Computational Economics

, Volume 25, Issue 1–2, pp 189–205 | Cite as

Aggregation of Dependent Risks Using the Koehler–Symanowski Copula Function



This study examines the Koehler and Symanovski copula function with specific marginals, such as the skew Student-t, the skew generalized secant hyperbolic, and the skew generalized exponential power distributions, in modelling financial returns and measuring dependent risks. The copula function can be specified by adding interaction terms to the cumulative distribution function for the case of independence. It can also be derived using a particular transformation of independent gamma functions. The advantage of using this distribution relative to others lies in its ability to model complex dependence structures among subsets of marginals, as we show for aggregate dependent risks of some market indices.


asset returns IFM method measures of dependence minimum distance estimation skew distributions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ayebo, A. and Kozubowski, T.J. (2003). An asymmetric generalization of gaussian and Laplace laws. Journal of Probability and Statistical Science, 1(2), 187–210.Google Scholar
  2. Caputo, A. (1998). Some properties of the family of Koehler-Symanowski distributions. The Collaborative Research Center (SBF) 386, Discussion Paper No. 103, University of Munich.Google Scholar
  3. Cherubini, U., Luciano, E. and Vecchiato, W. (2004). Copula Methods in Finance. Wiley, New York.Google Scholar
  4. Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance, 1, 223–236.CrossRefGoogle Scholar
  5. Cook, R.D. and Johnson, M.E. (1981). A family of distributions for modelling non-elliptically symmetric multivariate data. Journal of the Royal Statististical Society B, 43, 210–218.Google Scholar
  6. Cook, R.D. and Johnson, M.E. (1987). Generalized Burr-Pareto-logistic distributions with applications to a uranium exploration data set. Technometrics, 28, 123–131.Google Scholar
  7. Embrechts, P., McNeil, A. and Straumann, D. (1999). Correlation: Pitfalls and alternatives. RISK Magazine, May, 69–71.Google Scholar
  8. Embrechts, P., McNeil, A. and Straumann, D. (2002). Correlation and dependence in risk management: Properties and pitfalls. In: M.A.H. Dempster (ed.), Risk Management: Value at Risk and Beyond, pp. 176–223. Cambridge University Press, Cambridge.Google Scholar
  9. Embrechts, P., Lindskog, F. and McNeil, A. (2003). Modelling dependence with copulas and applications to risk management. In S. Rachev (ed.), Handbook of Heavy Tailed Distributions in Finance, Chapter 8, 329–384. Elsevier.Google Scholar
  10. Fernández, C. and Steel, M.F.J. (1998). On bayesian modeling of fat tails and skewness. Journal of the American Statistical Association, 93, 359–371.Google Scholar
  11. Johnson, M.E. (1987). Multivariate Statistical Simulation. Wiley, New York.Google Scholar
  12. Koehler, K.J. and Symanowski, J.T. (1995). Constructing multivariate distributions with specific marginal distributions, Journal of Multivariate Analysis, 55, 261–282.Google Scholar
  13. Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.Google Scholar
  14. Kodde, D., Palm, F. and Pfann, G. (1990). Asymptotic least squares estimation efficiency considerations and applications. Journal of Applied Econometrics, 5, 229–243.Google Scholar
  15. Lambert, P. and Laurent, S. (2001). Modelling Skewness Dynamics in Series of Financial Data Using Skewed Location-Scale Distributions. Discussion Paper No. 01–25, Institut de Statistique, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.Google Scholar
  16. Manomaiphiboon, K. and Russel, A.G. (2003). Formulation of joint probability density functions of velocity for turbolent flows: An alternative approach. Atmospheric Environment, 37, 4917–4925.CrossRefGoogle Scholar
  17. Nelsen, R.B. (1999). An Introduction to Copulas. Springer, New York.Google Scholar
  18. Palmitesta, P. and Provasi, C. (2004). GARCH-type models with generalized secant hyperbolic innovations. Studies in Nonlinear Dynamics & Econometrics, 8(2), Article 7.
  19. Rosenberg, J.V. and Schuermann, T. (2004). A General Approach to Integrated Risk Management with Skewed, Fat-Tailed Risk. FRB of New York Staff Report No. 185.Google Scholar
  20. Sheather, S.J. and Jones, M.C. (1991). A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society, Series B, 53, 683–690.Google Scholar
  21. Siegel, S. (1956). Non-Parametric Statistics for the Behavioral Sciences. McGraw-Hill.Google Scholar
  22. Wolfram, S. (1999). The Mathematica Book, 4th edn. Wolfram Media/Cambridge University Press, Cambridge.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Quantitative MethodsUniversity of SienaItaly
  2. 2.Department of Statistical SciencesUniversity of PaduaItaly

Personalised recommendations