On the elicitability of range value at risk

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.

Sensitivity Measures Based on Scoring Functions

“…RVaR fails to be elicitable on its own in general (Wang and Wei 2020). However, Fissler and Ziegel (2021) show that the triplet (VaR α , VaR β , RVaR α,β ) is elicitable subject to mild conditions. The (strictly) M-consistent scoring functions satisfying Assumption 1 (i) such that S(y, y, y, y) = 0 are of the form…”

Measurability of functionals and of ideal point forecasts

“…The measurability of q − γ from Example 3.4 and a Riemann approximation argument yield that T w is A(P w ) − B( R)-measurable. This directly yields the measurability of the lower and upper Expected Shortfall (by choosing w = 1 [0,α] /α or w = 1 [α,1] /(1 − α)) and of the Range Value at Risk (Cont et al, 2010;Fissler and Ziegel, 2021)…”

Elicitability and Identifiability of Systemic Risk Measures

“…The formalisation of how many auxiliary components are needed to render a functional T a part of a higher dimensional elicitable T has led to the notion of forecast complexity; see again Frongillo and Kash (2018) and the discussion in . For example, the risk measure Range Value at Risk has an elicitation complexity of 3 (Fissler & Ziegel, 2019a), while the mode functional generally possesses a corresponding complexity of ∞ (Dearborn & Frongillo, 2019;Heinrich, 2014). This means one needs to report the entire probability distribution, which amounts to considering the infinite dimensional identity functional (Brier, 1950;Good, 1952;Matheson & Winkler, 1976).…”

“…For the sake of simplicity and following and Fissler and Ziegel (2019a) we choose to consider prediction-observation-sequences that are independent and identically distributed over time. Despite this simplification, there is still a variety of parameters to consider in the simulation study:…”