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112
Conditional valueatrisk for general loss distributions
 Journal of Banking and Finance
, 2002
"... Abstract. Fundamental properties of conditional valueatrisk, as a measure of risk with significant advantages over valueatrisk, are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence o ..."
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Cited by 356 (28 self)
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Abstract. Fundamental properties of conditional valueatrisk, as a measure of risk with significant advantages over valueatrisk, are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. Conditional valueatrisk is able to quantify dangers beyond valueatrisk, and moreover it is coherent. It provides optimization shortcuts which, through linear programming techniques, make practical many largescale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking. Key Words: Valueatrisk, conditional valueatrisk, mean shortfall, coherent risk measures, risk sampling, scenarios, hedging, index tracking, portfolio optimization, risk management
On the coherence of expected shortfall
 In: Szegö, G. (Ed.), “Beyond VaR” (Special Issue). Journal of Banking & Finance
, 2002
"... Expected Shortfall (ES) in several variants has been proposed as remedy for the deficiencies of ValueatRisk (VaR) which in general is not a coherent risk measure. In fact, most definitions of ES lead to the same results when applied to continuous loss distributions. Differences may appear when the ..."
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Cited by 203 (8 self)
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Expected Shortfall (ES) in several variants has been proposed as remedy for the deficiencies of ValueatRisk (VaR) which in general is not a coherent risk measure. In fact, most definitions of ES lead to the same results when applied to continuous loss distributions. Differences may appear when the underlying loss distributions have discontinuities. In this case even the coherence property of ES can get lost unless one took care of the details in its definition. We compare some of the definitions of Expected Shortfall, pointing out that there is one which is robust in the sense of yielding a coherent risk measure regardless of the underlying distributions. Moreover, this Expected Shortfall can be estimated effectively even in cases where the usual estimators for VaR fail.
Optimization of Convex Risk Functions
, 2004
"... We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functio ..."
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Cited by 102 (15 self)
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We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functions.
Expected Shortfall: A Natural Coherent Alternative to Value at Risk
 Economic Notes
"... We discuss the coherence properties of Expected Shortfall (ES) asafinancial risk measure. This statistic arises in a natural way from the estimation of the “average of the 100p% worst losses ” in a sample of returns to a portfolio. Here p is some fixed confidence level. We also compare several alter ..."
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Cited by 75 (9 self)
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We discuss the coherence properties of Expected Shortfall (ES) asafinancial risk measure. This statistic arises in a natural way from the estimation of the “average of the 100p% worst losses ” in a sample of returns to a portfolio. Here p is some fixed confidence level. We also compare several alternative representations of ES which turn out to be more appropriate for certain purposes. Key words: Expected Shortfall; Risk measure; worst conditional expectation; tail conditional expectation; valueatrisk (VaR); conditional valueatrisk (CVaR); coherence; subadditivity. 1 A four years impasse Risk professionals have been looking for a coherent alternative to Value at Risk (VaR) for four years. Since the appearance, in 1997, of Thinking Coherently by Artzner et al [3] followed by Coherent Measures of Risk [4], it was clear to risk practitioners and researchers that the gap between market practice and theoretical progress had suddenly widened enormously. These papers in fact faced for the first time the problem of defining in a clearcut way what properties a statistic of a portfolio should have in order to be considered a sensible risk measure. The answer to this question was given through a complete characterization of such properties via an axiomatic formulation of the concept of coherent risk measure. With this result, risk management became all of a sudden a science in itself with its own rules correctly definedinadeductiveframework. Surprisingly enough, however, VaR, the risk measure adopted as best practice by essentially all banks and regulators, happened to fail the exam for being admitted in this science. VaR is not a coherent risk measure because it simply doesn’t fulfill one of the axioms of coherence.
Expected shortfall and beyond
 Journal of Banking & Finance
, 2002
"... Financial institutions have to allocate socalled economic capital in order to guarantee solvency to their clients and counterparties. Mathematically speaking, any methodology of allocating capital is a risk measure, i.e. a function mapping random variables to the real numbers. Nowadays valueatris ..."
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Cited by 65 (8 self)
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Financial institutions have to allocate socalled economic capital in order to guarantee solvency to their clients and counterparties. Mathematically speaking, any methodology of allocating capital is a risk measure, i.e. a function mapping random variables to the real numbers. Nowadays valueatrisk, which is defined as a fixed level quantile of the random variable under consideration, is the most popular risk measure. Unfortunately, it fails to reward diversification, as it is not subadditive. In the search for a suitable alternative to valueatrisk, Expected Shortfall (or conditional valueatrisk or tail valueatrisk) has been characterized as the smallest coherent and law invariant risk measure to dominate valueatrisk. We discuss these and some other properties of Expected Shortfall as well as its generalization to a class of coherent risk measures which can incorporate higher moment effects. Moreover, we suggest a general method on how to attribute Expected Shortfall risk contributions to portfolio components. JEL classification D81, C13.
Polyhedral risk measures in stochastic programming
 SIAM JOURNAL ON OPTIMIZATION
, 2005
"... We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedra ..."
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Cited by 54 (18 self)
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We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties. Polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the righthand side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we propose multiperiod extensions of the ConditionalValueatRisk.
An oldnew concept of convex risk measures: The optimized certainty equivalent
 Mathematical Finance
"... The optimized certainty equivalent (OCE) is a decision theoretic criterion based on a utility function, that was first introduced by the authors in 1986. This paper reexamines this fundamental concept, studies and extends its main properties, and put it in perspective to recent concepts of risk mea ..."
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Cited by 45 (1 self)
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The optimized certainty equivalent (OCE) is a decision theoretic criterion based on a utility function, that was first introduced by the authors in 1986. This paper reexamines this fundamental concept, studies and extends its main properties, and put it in perspective to recent concepts of risk measures. We show that the negative of the OCE naturally provides a wide family of risk measures that fits the axiomatic formalism of convex risk measures. Duality theory is used to reveal the link between the OCE and the ϕdivergence functional (a generalization of relative entropy), and allows for deriving various variational formulas for risk measures. Within this interpretation of the OCE, we prove that several risk measures recently analyzed and proposed in the literature (e.g., conditional value of risk, bounded shortfall risk) can be derived as special cases of the OCE by using particular utility functions. We further study the relations between the OCE and other certainty equivalents, providing general conditions under which these can be viewed as coherent/convex risk measures. Throughout the paper several examples illustrate the flexibility and adequacy of the OCE for building risk measures.
Coherent approaches to risk in optimization under uncertainty
 In Tutorials in Operations Research INFORMS
, 2007
"... Keywords Decisions often need to be made before all the facts are in. A facility must be built to withstand storms, floods, or earthquakes of magnitudes that can only be guessed from historical records. A portfolio must be purchased in the face of only statistical knowledge, at best, about how marke ..."
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Cited by 39 (3 self)
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Keywords Decisions often need to be made before all the facts are in. A facility must be built to withstand storms, floods, or earthquakes of magnitudes that can only be guessed from historical records. A portfolio must be purchased in the face of only statistical knowledge, at best, about how markets will perform. In optimization, this implies that constraints may need to be envisioned in terms of safety margins instead of exact requirements. But what does that really mean in model formulation? What guidelines make sense, and what are the consequences for optimization structure and computation? The idea of a coherent measure of risk in terms of surrogates for potential loss, which has been developed in recent years for applications in financial engineering, holds promise for a far wider range of applications in which the traditional approaches to uncertainty have been subject to criticism. The general ideas and main facts are presented here with the goal of facilitating their transfer to practical work in those areas. optimization under uncertainty; safeguarding against risk; safety margins; measures of risk; measures of potential loss; measures of deviation; coherency; valueatrisk; conditional valueatrisk; probabilistic constraints; quantiles; risk envelopes; dual representations; stochastic programming 1.
Optimal dynamic trading strategies with risk limits
 SSRN Electronic Paper Collection
, 2001
"... Value at Risk (VaR) has emerged in recent years as a standard tool to measure and control the risk of trading portfolios. Yet, existing theoretical analyses of the optimal behavior of a trader subject to VaR limits have produced a negative viewof VaR as a riskcontrol tool. In particular, VaR limits ..."
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Cited by 30 (1 self)
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Value at Risk (VaR) has emerged in recent years as a standard tool to measure and control the risk of trading portfolios. Yet, existing theoretical analyses of the optimal behavior of a trader subject to VaR limits have produced a negative viewof VaR as a riskcontrol tool. In particular, VaR limits have been found to induce increased risk exposure in some states and an increased probability of extreme losses. However, these conclusions are based on models that are either static or dynamically inconsistent. In this paper we formulate a dynamically consistent model of optimal portfolio choice subject to VaR limits and showthat the conclusions of earlier papers are incorrect if, consistently with common practice, the VaR is reevaluated dynamically making full use of conditioning information. In particular, we find that the risk exposure of a trader subject to a VaR limit is always lower than that of an unconstrained trader and that the probability of extreme losses is also lower. We also consider the Tail Conditional Expectation (TCE), a coherent risk measure often advocated as an alternative to VaR, and showthat in our dynamic setting it is always possible to transform a TCE limit into an equivalent VaR limit, and conversely.