Odhad hodnoty Value at Risk lineárního portfolia aktiv pomocí kopula funkcí

Abstract

The thesis is focused on the portfolio returns modelling using copula functions. The goal of the thesis is to verify the applicability of parametric models based on the copula functions to estimate the VaR of linear portfolio of financial assets on real data of returns of stock indices and exchange rates. The thesis is divided into two parts, theoretical-methodological and application part. In the theoretical part of the thesis, parametric models of portfolio returns are described. These models consist of two components: marginal distribution models and copula functions. Copula functions are needed to capture the dependences among marginal probability distributions. Subsequently, the Value at Risk methodology is clarified. First the VaR is defined and then various methods of estimating this value are described. The rest of the theoretical part contains an overview of statistical tests for backtesting VaR estimates. First the procedure of VaR backtesting is explained and subsequently statistical tests based mainly on the works of Kupiec and Christoffersen are described. In the application part the suitability of selected parametric models is verified. First, there are defined input data, the composition of the portfolios under consideration and the models used for returns modelling. In the thesis we consider these marginal models: normal distribution and normal inverse Gaussian model, and following copula functions: Gaussian and Student's copula function. These models are used to model the portfolio returns using Monte Carlo simulation method and subsequently to determine the VaR. Comparison of different models is carried out both ex-post and also by backtesting as described in the theoretical part. Backtesting results are compared both in terms of the number of exceptions (Kupiec unconditional test) and the independence of exceptions over time (Christoffersen test and the test proposed by Haas). At the end of the fourth chapter the individual results are summarised.