VaR-Efficient Portfolios for a Class of Super- and Sub-Exponentially Decaying Assets Return Distributions

Details VaR-Efficient Portfolios for a Class of Super- and Sub-Exponentially Decaying Assets Return Distributions VaR-Efficient Portfolios for a Class of Super- and Sub-Exponentially Decaying Assets Return Distributions

2003-01-06

arxiv.org/abs/physics/0301009v1

Quantitative Finance 4 (1), 17-36 (2003)

Physics

Physics and Society

Latex document of 33 pages including 1 table and 2 eps figures

Scientific paper

10.1088/1469-7688/4/1/002

Using a family of modified Weibull distributions, encompassing both sub-exponentials and super-exponentials, to parameterize the marginal distributions of asset returns and their multivariate generalizations with Gaussian copulas, we offer exact formulas for the tails of the distribution $P(S)$ of returns $S$ of a portfolio of arbitrary composition of these assets. We find that the tail of $P(S)$ is also asymptotically a modified Weibull distribution with a characteristic scale $\chi$ function of the asset weights with different functional forms depending on the super- or sub-exponential behavior of the marginals and on the strength of the dependence between the assets. We then treat in details the problem of risk minimization using the Value-at-Risk and Expected-Shortfall which are shown to be (asymptotically) equivalent in this framework.

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