Value-at-Risk and Expected Shortfall Relationship

Value-at-Risk or VaR has been a widely-used risk measure in most financial institution. It plays important roles as an alarm for future potential risk and thus a preparation for capital allocation. An alternative measure considered is Expected Shortfall (ES). This may have more advantages than VaR due to its ability to cover magnitude of risk. In this paper, we investigate the VaR and ES relationship for a random loss sample. Analytically, both VaR and ES predict a loss by calculating a quantile of distribution. Therefore, VaR and ES forecasts may be compared in order to obtain an optimum risk measure. The accuracy of VaR-ES relationship perfomance may be assessed through its coverage probability. We found that, as an optimum risk measure, VaR outperforms than ES due its closeness of coverage probability to a given level of confidence. In addition, a numerical analysis is carried out to demonstrate how the optimization techniques can be implemented.