Generalized quantiles as risk measures
Generalized quantiles as risk measures
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DOI：
10.1016/j.insmatheco.2013.10.015
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发表时间：
2014-01-01
影响因子：
1.9
通讯作者：
Gianin, Emanuela Rosazza
中科院分区：
文献类型：
--
作者：
Bellini, Fabio;Klar, Bernhard;Gianin, Emanuela Rosazza
在统计和精算文献中，通过适当的非对称损失函数的最小化，已经考虑了分位数的几种推广。所有这些广义分位数都具有可引出性的重要性质，由于它对应于自然回测方法的存在，因此最近受到了广泛的关注。在本文中，我们研究了M分位数作为非对称凸损失函数的最小值的情况，与Bellini和Rosazza Gianin（2012）中考虑的Orlicz分位数相反。我们讨论了它们作为风险度量的性质，并指出了它们与零效用保费原则以及Follmer和Schied（2002）引入的短缺风险度量的联系。特别地，我们证明了唯一的M-分位数是一致的风险度量，是Newey和Powell（1987）作为非对称二次损失函数的最小化者引入的期望值。我们提供了他们的二元和Kusuoka表示，并讨论了他们与CVaR的关系。我们分析了α-> 1的渐近性质，并表明对于非常重尾分布，期望比通常的分位数更保守。最后，我们展示了他们的鲁棒性的意义上的lipschitzianity相对于Wasserstein度量。(C)2013爱思唯尔有限公司版权所有。
In the statistical and actuarial literature several generalizations of quantiles have been considered, by means of the minimization of a suitable asymmetric loss function. All these generalized quantiles share the important property of elicitability, which has received a lot of attention recently since it corresponds to the existence of a natural backtesting methodology. In this paper we investigate the case of M-quantiles as the minimizers of an asymmetric convex loss function, in contrast to Orlicz quantiles that have been considered in Bellini and Rosazza Gianin (2012). We discuss their properties as risk measures and point out the connection with the zero utility premium principle and with shortfall risk measures introduced by Follmer and Schied (2002). In particular, we show that the only M-quantiles that are coherent risk measures are the expectiles, introduced by Newey and Powell (1987) as the minimizers of an asymmetric quadratic loss function. We provide their dual and Kusuoka representations and discuss their relationship with CVaR. We analyze their asymptotic properties for alpha -> 1 and show that for very heavy tailed distributions expectiles are more conservative than the usual quantiles. Finally, we show their robustness in the sense of lipschitzianity with respect to the Wasserstein metric. (C) 2013 Elsevier B.V. All rights reserved.