On law invariant coherent risk measures
01 Jan 2001-pp 83-95
TL;DR: In this paper, a special class of coherent risk measures is defined and a characterization of it is given, where the probability space is defined as a probability space and the coherent risk measure is defined in terms of a probability vector.
Abstract: The idea of coherent risk measures has been introduced by Artzner, Delbaen, Eber and Heath [1] We think of a special class of coherent risk measures and give a characterization of it Let (Ω, ℱ, P) be a probability space We denote L ∞(Ω, ℱ, P) by L ∞ Following [1], we give the following definition
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Book•
16 Oct 2005
TL;DR: The most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management can be found in this paper, where the authors describe the latest advances in the field, including market, credit and operational risk modelling.
Abstract: This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation
2,580 citations
Book•
24 Sep 2009
TL;DR: The authors dedicate this book to Julia, Benjamin, Daniel, Natan and Yael; to Tsonka, Konstatin and Marek; and to the Memory of Feliks, Maria, and Dentcho.
Abstract: List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.
2,443 citations
Cites background from "On law invariant coherent risk meas..."
...40) was derived in [139] for L∞(Ω,F , P ) spaces....
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TL;DR: In this paper, the authors 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.
Abstract: Expected shortfall (ES) in several variants has been proposed as remedy for the deficiencies of value-at-risk (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 ES, 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 ES can be estimated effectively even in cases where the usual estimators for VaR fail.
1,276 citations
Cites background from "On law invariant coherent risk meas..."
...Moreover, in a certain sense any law invariant coherent risk measure has a representation with ES as the main building block (see [11])....
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...3) shows that ES is the coherent risk measure used in [11] as main building block for the representation of law invariant coherent risk measures....
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Posted Content•
TL;DR: In this article, the authors 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.
Abstract: Expected Shortfall (ES) in several variants has been proposed as remedy for the defi-ciencies of Value-at-Risk (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.
Key words: Expected Shortfall; Risk measure; worst conditional expectation; tail con-ditional expectation; value-at-risk (VaR); conditional value-at-risk (CVaR); tail mean; co-herence; quantile; sub-additivity.
1,189 citations
TL;DR: In this article, the authors study a space of coherent risk measures M/ obtained as certain expansions of coherent elementary basis measures and give necessary and sufficient conditions on / for M/ to be a coherent measure.
Abstract: We study a space of coherent risk measures M/ obtained as certain expansions of coherent elementary basis measures. In this space, the concept of ‘‘risk aversion function’’ / naturally arises as the spectral representation of each risk measure in a space of functions of confidence level probabilities. We give necessary and sufficient conditions on / for M/ to be a coherent measure. We find in this way a simple interpretation of the concept of coherence and a way to map any rational investor’s subjective risk aversion onto a coherent measure and vice-versa. We also provide for these measures their discrete versions M ðNÞ / acting on finite sets of N independent realizations of a r.v. which are not only shown to be coherent measures for any fixed N, but also consistent estimators of M/ for large N. 2002 Elsevier Science B.V. All rights reserved.
799 citations
Cites background from "On law invariant coherent risk meas..."
...[8]....
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...The class of spectral measures has also been studied in a different formalism in Kusuoka (2001)....
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References
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TL;DR: In this paper, the authors present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties "coherent", and demonstrate the universality of scenario-based methods for providing coherent measures.
Abstract: In this paper we study both market risks and nonmarket risks, without complete markets assumption, and discuss methods of measurement of these risks. We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties “coherent.” We examine the measures of risk provided and the related actions required by SPAN, by the SEC=NASD rules, and by quantile-based methods. We demonstrate the universality of scenario-based methods for providing coherent measures. We offer suggestions concerning the SEC method. We also suggest a method to repair the failure of subadditivity of quantile-based methods.
8,651 citations
"On law invariant coherent risk meas..." refers background in this paper
...Following [1], we give the following definition....
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...Theorem 5 If m1 and m2 are probability measures on [0,1], and if...
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...in [1]....
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...(1) There is a probability measure m on [0,1] such that for X E Loo p(X) = 11 Pa(X)m(do:),...
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..., 0: E [0, 1], is a law invariant coherent risk measure with the Fatou property....
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Book•
01 Jan 1991TL;DR: A branching-process example and an easy strong law: product measure using martingale theory and the central limit theorem are presented.
Abstract: Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital role. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.
2,265 citations
"On law invariant coherent risk meas..." refers background in this paper
...(3) If $X_{n}$ is a uniformly bounded sequence that decreases to $X$, then $\rho(X_{*}....
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...(3) We have Lemma 11 from Equations (1), (2) and (3)....
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...(3) Positive homogeneity:for $\lambda>0$ we have $\rho(\lambda X)=\lambda\rho(X)$ ....
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01 Jan 2002
TL;DR: In this paper, the authors extend the definition of coherent risk measures to general probability spaces and show how to define such measures on the space of all random variables, and give examples that relate the theory of coherent risks to game theory and to distorted probability measures.
Abstract: We extend the definition of coherent risk measures, as introduced by Artzner, Delbaen, Eber and Heath, to general probability spaces and we show how to define such measures on the space of all random variables. We also give examples that relates the theory of coherent risk measures to game theory and to distorted probability measures. The mathematics are based on the characterisation of closed convex sets Pσ of probability measures that satisfy the property that every random variable is integrable for at least one probability measure in the set Pσ.
835 citations
"On law invariant coherent risk meas..." refers background in this paper
...(2) Let $\tilde{\mathrm{Y}}_{m}(\omega)=\sum_{k=1}^{2^{m}}1_{[((k-1)2^{-m},k2^{-m})}(\omega)\mathrm{Y}_{Q}(\omega-k2^{-m}+\tau_{m}(\sigma_{m}^{-1}(k))2^{-m})$....
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...(2) SubaddUwity : $\rho(X_{1}+X_{2})\leq\rho(X_{1})+\rho(X_{2})$ ....
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...(2) $\rho$ satisfies the Fatou property, $i....
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...(2) $\rho$ is a law invariant coherent risk measure with the Fatou property....
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...(2) If $F_{n}\in D$ converges to $F$ weakly, then $Z(x, F_{n})$ converges to $Z(x, F)$ for $\mu$ -a....
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