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386
A RiskFactor Model Foundation for RatingsBased Bank Capital Rules
 Journal of Financial Intermediation
, 2003
"... When economic capital is calculated using a portfolio model of credit valueatrisk, the marginal capital requirement for an instrument depends, in general, on the properties of the portfolio in which it is held. By contrast, ratingsbased capital rules, including both the current Basel Accord and i ..."
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Cited by 294 (1 self)
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When economic capital is calculated using a portfolio model of credit valueatrisk, the marginal capital requirement for an instrument depends, in general, on the properties of the portfolio in which it is held. By contrast, ratingsbased capital rules, including both the current Basel Accord and its proposed revision, assign a capital charge to an instrument based only on its own characteristics. I demonstrate that ratingsbased capital rules can be reconciled with the general class of credit VaR models. Contributions to VaR are portfolioinvariant only if (a) there is only a single systematic risk factor driving correlations across obligors, and (b) no exposure in a portfolio accounts for more than an arbitrarily small share of total exposure. Analysis of rates of convergence to asymptotic VaR leads to a simple and accurate portfoliolevel addon charge for undiversified idiosyncratic risk. There is no similarly simple way to address violation of the single factor assumption.
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 217 (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.
2002a): Expected Shortfall: A Natural Coherent Alternative to Value at Risk, in: Economic Notes by Banca Monte dei Paschi di Siena SpA
"... Abstract We discuss the coherence properties of Expected Shortfall (ES) as a financial 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 ..."
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Cited by 75 (9 self)
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Abstract We discuss the coherence properties of Expected Shortfall (ES) as a financial 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.
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 68 (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.
Beyond Correlation: Extreme Comovements Between Financial Assets
, 2002
"... This paper inv estigates the potential for extreme comov ements between financial assets by directly testing the underlying dependence structure. In particular, a tdependence structure, deriv ed from the Student t distribution, is used as a proxy to test for this extremal behav#a(0 Tests in three ..."
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Cited by 61 (5 self)
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This paper inv estigates the potential for extreme comov ements between financial assets by directly testing the underlying dependence structure. In particular, a tdependence structure, deriv ed from the Student t distribution, is used as a proxy to test for this extremal behav#a(0 Tests in three di#erent markets (equities, currencies, and commodities) indicate that extreme comov ements are statistically significant. Moreov er, the "correlationbased" Gaussian dependence structure, underlying the multiv ariate Normal distribution, is rejected with negligible error probability when tested against the tdependencealternativ e. The economic significance of these results is illustratedv ia three examples: comov ements across the G5 equity markets; portfoliov alueatrisk calculations; and, pricing creditderiv ativ es. JEL Classification: C12, C15, C52, G11. Keywords: asset returns, extreme comov ements, copulas, dependence modeling, hypothesis testing, pseudolikelihood, portfolio models, risk management. # The authorsw ould like to thankAndrew Ang, Mark Broadie, Loran Chollete, and Paul Glasserman for their helpful comments on an earlier version of this manuscript. Both authors arewS; the Columbia Graduate School of Business, email: {rm586,assaf.zeevi}@columbia.edu, current version available at www.columbia.edu\# rm586 1 Introducti7 Specification and identification of dependencies between financial assets is a key ingredient in almost all financial applications: portfolio management, risk assessment, pricing, and hedging, to name but a few. The seminal work of Markowitz (1959) and the early introduction of the Gaussian modeling paradigm, in particular dynamic Brownianbased models, hav e both contributed greatly to making the concept of co rrelatio almost synony...
Optimization with stochastic dominance constraints
 SIAM Journal on Optimization
"... We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for the ..."
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Cited by 55 (6 self)
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We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.
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 49 (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.
STABILITY OF MULTISTAGE STOCHASTIC PROGRAMS
 SIAM J. OPTIM.
, 2006
"... Quantitative stability of linear multistage stochastic programs is studied. It is shown that the infima of such programs behave (locally) Lipschitz continuous with respect to the sum of an L_rdistance and of a distance measure for the filtrations of the original and approximate stochastic (input) ..."
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Cited by 45 (11 self)
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Quantitative stability of linear multistage stochastic programs is studied. It is shown that the infima of such programs behave (locally) Lipschitz continuous with respect to the sum of an L_rdistance and of a distance measure for the filtrations of the original and approximate stochastic (input) processes. Various issues of the result are discussed and an illustrative example is given. Consequences for the reduction of scenario trees are also discussed.