Hedging Strategy and Electricity Contract Engineering - IFOR

60 Risk management
of focusing on the size of the loss and not just the frequency of losses.
CVaR will therefore penalize large losses, which is a desired risk measure
characteristic. CVaR is together with VaR illustrated in Figure 3.1.
Variance has a well-documented motivation as a risk measure, since it comes
out as the natural risk measure from an expected utility maximization. Consider
an investment choice based on such maximization with an investor
specific utility uS V function . A portfolio with a random return X is then
preferred to one with random return Z if uS XV E uS ZV E . Variance
however, as mentioned, comes out as risk measure only if the portfolio returns
are elliptically distributed or if the utility is quadratic, where the appropriateness
of a quadratic utility function is highly questionable. For elliptical
distributions a CVaR minimizer will choose the same portfolio as a variance
minimizer [RU00], while for asymmetric distributions a portfolio based on a
true downside risk measure will be chosen. CVaR hence generalizes variance
as a risk measure.
A further motivation for CVaR as utility derived risk measure can be given. Let
U 2 be the commonly considered class of utility functions that are increasing
(more is preferred to less) and concave (risk aversion). Levy [Lev92] has shown
the following theorem connecting utility maximization and CVaR.
uS XV uS G R S ZV XV d
d R S ZV G 0G c 1V S .
Theorem 3.3
E E u U 2 if and only if E X X E Z Z
A minimization of the conditional value at risk may not be achieved simultaneously
for c S 0G 1V all with a single portfolio. 7 But Theorem 3.3 states
that a portfolio chosen to minimize CVaR for a c fixed and a given mean is
non-dominated, i. e. there exists no other portfolio with the same mean which
would be preferred to the chosen portfolio by all investors with utilities u U 2 .
Bertsimas et al. [BLS00] state that one thus is naturally led to minimize CVaR
for c some . As a matter of fact the CAPM 8 and the two-fund separation
derived from the mean variance problem can actually be performed also for
7 Observe n Eo that p X qsrut>v Xw=x X is equivalent Eo Xxzy{v to y|n}w 1 o E p X ~rut€v Xw=x X
X„ lƒ‚
8 Capital asset pricing model, see [Sha64].
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