Parameter Uncertainty and Residual Estimation Risk

The required solvency capital for a financial portfolio is typically given by a tail risk measure such as value-at-risk. Estimating the value of that risk measure from a limited, often small, sample of data gives rise to potential errors in the selection of the statistical model and the estimation of its parameters.We propose to quantify the effectiveness of a capital estimation procedure via the notions of residual estimation risk and estimated capital risk. It is shown that, for capital estimation procedures that do not require the specification of a model (eg, historical simulation) or for worst-case scenario procedures, the impact of model uncertainty is substantial, while capital estimation procedures that allowfor multiple candidate models using Bayesian methods partially eliminate model error. In the same setting, we propose a way of quantifying model error that allows us to disentangle the impact of model uncertainty from that of parameter uncertainty.We illustrate these ideas by simulation examples considering standard loss and return distributions used in banking and insurance.