The research on optimal reinsurance design dated back to the 1960’s. For nearly half a century, the quest for optimal reinsurance designs has remained a fascinating subject, drawing significant interests from both academicians and practitioners. Its fascination lies in its potential as an effective risk management tool for the insurers. There are many ways of formulating the optimal design of reinsurance, depending on the chosen objective and constraints. In this thesis, we address the problem of optimal reinsurance designs from an insurer’s perspective. For an insurer, an appropriate use of the reinsurance helps to reduce the adverse risk exposure and improve the overall viability of the underlying business. On the other hand, reinsurance incurs additional cost to the insurer in the form of reinsurance premium. This implies a classical risk and reward tradeoff faced by the insurer. The primary objective of the thesis is to develop theoretically sound and yet practical solution in the quest for optimal reinsurance designs. In order to achieve such an objective, this thesis is divided into two parts. In the first part, a number of reinsurance models are developed and their optimal reinsurance treaties are derived explicitly. This part focuses on the risk measure minimization reinsurance models and discusses the optimal reinsurance treaties by exploiting two of the most common risk measures known as the Value-at-Risk (VaR) and the Conditional Tail Expectation (CTE). Some additional important economic factors such as the reinsurance premium budget, the insurer’s profitability are also considered. The second part proposes an innovative method in formulating the reinsurance models, which we refer as the empirical approach since it exploits explicitly the insurer’s empirical loss data. The empirical approach has the advantage that it is practical and intuitively appealing. This approach is motivated by the difficulty that the reinsurance models are often infinite dimensional optimization problems and hence the explicit solutions are achievable only in some special cases. The empirical approach effectively reformulates the optimal reinsurance problem into a finite dimensional optimization problem. Furthermore, we demonstrate that the second-order conic programming can be used to obtain the optimal solutions for a wide range of reinsurance models formulated by the empirical approach.

The research on optimal reinsurance design dated back to the 1960’s. For nearly half a century, the quest for optimal reinsurance designs has remained a fascinating subject, drawing significant interests from both academicians and practitioners. Its fascination lies in its potential as an effective risk management tool for the insurers. There are many ways of formulating the optimal design of reinsurance, depending on the chosen objective and constraints. In this thesis, we address the problem of optimal reinsurance designs from an insurer’s perspective. For an insurer, an appropriate use of the reinsurance helps to reduce the adverse risk exposure and improve the overall viability of the underlying business. On the other hand, reinsurance incurs additional cost to the insurer in the form of reinsurance premium. This implies a classical risk and reward tradeoff faced by the insurer. The primary objective of the thesis is to develop theoretically sound and yet practical solution in the quest for optimal reinsurance designs. In order to achieve such an objective, this thesis is divided into two parts. In the first part, a number of reinsurance models are developed and their optimal reinsurance treaties are derived explicitly. This part focuses on the risk measure minimization reinsurance models and discusses the optimal reinsurance treaties by exploiting two of the most common risk measures known as the Value-at-Risk (VaR) and the Conditional Tail Expectation (CTE). Some additional important economic factors such as the reinsurance premium budget, the insurer’s profitability are also considered. The second part proposes an innovative method in formulating the reinsurance models, which we refer as the empirical approach since it exploits explicitly the insurer’s empirical loss data. The empirical approach has the advantage that it is practical and intuitively appealing. This approach is motivated by the difficulty that the reinsurance models are often infinite dimensional optimization problems and hence the explicit solutions are achievable only in some special cases. The empirical approach effectively reformulates the optimal reinsurance problem into a finite dimensional optimization problem. Furthermore, we demonstrate that the second-order conic programming can be used to obtain the optimal solutions for a wide range of reinsurance models formulated by the empirical approach.

In this thesis the performances of different approximations are compared for a standard actuarial and financial problem: the estimation of quantiles and conditional tail expectations of the final value of a series of discrete cash flows. To calculate the risk measures such as quantiles and Conditional Tail Expectations, one needs the distribution function of the final wealth. The final value of a series of discrete payments in the considered model is the sum of dependent lognormal random variables. Unfortunately, its distribution function cannot be determined analytically. Thus usually one has to use time-consuming Monte Carlo simulations. Computational time still remains a serious drawback of Monte Carlo simulations, thus several analytical techniques for approximating the distribution function of final wealth are proposed in the frame of this thesis. These are the widely used moment-matching approximations and innovative comonotonic approximations. Moment-matching methods approximate the unknown distribution function by a given one in such a way that some characteristics (in the present case the first two moments) coincide. The ideas of two well-known approximations are described briefly. Analytical formulas for valuing quantiles and Conditional Tail Expectations are derived for both approximations. Recently, a large group of scientists from Catholic University Leuven in Belgium has derived comonotonic upper and comonotonic lower bounds for sums of dependent lognormal random variables. These bounds are bounds in the terms of "convex order". In order to provide the theoretical background for comonotonic approximations several fundamental ordering concepts such as stochastic dominance, stop-loss and convex order and some important relations between them are introduced. The last two concepts are closely related. Both stochastic orders express which of two random variables is the "less dangerous/more attractive" one. The central idea of comonotonic upper bound approximation is to replace the original sum, presenting final wealth, by a new sum, for which the components have the same marginal distributions as the components in the original sum, but with "more dangerous/less attractive" dependence structure. The upper bound, or saying mathematically, convex largest sum is obtained when the components of the sum are the components of comonotonic random vector. Therefore, fundamental concepts of comonotonicity theory which are important for the derivation of convex bounds are introduced. The most wide-spread examples of comonotonicity which emerge in financial context are described. In addition to the upper bound a lower bound can be derived as well. This provides one with a measure of the reliability of the upper bound. The lower bound approach is based on the technique of conditioning. It is obtained by applying Jensen's inequality for conditional expectations to the original sum of dependent random variables. Two slightly different version of conditioning random variable are considered in the context of this thesis. They give rise to two different approaches which are referred to as comonotonic lower bound and comonotonic "maximal variance" lower bound approaches. Special attention is given to the class of distortion risk measures. It is shown that the quantile risk measure as well as Conditional Tail Expectation (under some additional conditions) belong to this class. It is proved that both risk measures being under consideration are additive for a sum of comonotonic random variables, i.e. quantile and Conditional Tail Expectation for a comonotonic upper and lower bounds can easily be obtained by summing the corresponding risk measures of the marginals involved. A special subclass of distortion risk measures which is referred to as class of concave distortion risk measures is also under consideration. It is shown that quantile risk measure is not a concave distortion risk measure while Conditional Tail Expectation (under some additional conditions) is a concave distortion risk measure. A theoretical justification for the fact that "concave" Conditional Tail Expectation preserves convex order relation between random variables is given. It is shown that this property does not necessarily hold for the quantile risk measure, as it is not a concave risk measure. Finally, the accuracy and efficiency of two moment-matching, comonotonic upper bound, comonotonic lower bound and "maximal variance" lower bound approximations are examined for a wide range of parameters by comparing with the results obtained by Monte Carlo simulation. It is justified by numerical results that, generally, in the current situation lower bound approach outperforms other methods. Moreover, the preservation of convex order relation between the convex bounds for the final wealth by Conditional Tail Expectation is demonstrated by numerical results. It is justified numerically that this property does not necessarily hold true for the quantile.

Au cours des dernières années, des changements importants dans le domaine des assurances et des finances attirent de plus en plus l’attention sur la nécessité d’élaborer un cadre normalisé pour la mesure des risques. Récemment, il y a eu un intérêt croissant de la part des experts en assurance sur l’utilisation de l’espérance conditionnelle des pertes (CTE) parce qu’elle partage des propriétés considérées comme souhaitables et applicables dans diverses situations. En particulier, il répond aux exigences d’une mesure de risque “cohérente”, selon Artzner [2]. Cette thèse représente des contributions à l’inférence statistique en développant des outils, basés sur la convergence des intégrales fonctionnelles, pour l’estimation de la CTE qui présentent un intérêt considérable pour la science actuarielle. Tout d’abord, nous développons un outil permettant l’estimation de la moyenne conditionnelle E[X|X > x], ensuite nous construisons des estimateurs de la CTE, développons la théorie asymptotique nécessaire pour ces estimateurs, puis utilisons la théorie pour construire des intervalles de confiance. Pour la première fois, l’approche de bootstrap non paramétrique est explorée dans cette thèse en développant des nouveaux résultats applicables à la valeur à risque (VaR) et à la CTE. Des études de simulation illustrent la performance de la technique de bootstrap.

In this thesis the performances of different approximations are compared for a standard actuarial and financial problem: the estimation of quantiles and conditional tail expectations of the final value of a series of discrete cash flows. To calculate the risk measures such as quantiles and Conditional Tail Expectations, one needs the distribution function of the final wealth. The final value of a series of discrete payments in the considered model is the sum of dependent lognormal random variables. Unfortunately, its distribution function cannot be determined analytically. Thus usually one has to use time-consuming Monte Carlo simulations. Computational time still remains a serious drawback of Monte Carlo simulations, thus several analytical techniques for approximating the distribution function of final wealth are proposed in the frame of this thesis. These are the widely used moment-matching approximations and innovative comonotonic approximations. Moment-matching methods approximate the unknown distribution function by a given one in such a way that some characteristics (in the present case the first two moments) coincide. The ideas of two well-known approximations are described briefly. Analytical formulas for valuing quantiles and Conditional Tail Expectations are derived for both approximations. Recently, a large group of scientists from Catholic University Leuven in Belgium has derived comonotonic upper and comonotonic lower bounds for sums of dependent lognormal random variables. These bounds are bounds in the terms of "convex order". In order to provide the theoretical background for comonotonic approximations several fundamental ordering concepts such as stochastic dominance, stop-loss and convex order and some important relations between them are introduced. The last two concepts are closely related. Both stochastic orders express which of two random variables is the "less dangerous/more attractive" one. The central idea of comonotonic upper bound approximation is to replace the original sum, presenting final wealth, by a new sum, for which the components have the same marginal distributions as the components in the original sum, but with "more dangerous/less attractive" dependence structure. The upper bound, or saying mathematically, convex largest sum is obtained when the components of the sum are the components of comonotonic random vector. Therefore, fundamental concepts of comonotonicity theory which are important for the derivation of convex bounds are introduced. The most wide-spread examples of comonotonicity which emerge in financial context are described. In addition to the upper bound a lower bound can be derived as well. This provides one with a measure of the reliability of the upper bound. The lower bound approach is based on the technique of conditioning. It is obtained by applying Jensen's inequality for conditional expectations to the original sum of dependent random variables. Two slightly different version of conditioning random variable are considered in the context of this thesis. They give rise to two different approaches which are referred to as comonotonic lower bound and comonotonic "maximal variance" lower bound approaches. Special attention is given to the class of distortion risk measures. It is shown that the quantile risk measure as well as Conditional Tail Expectation (under some additional conditions) belong to this class. It is proved that both risk measures being under consideration are additive for a sum of comonotonic random variables, i.e. quantile and Conditional Tail Expectation for a comonotonic upper and lower bounds can easily be obtained by summing the corresponding risk measures of the marginals involved. A special subclass of distortion risk measures which is referred to as class of concave distortion risk measures is also under consideration. It is shown that quantile risk measure is not a concave distortion risk measure while Conditional Tail Expectation (under some additional conditions) is a concave distortion risk measure. A theoretical justification for the fact that "concave" Conditional Tail Expectation preserves convex order relation between random variables is given. It is shown that this property does not necessarily hold for the quantile risk measure, as it is not a concave risk measure. Finally, the accuracy and efficiency of two moment-matching, comonotonic upper bound, comonotonic lower bound and "maximal variance" lower bound approximations are examined for a wide range of parameters by comparing with the results obtained by Monte Carlo simulation. It is justified by numerical results that, generally, in the current situation lower bound approach outperforms other methods. Moreover, the preservation of convex order relation between the convex bounds for the final wealth by Conditional Tail Expectation is demonstrated by numerical results. It is justified numerically that this property does not necessarily hold true for the quantile.

本篇論文主要討論在死亡率改善不確定性之下的避險策略。當保險公司負債面的人壽保單是比年金商品來得多的時候，公司會處於死亡率的風險之下。我們假設死亡率和利率都是隨機的情況，部分的死亡率風險可以經由自然避險而消除，而剩下的死亡率風險和利率風險則由零息債券和保單貼現商品來達到最適避險效果。我們考慮mean variance、VaR和CTE當成目標函數時的避險策略，其中在mean variance的最適避險策略可以導出公式解。由數值結果我們可以得知保單貼現的確是死亡率風險的有效避險工具。 / This paper proposes hedging strategies to deal with the uncertainty of mortality improvement. When insurance company has more life insurance contracts than annuities in the liability, it will be under the exposure of mortality risk. We assume both mortality and interest rate risk are stochastic. Part of mortality risk is eliminated by natural hedging and the remaining mortality risk and interest rate risk will be optimally hedged by zero coupon bond and life settlement contract. We consider the hedging strategies with objective functions of mean variance, value at risk and conditional tail expectation. The closed-form optimal hedging formula for mean variance assumption is derived, and the numerical result show the life settlement is indeed a effective hedging instrument against mortality risk.

Value at Risk (VaR) and Expected Shortfall (ES) are methods often used to measure market risk. Inaccurate and unreliable Value at Risk and Expected Shortfall models can lead to underestimation of the market risk that a firm or financial institution is exposed to, and therefore may jeopardize the well-being or survival of the firm or financial institution during adverse markets. The objective of this study is therefore to examine various Value at Risk and Expected Shortfall models, including fatter tail models, in order to analyze the accuracy and reliability of these models. Thirteen VaR and ES models under three main approaches (Parametric, Non-Parametric and Semi-Parametric) are examined in this study. The results of this study show that the proposed model (ARMA(1,1)-GJR-GARCH(1,1)-SGED) gives the most balanced Value at Risk results. The semi-parametric model (Extreme Value Theory, EVT) is the most accurate Value at Risk model in this study for S&P 500. / October 2014