The role of the loss function in value-at-risk comparisons

This paper examines whether the comparison of value-at-risk (VaR) models depends on the loss function used for such a purpose. We showa detailed comparison for several VaR models for two groups of loss functions: designed for regulators/supervisors and for risk managers/firms. Additionally, we propose a firm's loss function (FLF) that exactly measures the opportunity capital cost of the firm when the losses are covered. We find that theVaR model that minimizes the total losses is robust to the regulator's loss function (RLF) but differs from FLF, although not in an arbitrary way. The results indicate that we must distinguish between two subgroups of FLFs: functions that proxy the opportunity cost of the firm when losses are covered, and functions that penalize in the same way when losses are covered or noncovered. The best VaR model is robust to these subgroups. In any case, what is clear is that the best VaR model depends on the family of functions used: RLFs and/or FLFs, as these families provide different results.