Show simple item record

Risk measures: A generalization from the univariate to the matrix-variate

dc.contributor.authorArias-Serna M.A
dc.contributor.authorCaro-Lopera F.J
dc.contributor.authorLoubes J.-M.
dc.date.accessioned2022-09-14T14:34:00Z
dc.date.available2022-09-14T14:34:00Z
dc.date.created2021
dc.identifier.issn14651211
dc.identifier.urihttp://hdl.handle.net/11407/7554
dc.descriptionThis paper develops a method for estimating value-at-risk and conditional value-at-risk when the underlying risk factors follow a beta distribution in a univariate and a matrix-variate setting. For this purpose, we connect the theory of the Gaussian hypergeometric function of matrix argument and integration over positive definite matrixes. For certain choices of the shape parameters, a and b, analytical expressions of the risk measures are developed. More generally, a numerical solution for the risk measures for any parameterization of beta-distributed loss variables is presented. The proposed risk measures are finally used for quantifying the potential risk of economic loss in credit risk. © 2021 Infopro Digital Risk (IP) Limited.eng
dc.language.isoeng
dc.publisherInfopro digital
dc.relation.isversionofhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85107435358&doi=10.21314%2fJOR.2021.003&partnerID=40&md5=eb87d780c78b49ca91526073f2c86679
dc.sourceJournal of Risk
dc.titleRisk measures: A generalization from the univariate to the matrix-variate
dc.typeArticle
dc.rights.accessrightsinfo:eu-repo/semantics/restrictedAccess
dc.publisher.programCiencias Básicas
dc.publisher.programIngeniería Financiera
dc.type.spaArtículo
dc.identifier.doi10.21314/JOR.2021.003
dc.subject.keywordBeta distributioneng
dc.subject.keywordGaussian hypergeometric function of matrix argu-menteng
dc.subject.keywordPositive definite matrixeseng
dc.subject.keywordRisk measureseng
dc.relation.citationvolume23
dc.relation.citationissue4
dc.relation.citationstartpage1
dc.relation.citationendpage20
dc.publisher.facultyFacultad de Ingenierías
dc.publisher.facultyFacultad de Ciencias Básicas
dc.affiliationArias-Serna, M.A., Faculty of Engineering, University of Medellín, Cra. 87, Medellín, 30-65, Colombia
dc.affiliationCaro-Lopera, F.J., Faculty of Basic Sciences, University of Medellín, Cra. 87, Medellín, 30-65, Colombia
dc.affiliationLoubes, J.-M., Toulouse Mathematics Institute, University Paul Sabatier, Bat 1R1, Bureau 109, Toulouse, 31062, France
dc.relation.referencesAlexander, C., (2008) Market Risk Analysis IV: Value at Risk Models, , Wiley
dc.relation.referencesAndrews, L. C., (1998) Special Functions of Mathematics for Engineers, , https://doi.org/10.1117/3.270709, Oxford University Press
dc.relation.referencesArias-Serna, M. A., Puerta-Yépes, M. E., Escalante-Coterio, C. E., Arango-Ospina, G., Q
dc.relation.referencesr/ model with CVaR˛ of costs minimization (2016) Journal of Industrial and Management Optimization, 13 (1), pp. 135-146. , https://doi.org/10.3934/jimo.2016008
dc.relation.referencesArtzner, P., Delbaen, F., Eber, J., Heath, D., Coherent measures of risk (1999) Mathematical Finance, 9, pp. 203-228. , https://doi.org/10.1080/10920277.1999.10595795
dc.relation.referencesBall, J. S., Automatic computation of zeros of Bessel functions and other special functions (2000) SIAM Journal on Scientific Computing, 21 (4), pp. 1458-1464. , https://doi.org/10.1137/s1064827598339074
dc.relation.referencesCousin, A., Di Bernardino, E., On multivariate extensions of value-at-risk (2013) Journal of Multivariate Analysis, 119, pp. 32-46. , https://doi.org/10.1016/j.jmva.2013.03.016
dc.relation.referencesCrouhy, M., Galai, D., Mark, R., A comparative analysis of current credit risk models (2000) Journal of Banking and Finance, 24 (1), pp. 59-117. , https://doi.org/10.1016/s0378-4266(99)00053-9
dc.relation.referencesDominici, D., Johnston, S. J., Jordaan, K., Real zeros of 2F1 hypergeometric polynomials (2013) Journal of Computational and Applied Mathematics, 247, pp. 152-161. , https://doi.org/10.1016/j.cam.2012.12.024
dc.relation.referencesDriver, K., Möller, M., Zeros of the hypergeometric polynomials F . n
dc.relation.referencesbI 2nI z (2001) Journal of Approximation Theory, 110 (1), pp. 74-87. , https://doi.org/10.1006/jath.2000.3548
dc.relation.referencesDuren, P. L., Guillou, B. J., Asymptotic properties of zeros of hypergeo-metric polynomials (2001) Journal of Approximation Theory, 111 (2), pp. 329-343. , https://doi.org/10.1006/jath.2001.3580
dc.relation.referencesEmbrechts, P., Puccetti, G., Bounds for functions of multivariate risks (2006) Journal of Multivariate Analysis, 97 (5), pp. 26-547. , https://doi.org/10.1016/j.jmva.2005.04.001
dc.relation.referencesErdelyi, A., (1955) Higher Transcendental Functions, , Bateman Manuscript Project. Mc-Graw Hill
dc.relation.referencesFarnum, N., Stanton, L., Some results concerning the estimation of beta distribution parameters in PERT (1987) Journal of the Operational Research Society, 38 (3), pp. 287-290. , https://doi.org/10.1057/jors.1987.45
dc.relation.referencesGabrielsen, A., Kirchiner, A., Liu, Z., Zagagli, P., Forecasting value-at-risk with time-varying variance, skewness and kurtosis in an exponential weighted moving aver-age framework (2015) Annals of Financial Economics, 10 (1), pp. 1-29. , https://doi.org/10.1142/S2010495215500050
dc.relation.referencesGamboa, F., Garivier, A., Iooss, B., Labopin-Richard, T., Bregman superquan-tiles: estimation methods and applications (2016) Dependence Modeling, 2016 (4), pp. 76-108. , https://doi.org/10.1515/demo-2016-0004
dc.relation.referencesGlasserman, P., Heidelberger, P., Shahabuddin, P., Portfolio value-at-risk with heavy tailed risk factors (2002) Mathematical Finance, 12 (3), pp. 239-269. , https://doi.org/10.1111/1467-9965.00141
dc.relation.referencesGuégan, D., Tarrant, W., On the necessity of five risk measures (2012) Annals of Finance, 8 (4), pp. 533-552. , https://doi.org/10.1007/s10436-012-0205-2
dc.relation.referencesGupta, A., Nagar, D., Matrix Variate Distributions (2000) Monographs and Surveys in Pure and Applied Mathematics, , Chapman & Hall/CRC
dc.relation.referencesHan, C., Liu, W., Chen, T., VaR/CVaR estimation under stochastic volatility models (2014) International Journal of Theoretical and Applied Finance, 17 (2), pp. 1-35. , https://doi.org/10.1142/S0219024914500095
dc.relation.referencesJohnson, D., The triangular distribution as a proxy for the beta distribution in risk analysis (1997) Journal of the Royal Statistical Society D, 46, pp. 387-398. , https://doi.org/10.1111/1467-9884.00091
dc.relation.referencesJohnson, N. L., Kotz, S., (1970) Continuous Univariate Distributions, , Houghton Mifflin Harcourt, Boston, MA
dc.relation.referencesJorion, P., (2007) Financial Risk Manager Handbook, , 4th edn. Wiley
dc.relation.references(1996) RiskMetrics™, , https://bit.ly/3uECVWa, JPMorgan Technical Document (fourth edition), December 17, JPMorgan/Reuters
dc.relation.referencesKhatri, C. G., On the mutual independence of certain statistics (1959) Annals of Mathematical Statistics, 30 (4), pp. 1258-1262. , https://doi.org/10.1214/aoms/1177706112
dc.relation.referencesKoev, P., Edelman, A., The efficient evaluation of the hypergeometric function of a matrix argument (2006) Mathematics of Computation, 75, pp. 833-846. , https://doi.org/10.1090/S0025-5718-06-01824-2
dc.relation.referencesLin, C., Shen, S., Can the student-t distribution provide accurate value at risk? (2006) Journal of Risk Finance, 7 (3), pp. 292-300. , https://doi.org/10.1108/15265940610664960
dc.relation.referencesMarinelli, C., Dáddona, E., Rachev, S., Multivariate heavy-tailed models for value-at-risk estimation (2012) International Journal of Theoretical and Applied Finance, 15 (4), pp. 1-32. , https://doi.org/10.1142/S021902491250029
dc.relation.referencesMcNeil, A. J., Frey, R., Embrechts, P., (2015) Quantitative Risk Management: Concepts, Techniques and Tools, , Princeton University Press, Princeton, NJ
dc.relation.referencesMuirhead, R. J., Aspects of Multivariate Statistical Theory (2005), Wiley Series in Proba-bility and Mathematical Statistics. Wiley
dc.relation.referencesOlkin, I., Rubin, H., Multivariate beta distributions and independence properties of the Wishart distribution (1964) Annals of Mathematical Statistics, 35 (1), pp. 261-269. , https://doi.org/10.1214/aoms/1177703748
dc.relation.referencesPflug, G., Some remarks on the value at risk and the conditional value at risk (2000) Probabilistic Constrained Optimization: Methodology and Applications, , https://doi.org/10.1007/978-1-4757-3150-715, Uryasev, S. (ed). Kluwer, Dordrecht
dc.relation.referencesRockafellar, R. T., Uryasev, S., Optimization of conditional value-at-risk (2000) The Journal of Risk, 2, pp. 21-41. , https://doi.org/10.21314/JOR.2000.038
dc.relation.referencesRockafellar, R. T., Uryasev, S., Conditional value-at-risk for general loss dis-tributions (2002) Journal of Banking and Finance, 26 (7), pp. 1443-1471. , https://doi.org/10.1016/S0378-4266(02)00271-6
dc.relation.referencesSharon, N., Itai, U., Approximation schemes for functions of positive-definite matrix values (2013) IMA Journal of Numerical Analysis, 33 (4), pp. 1436-1468. , https://doi.org/10.1093/imanum/drs049
dc.relation.referencesSrivastava, H., Zhou, J., Wang, Z., Asymptotic distribution of the zeros of certain classes of hypergeometric functions and polynomials (2011) Mathematics of Computation, 80 (2), pp. 1769-1784. , https://doi.org/10.1090/S0025-5718-2011-02409-9
dc.relation.referencesStavroyiannis, S., Makris, I., Nikolaidis, V., Zarangas, L., Econometric modeling and value-at-risk using the Pearson type-IV distribution (2012) International Review of Financial Analysis, 22, pp. 10-17. , https://doi.org/10.1016/j.irfa.2012.02.003
dc.relation.references(2008) Reglas relativas a la gestión del riesgo crediticio, , Superintendencia Financiera de Colombia Article, November 11, SFC
dc.relation.referencesTorres, R., Lillo, R. E., Laniado, H., A directional multivariate value at risk (2015) Insurance: Mathematics and Economics, 65 (C), pp. 111-123. , https://doi.org/10.1016/j.insmatheco.2015.09.002
dc.relation.referencesWagalath, L., Zubelli, J., A liquidation risk adjustment for value at risk and expected shortfall (2018) International Journal of Theoretical and Applied Finance, , https://doi.org/10.1142/S0219024918500103
dc.relation.referencesWang, J. Z., A note on estimation in the four-parameter beta distribution (2005) Maxi-mum, Communications in Statistics – Simulation and Computation, 34, pp. 495-500. , https://doi.org/10.1081/SAC-200068514
dc.relation.referencesWard, L., Lee, D., Practical application of risk-adjusted return on capital framework (2002), Dynamic Financial Analysis Discussion Paper, Casualty Actuarial Society
dc.relation.referencesWelbers, K., Van Atteveldt, W., Benoit, K., Text analysis in R (2017) Communica-tion Methods and Measures, 11 (4), pp. 245-265. , https://doi.org/10.1080/19312458.2017.1387238
dc.type.coarhttp://purl.org/coar/resource_type/c_6501
dc.type.versioninfo:eu-repo/semantics/publishedVersion
dc.type.driverinfo:eu-repo/semantics/article
dc.identifier.reponamereponame:Repositorio Institucional Universidad de Medellín
dc.identifier.repourlrepourl:https://repository.udem.edu.co/
dc.identifier.instnameinstname:Universidad de Medellín


Files in this item

FilesSizeFormatView

There are no files associated with this item.

This item appears in the following Collection(s)

Show simple item record