/

### 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.