Large-sample confidence intervals for risk measures of location–scale families

For a loss distribution belonging to a location–scale family, Fμ,σFμ,σ, the risk measures, Value-at-Risk and Expected Shortfall are linear functions of the parameters: μ+τσμ+τσ where ττ is the corresponding risk measure of the mean-zero and unit-variance member of the family. For each risk measure, we consider a natural estimator by replacing the unknown parameters μμ and σσ by the sample mean and (bias corrected) sample standard deviation, respectively. The large-sample parametric confidence intervals for the risk measures are derived, relying on the asymptotic joint distribution of the sample mean and sample standard deviation. Simulation studies with the Normal, Laplace and Gumbel families illustrate that the derived asymptotic confidence intervals for Value-at-Risk and Expected Shortfall outperform those of Bahadur (1966) and Brazauskas et al. (2008), respectively. The method can also be effectively applied to Log-location-scale families whose supports are positive reals; an illustrative example is given in the area of financial credit risk.