Taking Uncertainty Into Account: Bias Issues Arising from Parameter Uncertainty in Risk Models

Given a random variable of interest, a historical sample of its realized values, and the desire to model its possible future values, actuarial training provides many methods for selecting a family of probability models (distributions) and determining specific parameter values that best represent it. But how should one take parameter uncertainty (parameter risk) into account? In particular, uncertainty can lead to bias in estimators commonly used by actuaries. This paper examines the problem of adjusting estimated distributions (risk curves) to remove the undesirable bias effects of parameter risk, and shows several solutions. It goes on, however, to critique the very notion of uncertainty-adjusted risk curves, emphasizing that this is an ambiguous concept. The form of the adjustment depends crucially on details of the specific questions being addressed, so much so that an estimator can seem to be simultaneously overestimating and underestimating risk. Parameter uncertainty therefore cannot be "taken into account" in an unequivocal manner. It is recommended that parameter risk be held apart from process risk and presented in terms of confidence intervals; only with that as background - and with great care - should bias corrections be attempted.