The value-at- risk (VaR) is considered one of the most popular risk measure. The VaR is known for each risk and one has to manage the VaR for a joint position resulting from combination of different dependent risks. Since the VaR is the p-th quantile, it is important to know the dependence structure of the risks in order to find the VaR of functions of these risks. Embrechts and his collaborators have shown that the linear correlation is insufficient as a measure of dependence for studying VaR across a wide range of portfolio structures (Embrechts et al. (2002)). The problem mentioned above was first attacked by W. Hoeffding in 1941 (see collected works of Wassily Hoeffding, edited by Fisher and Sen (1994)) using what is now known as copula (see Nelsen (1999)). In Hoeffding's work, he studies different scale invariant measures of dependence using approximation to “copula” by a finite series in Legendre polynomials. He uses insurance data to compute certain correlations (e.g. Kendall's Tau and Spearman's rank correlation).
In this article, we follow the ideas of Sancette and Satchell (2004) to consider approximations to “copula” by Bernstein copula. We note the approximations for the measures of dependence: Kendall's Tau, Spearman's rank correlation, Hoeffding's dependence index and Pearson's coefficient of mean square contingency. We give the approximations using elementary probabilistic tools. Further, we establish that the empirical Bernstein copula converges to the true copula. Based on the approximating (empirical) Bernstein copula, we give a numerical procedure to determine the VaR and the Expected Shortfall. Finally, we compute non-parametric estimates of Kendall's tau and the Spearmans' rank correlations between well known international indexes. It is observed that the sample versions of these two measures are close to the approximations based on the empirical Bernstein copula when the data consist of monthly closing values. However when the data consist of bimonthly closing values, the estimates based on the empirical Bernstein copula are close to the above estimates but the sample versions are not. Thus estimates based on the empirical Bernstein copula seem to be better than the estimates based on the empirical copula.
Embrechts P., McNeil A. and Straumann D. (2002), Correlation and dependence in risk management: properties and pitfalls, Risk Management: Value at Risk and Beyond, in: M.A.H. Dempster, (ed.), Cambridge University Press, Cambridge, 176-223.
Hoeffding W. (1994), Scale-Invariant Correlation Theory, The Collected Works of Wassily Hoeffding, N.I. Fisher. and P.K. Sen, (eds.), Springer Series in Statistics, 57-107.
Nelsen B. (1999), An Introduction to Copulas, Lecture Notes in Statistics, 139, Springer, Verlag New York.
Sancette A. and Satchell S. (2004). The Bernstein Copula and its Applications to Modelling and Approximations of Multivariate distributions, Econometric Theory, EconometricTheory, 20, 535-652.
Keywords: Copula; Uniform convergence; Dependence measures; Value-at-risk
Biography: Uttara is a Professor at the Department of Statistics, University of Pune, India. Her present interests are in survival analysis, reliability and applications of statistics and probability in finance. She obtained PhD from Michigan State University in 1979. She has been in Pune University since 1981 with short term visiting appointments at other places.