On elicitable risk measures

Abstract

In the\ud present contribution, under weak\ud technical assumptions on the elicitability scoring function S we fully characterize the class of convex and\ud coherent risk measures that are elicitable with an accuracy rewarding\ud scoring function. In particular, we answer the question\ud posed by Ziegel (2013), showing that expectiles are indeed the\ud only elicitable coherent risk measure

Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations and Forecast Rankings

“…The foregoing discussion may also bear on the debate about the revision of the Basel protocol for banking regulation, which involves contention about the choice of the functional of in‐house risk distributions that banks are supposed to report to regulators (Embrechts et al ., ). Recently, expectiles have been proposed as potential candidates, as it has been proved that expectiles at level are the only elicitable law invariant coherent risk measures (Ziegel, ; Bellini and Bignozzi, ; Delbaen et al ., ). See McNeil et al .…”

“…Recent studies of elicitability have revealed that (strict) order sensitivity and (strict) consistency are equivalent in quite general settings (Nau, ; Lambert, ; Steinwart et al ., ; Bellini and Bignozzi, ). These results rely on continuity conditions on the scoring function and do not readily apply in our framework.…”

Reducing model risk via positive and negative dependence assumptions

“…We compute the bounds in (4.3) using expectiles as a risk measure, ρ = e p , for a portfolio of Gamma distributed random variables. Expectiles are attracting increasing interest in the literature on risk measures since they are the only coherent risk measure with the additional property of being elicitable; for more details on this see for instance Ziegel (2014) and Bellini and Bignozzi (2014). For a random variable X with E[|X|] < ∞, the expectile e p , computed at confidence level p ∈ (0, 1), is defined as the unique solution to…”

Downside Risk Evaluation with the R Package GAS

“…The quantile loss function in (2) for VaR assessment is appropriate since quantiles are elicited by it, that is, when the conditional distribution is static over the sample, the VaR can be estimated by minimizing the average quantile loss function. Unfortunately, there is no loss function available for which the ES risk measure is elicitable; see, for instance, Bellini and Bignozzi (2015) and Ziegel (2016). However, it has been recently shown by Fissler and Ziegel (2016) (FZ) that the couple (VaR, ES) is jointly elicitable,…”