European Actuarial Journal

, Volume 7, Issue 1, pp 277–296 | Cite as

Skew-elliptical distributions with applications in risk theory

  • Tomer Shushi
Original Research Paper


In this paper we derive important properties of the well-known skew-elliptical (SE) distributions which was introduced in Azzalini and Capitanio (J R Stat Soc Ser B (Stat Methodol) 65:367–389, 2003), and includes the more familiar skew-normal, skew-Student-t and skew-logistic distributions. We then derive the tail value at risk (TVaR) for a portfolio of SE risks. We provide the portfolio risk decomposition with TVaR. Furthermore, we obtain the Esscher premium principle, the weighted-premium principle, and the entropic risk measure with the underlying SE distributions. We also provide an explicit closed-form solution to the optimal portfolio selection with the SE distributions, and provide a numerical simulation of the results.


Esscher premium Loss distributions Optimal portfolio selection Skew-elliptical distributions Tail value at risk 



I thank the anonymous referees for their very useful comments.


  1. 1.
    Aas K, Haff IH (2006) The generalized hyperbolic skew student’s t-distribution. J Financ Econom 4:275–309CrossRefGoogle Scholar
  2. 2.
    Adcock CJ (2010) Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution. Ann Oper Res 176:221–234MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Adcock C, Eling M, Loperfido N (2015) Skewed distributions in finance and actuarial science: a review. Eur J Finance 21:1253–1281CrossRefGoogle Scholar
  4. 4.
    Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Finance 9:203–228MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83:715–726MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew-normal distribution. J R Stat Soc B 61:579–602MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J R Stat Soc Ser B (Stat Methodol) 65:367–389MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bernardi M, Maruotti A, Petrella L (2012) Skew mixture models for loss distributions: a Bayesian approach. Insur Math Econ 51:617–623MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bühlmann H (1980) An economic premium principle. ASTIN Bull 11:52–60MathSciNetCrossRefGoogle Scholar
  10. 10.
    Branco MD, Dey DK (2001) A general class of multivariate skew-elliptical distributions. J Multivar Anal 79:99–113MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Deniz EG, Polo FV, Bastida AH (2000) Robust Bayesian premium principles in actuarial science. Statistician 49:241–252Google Scholar
  12. 12.
    Eling M (2012) Fitting insurance claims to skewed distributions: are the skew-normal and skew-student good models? Insur Math Econ 51:239–248MathSciNetCrossRefGoogle Scholar
  13. 13.
    Eling M (2014) Fitting asset returns to skewed distributions: are the skew-normal and skew-student good models? Insur Math Econ 59:45–56MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fabozzi FJ, Gupta F, Markowitz HM (2002) The legacy of modern portfolio theory. J Invest 11:7–22CrossRefGoogle Scholar
  15. 15.
    Föllmer H, Knispel T (2011) Entropic risk measures: coherence vs. convexity, model ambiguity and robust large deviations. Stoch Dyn 11:333–351MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Framstad NC (2011) Portfolio separation properties of the skew-elliptical distributions, with generalizations. Stat Probab Lett 81:1862–1866MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Furman E, Zitikis R (2008) Weighted premium calculation principles. Insur Math Econ 42:459–465MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Francis JC, Kim D (2013) Modern portfolio theory: foundations, analysis, and new developments, vol 795. Wiley, New YorkGoogle Scholar
  19. 19.
    Genton MG (ed) (2004) Skew-elliptical distributions and their applications: a journey beyond normality. CRC Press, Boca RatonMATHGoogle Scholar
  20. 20.
    Genton MG, Loperfido NM (2005) Generalized skew-elliptical distributions and their quadratic forms. Ann Inst Stat Math 57:389–401MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Goh JW, Lim KG, Sim M, Zhang W (2012) Portfolio value-at-risk optimization for asymmetrically distributed asset returns. Eur J Oper Res 221:397–406MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Goovaerts M, De Vylder FE, Haezendonck J (1984) Insurance premiums, North-HollandGoogle Scholar
  23. 23.
    Kamps U (1998) On a class of premium principles including the Esscher principle. Scand Actuar J 1:75–80MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Landsman Z (2008) Minimization of the root of a quadratic functional under an affine equality constraint. J Comput Appl Math 216:319–327MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Landsman Z (2004) On the generalization of Esscher and variance premiums modified for the elliptical family of distributions. Insur Math Econ 35:563–579MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Landsman ZM, Valdez EA (2003) Tail conditional expectations for elliptical distributions. N Am Actuar J 7:55–71MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Landsman Z, Makov U, Shushi T (2013) Tail conditional expectations for generalized skew—elliptical distributions. Available at SSRN 2298265Google Scholar
  28. 28.
    Landsman Z, Makov U, Shushi T (2016) Multivariate tail conditional expectation for elliptical distributions. Insur Math Econ 70:216–223MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lane MN (2000) Pricing risk transfer transactions. ASTIN Bull 30:259–293MathSciNetCrossRefGoogle Scholar
  30. 30.
    Liu M, Lin TI (2015) Skew-normal factor analysis models with incomplete data. J Appl Stat 42:789–805MathSciNetCrossRefGoogle Scholar
  31. 31.
    Panjer HH (2002) Measurement of risk, solvency requirements and allocation of capital within financial conglomerates. University of Waterloo, Institute of Insurance and Pension Research, WaterlooGoogle Scholar
  32. 32.
    Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Finance 26:1443–1471CrossRefGoogle Scholar
  33. 33.
    Rubinstein M (2002) Markowitz’s portfolio selection: a fifty-year retrospective. J Finance 57:1041–1045CrossRefGoogle Scholar
  34. 34.
    Shaw RA, Smith AD, Spivak GS (2011) Measurement and modelling of dependencies in economic capital. Br Actuar J 16:601CrossRefGoogle Scholar
  35. 35.
    Smith MS, Gan Q, Kohn RJ (2012) Modelling dependence using skew t copulas: Bayesian inference and applications. J Appl Econom 27:500–522MathSciNetCrossRefGoogle Scholar
  36. 36.
    Theiler J, Scovel C (2009) Uncorrelated versus independent elliptically-contoured distributions for anomalous change detection in hyperspectral imagery. In: IS&T/SPIE electronic imaging, vol 7246. International Society for Optics and Photonics, pp 1–12. doi: 10.1117/12.814325
  37. 37.
    Van Heerwaarden AE, Kaas R, Goovaerts MJ (1989) Properties of the Esscher premium calculation principle. Insur Math Econ 8:261–267MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Vernic R (2005) On the multivariate skew-normal distribution and its scale mixtures. An St Univ Ovidius Constanta 2:83–96MathSciNetMATHGoogle Scholar
  39. 39.
    Vernic R (2006) Multivariate skew-normal distributions with applications in insurance. Insur Math Econ 38:413–426MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Wang S, Dhaene J (1998) Comonotonicity, correlation order and premium principles. Insur Math Econ 22:235–242MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© EAJ Association 2017

Authors and Affiliations

  1. 1.Ben-Gurion University of the NegevBeershebaIsrael

Personalised recommendations