2. Applications of Copulas for the Calculation of Value-at-Risk

Jörn Rank and Thomas Siegl
July 9, 2002

We will focus on the computation of the Value-at-Risk (VaR) from the perspective of the dependency structure between the risk factors. Apart from historical simulation, most VaR methods assume a multivariate normal distribution of the risk factors. Therefore, the dependence structure between different risk factors is defined by the correlation between those factors. It is shown in Embrechts, McNeil and Straumann (1999) that the concept of correlation entails several pitfalls. The authors therefore propose the use of copulas to quantify dependence.

For a good overview of copula techniques we refer to Nelsen (1999). Copulas can be used to describe the dependence between two or more random variables with arbitrary marginal distributions. In rough terms, a copula is a function $ C : [0,1]^n \to [0,1]$ with certain special properties. The joint multidimensional cumulative distribution can be written as

$\displaystyle \textrm{P}(X_1 \le x_1 , \ldots, X_n \le x_n)$ $\displaystyle =$ $\displaystyle C \left( \textrm{P}(X_1 \le x_1), \ldots, \textrm{P}(X_n \le x_n) \right)$  
  $\displaystyle =$ $\displaystyle C \left( F_1(x_1), \ldots, F_n(x_n) \right),$  

where $ F_1, \ldots, F_n$ denote the cumulative distribution functions of the $ n$ random variables $ X_1,\dots,X_n$. In general, a copula $ C$ depends on one or more copula parameters $ p_1, \ldots, p_k$ that determine the dependence between the random variables $ X_1,\dots,X_n$. In this sense, the correlation $ \rho(X_i,X_j)$ can be seen as a parameter of the so-called Gaussian copula.

Here we demonstrate the process of deriving the VaR of a portfolio using the copula method with XploRe , beginning with the estimation of the selection of the copula itself, estimation of the copula parameters and the computation of the VaR. Backtesting of the results is performed to show the validity and relative quality of the results. We will focus on the case of a portfolio containing two market risk factors only, the FX rates USD/EUR and GBP/EUR. Copulas in more dimensions exist, but the selection of suitable $ n$-dimensional copulas is still quite limited. While the case of two risk factors is still important for applications, e.g. spread trading, it is also the case that can be best described.

As we want to concentrate our attention on the modelling of the dependency structure, rather than on the modelling of the marginal distributions, we restrict our analysis to normal marginal densities. On the basis of our backtesting results, we find that the copula method produces more accurate results than ``correlation dependence''.