The presented examples clearly show that we not only can, but must use heavy tailed alternatives to the Gaussian law in order to obtain acceptable estimates of market losses. But can we substitute the Gaussian distribution with other distributions in Value at Risk (Expected Shortfall) calculations for whole portfolios of assets? Recall, that the definition of VaR utilizes the quantiles of the portfolio returns distribution and not the returns distribution of individual assets in the portfolio. If all asset return distributions are assumed to be Gaussian then the portfolio distribution is multivariate normal and well known statistical tools can be applied (). However, when asset returns are distributed according to a different law (or different laws!) then the multivariate distribution may be hard to tackle. In particular, linear correlation may no longer be a meaningful measure of dependence.
Luckily for us multivariate statistics offers the concept of copulas, for a review see  and . In rough terms, a copula is a function with certain special properties. Alternatively we can say that it is a multivariate distribution function defined on the unit cube . The technical definitions of copulas that can be found in the literature often look more complicated, but to a financial modeler, this definition is enough to build an intuition from. What is important for VaR calculations is that a copula enables us to construct a multivariate distribution function from the marginal (possibly different) distribution functions of individual asset returns in a way that takes their dependence structure into account. This dependence structure may be no longer measured by correlation, but by other adequate functions like rank correlation, comonotonicity and, especially, tail dependence (). Moreover, it can be shown that for every multivariate distribution function there exists a copula which contains all information on dependence. For example, if the random variables are independent, then the independence copula (also known as the product copula) is just the product of variables: . If the random variables have a multivariate normal distribution with a given covariance matrix then the Gaussian copula is obtained.
Copula functions do not impose any restrictions on the model, so in order to reach a model that is to be useful in a given risk management problem, a particular specification of the copula must be chosen. From the wide variety of copulas that exist probably the elliptical and Archimedean copulas are the ones most often used in applications. Elliptical copulas are simply the copulas of elliptically contoured (or elliptical) distributions, e.g. (multivariate) normal, , symmetric stable and symmetric generalized hyperbolic (). Rank correlation and tail dependence coefficients can be easily calculated for elliptical copulas. There are, however, drawbacks - elliptical copulas do not have closed form expressions, are restricted to have radial symmetry and have all marginal distributions of the same type. These restrictions may disqualify elliptical copulas from being used in some risk management problems. In particular, there is usually a stronger dependence between big losses (e.g. market crashes) than between big gains. Clearly, such asymmetries cannot be modeled with elliptical copulas. In contrast to elliptical copulas, all commonly encountered Archimedean copulas have closed form expressions. Their popularity also stems from the fact that they allow for a great variety of different dependence structures ([41,50]). Many interesting parametric families of copulas are Archimedean, including the well known Clayton, Frank and Gumbel copulas.
After the marginal distributions of asset returns are estimated and a particular copula type is selected, the copula parameters have to be estimated. The fit can be performed by least squares or maximum likelihood. Note, however, that for some copula types it may not be possible to maximize the likelihood function. In such cases the least squares technique should be used. A review of the estimation methods - including a description of the relevant XploRe quantlets - can be found in .
For risk management purposes, we are interested in the Value at Risk of a portfolio of assets. While analytical methods for the computation of VaR exist for the multivariate normal distribution (i.e. for the Gaussian copula), in most other cases we have to use Monte Carlo simulations. A general technique for random variate generation from copulas is the conditional distributions method (). A random vector having a joint distribution function can be generated by the following algorithm:
Copulas allow us to construct models which go beyond the standard notions of correlation and multivariate Gaussian distributions. As such, in conjunction with alternative asset returns distributions discussed earlier in this chapter, they yield an ideal tool to model a wide variety of financial portfolios and products. No wonder they are gradually becoming an element of good risk management practice.