Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T09:36:24.317Z Has data issue: false hasContentIssue false
  • English
  • Français

ON SOME PROPERTIES OF TWO VECTOR-VALUED VAR AND CTE MULTIVARIATE RISK MEASURES FOR ARCHIMEDEAN COPULAS

Published online by Cambridge University Press:  16 June 2014

Werner Hürlimann
Werner Hürlimann*
Affiliation:
Wolters Kluwer Financial Services Switzerland AG, CH-8008 Zürich, Switzerland E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the multivariate Value-at-Risk (VaR) and Conditional-Tail-Expectation (CTE) risk measures introduced in Cousin and Di Bernardino (Cousin, A. and Di Bernardino, E. (2013) Journal of Multivariate Analysis, 119, 32–46; Cousin, A. and Di Bernardino, E. (2014) Insurance: Mathematics and Economics, 55(C), 272–282). For absolutely continuous Archimedean copulas, we derive integral formulas for the multivariate VaR and CTE Archimedean risk measures. We show that each component of the multivariate VaR and CTE functional vectors is an integral transform of the corresponding univariate VaR measures. For the class of Archimedean copulas, the marginal components of the CTE vector satisfy the following properties: positive homogeneity (PH), translation invariance (TI), monotonicity (MO), safety loading (SL) and VaR inequality (VIA). In case marginal risks satisfy the subadditivity (MSA) property, the marginal CTE components are also sub-additive and hitherto coherent risk measures in the usual sense. Moreover, the increasing risk (IR) or stop-loss order preserving property of the marginal CTE components holds for the class of bivariate Archimedean copulas. A counterexample to the (IR) property for the trivariate Clayton copula is included.

Keywords

multivariate riskcoherent risk measureincreasing riskArchimedean copulaKendall's process
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 44 , Issue 3 , September 2014 , pp. 613 - 633
Copyright
Copyright © ASTIN Bulletin 2014 

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

REFERENCES

Adam, A., Houkari, M. and Laurent, J.-P. (2008) Spectral risk measures and portfolio selection. Journal of Banking and Finance, 32 (9), 1870–1882.CrossRefGoogle Scholar
Albrecht, P. (2004) Risk measures. In Encyclopedia of Actuarial Science (eds. Teugels, J.L. and Sundt, B.), pp. 1493–1501. New York, NY: J. Wiley.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
Barbe, P., Genest, C., Ghoudi, K. and Rémillard, B. (1996) On Kendall's process. Journal of Multivariate Analysis, 58, 197–229.CrossRefGoogle Scholar
Bargès, M., Cossette, H. and Marceau, E. (2009) TVaR-based capital allocation with copulas. Insurance: Mathematics and Economics, 45 (3), 348–361.Google Scholar
Belzunce, F. (2010) An introduction to the theory of stochastic orders. Boletin de Estadistica e Investigacion Operativa (SEIO), 26 (1), 4–18.Google Scholar
Belzunce, F., Castaño, A., Olvera-Cervantes, A. and Suàrez-Llorens, A. (2007) Quantile curves and dependence structure for bivariate distributions. Computational Statistics & Data Analysis, 51 (10), 5112–5129.Google Scholar
Brechmann, E.C. (2014) Hierarchical Kendall copulas: Properties and inference. Canadian Journal of Statistics, 42 (1), 78–108.Google Scholar
Cai, J. and Li, H. (2005) Conditional tail expectations for multivariate phase-type distributions. Journal of Applied Probability, 42 (3), 810–825.CrossRefGoogle Scholar
Capéraà, P., Fougères, P.-L. and Genest, C. (1997) A stochastic ordering based on a decomposition of Kendall's tau. In Distributions with Given Marginals and Moment Problems (eds. Benes, V. and Stepân, J.), pp. 81–86. Dordrecht, Netherlands: Kluwer.CrossRefGoogle Scholar
Chakak, A. and Ezzerg, M. (2000) Bivariate contours of copulas. Communications in Statistics Simulation and Computation, 29 (1), 175–185.Google Scholar
Chakak, A. and Imlahi, L. (2001) Multivariate probability integral transformation: Application to maximum likelihood estimation. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas (RACSAM), 95 (2), 201–212.Google Scholar
Cousin, A. and Di Bernardino, E. (2013) On multivariate extensions of value-at-risk. Journal of Multivariate Analysis, 119, 32–46.Google Scholar
Cousin, A. and Di Bernardino, E. (2014) On multivariate extensions of conditional-tail-expectation. Insurance: Mathematics and Economics, 55(C), 272–282.Google Scholar
De Giorgi, E. (2005) Reward-risk portfolio selection and stochastic dominance. Journal of Banking and Finance, 29 (4), 895–926.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005) Actuarial Theory for Dependent Risks. New York, NY: J. Wiley.Google Scholar
Denuit, M. and Müller, A. (2004) Stochastic orderings. In Encyclopedia of Actuarial Science (eds. Teugels, J.L. and Sundt, B.), pp. 1606–1610. New York, NY: J. Wiley.Google Scholar
Embrechts, P. and Hofert, M. (2013) A note on genrealized inverses. Mathematical Methods of Operations Research, 77 (3), 423–432.Google Scholar
Embrechts, P., McNeil, A.J. and Straumann, D. (2002) Correlation and dependence in risk management: Properties and pitfalls. In Risk Management: Value-at-Risk and Beyond (ed. Dempster, M.), pp. 176–223. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Embrechts, P. and Puccetti, G. (2006) Bounds for functions of multivariate risks. Journal of Multivariate Analysis, 97 (2), 526–547.Google Scholar
Feng, M.B. (2011) Coherent Distortion Risk Measures in Portfolio Selection. Master's thesis, University of Waterloo, Waterloo, ON, Canada. URL: http://uwspace.uwaterloo.ca/bitstream/10012/6169/1/Feng_MingBin.pdfGoogle Scholar
Feng, M.B. and Tan, K.S. (2012) Coherent distortion risk measures in portfolio selection. Systems Engineering Procedia, 4, 25–34.Google Scholar
Föllmer, H. and Schied, A. (2002) Convex measures of risk and trading constraints. Finance and Stochastics, 6 (4), 429–447.Google Scholar
Föllmer, H. and Schied, A. (2010) Convex and coherent risk measures. In Encyclopedia of Quantitative Finance (ed. Cont, R.), pp. 355–363. New York, NY: J. Wiley.Google Scholar
Genest, C. and Boies, J.-C. (2003) Detecting dependence with Kendall plots. The American Statistician, 57, 275–284.Google Scholar
Genest, C. and MacKay, R.J. (1986a) Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. The Canadian Journal of Statistics, 14, 145–159.Google Scholar
Genest, C. and MacKay, R.J. (1986b) The joy of couplas: Bivariate distribution with uniform marginals. The American Statistician, 40 (4), 280–283.Google Scholar
Genest, C. and Rivest, L.-P. (1993) Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association, 88 (423), 1034–1043.Google Scholar
Genest, C., Quessy, J.-F. and Rémillard, B. (2002) Tests of serial independence based on Kendall's process. The Canadian Journal of Statistics, 30 (3), 1–21.CrossRefGoogle Scholar
Genest, C., Quessy, J.-F. and Rémillard, B. (2006) Goodness-of-fit procedures for copula models based on the probability integral transformation. Scandinavian Journal of Statistics, 33, 337–366.Google Scholar
Genest, C. and Rivest, L.-P. (2001) On the multivariate probability integral transform. Statistics & Probability Letters, 53, 391–399.CrossRefGoogle Scholar
Ghoudi, K., Khoudraji, K. and Rivest, L.-P. (1998) Propriétés statistiques de valeurs extrêmes bidimensionnelles. Canadian Journal of Statistics, 26, 187–197.CrossRefGoogle Scholar
Goovaerts, M., Linders, D., Van Weert, K. and Tank, F. (2012) On the interplay between distortion-, mean value- and Haezendonck-Goovaerts risk measures. Insurance: Mathematics and Economics, 51 (3), 10–18.Google Scholar
Hesselager, O. (1995) Order relations for some distributions. Insurance: Mathematics and Economics, 16 (2), 129–134.Google Scholar
Hürlimann, W. (1998) On stop-loss order and the distortion principle. ASTIN Bulletin 28 (1), 119–134.Google Scholar
Hürlimann, W. (2001) Analytical evaluation of economic risk capital for portfolios of Gamma risk. ASTIN Bulletin, 31 (1), 107–122.Google Scholar
Hürlimann, W. (2002a) On risk and price: Stochastic orderings and measures. Proceedings of the 27th International Congress of Actuaries, Cancun, Mexico.Google Scholar
Hürlimann, W. (2002b) An alternative approach to portfolio selection. Proceedings of the 12th International AFIR Colloquium, Cancun, Mexico.Google Scholar
Hürlimann, W. (2003) Conditional value-at-risk bounds for compound Poisson risks and a normal approximation. Journal of Applied Mathematics, 3 (3), 141–154.CrossRefGoogle Scholar
Hürlimann, W. (2004a) Multivariate Fréchet copulas and conditional value-at-risk. International Journal of Mathematics and Mathematical Sciences, 7 (5–8), 345–364.CrossRefGoogle Scholar
Hürlimann, W. (2004b) Distortion risk measures and economic capital. North American Actuarial Journal 8 (1), 86–95.Google Scholar
Hürlimann, W. (2005) A note on generalized distortion risk measures. Finance Research Letters, 3 (4), 267–272.Google Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability, vol. 73. London: Chapman & Hall.Google Scholar
Kaas, R., Heerwaarden, van A.E. and Goovaerts, M.J. (1994) Ordering of Actuarial Risks. CAIRE Education Series, vol. 1. Brussels, Belgium: CAIRE.Google Scholar
Karlin, S. and Novikoff, A. (1963) Generalized convex inequalities. Pacific Journal of Mathematics, 13, 1251–1279.Google Scholar
Kimberling, C.H. (1974) A probabilistic interpretation of complete monotonicity. Aequationes Mathematicae, 10, 152–164.Google Scholar
Kolev, N., Ulisses dos Anjos, U. and Mendes, B.V. de M. (2006) Copulas: A review and recent developments. Stochatic Models, 22 (4), 617–660.Google Scholar
Landsman, Z.M. and Valdez, E.A. (2003) Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7 (4), 55–71.Google Scholar
Lee, J. and Prékopa, A. (2013) Properties and calculation of multivariate risk measures: MVaR and MCVaR. Annals of Operations Research, 211 (1), 225–254.Google Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton, NJ: Princeton University Press.Google Scholar
McNeil, A.J. and Neslehova, J. (2009) Multivariate Archimedean copulas, d-monotone functions and ℓ 1-norm symmetric distributions. The Annals of Statistics, 37 (5b), 3059–3097.Google Scholar
Müller, A. and Stoyan, D. (2002) Comparison Methods for Stochastic Models and Risks. New York, NY: J. Wiley.Google Scholar
Nadarajah, S., Zhang, B. and Chan, S. (2013) Estimation methods for expected shortfall. Quantitative Finance 14 (2), 271–291.CrossRefGoogle Scholar
Nappo, G. and Spizzichino, F. (2009) Kendall distributions and level sets in bivariate exchangeable survival models. Information Sciences 179, 2878–2890.CrossRefGoogle Scholar
Nelsen, R.B. (2006) An Introduction to Copulas. Lecture Notes in Statistics, vol. 139, 2nd ed. New York, NY: Springer-Verlag.Google Scholar
Nelsen, R.B., Quesada-Molina, J.J., Rodriguez-Lallena, J.A. and Ùbeda-Flores, M. (2001) Distribution functions of copulas: A class of probability integral transforms. Statistics & Probability Letters, 54, 277–282.Google Scholar
Nelsen, R.B., Quesada-Molina, J.J., Rodriguez-Lallena, J.A. and Ùbeda-Flores, M. (2003) Kendall distribution functions. Statistics & Probability Letters, 65, 263–268.Google Scholar
Ohlin, J. (1969) On a class of measures of dispersion with application to optimal reinsurance. ASTIN Bulletin, 5, 249–266.Google Scholar
Roman, D. and Mitra, G. (2009) Portfolio selection models: A review and new directions. Willmott Journal 1 (2), 69–85.Google Scholar
Schweizer, B. and Sklar, A. (1961) Associative functions and statistical triangle inequalities. Publicationes Mathematicae-Debrecen, 8, 169–186.Google Scholar
Sereda, E., Bronshtein, E., Rachev, S., Fabozzi, F., Sun, W. and Stoyanov, S. (2009) Distortion risk measures in portfolio optimization. In The Handbook of Portfolio Construction: Contemporary Applications of Markowitz Techniques (ed. Guerard, J.), pp. 649–674. New York, NY: Springer.Google Scholar
Shaked, M. and Shanthikumar, J.G. (1994) Stochastic Orders and Their Applications. New York, NY: Academic Press.Google Scholar
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. Springer Series in Statistics. New York, NY: Springer.Google Scholar
Sklar, A. (1959) Fonctions de répartition et leurs marges. Publications de l'Institut Statistique de l'Université de Paris 8, 229–231.Google Scholar
Tibiletti, L. (1993) On a new notion of multidimensional quantile. Metron International Journal of Statistics, 51(3–4), 77–83.Google Scholar
Tibiletti, L. (1995) Quasi-concavity property of multivariate distribution functions. Ratio Mathematica, 9, 27–36.Google Scholar
Wang, S., Young, V.R. and Panjer, H.H. (1997) Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics, 21, 173–183.Google Scholar
Wirch, J.L. and Hardy, M.R. (1999) A synthesis of risk measures for capital adequacy. Insurance: Mathematics and Economics, 25, 337–347.Google Scholar
Young, V.R. (2004) Premium principles. In Encyclopedia of Actuarial Science (eds. Teugels, J.L. and Sundt, B.), pp. 1322–1331. New York, NY: J. Wiley.Google Scholar