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34
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 109 (7 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.
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 43 (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 as a Tool for Financial Risk Management. Working paper. http://www.gloriamundi.org/var/wps.html
, 2001
"... We study the properties of Expected Shortfall from the point of view of financial risk management. This measure — which emerges as a natural remedy in some cases where VaR is not able to distinguish portfolios which bear different levels of risk — is indeed shown to have much better properties than ..."
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Cited by 38 (4 self)
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We study the properties of Expected Shortfall from the point of view of financial risk management. This measure — which emerges as a natural remedy in some cases where VaR is not able to distinguish portfolios which bear different levels of risk — is indeed shown to have much better properties than VaR. We show in fact that unlike VaR this variable is in general subadditive and therefore it is a Coherent Measure of Risk in the sense of reference [6] In this paper we review some classical arguments which arose in the last years in the debate on Value at Risk (VaR) as a measure for assessing the financial risks of a portfolio and we analyze an alternative measure of risk, which is a version of the Expected Shortfall used in Extreme Value Theory. In the second part of the paper the comparison between the two risk measures will be made on a more technical ground by analyzing some mathematical properties that play a crucial role in the definition of a risk measure. We begin with a paradox assuming that the reader is familiar with the concept of VaR. 1 A paradox Consider a portfolio A (made for instance of long option positions) of value 1000 Euro with a maximum downside level of 100 Euro and suppose that the worst 5 % cases on a fixed time horizon T are all of maximum downside. VaR at 5 % on this time horizon would then be 100 Euro. Consider now another portfolio B again of 1000 Euro which on the other hand invests also in strong short futures positions that allow for a potential unbounded maximum loss. We could easily choose B in such a way that its VaR is still 100 Euro on the time horizon T.
High volatility, thick tails and extreme value theory in valueatrisk estimation
 Insurance: Mathematics and Economics
, 2003
"... In this paper, the performance of the extreme value theory in ValueatRisk calculations is compared to the performances of other wellknown modeling techniques, such as GARCH, variancecovariance method and historical simulation in a volatile stock market. The models studied can be classified into ..."
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Cited by 11 (1 self)
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In this paper, the performance of the extreme value theory in ValueatRisk calculations is compared to the performances of other wellknown modeling techniques, such as GARCH, variancecovariance method and historical simulation in a volatile stock market. The models studied can be classified into two groups. The first group consists of GARCH(1,1) and GARCH(1,1)t models which yield highly volatile quantile forecasts. The other group, consisting of historical simulation, variancecovariance approach, adaptive generalized pareto distribution (GPD) and nonadaptive GPD models leads to more stable quantile forecasts. The quantile forecasts of GARCH(1,1) models are excessively volatilite relative to the GPD quantile forecasts. This makes the GPD model to be a robust quantile forecasting tool which is practical to implement and regulate for VaR measurements. Key Words: ValueatRisk, financial risk management, extreme value theory.
Financial Risk and Heavy Tails
 HEAVYTAILED DISTRIBUTIONS IN FINANCE , SVETLOZAR T. RACHEV (ED.)
, 2001
"... It is of great importance for those in charge of managing risk to understand how financial asset returns are distributed. Practitioners often assume for convenience that the distribution is normal. Since the 1960s, however, empirical evidence has led many to reject this assumption in favor of variou ..."
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Cited by 8 (0 self)
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It is of great importance for those in charge of managing risk to understand how financial asset returns are distributed. Practitioners often assume for convenience that the distribution is normal. Since the 1960s, however, empirical evidence has led many to reject this assumption in favor of various heavytailed alternatives. In a heavytailed distribution the likelihood that one encounters significant deviations from the mean is much greater than in the case of the normal distribution. It is now commonly accepted that financial asset returns are, in fact, heavytailed. The goal of this survey is to examine how these heavy tails affect several aspects of financial portfolio theory and risk management. We describe some of the methods that one can use to deal with heavy tails and we illustrate them using the NASDAQ composite index.
The Generalized Extreme Value (GEV) Distribution, Implied Tail Index and Option Pricing 1
, 2005
"... Crisis events such as the 1987 stock market crash, the Asian Crisis and the bursting of the DotCom bubble have radically changed the view that extreme events in financial markets have negligible probability. This paper argues that the use of the Generalized Extreme Value (GEV) distribution to model ..."
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Cited by 5 (2 self)
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Crisis events such as the 1987 stock market crash, the Asian Crisis and the bursting of the DotCom bubble have radically changed the view that extreme events in financial markets have negligible probability. This paper argues that the use of the Generalized Extreme Value (GEV) distribution to model the Risk Neutral Density (RND) function provides a flexible framework that captures the negative skewness and excess kurtosis of returns, and also delivers the market implied tail index of asset returns. We obtain an original analytical closed form solution for the Harrison and Pliska (1981) no arbitrage equilibrium price for the European option in the case of GEV asset returns. The GEV based option prices successfully remove the well known pricing bias of the BlackScholes model. We explain how the implied tail index is efficacious at identifying the fat tailed behaviour of losses and hence the left skewness of the price RND functions, particularly around crisis events.
Large deviations bounds for estimating conditional valueatrisk
, 2006
"... www.elsevier.com/locate/orl ..."
VargaHaszonits I, Feasibility of portfolio optimization under coherent risk measures, 2008 arXiv:0803.2283v3 [physics.socph
"... Abstract. It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio cannot be optimized under any coherent measure on that ..."
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Cited by 4 (4 self)
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Abstract. It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio cannot be optimized under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability was first discovered on the special example of Expected Shortfall which is used here both as an illustration and as a prompt for generalization.
Economic Risk Capital and Reinsurance: An Application to Fire Claims of an Insurance Company,” RAS, Pianificazione Redditività diGruppo
, 2001
"... The viability of an insurance company depends critically on the size and frequency of large claims. An accurate modelling of the distribution of large claims contributes to correct pricing’s and reserving’s decisions while maintaining, at an acceptable level, the unexpected fluctuations in the resul ..."
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Cited by 2 (0 self)
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The viability of an insurance company depends critically on the size and frequency of large claims. An accurate modelling of the distribution of large claims contributes to correct pricing’s and reserving’s decisions while maintaining, at an acceptable level, the unexpected fluctuations in the results through reinsurance. We provide a model for large losses and we extrapolate through a simulation a scenario based on separate estimation of loss frequency and loss severity according to extreme value theory, with particular reference to generalized Pareto approximations. We present an application to the fire claims of an insurance company. One conclusion is that the distribution of fire claims is longtailed and an accurate exploratory data analysis is done to detect heavy tailed behavior and stability of parameters and statistics across different thresholds. We simulate the impact of a quota share and an excess of loss reinsurance structure on the distribution of total loss and on economic risk capital. We provide also a tool to price and investigate how different reinsurance programs can affect economic risk capital and explain the rationale of the choice of the optimal reinsurance programmes to smooth economic results.