Value at Risk bounds for portfolios of non-normal returns

This paper studies Value at Risk (VaR) bounds for sums of stochastically dependent random variables, i.e. portfolios of correlated financial assets. The bounds hold under no restrictions on the dependence or on the marginal distributions of returns. An improvement of the bounds is given for positive (quadrant) dependent rvs. Both sets of bounds are computed for portfolios of 6 international indices. Backtesting confirms the usefulness of the approach, even with respect to other shortcuts, such as the normality assumption. For small portfolios, bounds are not over conservative. This paper studies Value at Risk (VaR) bounds for sums of stochastically dependent random variables, i.e. portfolios of correlated financial assets. The bounds are quickly computed, suitable also for fat-tailed distributions, and hold under no or very general assumptions on the dependence and marginal distributions of the portfolio components. From the probabilistic point of view, this amounts to bounding the quantile of a sum without the independence assumption. The problem of determining the quantile of a sum exactly is not trivial, and, as far as we know, is solved in closed form only in very specific cases, such as the perfect linear dependence or the normal one. Instead of an exact result, we exploit recent contributions on distribution functions of sums (by Denuit, Genest and Marceau (1999)) in order to provide bounds for the VaR quantile. For the bivariate case, analogous theoretical results have been obtained by Embrecht, McNeil and Straumann (1999), Durrleman, Nikeghbali and Roncalli (2000)1 . From the financial point of view, the question boils down to determining the riskiness of portfolios, whenever their assets do not have independent returns. In the financial practice, the problem is usually solved either by assuming that returns are normally distributed, or by resorting to historical or Monte Carlo 1The latter also provide a bivariate application to the London Metal Exchange Market.