DOI QR코드

DOI QR Code

Multivariate CTE for copula distributions

  • Received : 2017.02.27
  • Accepted : 2017.03.24
  • Published : 2017.03.31

Abstract

The CTE (conditional tail expectation) is a useful risk management measure for a diversified investment portfolio that can be generally estimated by using a transformed univariate distribution. Hong et al. (2016) proposed a multivariate CTE based on multivariate quantile vectors, and explored its characteristics for multivariate normal distributions. Since most real financial data is not distributed symmetrically, it is problematic to apply the CTE to normal distributions. In order to obtain a multivariate CTE for various kinds of joint distributions, distribution fitting methods using copula functions are proposed in this work. Among the many copula functions, the Clayton, Frank, and Gumbel functions are considered, and the multivariate CTEs are obtained by using their generator functions and parameters. These CTEs are compared with CTEs obtained using other distribution functions. The characteristics of the multivariate CTEs are discussed, as are the properties of the distribution functions and their corresponding accuracy. Finally, conclusions are derived and presented with illustrative examples.

Keywords

References

  1. Acerbi, C. and Tasche, D. (2002). Expected shortfall: A natural coherent alternative to VaR. Economic Notes, 31, 379-388. https://doi.org/10.1111/1468-0300.00091
  2. Andersson, F., Mausser, H., Rosen, D. and Uryasev, S. (2001). Credit risk optimization with conditional value-at-risk criterion. Mathematical Programming, 89, 273-291. https://doi.org/10.1007/PL00011399
  3. Artzner, P., Delbaen, F., Eber, J. M. and Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203-228. https://doi.org/10.1111/1467-9965.00068
  4. Cai, J. and Li, H. (2005). Conditional tail expectations for multivariate phase-type distributions. Journal of Applied Probability, 42, 810-825. https://doi.org/10.1017/S0021900200000796
  5. Cousin, A. and Di Bernardino, E. (2013). On multivariate extensions of value-at-risk. Journal of Multivariate Analysis, 119, 32-46. https://doi.org/10.1016/j.jmva.2013.03.016
  6. Cousin, A. and Di Bernardino, E. (2014). On multivariate extensions of conditional-tail-expectation. Insur-ance: Mathematics and Economics, 55, 272-282. https://doi.org/10.1016/j.insmatheco.2014.01.013
  7. Embrechts, P., Mcneil, A.J. and Straumann, D. (1999). Correlation: Pitfalls and alternatives. Risk, 5, 69-71.
  8. Embrechts, P. and Puccetti, G. (2006). Bounds for functions of multivariate risks. Journal of Multivariate Analysis, 97, 526-547. https://doi.org/10.1016/j.jmva.2005.04.001
  9. Genest, C., Ghoudi, K. and Rivest, L. P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82, 543-552. https://doi.org/10.1093/biomet/82.3.543
  10. Hong, C. S. and Kim, T. W. (2016). Multivariate conditional tail expectations. Korean Journal of Applied Statistics, 29, 1201-1212.
  11. Hong, C. S. and Kwon, T. W. (2010). Distribution fitting for the rate of return and value at risk. Journal of the Korean Data & Information Science Society, 21, 219-229.
  12. Hong, C. S., Han, S. J. and Lee, G. P. (2016). Value at risk and alternative value at risk. Korean Journal of Applied Statistics, 29, 689-697. https://doi.org/10.5351/KJAS.2016.29.4.689
  13. Hong, C. S. and Lee, J. H. (2011a). VaR estimation of multivariate distribution using copula functions. Korean Journal of Applied Statistics, 24, 523-533. https://doi.org/10.5351/KJAS.2011.24.3.523
  14. Hong, C. S. and Lee, W. Y. (2011b). VaR estimation with multiple copula functions. Korean Journal of Applied Statistics, 24, 809-820. https://doi.org/10.5351/KJAS.2011.24.5.809
  15. Jorion, P. (2006). Value at risk, the new benchmark for market risk, 3rd Ed., McGraw-Hill, New York.
  16. Ko, K. Y. and Son, Y. S. (2015). Optimal portfolio and VaR of KOSPI200 using One-factor model. Journal of the Korean Data & Information Science Society, 26, 323-334. https://doi.org/10.7465/jkdi.2015.26.2.323
  17. Li, D. X. (1999). Value at Risk based on the volatility skewness and kurtosis, RiskMetrics Group, New York.
  18. Nelsen, R. B. (2006). An Introduction to Copulas, Springer, New York.
  19. Park, S. and Baek, C. (2014). On multivariate GARCH model selection based on risk management. Journal of the Korean Data & Information Science Society, 25, 1333-1343. https://doi.org/10.7465/jkdi.2014.25.6.1333
  20. Rockafellar, R. T. and Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21-41. https://doi.org/10.21314/JOR.2000.038
  21. Rockafellar, R. T. and Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of banking & finance, 26, 1443-1471. https://doi.org/10.1016/S0378-4266(02)00271-6
  22. Shih, J. H. and Louis, T. A. (1995). Inferences on the association parameter in copula models for bivariate survival data. Biometrics, 51, 1384-1399. https://doi.org/10.2307/2533269
  23. Sklar, A. (1959). Fonctions de repartition a n dimensions et leurs marges. l'Institut de Statistique de l'Universite de Paris, 8, 229-231.
  24. Zangari, P. (1996). An improved methodology for measuring VaR. RiskMetrics Monitor, 2, 7-25.

Cited by

  1. 이변량 왜도, 첨도 그리고 표면그림 vol.28, pp.5, 2017, https://doi.org/10.7465/jkdi.2017.28.5.959