Conditional Value At Risk Measure Research Articles

Convex and Nonconvex Risk-Based Linear Regression at Scale

The value at risk (VaR) and the conditional value at risk (CVaR) are two popular risk measures to hedge against the uncertainty of data. In this paper, we provide a computational toolbox for solving high-dimensional sparse linear regression problems under either VaR or CVaR measures, the former being nonconvex and the latter convex. Unlike the empirical risk (neutral) minimization models in which the overall losses are decomposable across data, the aforementioned risk-sensitive models have nonseparable objective functions so that typical first order algorithms are not easy to scale. We address this scaling issue by adopting a semismooth Newton-based proximal augmented Lagrangian method of the convex CVaR linear regression problem. The matrix structures of the Newton systems are carefully explored to reduce the computational cost per iteration. The method is further embedded in a majorization–minimization algorithm as a subroutine to tackle the nonconvex VaR-based regression problem. We also discuss an adaptive sieving strategy to iteratively guess and adjust the effective problem dimension, which is particularly useful when a solution path associated with a sequence of tuning parameters is needed. Extensive numerical experiments on both synthetic and real data demonstrate the effectiveness of our proposed methods. In particular, they are about 53 times faster than the commercial package Gurobi for the CVaR-based sparse linear regression with 4,265,669 features and 16,087 observations. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported in part by the NSF, the Division of Computing and Communication Foundations [Grant 2153352], the National Natural Science Foundation of China [Grant 12271187], and the Hong Kong Research Grant Council [Grant 15304019]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.1282 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0012 ) at ( http://dx.doi.org/10.5281/zenodo.7483279 ).

Supply option purchasing decisions via mismatch cost minimization

In this paper, the optimal purchasing decisions of the option procurement of newsvendor-type commodities are studied. Based on the mismatch cost minimization criterion, the present study analyzes the option purchasing decisions of a risk-averse newsvendor under the measure of Conditional Value-at-Risk (CVaR). For the purpose of controlling the procurement risk, it is found that a newsvendor’s optimal option purchasing decision under the CVaR measure increases at the quantile level under certain conditions. The result provides alternative explanations for a newsvendor’s over-purchasing behaviors in reality. To mitigate the conservativeness of the CVaR measure, a combined objective defined as the mean-CVaR on mismatch cost is introduced, and the optimal option purchasing decision of a newsvendor minimizing the combined objective is derived. The combined objective can strike a balance between the minimization of CVaR of mismatch cost as well as the minimization of expected mismatch cost. In addition, a sensitivity analysis is performed to exhibit the results and managerial insights are presented to hedge against the potential risks for newsvendors in the option purchasing decision-making.

Portfolio constructions in cryptocurrency market: A CVaR-based deep reinforcement learning approach

Cryptocurrency markets have much larger tail risk than traditional financial markets, and constructing portfolios with such large tail risk assets would be challenging. Therefore, cryptocurrency funds demand new superior risk management models and Conditional Value at Risk (CVaR) is a prevailing risk measure for constructing portfolios in stock markets with large tail risk. Consequently, our paper contributes to the literature by developing a new cryptocurrency portfolio model framework based on the CVaR risk measure and a deep reinforcement learning optimization framework. We use the data from cryptocurrency market starting 2015 to 2021, unfolding that CVaR measure with deep learning outperforms the traditional portfolio construction technique. Compared with traditional economic parameter-based portfolio models, our model free based approach can capture the nonlinear compounding effect of multiple risk shocks by deep reinforcement learning on the risk distribution with economic structural breakdown. It can guide investments in financial markets with high tail risks.

New Robust Reward-Risk Ratio Models with CVaR and Standard Deviation

In this paper, we present two robust reward-risk ratio optimization models. Two new models contain the worst case of not only conditional value-at-risk (CVaR), but also standard deviation (SD). Using properties of reward measure, CVaR measure, and standard deviation measure, new models can be proved to equivalent to min-max problems. When the uncertainty set is an ellipsoid, new models can be further rewritten as second-order cone problems step by step. Finally, we implement new models to portfolio problems. It shows that new models are robust and comparable with mean-CVaR ratio model. Since considering standard deviation, allocation decision obtained by new models can give reasonable rewards according to personal preferences.

Market-Risk Optimization among the Developed and Emerging Markets with CVaR Measure and Copula Simulation

In this paper, the generalized Pareto distribution (GPD) copula approach is utilized to solve the conditional value-at-risk (CVaR) portfolio problem. Particularly, this approach used (i) copula to model the complete linear and non-linear correlation dependence structure, (ii) Pareto tails to capture the estimates of the parametric Pareto lower tail, the non-parametric kernel-smoothed interior and the parametric Pareto upper tail and (iii) Value-at-Risk (VaR) to quantify risk measure. The simulated sample covers the G7, BRICS (association of Brazil, Russia, India, China and South Africa) and 14 popular emerging stock-market returns for the period between 1997 and 2018. Our results suggest that the efficient frontier with the minimizing CVaR measure and simulated copula returns combined outperforms the risk/return of domestic portfolios, such as the US stock market. This result improves international diversification at the global level. We also show that the Gaussian and t-copula simulated returns give very similar but not identical results. Furthermore, the copula simulation provides more accurate market-risk estimates than historical simulation. Finally, the results support the notion that G7 countries can provide an important opportunity for diversification. These results are important to investors and policymakers.

A newsvendor model with autocorrelated demand under a time-consistent dynamic CVaR measure

As a result of autocorrelation, static risk measures such as value at risk and Conditional Value at Risk (CVaR) are time inconsistent and can thus result in inconsistent decisions over time. In this article, we present a time-consistent dynamic CVaR measure and examine it in the context of a newsvendor problem with autocorrelated demand. Due to the concavity of our CVaR measure, the dynamic program formulation associated with our dynamic newsvendor problem is not immediately separable. However, by exploring certain properties of the dynamic CVaR measure and underlying profit function, our dynamic program can be transformed into a sequence of (single-period) risk-averse newsvendor problems that depend on the observed demand history. By examining the structure of the optimal order quantities, we find both intuitive and counterintuitive results. When demands are positively correlated, the optimal order quantity is monotonically increasing in the degree of risk aversion. However, when demands are negatively correlated and the underlying cost structure satisfies certain conditions, the optimal order quantity is no longer monotonically increasing in the degree of risk aversion. Instead, the optimal order quantity is a decreasing (increasing) function of the degree of risk aversion when it is below (above) a certain threshold. We also show that these results continue to hold when demands follow a general ARMA process, and when inventory carryover is considered.

Risk aversion based interval stochastic programming approach for agricultural water management under uncertainty

In this study, a risk aversion based interval stochastic programming (RAIS) method is proposed through integrating interval multistage stochastic programming and conditional value at risk (CVaR) measure for tackling uncertainties expressed as probability distributions and intervals within a multistage context. The RAIS method can reflect dynamic features of the system conditions through transactions at discrete points in time over the planning horizon. Using the CVaR measure, RAIS can effectively reflect system risk resulted from random parameters. When random events are occurred, the adjustable alternatives can be achieved by setting desired targets according to the CVaR, which could make the revised decisions to minimize the economic penalties. Then, the RAIS method is applied to planning agricultural water management in the Zhangweinan River Basin that is plagued by drought due to serious water scarcity. A set of decision alternatives with different combinations of risk levels employed to the objective function and constraints are generated for planning water resources allocation. The results can not only help decision makers examine potential interactions between risks under uncertainty, but also help generate desired policies for agricultural water management with a maximized payoff and a minimized loss.

Coping with risk management and fill rate in the loss-averse newsvendor model

In this paper, we introduce the CVaR (Conditional Value-at-Risk) measure to hedge against the risks for the loss-averse newsvendor. We derive the optimal order quantities for the loss-averse newsvendor to maximize the CVaR objective of utility with and without the shortage cost, respectively. It is shown that when shortage cost is not considered, an increase in the degree of loss aversion leads to a decrease in the fill rate. Therefore the newsvendor should strike a balance between the two conflicting targets in selecting the optimal order quantity. When the shortage cost is taken into account and sufficiently large, it is proved that an increase in the degree of loss aversion leads to an increase in the fill rate. To manage both of the two targets, the newsvendor should order more items in this case. Our sensitivity analysis further verifies that high risk implies high return and low risk comes with low return. Based on the analysis, some management insights for the risk management of the loss-averse newsvendor are presented.

Are Islamic stock indexes exposed to systemic risk? Multivariate GARCH estimation of CoVaR

The main objective of this paper is to assess the exposure of Islamic stock indexes to systemic tail events. We use Conditional Value-at-Risk (CoVaR) and Delta CoVaR measures as developed by Adrian and Brunnermeier (2011) and a sample of Islamic and conventional stock indexes, from various developed and emerging markets, during the period September 2005 to March 2015. The empirical results reveal that the systemic risk has a moderate adverse effect on Islamic indexes, with a lower level in Gulf Cooperation Council countries (GCC hereafter). The findings also show the Asian stock indexes can be considered as effective hedge assets, after the global financial crisis (GFC hereafter). Furthermore, the empirical reveal that portfolio including Islamic stock indexes performs better than a benchmark portfolio in turmoil periods. These findings have several implications in financial decisions including the strategy of stability and asset allocation.

A Risk-Averse Newsvendor Model Under Trade Credit Contract with CVaR

A supply chain with a supplier and a risk-averse retailer is considered in the paper under trade credit contract. The retailer as newsvendor faces a non-negative random demand and the supplier provides the trade credit for the risk-averse retailer with budget constraints. Different from the existing research, in a conditional value-at-risk (CVaR) framework, the optimal ordering quantity and wholesale price are obtained. Analytical results are obtained for the newsvendor retailer’s optimal ordering quantity and supplier’s optimal wholesale price under CVaR measure. Sensitivity analysis is also yielded. It is found that the optimal ordering quantity decreases as the degree of risk aversion increases. Furthermore, we analyze the effect of the initial budget of retailer and the wholesale price on the order quantity decision. This paper also finds that the trade credit contract could create value for a risk-averse supply chain with budget constraints. Finally, to compare with the existing results the theoretical analysis and numerical examples are illustrated.

Analysing Systemic Risk in the Chinese Banking System

AbstractWe examine systemic risk in the Chinese banking system by estimating the conditional value at risk (CoVaR), the marginal expected shortfall (MES), the systemic impact index (SII) and the vulnerability index (VI) for 16 listed banks in China for the 2007–2014 period. We find that these measures show different patterns, capturing different aspects of systemic risk of Chinese banks. However, rankings of banks based on these measures are significantly correlated. The time‐series results for the CoVaR and MES measures suggest that systemic risk in the Chinese banking system decreased after the global financial crisis but started rising in 2014.

Precio internacional de los metales y riesgo de mercado en la Bolsa de Valores de Lima

In this paper we use the conditional Value at Risk (CoVaR) and CoVaR variation (ΔCoVaR) proposed by Adrian and Brunnermeier (2008, 2011, 2016) to estimate the Peruvian stock market risk (through the IGBVL) conditioned on the international financial market (given that the S&P500) and conditioned on three of the main commodities exported by Peru: copper, silver and gold. Moreover, the CoVaR measures are compared with the VaR of the IGBVL to understand the differences using conditional and unconditional risk measure estimators. The results show that both CoVaR and ΔCoVaR are useful indicators to measure the Peruvian stock market risk.

Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer

This paper considers a dual channel supply chain consisting of a risk-neutral supplier and a risk-averse retailer, in which the market demand is uncertain and the supplier opens an e-channel, thus directly participating in the market. At the beginning of the sales season, the supplier and the retailer construct their initial stocks, and they follow a consistent pricing strategy. Under the Value-at-Risk (VaR) criterion and the Conditional Value-at-Risk (CVaR) criterion, we formulate the problem as a Stackelberg game model and obtain the equilibrium solutions in the decentralized and centralized situations. Based on the advantage of the CVaR measure, we further explore the effects of the retailer's risk indicator on the retail price, the ordering quantities of the two channels and the profits of the two members, and the total profits of the supply chain. Further, an improved risk-sharing contract is presented to coordinate the dual-channel supply chain and ensures that both supply chain members achieve a win–win outcome. In addition, we make an extension to the case of the inconsistent prices in the two channels, and we also verify that the dual-channel supply chain can be coordinated by the similar improved risk-sharing contract.

CVaR度量在极值理论中的应用

近半个世纪以来，随着经济的全球化和多元化，金融风险的度量逐渐受到金融界以及经济学者的关注。90年代后，新型风险管理工具VaR(在险价值)测量方法逐步发展起来，以它能够科学、准确、综合的度量风险值而倍受国际金融界的青睐。但在极端事件发生期，VaR的度量准确性不如CVaR(条件在险价值)。本文意在研究CVaR度量在极值理论上的应用。 Since the last half a century, with the globalization and diversification of economy, the financial risk measurement has gradually been concerned by the financial and economic scholars. After the 1990s, the new risk management tool, VaR (value at risk) measurement method has been devel-oped gradually, which can measure risk value scientifically, accurately and comprehensively, and it is welcomed in the international financial community, but in extreme event, the accuracy of VaR is less than that of CVaR (conditional value at risk). This paper is intended to study the application of CVaR measure in extreme value theory.

Economically Efficient Power Storage Operation by Dealing with the Non-Normality of Power Prediction

Various predictive models about the residential energy demand and residential renewable energy production have been proposed. Recent studies have confirmed that they are not normally distributed over time. The increase in renewable energy installation has brought the issue of energy storage charge and discharge control. Thus, storage control methods that properly address non-normality are required. In this paper, we formulated the economically optimal storage control problem using Markov decision process (MDP) and the conditional value at risk (CVaR) measure to deal with the non-normality of predictive distribution about the household’s net load. The CVaR measure was employed to treat with the chance constraint on the battery capacitor, in other words, overcharge risk and over-discharge risk. We conducted a simulation to compare the annual economic saving performances between two MDPs: one is the MDP with a Gaussian predictive distribution and the other is the MDP with a normalized frequency distribution (non-normal). We used the real time series of 35 residential energy consumption and PV generation data in Japan. The importance of addressing the non-normality of random variables was shown by our simulation.

On the newsvendor model with conditional Value-at-Risk of opportunity loss

To manage the risk arising from uncertainty in market demand, this paper introduces the Conditional Value-at-Risk (CVaR) measure into the decision framework of the newsvendor who aims to minimise his opportunity loss. It is found under the CVaR measure that the newsvendor’s optimal order quantity is increasing in the confidence level when the understock loss is bigger than the overstock loss. This implies that an over-ordering may be even more caused by the newsvendor’s risk aversion about opportunity loss than risk seeking behaviour. Under this optimal order quantity, it is proved that the newsvendor’s expected profit and expected opportunity loss are decreasing and increasing in the confidence level, respectively. Furthermore, some management insights are presented to facilitate the risk management of the newsvendor model.

Security Games for Risk Minimization in Automatic Generation Control

The power grid is a critical infrastructure that must be protected against potential threats. While modern technologies at the center of the ongoing smart grid evolution increase its operational efficiency, they also make it more susceptible to malicious attacks such as false data injection to electronic monitoring systems. This paper presents a game-theoretic approach to smart grid security by combining quantitative risk management techniques with decision making on protective measures. The consequences of data injection attacks are quantified using a risk assessment process where the well-known conditional value-at-risk (CVaR) measure provides an estimate of the defender's loss due to load shed in simulated scenarios. The calculated risks are then incorporated into a stochastic security game model as input parameters. The decisions on defensive measures are obtained by solving the game using dynamic programming techniques which take into account resource constraints. Thus, the formulated security game provides an analytical framework for choosing the best response strategies against attackers and minimizing potential risks. The theoretical results obtained are demonstrated through numerical examples. Simulation results show that different risk measures lead to different defense strategies, but the CVaR measure prioritizes high-loss tail events.

Production planning problem with pricing under random yield: CVaR criterion

In this paper, we address a basic production planning problem with price dependent demand and stochastic yield of production. We use price and target quantity as decision variables to lower the risk of low yield. The value of risk control becomes more important especially for products with short life cycle. This is because, the profit implications of low yield might be unbearable in the short run. We apply Conditional Value at Risk (CVaR) to model the risk. CVaR measure is a coherent risk measure and thereby having nice conceptual and mathematical underpinnings. It is also widely used in practice. We consider the problem under general demand function and general distribution function of yield and find sufficient conditions under which the problem has a unique local maximum. We also both analytically and numerically analyze the impact of parameter change on the optimal solution. Among our results, we analytically show that with increasing risk aversion, the optimal price increases. This relation is opposite to that of in Newsvendor problem where the uncertainty lies in demand side.

Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity

We consider the problem of optimal portfolio choice using the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures for a market consisting of n risky assets and a riskless asset and where short positions are allowed. When the distribution of returns of risky assets is unknown but the mean return vector and variance/covariance matrix of the risky assets are fixed, we derive the distributionally robust portfolio rules. Then, we address uncertainty (ambiguity) in the mean return vector in addition to distribution ambiguity, and derive the optimal portfolio rules when the uncertainty in the return vector is modeled via an ellipsoidal uncertainty set. In the presence of a riskless asset, the robust CVaR and VaR measures, coupled with a minimum mean return constraint, yield simple, mean-variance efficient optimal portfolio rules. In a market without the riskless asset, we obtain a closed-form portfolio rule that generalizes earlier results, without a minimum mean return restriction.

Stochastic Optimization Problems with CVaR Risk Measure and Their Sample Average Approximation

We provide a refined convergence analysis for the SAA (sample average approximation) method applied to stochastic optimization problems with either single or mixed CVaR (conditional value-at-risk) measures. Under certain regularity conditions, it is shown that any accumulation point of the weak GKKT (generalized Karush-Kuhn-Tucker) points produced by the SAA method is almost surely a weak stationary point of the original CVaR or mixed CVaR optimization problems. In addition, it is shown that, as the sample size increases, the difference of the optimal values between the SAA problems and the original problem tends to zero with probability approaching one exponentially fast.