More than you ever wanted to know about conditional value at risk optimization 285
(portfolio risk is lower than the sum of stand-alone risks). The last axiom is the
most intuitive as diversification should decrease risk. However, this need not be
the case with VaR. Suppose we add two marginal distributions with both long
left tails (i.e., extreme losses) together. Looking at each marginal distribution in
isolation, these extreme events will not be picked up by value of risk owing to
its ignorance on what is looming in the tail of a distribution. Extreme losses are
simply not frequent enough to identify themselves at the 95% confidence level.
However, when combined, the losses diversify into the joint distribution and so
the VaR of the combined portfolio will rise. In other words, VaR will provide
the wrong diversification advice. Given that VaR so blatantly fails when distri-
butions are highly skewed (it works fine if distributions are elliptical, but then
it adds no additional informational value to variance), one should not call it
a risk measure in the first place. This led many to the adoption of conditional
value at risk ( CVaR ), which offers not only computational advantages (it can be
optimized with a linear program, while VaR can only be dealt with heuristically
due to its nonconvex nature as we have seen in the diversification example).
Contrary to VaR , CVaR is a coherent risk measure, i.e., it passes ARTZNER’s
statistical axioms. It does so by looking into the tail of a distribution, i.e., it
averages across all losses that exceed a 95% confidence bound. While this looks
like good news at first glance, the remainder of this text will confront the reader
with implementation challenges and conceptional problems for CVaR.
Implementation problems center around the fact that CVaR is extremely
sensitive to estimation error (more than other risk measures) and approxima-
tion error (this issue is unique to all scenario optimization problems and not
existent in mean – variance investing) to name only two. 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 naive
use of CVaR. Conceptional problems are in our view more limiting. What we
point out here is the inability of CVaR to integrate well into the way investors
think about risk. Averaging across small and extremely large losses, i.e., giv-
ing them the same weight in your risk calculation does not reflect rising risk
aversion against extreme losses, which is probably the most agreeable part of
economic decision theory.
12.2 A simple algorithm for CVaR optimization
We start with a brief description around the technicalities of CVaR opti-
mization. Let us first define an auxiliary variable, e s , for each of s 1, ... , S
scenarios.
eVsiaRwri is
nmax ,⎡ (^0) ∑ 1
⎣⎢
⎤
⎦⎥
(12.1)
It measures the excess loss of a portfolio consisting of i 1, ... , n securities
with respective weights w i that pay off r is in scenario s. There are no restrictions