Coherent Risk Measures and Convex Combinations of the Conditional Value at Risk (C V@R)

Georg Ch Pflug

doi:10.17713/ajs.v31i1.471

Abstract

The conditional-value-at-risk (C V@R) has been widely used as a risk measure. It is well known, that C V@R is coherent in the sense of Artzner, Delbaen, Eber, Heath (1999). The class of coherent risk measures is convex. It was conjectured, that all coherent risk measures can be represented as convex combinations of C V@R’s. In this note we show that this conjecture is wrong.

Highlights

Let the random variable represent the future value of a portfolio
The conditional value-at-risk Î@R is defined as follows
It was conjectured that the class Î@RÀ coincides with the class of coherent risk measures

Summary

Introduction

Let the random variable represent the future value of a portfolio. To measure the risk contained in is an important task in stochastic finance. Among the enormous group of statistical parameters, which can be associated to , like expectation, median, variance, mean absolute deviation, coefficient of variation, Gini-measure etc., only some qualify as acceptable risk measures. Delbaen, Eber, Heath (1999) call a statistical parameterμ coherent, if it has the following properties: (i) First order stochastic monotonicity. È 1⁄2 Ù È 3⁄4 Ù for all Ù implies 1⁄2 μ 3⁄4 μ (ii) Positive homogeneity.

Results

Conclusion