensitivity of the hedging expense to various parameters. The costly nature of the death benefit is documented. It is also shown that for a typical contract, the fee required to fund the cost of hedging calculated under the assumption that the policyholder withdraws at the contract rate is an appropriate approximation to the fee calculated assuming optimal consumption.

Title: Financial Mathematics in Czech textbooks from the Marchet's Reform Author: Martin Melcer Department: Department of Mathematics Education Supervisor: doc. RNDr. Martina Bečvářová, Ph.D. Abstract: The PhD thesis presents a comprehensive view of the development and the position of financial mathematics in Czech textbooks and collections particularly those used in high schools with consideration of the political situation in our country. The analysed period from 1908, i.e. from the Marchet's Reform, to the present is divided into five principal stages. In each stage textbooks covering financial mathematics instruction in all types of secondary schools are chosen. This shows the level and extent of the presentation, the complexity of the given tasks and problems and the way in which they are integrated into the textbooks or the syllabus, more precisely. At first the basic general characteristics of the textbooks are presented followed by a detailed description of partial topics of study which are then analysed thoroughly and mutually compared. The conclusions of the PhD thesis reflect the current situation of financial mathematics instruction and offer ways leading to the improvement in the level of financial literacy of our citizens. Keywords: financial mathematics, principal, interest, debt, annuity,...

[pt] Criado em 1999, o Fundo de Financiamento Estudantil (FIES) é um programa do Governo Federal destinado à concesão de financiamentos a alunos matriculados no ensino superior não gratuito. O programa adquiriu relevância após sofrer substanciais alterações normativas e operacionais ao longo do ano de 2010. Como resultado, a procura pelo financiamento registrou aumento significativo. Esse fluxo de novos alunos financiados atingiu de maneira heterogênea diferentes Instituições de Ensino. No trabalho, exploramos essa heterogeneidade com o objetivo de investigar se esse relaxamento de crédito estudantil via FIES teve algum efeito sobre taxas escolares. Por meio de uma estratégia de diferença em diferenças, mostrando que o FIES causou aumento para aferir se o aumento é devido a uma diminuição de elasticidade ou a um deslocamento da curva de demanda. / [en] Created in 1999, the fundo de Financiamento Estudantil (FIES) is a Brazilian Federal Government program destined to offer tuition fee loans to students in higher education institutions. The program acquired relevance after substancial operational and normative changes throughout 2010. AS a result, the demand for student loans increased significantly. This flow of new students affected the higher education institutions in a heterogeneous way. In this paper, we explore this heterogeneity to investigate if this easing of student loan generated by FIES affected tuition fees. Through a difference in differences approach, we conclude that FIES caused an increase in tuition costs. Then, we apply a structural model of demand to investigate if this increase is due to a reduction in demand elasticity or a shift of the demand curve itself.

Submitted by Regis Baratti Lima Salgado (regiss@fgvmail.br) on 2012-10-09T13:30:32Z No. of bitstreams: 1 TeseRegisBarattiLSalgado_VFim.pdf: 861409 bytes, checksum: 48cbf91a3e1e3ed3d3c91940bfde85ac (MD5) / Approved for entry into archive by ÁUREA CORRÊA DA FONSECA CORRÊA DA FONSECA (aurea.fonseca@fgv.br) on 2012-10-22T19:15:00Z (GMT) No. of bitstreams: 1 TeseRegisBarattiLSalgado_VFim.pdf: 861409 bytes, checksum: 48cbf91a3e1e3ed3d3c91940bfde85ac (MD5) / Made available in DSpace on 2012-10-23T18:23:36Z (GMT). No. of bitstreams: 1 TeseRegisBarattiLSalgado_VFim.pdf: 861409 bytes, checksum: 48cbf91a3e1e3ed3d3c91940bfde85ac (MD5) Previous issue date: 2012-08-22 / This thesis extend the theoretical dominance of annuities over non-contingent discount notes; under standard assumptions, we show that full annuitization is optimal even in incomplete annuity markets. Through numerical simulations, we scrutinize factors a ecting annuitization decision, consolidating and extending previous research by taking into account unfair prices, bequest motives, and out-of-pocket medical expenses. We also take into consideration the insurer's risk of default, and relax an implicit assumption in most past models and detach annuitization from retirement, i.e.: we do not presume that consumers are already retired from work when they decide whether or not to annuitize. In line with previous literature, our results originate very high levels of annuitization. Yet, we show that the demand for annuities drops sharply, if preferences are such that the implied optimal consumption path decays with age. We also show that optimal annuitization timing is closely related to the endowments pattern. / A inconsistência entre a teoria e o comportamento empírico dos agentes no que tange ao mercado privado de pensões tem se mostrado um dos mais resistentes puzzles presentes na literatura econômica. Em modelos de otimização intertemporal de consumo e poupança sob incerteza em relação ao tempo de vida dos agentes, anuidades são ativos dominantes, anulando ou restringindo fortemente a demanda por ativos cujos retornos não estão relacionados à probabilidade de sobrevivência. Na prática, entretanto, consumidores são extremamente céticos em relação às anuidades. Em oposição ao seguro contra longevidade oferecido pelas anuidades, direitos sobre esses ativos - essencialmente ilíquidos - cessam no caso de morte do titular. Nesse sentido, choques não seguráveis de liquidez e a presença de bequest motives foram consideravelmente explorados como possíveis determinantes da baixa demanda verificada. Apesar dos esforços, o puzzle persiste. Este trabalho amplia a dominância teórica das anuidades sobre ativos não contingentes em mercados incompletos; total na ausência de bequest motives, e parcial, quando os agentes se preocupam com possíveis herdeiros. Em linha com a literatura, simulações numéricas atestam que uma parcela considerável do portfolio ótimo dos agentes seria constituída de anuidades mesmo diante de choques de liquidez, bequest motives, e preços não atuarialmente justos. Em relação a um aspecto relativamente negligenciado pela academia, mostramos que o tempo ótimo de conversão de poupança em anuidades está diretamente relacionado à curva salarial dos agentes. Finalmente, indicamos que, caso as preferências dos agentes sejam tais que o nível de consumo ótimo decaia com a idade, a demanda por anuidades torna-se bastante sensível ao sobrepreço (em relação àquele atuarialmente justo) praticado pela indústria, chegando a níveis bem mais compatíveis com a realidade empírica.

This thesis is focused on practical application of two methods used in non-life insurance, Nested Monte Carlo and Least squares Monte Carlo. Best estimate and 99.5% quantile was calculated using both methods and results was compared. Both methods are similar in estimates and therefore can be used for computation of capital requirement. Least squares Monte Carlo seem more favourable, because it significantly reduces computation time.

Cette thèse est consacrée à l’étude de certains processus affines d’un point de vue théorique ainsi que appliqué par le biais d’applications en mathématiques actuarielles. Les processus affines sont des processus markoviens homogènes dont la transformée de Laplace du semi-groupe admet une structure exponentielle affine en son état initial. En vertu de leur riche structure, ces processus trouvent de nombreuses applications en biologie, physique et en mathématique de l’économie. Dans le chapitre 1, nous considérons les processus CBI (Continuous state Branching with Immigration) qui correspondent à un exemple de processus affines à valeurs dans la demi- droite réelle positive. Notre objectif dans ce chapitre consiste à étudier la transformée de Laplace du temps d’atteinte de ces processus à un niveau donné. Notre approche est basée sur un récent développement de Patie and Vigon [2015] concernant la théorie du potentiel pour la classe des processus de Markov complètement asymétriques à valeurs dans la droite réelle. Dans un premier temps, cette approche originale nous permet de donner une expression pour la transformée de Laplace du premier temps d’atteinte d’un niveau qui se trouve en dessous du point de départ du processus et ainsi de retrouver par une approche plus générale le résultat récemment obtenu par Duhalde et al. [2014]. Dans un deuxième temps, nous nous intéressons au cas où le niveau à atteindre se trouve au-dessus du point de départ du processus. Dans ce cadre, la situation est plus compliquée puisque le processus admet des sauts positifs. Néanmoins, nous sommes capable de donner une expression pour la transformée de Laplace de ce temps d’atteinte. Dans le chapitre 2, nous considérons les processus Wishart qui correspondent à un exemple de processus affines à valeurs dans le cône des matrices symétriques semi-définies positives. Plus précisément, nous considérons des processus de Wishart sans retour à la moyenne et nous étudions le problème de déterminer le plus petit instant tel que la transformée de Laplace du processus et/ou son intégrale devienne infinie. Le problème de déterminer le domaine maximal d’existence d’une transformée de Laplace est une préoccupation importante et cruciale, pas seulement d’un point de vue théorique, mais surtout pour des raisons pratiques liées à l’implémentation numérique. Sous une faible hypothèse de commutativité, nous obtenons le temps d’explosion de la transformée de Laplace jointe. Cette hypothèse disparaît lorsque l’on s’intéresse aux temps d’explosion de la transformée de Laplace de l’intégrale du processus ou du processus seul. De plus, nous expliquons le comportement du temps d’explosion en terme du rôle de la corrélation entre les facteurs positifs, ce qui va au-delà des résultats unidimensionnels que l’on peut retrouver dans la littérature. Dans le chapitre 3, nous considérons des modèles affines où une structure de dépendance entre les taux de mortalité et d’intérêt existe. Nous nous concentrons sur deux sortes de modèles. Le premier modèle suppose que les dynamiques de la mortalité et du processus de taux d’intérêt sont conduits par un processus de Cox-Ingersoll-Ross (CIR) multidimensionnel et le second par un processus de Wishart. Dans ce contexte, nos objectifs consistent à déterminer des formules pour les prix de certains contrats d’assurance tels que les options annuités garanties ainsi que d’étudier l’influence d’une structure de dépendance sur ces prix. En suivant la méthodologie introduite par Jalen and Mamon [2009], nous commençons par dériver une formule pour le prix de ces contrats. Ensuite, nous étudions la sensibilité du prix de ces contrats par rapport à la structure de dépendance entre les taux de mortalité et d’intérêt. Nous observons que dans un modèle affine général comme le modèle de Wishart qui permet de reproduire une structure de dépendance stochastique et plus riche entre les taux de mortalité et d’intérêt, plusieurs scénarios pour les prix peuvent être reproduits. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This paper aims to introduce the reader to the premium setting of annuities within life insurance. This is done using a hypothetical annuity contract offered to 36-year-olds in Sweden. The contract provides an annual pension from age 65 until either the individual's death or age 90, after which payouts cease. The analysis employs life tables using real-life data to estimate mortality, discounting to decide present values, and calculates fair and risk-adjusted premia for lump sum and annual payment options using theory and simulations. Ultimately, we found that the method used was insufficient given the data. This is due to the last decades' rise in life expectancy, requiring us to use other methods to acquire accurate premia. / <p>Detta arbete omfattar 3,0 hp och är en del av kursen Matematisk kommunikation (MM7020), 7,5 hp.</p>