Robustness and sensitivity analysis of risk measurement procedures

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a “risk measure” which summarizes the risk of the portfolio. We define the notion of “risk measurement procedure”, which includes both of these steps and study the robustness of risk measurement procedures and their sensitivity to a change in the data set. After introducing a rigorous definition of ’robustness’ of a risk measurement procedure, we illustrate the presence of a conflict between subadditivity and robustness of risk measurement procedures. We propose a measure of sensitivity for risk measurement procedures and compute the sensitivity function of various examples of risk estimators used in financial risk management, showing that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate in particular that using historical Value at Risk leads to a more robust procedure for risk measurement than recently proposed alternatives like CVaR. We also propose other risk measurement procedures which possess the robustness property. ∗We thank Hans Föllmer, Peter Bank, Paul Embrechts, Gerhard Stahl and seminar participants at QMF 2006 (Sydney), Humboldt University (Berlin), Torino, Lecce, INFORMS Applied Probability Days (Eindhoven), Cornell ORIE seminar and the Harvard Statistics Seminar for helpful comments. This project has benefited from partial funding by the European Research Network “Advanced Mathematical Methods for Finance” (AMAMEF).