17. Copulas and Value-at-Risk

The conventional procedure of approximating risk factors with multivariate normal distributions implies that the risk's dependence structure is reduced to a fixed, predetermined type. Even if the autocorrelation structure of the risk factors is neglected, stipulating a multivariate normal distribution means that the following assumptions hold:

  1. symmetric distribution of returns
  2. the tails of the distribution are not too heavy
  3. linear dependence

The first point is discussed in Chapter 3. The second point is not empirically proven, since in general the tails of the distribution display leptokurtic. The third point deals with the properties of the covariance (correlation): high correlation means that there is almost a linear relationship among the risk factors. Although there are many reasons against this procrustes type assumption of a normal distribution, there are a number of practical reasons, which in particular are associated with performing the calculations, Embrechts et al. (1999a) and Embrechts et al. (1999b).

A generalized representation of the dependence of the risk factors can be obtained using a Copula. Nelsen (1999) gives a good overview of this mathematical concept. In this chapter we will concentrate on the representation of two-dimensional copulas and we will discuss their application in calculating the Value-at-Risk (VaR).

A copula is a function $ C : [0,1]^p \rightarrow [0,1]$ with particular properties. The basic idea is to describe the joint distribution of a random variable $ X=(X_1, \dots, X_p)^\top $ using a copula $ C$:

$\displaystyle {\P}(X_1 \le x_1 , \ldots, X_p \le x_p)$ $\displaystyle =$ $\displaystyle C \left( {\P}(X_1 \le x_1), \ldots, {\P}(X_p \le x_p) \right)$  
  $\displaystyle =$ $\displaystyle C \left( F_1(x_1), \ldots, F_p(x_p) \right),$  

where $ F_1, \ldots, F_p$ represent the cumulative distributions function of the variables $ X_j, \; j=1, \cdots, p$. A copula $ C$ is in general dependent on parameters that determine the respective form of the dependence. The correlation of two variables represents a special dependence structure which can be described with the help of the Gaussian copula (which will be introduced shortly).

This chapter introduces how to calculate the VaR with the help of the copula technique. For clarity of presentation we will first focus on cases with two risk factors: the FX time series USD/EUR and GBP/EUR, see also Rank and Siegl (2002).