[pt] O Modelo Média-Variância, proposto por Markowitz para resolver o problema de estruturação ótima de carteiras de investimentos, utiliza uma medida simétrica de risco, o desvio padrão dos retornos. Contudo, a grande utilização no mercado financeiro de ativos com retornos assimétricos levou ao desenvolvimento de medidas de risco assimétricas, como a semivariância e o downside risk, buscando quantificar de forma mais precisa a percepção de risco investidor. Neste trabalho, comparamos algumas metodologias para estruturar carteiras de investimentos contendo ativos com retornos assimétricos. / [en] The Mean-Variance model for asset allocation, proposed by Markowitz, use a symmetric risk measure, the standard deviation of returns. Although, because of the increasing use of financial instruments with asymmetric payoffs, asymmetric risk measures, as semivariance and downside risk, have been required in order to quantify the investor´s perception of risk as accurately as possible. So, we compare in this work several methodologies to structures optimal portfolios containing securities with asymmetric returns.

Title: Reinsurance optimization using stochastic programming and risk measures Author: Jan Došel Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Martin Branda, Ph.D., Department of Probability and Mathe- matical Statistics Abstract: The diploma thesis deals with an application of a stochastic progra- mming in a reinsurance optimization problem in terms of a present regulatory framework of the insurance companies within the European Union, i.e. Solvency II. In this context, the reinsurance does not only transfer a portion of the risk to the reinsurer but also reduces an amout of required capital. The thesis utilizes certain risk measures and their properties, premium principles and non-linear in- teger programming. In the theoretical part, there are basic terms from Solvency II, reinsurance, risk measures and the comonotonicity of random variables descri- bed and the optimization problem itself is derived. The approach is then applied in the practical part on data of Czech Insurers' Bureau using the GAMS software. Finally, a stability of the solution is tested depending on several parameters. Keywords: reinsurance optimization, stochastic programming, Solvency II, risk measures 1

Title: Bilevel optimization problems and their applications to portfolio selection Author: Lenka Godul'ová Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D. Abstract: This work deals with the problem of bilevel tasks. First, it recalls the basic knowledge of mean-risk models, risk measure in singlelevel problems, and second degree stochastic dominance. Then it presents basic knowledge of bilevel tasks. bilevel problems have several advantages over singlelevel. In one process, it is possible to analyze two different or even conflicting situations. The bilevel role can better capture the relationship between the two objects. The main focus of the thesis is the formulation of various bilevel tasks and their reformulation into the simplest form. The numerical part deals with four types of formulated bilevel problems at selected risk measures. Keywords: Bilevel problems, Second degree stochastic dominance, Risk measures 1

Title: Bilevel optimization problems and their applications to portfolio selection Author: Lenka Godul'ová Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D. Abstract: This work deals with the problem of bilevel tasks. First, it recalls the basic knowledge of mean-risk models, risk measure in singlelevel problems, and second degree stochastic dominance. Then it presents basic knowledge of bilevel tasks. bilevel problems have several advantages over singlelevel. In one process, it is possible to analyze two different or even conflicting situations. The bilevel role can better capture the relationship between the two objects. The main focus of the thesis is the formulation of various bilevel tasks and their reformulation into the simplest form. The numerical part deals with four types of formulated bilevel problems at selected risk measures. Keywords: Bilevel problems, Second degree stochastic dominance, Risk measures 1

Cette thèse se compose de deux parties indépendantes qui portent sur le contrôle stochastique avec des espérances non linéaires et les équations stochastiques rétrogrades (EDSR), ainsi que sur les méthodes numériques de résolution de ces équations. Dans la première partie on étudie une nouvelle classe d'équations stochastiques rétrogrades, dont la particularité est que la condition terminale n'est pas fixée mais vérifie une contrainte non linéaire exprimée en termes de "f-espérances". Ce nouvel objet mathématique est étroitement lié aux problèmes de couverture approchée des options européennes où le risque de perte est quantifié en termes de mesures de risque dynamiques, induites par la solution d'une EDSR non linéaire. Dans le chapitre suivant on s'intéresse aux problèmes d'arrêt optimal pour les mesures de risque dynamiques avec sauts. Plus précisément, on caractérise dans un cadre markovien la mesure de risque minimale associée à une position financière comme l'unique solution de viscosité d'un problème d'obstacle pour une équation intégro-différentielle. Dans le troisième chapitre, on établit un principe de programmation dynamique faible pour un problème mixte de contrôle stochastique et d'arrêt optimal avec des espérances non linéaires, qui est utilisé pour obtenir les EDP associées.La spécificité de ce travail réside dans le fait que la fonction de gain terminal ne satisfait aucune condition de régularité (elle est seulement considérée mesurable), ce qui n'a pas été le cas dans la littérature précédente. Dans le chapitre suivant, on introduit un nouveau problème de jeux stochastiques, qui peut être vu comme un jeu de Dynkin généralisé (avec des espérances non linéaires). On montre que ce jeu admet une fonction valeur et on obtient des conditions suffisantes pour l'existence d'un point selle. On prouve que la fonction valeur correspond à l'unique solution d'une équation stochastique rétrograde doublement réfléchie avec un générateur non linéaire général. Cette caractérisation permet d'obtenir de nouveaux résultats sur les EDSR doublement réfléchies avec sauts. Le problème de jeu de Dynkin généralisé est ensuite étudié dans un cadre markovien.Dans la deuxième partie, on s'intéresse aux méthodes numériques pour les équations stochastiques rétrogrades doublement réfléchies avec sauts et barrières irrégulières, admettant des sauts prévisibles et totalement inaccessibles. Dans un premier chapitre, on propose un schéma numérique qui repose sur la méthode de pénalisation et l'approximation de la solution d'une EDSR par une suite d'EDSR discrètes dirigées par deux arbres binomiaux indépendants (un qui approxime le mouvement brownien et l'autre le processus de Poisson composé). Dans le deuxième chapitre, on construit un schéma en discrétisant directement l'équation stochastique rétrograde doublement réfléchie, schéma qui présente l'avantage de ne plus dépendre du paramètre de pénalisation. On prouve la convergence des deux schémas numériques et on illustre avec des exemples numériques les résultats théoriques. / This thesis consists of two independent parts which deal with stochastic control with nonlinear expectations and backward stochastic differential equations (BSDE), as well as with the numerical methods for solving these equations.We begin the first part by introducing and studying a new class of backward stochastic differential equations, whose characteristic is that the terminal condition is not fixed, but only satisfies a nonlinear constraint expressed in terms of "f - expectations". This new mathematical object is closely related to the approximative hedging of an European option, when the shortfall risk is quantified in terms of dynamic risk measures, induced by the solution of a nonlinear BSDE. In the next chapter we study an optimal stopping problem for dynamic risk measures with jumps.More precisely, we characterize in a Markovian framework the minimal risk measure associated to a financial position as the unique viscosity solution of an obstacle problem for partial integrodifferential equations. In the third chapter, we establish a weak dynamic programming principle for a mixed stochastic control problem / optimal stopping with nonlinear expectations, which is used to derive the associated PDE. The specificity of this work consists in the fact that the terminal reward does not satisfy any regularity condition (it is considered only measurable), which was not the case in the previous literature. In the next chapter, we introduce a new game problem, which can be seen as a generalized Dynkin game (with nonlinear expectations ). We show that this game admits a value function and establish sufficient conditions ensuring the existence of a saddle point . We prove that the value function corresponds to the unique solution of a doubly reected backward stochastic equation (DRBSDE) with a nonlinear general driver. This characterization allows us to obtain new results on DRBSDEs with jumps. The generalized Dynkin game is finally addressed in a Markovian framework.In the second part, we are interested in numerical methods for doubly reected BSDEs with jumps and irregular barriers, admitting both predictable and totally inaccesibles jumps. In the first chapter we provide a numerical scheme based on the penalisation method and the approximation of the solution of a BSDE by a sequence of discrete BSDEs driven by two independent random walks (one approximates the Brownian motion and the other one the compensated Poisson process). In the second chapter, we construct an alternative scheme based on the direct discretisation of the DRBSDE, scheme which presents the advantage of not depending anymore on the penalization parameter. We prove the convergence of the two schemes and illustrate the theoretical results with some numerical examples.

Insurance companies or other financial institutions face financial risks during their various activites. Risk capital is allocated in order to cover these risks. The goal of capital allocation is to redistribute this capital to various constituents of the firm with respect to their riskiness. The thesis deals with risk measures and allocation methods. Special emphasis is put on the notions of coherent risk measures and coherent allocation methods. Conditions of coherence are checked for certain allocation methods. The thesis also deals with practical calculation of allocations to individual risks using allocation methods. 1

01 Abstract: This thesis is concerned with the robust methods in portfolio theory. Different risk measures used in portfolio management are introduced and the corresponding robust portfolio optimization problems are formulated. The analytical solutions of the robust portfolio optimization problem with the lower partial moments (LPM), value-at-risk (VaR) or conditional value-at-risk (CVaR), as a risk measure, are presented. The application of the worst-case conditional value-at-risk (WCVaR) to robust portfolio management is proposed. This thesis considers WCVaR in the situation where only partial information on the underlying probability distribution is available. The minimization of WCVaR under mixture distribution uncertainty, box uncertainty, and ellipsoidal uncertainty are investigated. Several numerical examples based on real market data are presented to illustrate the proposed approaches and advantage of the robust formulation over the corresponding nominal approach.

Responding to the changes in the insurance environment of the past decade, insurance regulators globally have been revamping the valuation and capital regulations. This thesis is concerned with the design and analysis of statistical inference procedures that are used to implement these new and upcoming insurance regulations, and their analysis in a more general setting toward lending further insights into their performance in practical situations. The quantitative measure of risk that is used in these new and upcoming regulations is the risk measure known as the Tail Value-at-Risk (T-VaR). In implementing these regulations, insurance companies often have to estimate the T-VaR of product portfolios from the output of a simulation of its cash flows. The distributions for the underlying economic variables are either estimated or prescribed by regulations. In this situation the computational complexity of estimating the T-VaR arises due to the complexity in determining the portfolio cash flows for a given realization of economic variables. A technique that has proved promising in such settings is that of importance sampling. While the asymptotic behavior of the natural non-parametric estimator of T-VaR under importance sampling has been conjectured, the literature has lacked an honest result. The main goal of the first part of the thesis is to give a precise weak convergence result describing the asymptotic behavior of this estimator under importance sampling. Our method also establishes such a result for the natural non-parametric estimator for the Value-at-Risk, another popular risk measure, under weaker assumptions than those used in the literature. We also report on a simulation study conducted to examine the quality of these asymptotic approximations in small samples. The Haezendonck-Goovaerts class of risk measures corresponds to a premium principle that is a multiplicative analog of the zero utility principle, and is thus of significant academic interest. From a practical point of view our interest in this class of risk measures arose primarily from the fact that the T-VaR is, in a sense, a minimal member of the class. Hence, a study of the natural non-parametric estimator for these risk measures will lend further insights into the statistical inference for the T-VaR. Analysis of the asymptotic behavior of the generalized estimator has proved elusive, largely due to the fact that, unlike the T-VaR, it lacks a closed form expression. Our main goal in the second part of this thesis is to study the asymptotic behavior of this estimator. In order to conduct a simulation study, we needed an efficient algorithm to compute the Haezendonck-Goovaerts risk measure with precise error bounds. The lack of such an algorithm has clearly been noticed in the literature, and has impeded the quality of simulation results. In this part we also design and analyze an algorithm for computing these risk measures. In the process of doing we also derive some fundamental bounds on the solutions to the optimization problem underlying these risk measures. We also have implemented our algorithm on the R software environment, and included its source code in the Appendix.

The Heath-Jarrow-Morton (HJM) model represents the latest in powerful arbitrage-free technology for modeling the term structure and managing interest rate risk. Yet risk management strategies in the form of immunization portfolios using duration, convexity, and M-square are still widely used in bond portfolio management today. This study addresses the question of how traditional risk measures and immunization strategies perform when the term structure evolves in the HJM manner. Using Monte Carlo simulation, I analyze four HJM volatility structures, four initial term structure shapes, three holding periods, and two traditional immunization approaches (duration-matching and duration-and-convexity-matching). I also examine duration and convexity measures derived specifically for the HJM framework. In addition I look at whether portfolios should be constructed randomly, by minimizing their M-squares or using barbell or bullet structures. I assess immunization performance according to three criteria. One of these criteria corresponds to active portfolio management, and the other two correspond to passive portfolio management. Under active portfolio management, an asset portfolio is successfully immunized if its holding period return is greater than or equal to the holding period return of the liability portfolio. Under passive portfolio management, the closer the returns of the asset portfolio to the returns of the liability portfolio, the better the immunization performance. The results of the study suggest that, under the active portfolio management criterion, and with the duration matching strategy, HJM and traditional duration measures have similar immunization performance when forward rate volatilities are low. There is a substantial deterioration in the immunization performance of traditional risk measures when there is high volatility. This deterioration is not observed with HJM duration measures. These results could be due to two factors. Traditional risk measures could be poor risk measures, or the duration matching strategy is not the most appropriate immunization approach when there is high volatility because yield curve shifts would often be large. Under the active portfolio management criterion and with the duration and convexity matching strategy, the immunization performance of traditional risk measures improves considerably at the high volatility segments of the yield curve. The improvement in the performance of the HJM risk measures is not as dramatic. The immunization performance of traditional duration and convexity measures, however, deteriorates at the low volatility segments of the yield curve. This deterioration is not observed when HJM risk measures are used. Overall, with the duration and convexity matching strategy, the immunization performance of portfolios matched with traditional risk measures is very close to that of portfolios matched with the HJM risk measures. This result suggests that the duration and convexity matching approach should be preferred to duration matching alone. Also the result shows that the underperformance of traditional risk measures under high volatility is not due to their being poor risk measures, but rather due to the reason that the duration matching strategy is not an appropriate immunization approach when there is high volatility in the market. Under the passive portfolio management criteria, the performances of traditional and HJM measures are similar with the duration matching strategy. Less than 29% of the duration matched portfolios have returns within one basis point of the target yield, whereas almost all are within 100 basis points of the target yield. These results suggest that the duration matching strategy might not be sufficient to generate cash flows close to those of the target bond. The duration measure assumes a linear relation between the bond price and the yield change, and the nonlinearities that are not captured by the duration measure might be important. When the duration and convexity matching strategy is used, more than 36% of the portfolios are within one basis point of the target with HJM risk measures. This dramatic improvement in the immunization performance of HJM measures is not guaranteed for traditional risk measures. In fact, there are certain cases in which the performance of traditional risk measures deteriorates with the duration and convexity matching strategy. In this respect, choosing the correct risk measure is more important than the immunization strategy when passive portfolio management is pursued. Under active portfolio management criterion, there is no significant difference among bullet, barbell, minimum M-square, and random portfolios with both duration matching and duration and convexity matching strategies. Under the passive portfolio management criterion, bullet portfolios produce closer returns to the target for short holding periods when the duration matching strategy is used. With the duration and convexity matching strategy, bullet, barbell and minimum M-square portfolios produce closer returns to the target for short holding periods. Random portfolios perform as well as bullet, barbell and minimum M-square portfolios for medium to long holding periods. These results suggest that when the duration matching strategy is used, bullet portfolios are preferable to other portfolio formation strategies for short holding periods. When the duration and convexity matching strategy is used, no portfolio formation strategy is better than the other. Under the active portfolio management criterion, minimum M-square portfolios are successfully immunized under each yield curve shape and volatility structure considered. Under the passive portfolio management criterion, minimum M-square portfolios perform better for short holding periods, and their performance deteriorates as the holding period increases, irrespective of the volatility level. This suggests that the performance of minimum M-square portfolios is more sensitive to the holding period rather than the volatility. Therefore, minimum M-square portfolios would be preferred in the markets when there are large changes in volatility. Overall, the results of the study suggest that, under the active portfolio management criterion and with the duration matching strategy, traditional duration measures underperform their HJM counterparts when forward rate volatilities are high. With the duration and convexity matching strategy, this underperformance is not as dramatic. Also no particular portfolio formation strategy is better than the other under the active portfolio management criterion. Under the passive portfolio management criterion, the duration matching strategy is not sufficient to generate cash flows closer to those of the target bond. The duration and convexity matching strategy, however, leads to substantial improvement in the immunization performance of the HJM risk measures. This improvement is not guaranteed for the traditional risk measures. Under the passive portfolio management criterion, bullet portfolios are preferred to other portfolio formation strategies for short holding periods. For medium to long holding periods, however, the portfolio formation strategy does not significantly affect immunization performance. Also, the immunization performance of minimum M-square portfolios is more sensitive to the holding period rather than the volatility. / Ph. D.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / We present the Shortfall Deviation Risk (SDR), a risk measure that represents the expected loss of results that occur with certain probability penalized by the dispersion of results worse than such expectation. The SDR combines the Expected Shortfall (ES) and the Shortfall Deviation (SD), which we also introduce, contemplating the two fundamental pillars of the risk concept the probability of adverse events (ES) and the variability of an expectation (SD) and considers extreme results. We demonstrate that the SD is a generalized deviation measure, whereas the SDR is a coherent risk measure. We achieve the dual representation of the SDR, and we discuss issues such as its representation by a weighted ES, acceptance sets, convexity, continuity and the relationship with stochastic dominance. Illustrations using Monte Carlo simulation and real data indicate that the SDR offers greater protection to measure risk than other measures, especially in turbulent times. / Esse trabalho apresenta o Risco do Desvio da Perda (Shortfall Deviation Risk SDR), uma medida de risco que representa a perda esperada de resultados que ocorrem com determinada probabilidade penalizada pela dispersão de resultados piores que essa expectativa. O SDR combina a Perda Esperada (Expected Shortfall ES) com o Desvio da Perda (Shortfall Deviation SD), introduzido nesse trabalho, de modo a contemplar os dois pilares fundamentais do conceito de risco, que são a possibilidade de eventos ruins (ES) e a variabilidade sobre uma expectativa (SD), além de levar em conta resultados extremos. Neste estudo é demonstrado que o SD é uma medida de desvio generalizado, ao passo que o SDR é uma medida de risco coerente. A representação dual do SDR é obtida, e questões como sua representação por meio de uma ponderação da ES, conjuntos de aceitação, convexidade, continuidade e relação com dominância estocástica são discutidas. Ilustrações com simulação Monte Carlo e dados reais indicam que o SDR oferece maior proteção na mensuração do risco que outras medidas, especialmente em momentos de turbulência.