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Register a new userOn the properties of the Lambda value at risk: robustness, elicitability and consistency
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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 (Lambda VaR), has been recently proposed by Frittelli et al. (2014), 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 Lambda VaR also satisfies the properties of robustness, elicitability and consistency under some conditions.
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A new risk measure, the lambda value at risk (Lambda VaR), has been recently proposed from a theoretical point of view as a generalization of the value at risk (VaR). The Lambda VaR appears attractive for its potential ability to solve several problems of the VaR. In this paper we propose three nonparametric backtesting methodologies for the Lambda VaR which exploit different features. Two of these tests directly assess the correctness of the level of coverage predicted by the model. One of these tests is bilateral and provides an asymptotic result. A third test assess the accuracy of the Lambda VaR that depends on the choice of the P&L distribution. However, this test requires the storage of more information. Finally, we perform a backtesting exercise and we compare our results with the ones from Hitaj and Peri (2015)
We derive bounds on the distribution function, therefore also on the Value-at-Risk, of $varphi(mathbf X)$ where $varphi$ is an aggregation function and $mathbf X = (X_1,dots,X_d)$ is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three types of available information on the dependence structure: First, we consider the case where extreme value information, such as the distributions of partial minima and maxima of $mathbf X$, is available. In order to include this information in the computation of Value-at-Risk bounds, we utilize a reduction principle that relates this problem to an optimization problem over a standard Frechet class, which can then be solved by means of the rearrangement algorithm or using analytical results. Second, we assume that the copula of $mathbf X$ is known on a subset of its domain, and finally we consider the case where the copula of $mathbf X$ lies in the vicinity of a reference copula as measured by a statistical distance. In order to derive Value-at-Risk bounds in the latter situations, we first improve the Frechet--Hoeffding bounds on copulas so as to include this additional information on the dependence structure. Then, we translate the improved Frechet--Hoeffding bounds to bounds on the Value-at-Risk using the so-called improved standard bounds. In numerical examples we illustrate that the additional information typically leads to a significant improvement of the bounds compared to the marginals-only case.
In economics, insurance and finance, value at risk (VaR) is a widely used measure of the risk of loss on a specific portfolio of financial assets. For a given portfolio, time horizon, and probability $alpha$, the $100alpha%$ VaR is defined as a threshold loss value, such that the probability that the loss on the portfolio over the given time horizon exceeds this value is $alpha$. That is to say, it is a quantile of the distribution of the losses, which has both good analytic properties and easy interpretation as a risk measure. However, its extension to the multivariate framework is not unique because a unique definition of multivariate quantile does not exist. In the current literature, the multivariate quantiles are related to a specific partial order considered in $mathbb{R}^{n}$, or to a property of the univariate quantile that is desirable to be extended to $mathbb{R}^{n}$. In this work, we introduce a multivariate value at risk as a vector-valued directional risk measure, based on a directional multivariate quantile, which has recently been introduced in the literature. The directional approach allows the manager to consider external information or risk preferences in her/his analysis. We have derived some properties of the risk measure and we have compared the univariate textit{VaR} over the marginals with the components of the directional multivariate VaR. We have also analyzed the relationship between some families of copulas, for which it is possible to obtain closed forms of the multivariate VaR that we propose. Finally, comparisons with other alternative multivariate VaR given in the literature, are provided in terms of robustness.
We propose a generalization of the classical notion of the $V@R_{lambda}$ 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 $V@R_{lambda}$ 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 $mathcal{P}(% mathbb{R}).$
Several well-established benchmark predictors exist for Value-at-Risk (VaR), a major instrument for financial risk management. Hybrid methods combining AR-GARCH filtering with skewed-$t$ residuals and the extreme value theory-based approach are particularly recommended. This study introduces yet another VaR predictor, G-VaR, which follows a novel methodology. Inspired by the recent mathematical theory of sublinear expectation, G-VaR is built upon the concept of model uncertainty, which in the present case signifies that the inherent volatility of financial returns cannot be characterized by a single distribution but rather by infinitely many statistical distributions. By considering the worst scenario among these potential distributions, the G-VaR predictor is precisely identified. Extensive experiments on both the NASDAQ Composite Index and S&P500 Index demonstrate the excellent performance of the G-VaR predictor, which is superior to most existing benchmark VaR predictors.
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