Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-30T05:41:31.915Z Has data issue: false hasContentIssue false
  • English
  • Français

MULTIVARIATE GEOMETRIC TAIL- AND RANGE-VALUE-AT-RISK

Published online by Cambridge University Press:  21 October 2019

Klaus Herrmann Open the ORCID record for Klaus Herrmann [Opens in a new window] ,
Marius Hofert  and
Mélina Mailhot
Klaus Herrmann*
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada, E-Mail: klaus.herrmann@uwaterloo.ca
Marius Hofert
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada, E-Mail: marius.hofert@uwaterloo.ca
Mélina Mailhot
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada, E-Mail: melina.mailhot@concordia.ca
*
E-Mail: klaus.herrmann@uwaterloo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

A generalization of range-value-at-risk (RVaR) and tail-value-at-risk (TVaR) for d-dimensional distribution functions is introduced. Properties of these new risk measures are studied and illustrated. We provide special cases, applications and a comparison with traditional univariate and multivariate versions of the TVaR and RVaR.

Keywords

Multivariate risk measurestail-value-at-riskrange-value-at-riskgeometric quantilesdependence
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 1 , January 2020 , pp. 265 - 292
Copyright
© Astin Bulletin 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, I. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th. Mineola: Dover Publications.Google Scholar
Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9(3), 203–228.CrossRefGoogle Scholar
Ben Tahar, I. (2006) Tail conditional expectation for vector-valued risks. SFB 649 Discussion Papers.Google Scholar
Chaudhuri, P. (1996) On a geometric notion of quantiles for multivariate data. Journal of the American Statistical Assosiation, 91(434), 862–872.CrossRefGoogle Scholar
Cherubini, U., Luciano, E. and Vecchiato, W. (2004) Copula Methods in Finance. Hoboken: John Wiley & Sons.CrossRefGoogle Scholar
Cont, R., Deguest, R. and Scandolo, G. (2010) Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance, 10(6), 593–606.CrossRefGoogle Scholar
Cossette, H., Mailhot, M. and Marceau, É. (2012) TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts. Insurance: Mathematics and Economics, 50(2), 247–256.Google Scholar
Cossette, H., Mailhot, M., Marceau, É. and Mesfioui, M. (2013) Bivariate lower and upper orthant value-at-risk. European Actuarial Journal, 3(2), 321–357.CrossRefGoogle Scholar
Cossette, H., Mailhot, M., Marceau, É. and Mesfioui, M. (2016) Vector-valued tail value-at-risk and capital allocation. Methodology and Computing in Applied Probability, 18(3), 653–674.CrossRefGoogle Scholar
Di Bernardino, E., Fernández-Ponce, J.M., Palacios-Rodríguez, F. and Rodríguez-Griñolo, M.R. (2015) On multivariate extensions of the conditional value-at-risk measure. Insurance: Mathematics and Economics, 61, 1–16.Google Scholar
Di Bernardino, E., Laloë, T., Maume-Deschamps, V. and Prieur, C. (2013) Plug-in estimation of level sets in a non-compact setting with applications in multivariable risk theory. ESAIM: Probability and Statistics, 17, 236–256.CrossRefGoogle Scholar
Embrechts, P. and Puccetti, G. (2006) Bounds for functions of multivariate risks. Journal of Multivariate Analysis, 97(2), 526–547.CrossRefGoogle Scholar
Fang, K.T., Kotz, S. and Ng, K.W. (1990) Symmetric Multivariate and Related Distributions. London: Chapman & Hall.CrossRefGoogle Scholar
Fishman, G. (1996) Monte Carlo: Concepts, Algorithms, and Applications, Springer.CrossRefGoogle Scholar
Girard, S. and Stupfler, G. (2015) Extreme geometric quantiles in a multivariate regular variation framework. Extremes, 18, 629–663.CrossRefGoogle Scholar
Girard, S. and Stupfler, G. (2017) Intriguing properties of extreme geometric quantiles. Revstat, 15, 107–139.Google Scholar
Harman, R. and Lacko, V. (2010) On decompositional algorithms for uniform sampling from n-spheres and n-balls. Journal of Multivariate Analysis, 101, 2297–2304.CrossRefGoogle Scholar
Herrmann, K., Hofert, M. and Mailhot, M. (2018) Multivariate geometric expectiles. Scandinavian Actuarial Journal, 7, 629–659.CrossRefGoogle Scholar
Johnson, N.L. and Kotz, S. (1970) Continuous Univariate Distributions - 2, 1st edition. Boston: Houghton Mifflin Company.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008) Modern Actuarial Risk Theory, 2nd edition. New York: Springer.CrossRefGoogle Scholar
Kim, J. and Hardy, M. (2009) A capital allocation based on a solvency exchange option. Insurance: Mathematics and Economics, 44(3), 357–366.Google Scholar
Koenker, R. and Bassett, J.R. (1978) Regression quantiles. Econometrica, 46(1), 33–50.CrossRefGoogle Scholar
Koltchinskii, V.I. (1997) M-estimation, convexity and quantiles. Annals of Statistics, 25(2), 435–477.CrossRefGoogle Scholar
Lemieux, C. (2009) Monte Carlo and Quasi-Monte Carlo Sampling, New York: Springer-Verlag.Google Scholar
Maume-Deschamps, V., Rullière, D. and Said, K. (2017) Multivariate extensions of expectiles risk measures. Dependence Modeling, 5(1), 20–44.CrossRefGoogle Scholar
Mcneil, A., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management: Concepts, Techniques, and Tools, 2nd edition. Princeton: Princeton University Press.Google Scholar
Nelsen, R.B. (2006) An Introduction to Copulas, New York: Springer-Verlag.CrossRefGoogle Scholar
Newey, W. and Powell, J. (1987) Asymmetric least squares estimation and testing. Econometrica, 55(4), 819–847.CrossRefGoogle Scholar
Nuttal, A.H. (1975) Some integrals involving the QM function. IEEE Transactions on Information Theory, 21(1), 95–96.CrossRefGoogle Scholar
Rubin, L., Ranson, N. and Shi, X. (2009) Analysis of Methods for Determining Margins for Uncertainty under a Principle-Based Framework for Life Insurance and Annuity Products. Society of Actuaries, March.CrossRefGoogle Scholar
Small, C. (1990) A survey of multidimensional medians. International statistical review, 58(3), 263–277.Google Scholar
Stoer, J. and Bulirsch, R. (2002) Introduction to Numerical Analysis, 3rd edition. New York: Springer.Google Scholar