On the properties of the Lambda value at risk: robustness, elicitability and consistency

Recently, the financial industry and regulators have enhanced the debate on the good properties of a risk measure. A fundamental issue is the evaluation of the quality of a risk estimation. On the one hand, a backtesting procedure is desirable for assessing the accuracy of such an estimation and this can be naturally achieved by elicitable risk measures. For the same objective, an alternative approach has been introduced by Davis [Stat. Risk Model. Appl. Finance Insurance, 2016, 33, 67–93] through the so-called consistency property. On the other hand, a risk estimation should be less sensitive with respect to small changes in the available data-set and exhibit qualitative robustness. A new risk measure, the Lambda value at risk (ΛVaR$ \Lambda VaR $), has been recently proposed by Frittelli et al. [Math. Finance, 2014, 24, 442–463], as a generalization of VaR with the ability to discriminate the risk among P&L distributions with different tail behaviour. In this article, we show that ΛVaR$ \Lambda VaR $ also satisfies the properties of robustness, elicitability and consistency under some conditions.