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.

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.