Value at Risk trade-off and capital allocation with copulas

This paper uses copula functions in order to evaluate tail probabilities and market risk trade-offs at a given confidence level, dropping the joint normality assumption on returns. Copulas enable to represent distribution functions separating the marginal distributions from the association structure. We present an application to two stock market indices: for each market we recover the marginal probability distribution. We then calibrate copula functions and recover the joint distribution. The estimated copulas directly give the joint probabilities of extreme losses. Their level curves measure the trade-off between losses over different desks. This trade off can be exploited for capital allocation and is shown to depend on fat-tails. ∗Part of the material in this paper was presented at the Conference ”Managing credit and market risk”, Verona, May 25-26, 2000.