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 ( ), 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 also satisfies the properties of robustness, elicitability and consistency under some conditions.

A new risk measure, lambda value at risk ( ), has been recently proposed as a generalization of value at risk (VaR). appears attractive for its potential ability to solve several problems of VaR. This paper provides the first study on the backtesting of . We propose three nonparametric tests which exploit different features. Two tests are based on simple results of probability theory. One test is unilateral and is more suitable for small samples of observations. A second test is bilateral and provides an asymptotic result. A third test is based on simulations and allows for a more accurate comparison among computed with different assumptions on the asset return distribution. Finally, we perform a backtesting exercise that confirms a higher performance of in respect to VaR especially when it is estimated with distributions that better capture tail behavior.

This paper presents the first methodological proposal of estimation of the $\mathsf{\Lambda }VaR$ . Our approach is dynamic and calibrated to market extreme scenarios, incorporating the need of regulators and financial institutions in more sensitive risk measures. We also propose a simple backtesting methodology by extending the $VaR$ hypothesis-testing framework. Hence, we test our $\mathsf{\Lambda }VaR$ proposals under extreme downward scenarios of the financial crisis and different assumptions on the profit and loss distribution. The findings show that our $\mathsf{\Lambda }VaR$ estimations are able to capture the tail risk and react to market fluctuations significantly faster than the $VaR$ and expected shortfall. The backtesting exercise displays a higher level of accuracy for our $\mathsf{\Lambda }VaR$ estimations.

We give an axiomatic foundation to $\Lambda$-quantiles, a family of generalized quantiles introduced in [M. Frittelli, M. Maggis, and I. Peri, Math. Finance, 24 (2014), pp. 442--463] under the name...

Risk contributions of portfolios form an indispensable part of risk-adjusted performance measurement. The risk contribution of a portfolio, e.g. in the Euler or Aumann-Shapley framework, is given by the partial derivatives of a risk measure applied to the portfolio profit and loss in the direction of the asset units. For risk measures that are not positively homogeneous of degree 1, however, known capital allocation principles do not apply. We study the class of lambda quantile risk measures that includes the well-known Value-at-Risk as a special case but for which no known allocation rule is applicable. We prove differentiability and derive explicit formulae of the derivatives of lambda quantiles with respect to their portfolio composition, that is, their risk contribution. For this purpose, we define lambda quantiles on the space of portfolio compositions and consider generic (also non-linear) portfolio operators. We further derive the Euler decomposition of lambda quantiles for generic portfolios and show that lambda quantiles are homogeneous in the space of portfolio compositions, with a homogeneity degree that depends on the portfolio composition and the lambda function. This result is in stark contrast to the positive homogeneity properties of risk measures defined in the space of random variables, which admit a constant homogeneity degree. We introduce a generalised version of Euler contributions and Euler allocation rule, which are compatible with risk measures of any homogeneity degree and non-linear but homogeneous portfolios. These concepts are illustrated by a non-linear portfolio using financial market data.