A quantitative comparison of risk measures

The choice of a risk measure reflects a subjective preference of the decision maker in many managerial, or real world economic problem formulations. To evaluate the impact of personal preferences it is thus of interest to have comparisons with other risk measures at hand. This paper develops a framework for comparing different risk measures. We establish a one-to-one relationship between norms and risk measures, that is, we associate a norm with a risk measure and conversely, we use norms to recover a genuine risk measure. The methods, which are developed for norms first, allows tight comparisons of risk measures, and tight lower and upper bounds for risk measures are made available. In this way we present a general framework for comparing risk measures, with applications in numerous directions.