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### Enterprise Risk Management Formula Book 12. Copulas

12.1        Definition

A copula is a multivariate cumulative distribution function for an  dimensional random vector  in the unit hypercube () that has uniform marginals, , each distributed according to  but not in general independent of each other. Let  also be restricted to the unit hypercube . Then a copula is defined as a function of the form:

Equivalently  is the joint cumulative distribution function for the random vector .

The copula density (for a continuous copula) is the pdf for which the cdf is the copula.

12.2        Properties

In the bivariate case () for a general function  to be a copula it must satisfy the following properties:

1.             for all

2.             must be increasing in both  and

3.             for all  and

4.

5.

12.3        Sklar’s theorem

If  is a joint (cumulative) distribution with marginal cdf’s  then there exists a copula  which maps the unit hypercube  onto the interval  such that for all  we have:

Moreover, if the  are continuous functions then the copula is unique and

Conversely, suppose  is a copula and  are univariate cdf’s. Then the function  is a joint distribution function with marginal cdf’s .

12.4        Example copulas

The Archimedean family involves copulas of the following form, where , , ,  is continuous and strictly decreasing and

Special cases include the Clayton copula which has  (for some suitable value of ) and the independence or product copula which has .

12.5        Tail dependence

If  and  are continuous random variables with copula  then their coefficient of (joint lower) tail dependence (if it exists) is:

For continuous random variables  and  each with lower limit of  the coefficient of (lower) tail dependence is also:

12.6        Simulating copulas

Correlated Gaussian (i.e. multivariate normal) random variables (i.e. random variables with a Gaussian copula and Gaussian marginals) can be generated using Cholesky decomposition.

For random variables that have a Gaussian copula but non-normal marginal (with cdfs ) we can generate a vector  of correlated Gaussian random variables as above and then transform as per .

In general, for non-Gaussian copulas we may need to generate a vector of unit uniform random variables  and then transform them using ,  etc.