Parameter Uncertainty and Residual Estimation Risk

The notion of residual estimation risk is introduced in order to study the impact of parameter uncertainty on capital adequacy, for a given risk measure and capital estimation procedure. Residual estimation risk is derived by applying the risk measure on a portfolio consisting of a random loss and a capital estimator, reflecting the randomness inherent in the data. Residual risk thus equals the additional amount of capital that needs to be added to the portfolio to make it acceptable. We propose modified capital estimation procedures, based on parametric bootstrapping and on predictive distributions, which tend to increase capital requirements, by compensating for parameter uncertainty and leading to a residual risk close to zero. In the particular case of location-scale families of distributions, the analysis simplifies substantially and a capital estimator can always be found that leads to a residual risk of exactly zero