Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation

Conditional value-at-risk (CVaR) and value-at-risk, also called the superquantile and quantile, are frequently used to characterize the tails of probability distributions and are popular measures of risk in applications where the distribution represents the magnitude of a potential loss. buffered pr...

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Veröffentlicht in:	Annals of operations research Jg. 299; H. 1-2; S. 1281 - 1315
Hauptverfasser:	Norton, Matthew, Khokhlov, Valentyn, Uryasev, Stan
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.04.2021 Springer Springer Nature B.V
Schlagworte:	Business and Management Combinatorics Confidence intervals Convex analysis Density Distribution (Probability theory) Engineering Financial risk Functions, Inverse Linear programming Logistics Mathematical analysis Measurement Method of moments Methods Operations research Operations Research/Decision Theory Optimization Portfolio management Quantiles Random variables Risk S.I.: Recent Developments in Financial Modeling and Risk Management Statistical analysis Theory of Computation United States Buffered probability of exceedance Portfolio optimization Conditional value-at-risk Density estimation Superquantile
ISSN:	0254-5330, 1572-9338
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Abstract	Conditional value-at-risk (CVaR) and value-at-risk, also called the superquantile and quantile, are frequently used to characterize the tails of probability distributions and are popular measures of risk in applications where the distribution represents the magnitude of a potential loss. buffered probability of exceedance (bPOE) is a recently introduced characterization of the tail which is the inverse of CVaR, much like the CDF is the inverse of the quantile. These quantities can prove very useful as the basis for a variety of risk-averse parametric engineering approaches. Their use, however, is often made difficult by the lack of well-known closed-form equations for calculating these quantities for commonly used probability distributions. In this paper, we derive formulas for the superquantile and bPOE for a variety of common univariate probability distributions. Besides providing a useful collection within a single reference, we use these formulas to incorporate the superquantile and bPOE into parametric procedures. In particular, we consider two: portfolio optimization and density estimation. First, when portfolio returns are assumed to follow particular distribution families, we show that finding the optimal portfolio via minimization of bPOE has advantages over superquantile minimization. We show that, given a fixed threshold, a single portfolio is the minimal bPOE portfolio for an entire class of distributions simultaneously. Second, we apply our formulas to parametric density estimation and propose the method of superquantiles (MOS), a simple variation of the method of moments where moments are replaced by superquantiles at different confidence levels. With the freedom to select various combinations of confidence levels, MOS allows the user to focus the fitting procedure on different portions of the distribution, such as the tail when fitting heavy-tailed asymmetric data.
AbstractList	Conditional value-at-risk (CVaR) and value-at-risk, also called the superquantile and quantile, are frequently used to characterize the tails of probability distributions and are popular measures of risk in applications where the distribution represents the magnitude of a potential loss. buffered probability of exceedance (bPOE) is a recently introduced characterization of the tail which is the inverse of CVaR, much like the CDF is the inverse of the quantile. These quantities can prove very useful as the basis for a variety of risk-averse parametric engineering approaches. Their use, however, is often made difficult by the lack of well-known closed-form equations for calculating these quantities for commonly used probability distributions. In this paper, we derive formulas for the superquantile and bPOE for a variety of common univariate probability distributions. Besides providing a useful collection within a single reference, we use these formulas to incorporate the superquantile and bPOE into parametric procedures. In particular, we consider two: portfolio optimization and density estimation. First, when portfolio returns are assumed to follow particular distribution families, we show that finding the optimal portfolio via minimization of bPOE has advantages over superquantile minimization. We show that, given a fixed threshold, a single portfolio is the minimal bPOE portfolio for an entire class of distributions simultaneously. Second, we apply our formulas to parametric density estimation and propose the method of superquantiles (MOS), a simple variation of the method of moments where moments are replaced by superquantiles at different confidence levels. With the freedom to select various combinations of confidence levels, MOS allows the user to focus the fitting procedure on different portions of the distribution, such as the tail when fitting heavy-tailed asymmetric data.
Audience	Academic
Author	Khokhlov, Valentyn Norton, Matthew Uryasev, Stan
Author_xml	– sequence: 1 givenname: Matthew surname: Norton fullname: Norton, Matthew email: mnorton@nps.edu organization: Operations Research Department, Naval Postgraduate School – sequence: 2 givenname: Valentyn surname: Khokhlov fullname: Khokhlov, Valentyn organization: CFA Society – sequence: 3 givenname: Stan surname: Uryasev fullname: Uryasev, Stan organization: Department of Industrial and Systems Engineering, Risk Management and Financial Engineering Laboratory, University of Florida
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Cites_doi	10.1016/j.ress.2010.01.001 10.1007/s11069-016-2324-y 10.1137/15M1042644 10.21314/JOR.2000.038 10.1016/S0378-4266(02)00271-6 10.1080/01621459.2014.929522 10.1111/1467-9965.00068 10.1080/10920277.2003.10596118 10.1007/s10107-014-0801-1 10.1007/s10479-016-2354-6 10.1016/j.ejor.2018.01.021 10.1080/03610919908813579
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Keywords	Buffered probability of exceedance Portfolio optimization Conditional value-at-risk Density estimation Superquantile
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References_xml	– reference: RockafellarRUryasevSOptimization of conditional value-at-riskThe Journal of Risk200023214110.21314/JOR.2000.038 – reference: RockafellarRTUryasevSConditional value-at-risk for general loss distributionsJournal of Banking & Finance20022671443147110.1016/S0378-4266(02)00271-6 – reference: Andreev, A., Kanto, A., & Malo, P. (2005). On closed-form calculation of CVaR. Helsinki School of Economics working paper W-389. – reference: ShangDKuzmenkoVUryasevSCash flow matching with risks controlled by buffered probability of exceedance and conditional value-at-riskAnnals of Operations Research20182601–250151410.1007/s10479-016-2354-6 – reference: KarianZADudewiczEJFitting the generalized lambda distribution to data: A method based on percentilesCommunications in Statistics: Simulation and Computation199928379381910.1080/03610919908813579 – reference: SgouropoulosNYaoQYastremizCMatching a distribution by matching quantiles estimationJournal of the American Statistical Association201511051074275910.1080/01621459.2014.929522 – reference: DavisJRUryasevSAnalysis of tropical storm damage using buffered probability of exceedanceNatural Hazards201683146548310.1007/s11069-016-2324-y – reference: RockafellarRRoysetJOn buffered failure probability in design and optimization of structuresReliability Engineering & System Safety20109549951010.1016/j.ress.2010.01.001 – reference: Norton, M., & Uryasev, S. (2016). Maximization of AUC and buffered AUC in binary classification. Mathematical Programming, 174(1–2), 575–612. – reference: MafusalovAUryasevSBuffered probability of exceedance: Mathematical properties and optimizationSIAM Journal on Optimization20182821077110310.1137/15M1042644 – reference: RockafellarRTRoysetJORandom variables, monotone relations, and convex analysisMathematical Programming20141481–229733110.1007/s10107-014-0801-1 – reference: Uryasev, S. (2014). Buffered probability of exceedance and buffered service level: Definitions and properties. Department of Industrial and Systems Engineering, University of Florida, research report 3. – reference: EverittBSThe Cambridge dictionary of statistics2006CambridgeCambridge University Press – reference: LandsmanZMValdezEATail conditional expectation for elliptical distributionsNorth American Actuarial Journal200374557110.1080/10920277.2003.10596118 – reference: MafusalovAShapiroAUryasevSEstimation and asymptotics for buffered probability of exceedanceEuropean Journal of Operational Research2018270382683610.1016/j.ejor.2018.01.021 – reference: NortonMMafusalovAUryasevSSoft margin support vector classification as buffered probability minimizationThe Journal of Machine Learning Research201718122852327 – reference: ArtznerPDelbaenFEberJMHeathDCoherent measures of riskMathematical Finance1999920322810.1111/1467-9965.00068 – volume-title: The Cambridge dictionary of statistics year: 2006 ident: 3373_CR4 – volume: 95 start-page: 499 year: 2010 ident: 3373_CR11 publication-title: Reliability Engineering & System Safety doi: 10.1016/j.ress.2010.01.001 – ident: 3373_CR1 – volume: 18 start-page: 2285 issue: 1 year: 2017 ident: 3373_CR9 publication-title: The Journal of Machine Learning Research – volume: 83 start-page: 465 issue: 1 year: 2016 ident: 3373_CR3 publication-title: Natural Hazards doi: 10.1007/s11069-016-2324-y – volume: 28 start-page: 1077 issue: 2 year: 2018 ident: 3373_CR8 publication-title: SIAM Journal on Optimization doi: 10.1137/15M1042644 – volume: 2 start-page: 21 issue: 3 year: 2000 ident: 3373_CR12 publication-title: The Journal of Risk doi: 10.21314/JOR.2000.038 – volume: 26 start-page: 1443 issue: 7 year: 2002 ident: 3373_CR14 publication-title: Journal of Banking & Finance doi: 10.1016/S0378-4266(02)00271-6 – ident: 3373_CR10 – ident: 3373_CR17 – volume: 110 start-page: 742 issue: 510 year: 2015 ident: 3373_CR15 publication-title: Journal of the American Statistical Association doi: 10.1080/01621459.2014.929522 – volume: 9 start-page: 203 year: 1999 ident: 3373_CR2 publication-title: Mathematical Finance doi: 10.1111/1467-9965.00068 – volume: 7 start-page: 55 issue: 4 year: 2003 ident: 3373_CR6 publication-title: North American Actuarial Journal doi: 10.1080/10920277.2003.10596118 – volume: 148 start-page: 297 issue: 1–2 year: 2014 ident: 3373_CR13 publication-title: Mathematical Programming doi: 10.1007/s10107-014-0801-1 – volume: 260 start-page: 501 issue: 1–2 year: 2018 ident: 3373_CR16 publication-title: Annals of Operations Research doi: 10.1007/s10479-016-2354-6 – volume: 270 start-page: 826 issue: 3 year: 2018 ident: 3373_CR7 publication-title: European Journal of Operational Research doi: 10.1016/j.ejor.2018.01.021 – volume: 28 start-page: 793 issue: 3 year: 1999 ident: 3373_CR5 publication-title: Communications in Statistics: Simulation and Computation doi: 10.1080/03610919908813579
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Snippet	Conditional value-at-risk (CVaR) and value-at-risk, also called the superquantile and quantile, are frequently used to characterize the tails of probability...
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SubjectTerms	Business and Management Combinatorics Confidence intervals Convex analysis Density Distribution (Probability theory) Engineering Financial risk Functions, Inverse Linear programming Logistics Mathematical analysis Measurement Method of moments Methods Operations research Operations Research/Decision Theory Optimization Portfolio management Quantiles Random variables Risk S.I.: Recent Developments in Financial Modeling and Risk Management Statistical analysis Theory of Computation
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Title	Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation
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