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Risk quantification for life-cycle management of infrastructure considering the effect of maintenance behavior
Published in Jaap Bakker, Dan M. Frangopol, Klaas van Breugel, Life-Cycle of Engineering Systems, 2017
Another coherent risk measure named Entropic Value at Risk (EVaR) is proposed by Ahmadi-Javid (2011). EVaR is derived from Chernoff bound, which gives tight estimation of upper bound of random variable with an exponentially decreasing probability density function. EVaR can provide more strict evaluation than conventional risk measures, such as VaR and CVaR, and its value gives an upper bound of both VaR and CVaR with the same confidence levels. It means that EVaR is more risk averse than VaR and CVaR. EVaR is defined as EVaR1−α(X)=supQ≪P,DKL(Q||P)≤−lnαEQ[X]
Preference robust models in multivariate utility-based shortfall risk minimization
Published in Optimization Methods and Software, 2022
Yuan Zhang, Huifu Xu, Wei Wang
After the introduction of the coherent risk measure, many variations and extensions have been proposed and studied in the literature, see e.g. [3,18,36,38]. Conditional value-at-risk (CVaR), as one of the coherent risk measures, studied by [2,8,43] has gained popularity in recent years. CVaR has emerged as a better alternative to VaR because it measures the sizes of the potential losses beyond the threshold amount indicated by the VaR. Although CVaR is used in a wide range of applications such as supply chain [13], network design [10], healthcare [14], etc., it still suffers from severe deficiencies. One main drawback of CVaR is that it only captures a limited spectrum of risk attitudes and may not accurately represent the risk aversion of some decision makers. There are also many interesting coherent risk measures in finance such as spectral risk measures [1,4] defined as weighted average of VaR, entropic value-at-risk (EVaR) [5] which connects the VaR and the relative entropy.
Risk-averse real driving emissions optimization considering stochastic influences
Published in Engineering Optimization, 2020
A. Wasserburger, C. Hametner, N. Didcock
Let X be a continuous random variable on the probability space . Let . The Value-at-Risk (VaR) at the confidence level α is defined as The Conditional Value-at-Risk (CVaR) at the confidence level α is defined as The Entropic Value-at-Risk (EVaR) at the confidence level α is defined as where is the moment-generating function.
Avoiding momentum crashes using stochastic mean-CVaR optimization with time-varying risk aversion
Published in The Engineering Economist, 2023
To manage the risk of the investment strategies, the risk measure must be selected. Barroso and Santa-Clara (2015) and Daniel and Moskowitz (2016) both quantify the risk of momentum portfolios in terms of volatility (variance). Inspired by the risk parity approaches (Asness et al., 2012), Barroso and Santa-Clara (2015) scale the winner-minus-loser (WML) portfolio return by its realized volatility of past six-month returns to control the investment risk of the cross-sectional momentum strategy. Daniel and Moskowitz (2016) maximize the Sharpe ratio of the WML portfolio, which turns to be a variant of the traditional Markowitz mean-variance approach (Markowitz, 1952). Volatility and variance quantify risk in terms of both upside gains and downside losses and, thus, cannot distinguish the direction of investment movement. However, investors worry about the loss only. In addition, researchers have found that higher moments, such as skewness and kurtosis, are non-negligible in asset allocation when the asset returns are not normally distributed. Therefore, a downside risk measure, capable of being expressed as a function of skewness and kurtosis for non-normal investment returns and assessing the extent of losses only (Xiong & Idzorek, 2011), is more suitable to reduce the risk of momentum investing. By far, value at risk (VaR) and conditional value-at-risk (CVaR) are the two most commonly-used downside risk measures in finance (Banihashemi & Navidi, 2017; Benati & Conde, 2022; Guo et al., 2019; Kim & Park, 2021; Kim et al., 2012; Sharifi & Kwon, 2018). VaR estimates the maximum loss that could occur over a given period with a specified confidence level. CVaR is defined as the conditional expectation of portfolio losses beyond a prespecified percentile of the distribution of the loss (Rockafellar & Uryasev, 2000). Although both VaR and CVaR are superior to variance for asymmetric asset return distributions, there are two reasons for the preference in portfolio and risk management of CVaR relative to VaR. First, CVaR takes into account not only the probability but also the size of the loss (Rockafellar & Uryasev, 2000). Second, Pflug (2000) proves that CVaR satisfies the properties for a coherent risk measure while VaR fails due to the violation of the sub-additivity property, which posits that diversification reduces risk. More recently, Ahmadi-Javid (2012) proposes a new risk measure, entropic value-at-risk (EVaR), which uses the Chernoff inequality to create an upper bound for the CVaR and is proven to be coherent. Compared to CVaR, EVaR has stronger monotonicity and is more computationally efficient when solving large-scale optimization problems due to the number of variables and constraints of the EVaR formulation being independent of the sample size (Ahmadi-Javid & Fallah-Tafti, 2019). However, EVaR is more risk-averse at the same confidence level than CVaR; that is, the incorporation of EVaR will result in allocating more resources to avoid risk, which may not be favored by investors who want to allocate the least amount of resources. This property makes EVaR less attractive than CVaR.