We propose a generalization of the classical notion of the V @R λ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The V @R λ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on P(R).

This paper presents the first methodological proposal of estimation of the ΛVaR. 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 ΛVaR proposals under extreme downward scenarios of the financial crisis and different assumptions on the profit and loss distribution. The findings show that our ΛVaR 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 ΛVaR estimations.

Recently, 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 (2016) 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 (ΛV aR), has been recently proposed by Frittelli et al. (2014), as a generalization of V aR with the ability to discriminate the risk among P&L distributions with different tail behaviour. In this article, we show that ΛV aR also satisfies the properties of robustness, elicitability and consistency under some conditions.

A new risk measure, Lambda value at risk (ΛV aR), has been recently proposed as a generalization of Value at risk (V aR). ΛV aR appears attractive for its potential ability to solve several problems of V aR. This paper provides the first study on the backtesting of ΛV aR. 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 ΛV aRs computed with different assumptions on the asset return distribution. Finally, we perform a backtesting exercise that confirms a higher performance of ΛV aR in respect to V aR especially when it is estimated with distributions that better capture tail behaviour.

We give an axiomatic foundation to Λ-quantiles, a family of generalized quantiles introduced in [7] under the name of Lambda Value at Risk. Under mild assumptions, we show that these functionals are characterized by a property that we call "locality", that means that any change in the distribution of the probability mass that arises entirely above or below the value of the Λ-quantile does not modify its value. We compare with a related axiomatization of the usual quantiles given by C. Chambers in [4], based on the stronger property of "ordinal covariance", that means that quantiles are covariant with respect to increasing transformations. Further, we present a systematic treatment of the properties of Λ-quantiles, refining some of the results of [7] and [3] and showing that in the case of a nonincreasing Λ the properties of Λ-quantiles closely resemble those of the usual quantiles.

We give an axiomatic foundation to Λ-quantiles, a family of generalized quantiles introduced by Frittelli et al. ( 2014) under the name of Lambda Value at Risk. Under mild assumptions, we show that these functionals are characterized by a property that we call "locality", that means that any change in the distribution of the probability mass that arises entirely above or below the value of the Λ-quantile does not modify its value. We compare with a related axiomatization of the usual quantiles given by Chambers ( 2009), based on the stronger property of "ordinal covariance", that means that quantiles are covariant with respect to increasing transformations. Further, we present a systematic treatment of the properties of Λ-quantiles, refining some of the results of Frittelli et al. (2014) and Burzoni et al. (2017) and showing that in the case of a nonincreasing Λ the properties of Λ-quantiles closely resemble those of the usual quantiles.

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 return in direction of the asset weights. 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 wellknown 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 on 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 portfolios. We further provide financial interpretations of the homogeneity degree of lambda quantiles and introduce the notion of event-specific homogeneity of portfolio operators.

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