Evaluating Range Value at Risk Forecasts.

Abstract

The debate of what quantitative risk measure to choose in practice has mainly focused on the dichotomy between Value at Risk (VaR) -- a quantile -- and Expected Shortfall (ES) -- a tail expectation. Range Value at Risk (RVaR) is a natural interpolation between these two prominent risk measures, which constitutes a tradeoff between the sensitivity of the latter and the robustness of the former, turning it into a practically relevant risk measure on its own. As such, there is a need to statistically validate RVaR forecasts and to compare and rank the performance of different RVaR models, tasks subsumed under the term 'backtesting' in finance. The predictive performance is best evaluated and compared in terms of strictly consistent loss or scoring functions. That is, functions which are minimised in expectation by the correct RVaR forecast. Much like ES, it has been shown recently that RVaR does not admit strictly consistent scoring functions, i.e., it is not elicitable. Mitigating this negative result, this paper shows that a triplet of RVaR with two VaR components at different levels is elicitable. We characterise the class of strictly consistent scoring functions for this triplet. Additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares in robust regression.