VaR-Efficient Portfolios for a Class of Super- and Sub-Exponentially Decaying Assets Return Distributions

Using a family of modified Weibull distributions, encompassing both sub-exponentials and super-exponentials, to parameterize the marginal distributions of asset returns and their multivariate generalizations with Gaussian copulas, we offer exact formulas for the tails of the distribution P (S) of returns S of a portfolio of arbitrary composition of these assets. We find that the tail of P (S) is also asymptotically a modified Weibull distribution with a characteristic scale χ function of the asset weights with different functional forms depending on the superor sub-exponential behavior of the marginals and on the strength of the dependence between the assets. We then treat in details the problem of risk minimization using the Value-at-Risk and Expected-Shortfall which are shown to be (asymptotically) equivalent in this framework. Introduction In recent years, the Value-at-Risk has become one of the most popular risk assessment tool (Duffie and Pan 1997, Jorion 1997). The infatuation for this particular risk measure probably comes from a variety of factors, the most prominent ones being its conceptual simplicity and relevance in addressing the ubiquitous large risks often inadequately accounted for by the standard volatility, and from its prominent role in the recommendations of the international banking authorities (Basle Commitee on Banking Supervision 1996, 2001). Moreover, down-side risk measures such as the Value-at-risk seem more in accordance with observed behavior of economic agents. For instance, according to prospect theory (Kahneman and Tversky 1979), the perception of downward market movements is not the same as upward movements. This may be reflected in the so-called leverage effect, first discussed by (Black 1976), who observed that the volatility of a stock tends to increase when its price drops (see (Fouque et al. 2000, Campbell, Lo and McKinley 1997, Bekaert and Wu 2000, Bouchaud et al. 2001) for reviews and recent works). Thus, it should be more natural to consider down-side risk measures like the VaR than the variance traditionally used in portfolio management (Markowitz 1959) which does not differentiate between positive and negative change in future wealth.