The predictive performance of point forecasts for a statistical functional, such as the mean, a quantile, or a certain risk measure, is commonly assessed in terms of scoring (or loss) functions. A scoring function should be (strictly) consistent for the functional of interest, that is, the expected score should be minimised by the correctly specified functional value. A functional is elicitable if it possesses a strictly consistent scoring function. In quantitative risk management, the elicitability of a risk measure is closely related to comparative backtesting procedures. As such, it has gained considerable interest in the debate about which risk measure to choose in practice . While this discussion has mainly focused on the dichotomy between Value at Risk (VaR) - a quantile - and Expected Shortfall (ES) - a tail expectation , this paper is concerned with Range Value at Risk (RVaR) . RVaR can be regarded as an interpolation of VaR and ES , which constitutes a tradeoff between the sensitivity of the latter and the robustness of the former . Recalling that RVaR is not elicitable , we show that a triplet of RVaR with two VaR components at different levels is elicitable. We characterise the class of strictly consistent scoring functions . Moreover, 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.
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.
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