The comonotonicity coefficient: A new measure of positive dependence in a multivariate setting
In financial and actuarial sciences, knowledge about the dependence structure is of a great importance. Unfortunately this kind of information is often scarce. Many research has already been done in this field e.g. through the theory of comonotonicity. It turned out that a comonotonic dependence structure provides a very useful tool when approximating an unknown but (preferably strongly) positive dependence structure. As a consequence of this evolution, there is a need for a measure which reflects how close a given dependence structure approaches the comonotonic one. In this contribution, we design a measure of (positive) association between n variables (X1,X2, · · · ,Xn) which is useful in this context. The proposed measure, the comonotonicity coefficient _(X) takes values in the range [0, 1]. As we want to quantify the degree of comonotonicity, _(X) is defined in such a way that it equals 1 in case (X1,X2, · · · ,Xn) is comonotonic and 0 in case (X1,X2, · · · ,Xn) is independent. It should be mentioned that both the marginal distributions and the dependence structure of the vector (X1,X2, · · · ,Xn) will have an effect on the resulting value of this comonotonicity coefficient. In a first part, we show how _(X) can be designed analytically, by making use of copulas for modeling the dependence structure. In the particular case where n = 2, we compare our measure with the classic dependence measures and find some remarkable relations between our measure and the Pearson and Spearman correlation coefficients. In a second part, we focus on the case of a discounting Gaussian process and we investigate the performance of our comonotonicity coefficient in such an environment. This provides us insight in the reason why the comonotonic structure is a good approximation for the dependence structure.
|Date of creation:||Jan 2006|
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- Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: theory," Insurance: Mathematics and Economics, Elsevier, vol. 31(1), pages 3-33, August.
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- Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
- Kaas, Rob & Dhaene, Jan & Goovaerts, Marc J., 2000. "Upper and lower bounds for sums of random variables," Insurance: Mathematics and Economics, Elsevier, vol. 27(2), pages 151-168, October.
- Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: applications," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 133-161, October.
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