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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
Correlation And Dependence In Risk Management: Properties And Pitfalls
 RISK MANAGEMENT: VALUE AT RISK AND BEYOND
, 1999
"... Modern risk management calls for an understanding of stochastic dependence going beyond simple linear correlation. This paper deals with the static (nontimedependent) case and emphasizes the copula representation of dependence for a random vector. Linear correlation is a natural dependence measure ..."
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Cited by 319 (37 self)
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Modern risk management calls for an understanding of stochastic dependence going beyond simple linear correlation. This paper deals with the static (nontimedependent) case and emphasizes the copula representation of dependence for a random vector. Linear correlation is a natural dependence measure for multivariate normally and, more generally, elliptically distributed risks but other dependence concepts like comonotonicity and rank correlation should also be understood by the risk management practitioner. Using counterexamples the falsity of some commonly held views on correlation is demonstrated; in general, these fallacies arise from the naive assumption that dependence properties of the elliptical world also hold in the nonelliptical world. In particular, the problem of finding multivariate models which are consistent with prespecified marginal distributions and correlations is addressed. Pitfalls are highlighted and simulation algorithms avoiding these problems are constructed. ...
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 283 (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.
Estimation of TailRelated Risk Measures for Heteroscedastic Financial Time Series: an Extreme Value Approach
 Journal of Empirical Finance
, 1998
"... We propose a method for estimating VaR and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return series. Our approach combines pseudomaximumlikelihood fitting of GARCH models to estimate the current volatility and extreme value theory (EVT) ..."
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Cited by 230 (6 self)
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We propose a method for estimating VaR and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return series. Our approach combines pseudomaximumlikelihood fitting of GARCH models to estimate the current volatility and extreme value theory (EVT) for estimating the tail of the innovation distribution of the GARCH model. We use our method to estimate conditional quantiles (VaR) and conditional expected shortfalls (the expected size of a return exceeding VaR), this being an alternative measure of tail risk with better theoretical properties than the quantile. Using backtesting of historical daily return series we show that our procedure gives better oneday estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. With the help of our fitted models we adopt a Monte Carlo approach to estimating the conditional quantiles of returns over multipleday horizons and find that t...
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.
TimeChanged Lévy Processes and Option Pricing
, 2002
"... As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return ..."
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Cited by 182 (21 self)
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As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that timechanged Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.
Convex approximations of chance constrained programs
 SIAM Journal of Optimization
, 2006
"... Abstract. We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given close to one probability, a system of randomly perturbed convex constraints. This problem may happen to be computationally intractable; our goal is to build its ..."
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Cited by 120 (9 self)
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Abstract. We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given close to one probability, a system of randomly perturbed convex constraints. This problem may happen to be computationally intractable; our goal is to build its computationally tractable approximation, i.e., an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem. We construct a general class of such convex conservative approximations of the corresponding chance constrained problem. Moreover, under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independentofeachother random variables, we build a large deviationtype approximation, referred to as “Bernstein approximation, ” of the chance constrained problem. This approximation is convex and efficiently solvable. We propose a simulationbased scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the Bernstein and wellknown scenario approximation approaches. Finally, we extend our construction to the case of ambiguous chance constrained problems, where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly, while the chance constraint should be satisfied for every distribution given by this set.
Some remarks on the valueatrisk and the conditional valueatrisk
 in Probabilistic Constrained Optimization: Methodology and Applications
, 2000
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Function spaces and capacity related to a sublinear expectation: application to GBrownian motion paths
 POTENTIAL ANALYSIS
, 2010
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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.