A New Robust Risk Measure: Quantile Shortfall
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
A New Robust Risk Measure: Quantile Shortfall You Li CHEN Law School, Wuhan University, Wuhan 430072, P. R. China E-mail : [email protected]
Yan Yan LIU1) School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China E-mail : [email protected]
Guang Cai MAO School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P. R. China E-mail : [email protected]
Yuan Shan WU School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, P. R. China E-mail : [email protected]
Fei YAN School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China E-mail : [email protected] Abstract Among recent measures for risk management, value at risk (VaR) has been criticized because it is not coherent and expected shortfall (ES) has been criticized because it is not robust to outliers. Recently, [Math. Oper. Res., 38, 393–417 (2013)] proposed a risk measure called median shortfall (MS) which is distributional robust and easy to implement. In this paper, we propose a more generalized risk measure called quantile shortfall (QS) which includes MS as a special case. QS measures the conditional quantile loss of the tail risk and inherits the merits of MS. We construct an estimator of the QS and establish the asymptotic normality behavior of the estimator. Our simulation shows that the newly proposed measures compare favorably in robustness with other widely used measures such as ES and VaR. Keywords
Nonparametric estimation, quantile shortfall, risk measure, robust
MR(2010) Subject Classification
62G08, 62G35
Received March 28, 2019, revised October 16, 2019, accepted December 5, 2019 Supported by the National Natural Science Foundation of China (Grant No. 11571263) and Fundamental Research Funds for the Central Universities (Grant No. 2042018kf0243) 1) Corresponding author
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Introduction
The extreme risk exists in various areas of financial investment, credit and insurance. For investors and risk managers, predicting the probability of the extreme risk loss is an important task. A common risk quantification index named as Value at Risk (VaR), measures the maximum potential loss of a given portfolio over a prescribed holding period at a given confidence level e. g. 1% or 5%. Assessing VaR amounts to estimating the tail quantiles of the conditional distribution of financial returns. The parametric, semiparametric and nonparametric models have been widely used to compute the VaR [9]. Although VaR has become one of the standard measure of financial market risk, it has been criticized for not being sub-additive [4]. This means that diversification does not necessarily reduce VaR therefore it is contrast to the framework of modern portfolio theory. In addition, VaR ignores the statistical properties of significant loss beyond the quantile point of interest [1–3]. To overcome the shortcomings of VaR, [2] proposed an al
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