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Modelling random vectors of dependent risks with different elliptical components

Published online by Cambridge University Press:  22 February 2021

Zinoviy Landsman  and
Tomer Shushi
Zinoviy Landsman
Affiliation:
Actuarial Research Center, Department of Statistics, University of Haifa, Haifa3498838, Israel Holon Institute of Technology, Holon5810201, Israel
Tomer Shushi*
Affiliation:
Department of Business Administration, Guilford Glazer Faculty of Business and Management, Ben-Gurion University of the Negev, Beer-Sheva8410501, Israel
*
*Corresponding author. E-mail: [email protected]
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Abstract

In Finance and Actuarial Science, the multivariate elliptical family of distributions is a famous and well-used model for continuous risks. However, it has an essential shortcoming: all its univariate marginal distributions are the same, up to location and scale transformations. For example, all marginals of the multivariate Student’s t-distribution, an important member of the elliptical family, have the same number of degrees of freedom. We introduce a new approach to generate a multivariate distribution whose marginals are elliptical random variables, while in general, each of the risks has different elliptical distribution, which is important when dealing with insurance and financial data. The proposal is an alternative to the elliptical copula distribution where, in many cases, it is very difficult to calculate its risk measures and risk capital allocation. We study the main characteristics of the proposed model: characteristic and density functions, expectations, covariance matrices and expectation of the linear regression vector. We calculate important risk measures for the introduced distributions, such as the value at risk and tail value at risk, and the risk capital allocation of the aggregated risks.

Keywords

Elliptical distributionsSpherical distributionsMulti-spherical distributionsRisk measuresTail value at riskValue at risk
Type
Original Research Paper
Information
Annals of Actuarial Science , Volume 16 , Issue 1 , March 2022 , pp. 6 - 24
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Institute and Faculty of Actuaries

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References

Balakrishna, N. & Shiji, K. (2014). On a class of bivariate exponential distributions. Statistics and Probability Letters, 85, 153–180.CrossRefGoogle Scholar
Cambanis, S., Steel, H. & Gordon, S. (1981). On the theory of elliptically contoured distributions. Journal of Multivariate Analysis, 11, 368–385.CrossRefGoogle Scholar
Efthimiou, C. (2010). Introduction to Functional Equations. Mathematical Sciences Research Institute (MSRI).Google Scholar
Fang, H.B., Fang, K.T. & Kotz, S. (2002). The Meta-elliptical distributions with given marginals. Journal of Multivariate Analysis, 82(1), 1–16.CrossRefGoogle Scholar
Fang, K.T, Kotz, S. & Ng, K.W. (1990). Symmetric Multivariate and Related Distributions. United Kingdom: Chapman and Hall.CrossRefGoogle Scholar
Frahm, G. (2004). Generalized Elliptical Distributions: Theory and Applications (Doctoral dissertation, Universitätsbibliothek).Google Scholar
GÓmez, H.W., Quintana, F.A. & Torres, F.J. (2007). A new family of slash-distributions with elliptical contours. Statistics & Probability Letters, 77(7), 717–725.CrossRefGoogle Scholar
Ignatieva, K., & Landsman, Z. (2015). Estimating the tails of loss severity via conditional risk measures for the family of symmetric generalized hyperbolic distributions. Insurance: Mathematics and Economics, 65, 172–186.Google Scholar
Kring, S., Rachev, S., HÖchstÖtter, M., Fabozzi, F. & Bianch, M.L. (2009). Multi-tail generalized elliptical distributions for asset returns. Econometrics Journal, 12, 272–291.CrossRefGoogle Scholar
Krishnamoorthy, A.S. & Parthasarathy, M.A. (1951). Multivariate gamma-type distribution. The Annals of Mathematical Statistics, 22(4), 549–557.CrossRefGoogle Scholar
Landsman, Z. (2004). On the generalization of Esscher and variance premiums modified for the elliptical family of distributions. Insurance: Mathematics and Economics, 35, 563–579.Google Scholar
Landsman, Z. & Makov, U. (2011). Translation-invariant and positive-homogeneous risk measures and optimal portfolio management. European Journal of Finance, 17(4), 307–320.CrossRefGoogle Scholar
Landsman, Z. & Neslehova, J. (2008). Stein’s Lemma for elliptical random vectors. Journal of Multivariate Analysis, 99(5), 912–927.CrossRefGoogle Scholar
Landsman, Z.M. & Valdez, E.A. (2003). Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7, 55–71.CrossRefGoogle Scholar
Landsman, Z.M. & Valdez, E.A. (2005). Tail conditional expectations for exponential dispersion models. Astin Bulletin, 35, 189–209.CrossRefGoogle Scholar
Landsman, Z., Makov, U. & Shushi, T. (2016). Multivariate tail conditional expectation for elliptical distributions. Insurance: Mathematics and Economics, 70, 216–223.Google Scholar
Landsman, Z., Makov, U. & Shushi, T. (2018). A Multivariate tail covariance measure for elliptical distributions. Insurance: Mathematics and Economics, 81, 27–35.Google Scholar
Masjed-Jamei, M. & Koepf, W. (2019). A new identity for generalized hypergeometric functions and applications. Axioms, 8, 12; doi: 10.3390/axioms8010012.CrossRefGoogle Scholar
Nadarajah, S. (2010). Simple expressions for a bivariate chi-square distribution. Statistics, 44(2), 189–201.CrossRefGoogle Scholar
Ollila, E. & Koivunen, V. (2004). Generalized complex elliptical distributions. In Processing Workshop Proceedings, 2004 Sensor Array and Multichannel Signal (pp. 460–464). IEEE.Google Scholar
Van Dan Berg, J., Roux, J. & Bekker, A. (2013). A bivariate generalization of gamma distribution. Communications in Statistics - Theory and Methods, 42(19), 3514–3527.CrossRefGoogle Scholar