Abstract

Abstract The debate of which quantitative risk measure to choose in practice has mainly focused on the dichotomy between value at risk (VaR) and expected shortfall (ES). Range value at risk (RVaR) is a natural interpolation between VaR and ES, constituting a tradeoff between the sensitivity of ES and the robustness of VaR, turning it into a practically relevant risk measure on its own. Hence, there is a need to statistically assess, compare and rank the predictive performance of different RVaR models, tasks subsumed under the term “comparative backtesting” in finance. This is best done in terms of strictly consistent loss or scoring functions, i.e., functions which are minimized in expectation by the correct risk measure forecast. Much like ES, RVaR does not admit strictly consistent scoring functions, i.e., it is not elicitable. Mitigating this negative result, we show that a triplet of RVaR with two VaR-components is elicitable. We characterize all 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 regression.

Highlights

  • In the field of quantitative risk management, the last one or two decades have seen a lively debate about which monetary risk measure [3] would be best in practice

  • The debate of which quantitative risk measure to choose in practice has mainly focused on the dichotomy between value at risk (VaR) and expected shortfall (ES)

  • This is best done in terms of strictly consistent loss or scoring functions, i.e., functions which are minimized in expectation by the correct risk measure forecast

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Summary

Introduction

In the field of quantitative risk management, the last one or two decades have seen a lively debate about which monetary risk measure [3] would be best in (regulatory) practice. Fissler and Ziegel [21] showed that the pair (VaRβ , ESβ) is elicitable and identifiable, with the structural difference that the revelation principle is not applicable in this instance. This is followed by the more general finding that the minimal expected score and its minimizer are always jointly elicitable [6, 25]. The main results are presented, establishing the elicitability of the triplet (VaRα , VaRβ , RVaRα,β) (Theorem 3.3) and characterizing the class of strictly consistent scoring functions (Theorem 3.7), exploiting the identifiability result of Proposition 3.1.

Definition of range value at risk
Elicitability and scoring functions
Elicitability and identifiability results
Translation invariance and homogeneity
Mixture representation of scoring functions
Simulations
Implications for regression
Trimmed least squares
Connections to Huber loss and Huber skipped mean
A Appendix

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