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 γξ = γξ(Ψ, α) of the tail index α, the spectral measure Ψ, and the vector ξ 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 α > 1 and negative for α < 1. Strong consistency and asymptotic normality are established for a semiparametric estimator of the mapping ξ 7→ γξ. Strong consistency is also established for the estimated optimal portfolio.