Asymptotics of Portfolio Tail Risk Metrics for Elliptically Distributed Asset Returns

We develop explicit asymptotic expansions of the portfolio Value-at-Risk (VaR) and portfolio Expected Shortfall (ES) for a large family of multivariate elliptical distributions. The family includes distributions of exponential type such as Kotz distributions, and power type such as the multivariate Student t-distribution. Our results imply that the difference between the portfolio ES and its VaR depends on the tail heaviness of the joint asset return distribution. For assets exhibiting exponential tail decay, the ratio between ES and VaR is asymptotically zero, whereas for assets exhibiting power type tail decay, the portfolio ES is strictly larger than its VaR. The amount of the risk reduction through merging subportfolios into a single portfolio depends solely on the dispersion of the joint asset return distribution