The sensitivity of a risk measure with respect to the parameters of the hypothesized distribution is a useful tool in investigating the impact of marginal rebalancing decisions on the portfolio return distribution and also in the analysis of the asymptotic variability of the risk estimator. We calculate the relative importance of the conditional value-at-risk (CVaR) sensitivity with respect to tail thickness and scale of the portfolio return distribution in the case of regularly varying tails and in the case of exponential and faster-than-exponential decay. We discuss the implications for asset return modeling and the asymptotic variability of the risk estimator.

In the paper, we generalize the classical benchmark tracking problem by introducing the class of relative deviation metrics. We introduce an axiomatic description of the benchmark tracking problem and a classification inspired by the theory of probability metrics. Two examples of such metrics are provided and their in-sample behaviour is compared to the classical tracking error in a numerical example.