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
with certain special properties.
The joint multidimensional cumulative distribution can be written as
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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 -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''.