Asymptotic equivalence of conservative value-at-risk- and expected shortfall-based capital charges

We show that the conservative estimate of the value-at-risk (VaR) for the sum of d random losses with given identical marginals and finite mean is equivalent to the corresponding conservative estimate of the expected shortfall in the limit, as the number of risks becomes arbitrarily large. Examples of interest in quantitative risk management show that the equivalence also holds for relatively small and inhomogeneous risk portfolios. When the individual random losses have infinite first moment, we show that VaR can be arbitrarily large with respect to the corresponding VaR estimate for comonotonic risks if the risk portfolio is large enough.