Conditional value‐at‐risk beyond finance: a survey

• Published date: 01 May 2020
• DOI: http://doi.org/10.1111/itor.12726
• Date: 01 May 2020

Intl. Trans. in Op. Res. 27 (2020) 1277–1319

DOI: 10.1111/itor.12726

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Conditional value-at-risk beyond finance: a survey

C. Filippi, G. Guastaroba and M.G. Speranza

Department of Economics and Management, C.da S. Chiara 50, University of Brescia, Brescia, Italy

E-mail: carlo.filippi@unibs.it [Filippi]; gianfranco.guastaroba@unibs.it [Guastaroba];

grazia.speranza@unibs.it [Speranza]

Received 11 February2019; received in revised form 5 July 2019; accepted 10 September 2019

Abstract

A large number of problemsinvolve making decisions in an uncertain environment and, hence,with unknown

outcomes. Optimization models aimed at controlling the trade-off between risk and return in finance have

been widely studied since the seminal work by Markowitz in 1952. In financial applications, shortfall or

quantile risk measures are receiving ever-increasing attention. Conditional value-at-risk (CVaR) is arguably

the most popular of such measures. In the last decades, optimization models aimed at controlling risk have

been applied to several application domains different from financial optimization. This survey provides an

overview of the main contributions where CVaR is incorporated into an optimization approach and applied

to a context different from financial engineering. The literature is classified following an application-oriented

perspective. The applications cover classical areas studied in operational research—such as supply chain

management, scheduling, and networks—and less classical areas such as energyand medicine. For each area,

concise paper excerpts are provided that convey the main ideas of the problems studied, and analyze how

the CVaR has been used to cope with differentsources of uncertainty. Finally,some open research directions

are outlined.

Keywords:risk management; literature review; conditional value-at-risk; optimization under uncertainty; applications

1. Introduction

The demand for mechanisms to control or limit the risk arises in several application domains

including finance, supply chain management (see Heckmann et al., 2015), energy, and medicine. To

stress on this issue, we quote from Eppen et al. (1989) where, while addressing a problem faced by

General Motors, the authors claim that

“In any decision under risk, expected profit is not the only objective. Management is also

concerned about the risk involved. This is especially true for a set of decisions like the

one faced by GM, where large amounts of money and the careers of many individuals are

involved.”

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1278 C. Filippi et al. / Intl. Trans. in Op. Res. 27 (2020) 1277–1319

Markowitz (1952) first formalized the problem of determining the optimal investment in a port-

folio of financial assets employing a risk-return framework. In this seminal work, a quadratic

optimization model was devised to control the trade-off between risk—measured by the variance—

and return. The paper by Markowitz has posed the fundamental basis for the development of a

large part of the modern theory of financial optimization. Note that the appropriateness of a risk

measure is guided by a decision maker’s attitude toward losses and gains. In these terms, although

the variance may be suitable when the outcome distribution is close to be symmetric, it is not

appropriate in the general case where the distribution may be asymmetric as it equally penalizes

losses and gains. In fact, a rational decision maker does not regard gains above some target level

as a risk to be hedged, but rather wishes to minimize losses. To overcome this limit, several other

risk measures have been consideredin financial applications.Shortfall-based or quantile-based risk

measures have rapidly gained wide popularity during the first decade of the 21st century. The most

used of such measures is the conditional value-at-risk, henceforth called CVaR, first developed by

Rockafellar and Uryasev (2000).

CVaR is strongly related to another risk measure called value at risk (VaR), which is used in

various financial and engineering problems, including military, nuclear, and airspace applications

(Sarykalin et al., 2008). The VaR of a portfolio of assets, given a specified probability level α,canbe

defined as the smallest threshold value ηsuch that the probability that the loss exceeds ηis 1 −α.

The value αis chosen by the decision maker, and is usually called confidence level. As pointed out by

Rockafellar and Uryasev (2002), a very serious shortcoming of VaR is that it does not provide any

indication about the severity of losses beyond its value. As highlighted by Sarykalin et al. (2008),

one can significantly increase the largest loss exceedingthe VaR without actually changing the VaR.

The CVaR overcomes this limit as it measures the conditional expectation of losses above η.

There are several reasons explaining the success of CVaR as a risk measure. From a practical

point of view, CVaR penaliz es only the negative deviations with respect to an efficiency target (it is

a so-called downside risk measure). It is sensitive to the worst outcomes but it is not as conservative

as a minimax approach. At the same time, it may be seen as a still rather conservative risk measure

as it considers only a fraction of the worst outcomes of a random variable, neglecting the others.

However, this fraction can be tuned according to the decision maker’s attitude toward risk. From

a theoretical viewpoint, CVaR is a coherent risk measure, as defined in Artzner et al. (1999). From

a computational point of view, CVaR can often be embedded in an optimization model by adding

linear constraints and continuous variables, that is, without significantly increasing the complexity

of the resulting optimization model. Therefore, it is not surprising thatin recent years several authors

incorporated CVaR as an additional criterion in their optimization models, while facing a number

of applications different from financial optimization.

1.1. Purpose and scope of the survey

The literature where CVaR is employed in financial and related applications is remarkably vast.

Over the last decade, the number of papers where CVaR is used in application areas different from

quantitative finance motivates the need for structuring and organizing the related literature.

The purpose of this survey is to guide the reader through the literaturewhere the CVaR is applied

to a context different fromquantitative finance, by providinga classification and systematic overview

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C. Filippi et al. / Intl. Trans. in Op. Res. 27 (2020) 1277–1319 1279

Fig. 1. The classification per area of the literature reviewed.

of the foremost contributions published. We have collected and reviewed only contributions from

international and peer-reviewedscientific journals. Our scope is to cover those papers whereCVaR is

employed in an optimization phase to support the decision maker. Hence, we do not include papers

where CVaR is computed only as a postoptimization statistic to validate or compare different

solutions. Interestingly, the CVaR has also been employed in deterministic contexts. The most

relevant example is when CVaR and closely related measures have been applied to optimization

problems where concern is given to a fair and equitable distribution of resources. Along this line of

research, some authors have studied location–allocation problems and allocation problems related

to communication networks. The recent survey by Ogryczak et al. (2014) provides an overview

of this class of problems. To the sake of brevity, we report in the following only those papers

incorporating the CVaR in a deterministic context, which are not mentioned in Ogryczak et al.

(2014).

1.2. Classification used

Figure 1 shows the classification of the papers reviewed in this survey. Classical topics in the Op-

erations Research and Management Science (OR/MS) literature are “Inventory management,”

“Supply chain management,” “Transportation and traffic control,” “Location and supply chain

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2019 The Authors.

International Transactionsin Operational Research C

2019 International Federation of OperationalResearch Societies