On optimal portfolio diversification with respect to extreme risks

Extreme losses of portfolios with heavy-tailed components are studied in the framework of multivariate regular variation. Asymptotic distributions of extreme portfolio losses are characterized by a functional gamma(xi) = gamma(xi) (alpha, Psi) of the tail index alpha, the spectral measure psi, and the vector xi of portfolio weights. Existence, uniqueness, and location of the optimal portfolio are analysed and applied to the minimization of risk measures. It is shown that diversification effects are positive for alpha > 1 and negative for alpha < 1 Strong consistency and asymptotic normality are established for a semiparametric estimator of the mapping xi -> gamma(xi). Strong consistency is also established for the estimated optimal portfolio.