© 2009 Elsevier Limited. All rights reserved.
Doi:10.1016/B978-0-12-374952-9.00012-9.

2010

More than you ever wanted to

know about conditional value at

risk optimization

Bernd Scherer

12

Executive Summary

We split our critique on conditional value at risk (CVaR) into implementation

and concept issues. While implementation issues can be overcome at the cost of

sophisticated statistical procedures that are not yet widely available, they pose

a strong objection against current na ï ve use of CVaR. Estimation error sensitiv-

ity amplified by approximation error and difficulties in modeling fast updating

scenario matrices for nonnormal multivariate return distributions will stop many

practitioners from applying CVaR. More limiting in our view, however, is the

inabi lity of CVaR to integrate well into the way investors think about risk.

Averaging across small and extremely large losses, i.e., giving them the same

weight, does not reflect rising risk aversion against extreme losses, which is

probably the most agreeable part of expected utility theory.

12.1 Introduction : Risk measures and their axiomatic foundations

The world of investments offers a puzzling variety of alternative risk measures.
It almost seems as if we could invent risk measures freely. However, it needs to
be stressed that we cannot. Risk measures must be supported by axioms (state-
ments that are generally believed to be true). These axioms can then be used to
build or criticize risk measures and, of course, a risk measure is only as con-
vincing as the axioms it is built on. Discussing risk measures without reference
to their axioms is a pointless exercise.
We distinguish between axioms on random variables (actuarial or statistical
axioms) and axioms on behavior (economic axioms). Economic axioms require
a risk measure or more generally an objective function to be consistent with
decision making under uncertainty formulated by Neumann and Morgenstern
(1944) , i.e., expected utility optimization. This is very familiar to economists
and builds the backbone of mean – variance optimization by Markowitz (1952).