146 Optimizing Optimization
In other words, we minimize the weighted squared deviations from the obser-
vations. Weights are asymmetric: if ω 0.5 then negative residuals receive a
larger weight than positive residuals. When ω 0.5, the problem boils down to
minimizing the sum of squared errors, yielding the sample mean as the solution.
It is easy to show that VaR can be characterized as the value that minimizes
asymmetric absolute deviations instead of square deviations. The function
ρ (^) ω ( x ) is replaced by ρα ( x ) | α I ( x 0)| | x |, 0 α 1. As a consequence,
the difference between VaR and EVaR can be stated in terms of the shape of
the objective function in the optimization problem (6.2).
A fundamental property of expectiles, which follows from the first-order
conditions of the minimization problem, is that the weighted sum of residuals
is equal to zero:
|( )|( )ωμμ
IYiiY
i
n
1
∑^0
EVaR has several advantages. First, when expectiles exist, they characterize the
shape of the distribution just like quantiles. In fact, each ω -expectile corresponds
to an α -quantile, although the relation between ω and α varies from distribu-
tion to distribution, as can be seen from Table 6.1. For example, a 5% EVaR
when returns are normally distributed corresponds to a 12.6% VaR. However,
for heavy tailed distributions like a t (5), the 5% EVaR corresponds to a 10%
VaR. If α is seen as an indication of how conservative the risk measure is, then it
can be seen that EVaR is more conservative when extreme losses are more likely.
A second advantage of EVaR is that it is a coherent measure of risk. 3 It also
admits an interpretation in terms of utility maximization. Manganelli (2007)
shows that an expectile can be seen as the expected utility of an agent having
asymmetric preferences on below target and above target returns. His argu-
ment is presented in more detail in the section on asset allocation. The resulting
portfolio allocations are not dominated in the second-order stochastic domi-
nance sense. In other words, a portfolio is second-order stochastic dominance
efficient if and only if it is optimal for some investor who is nonsatiable (i.e.,
the higher the profit, the better) and risk averse. It is interesting to note that
the mean – variance optimal portfolios do not satisfy this property: There may
3 It can be shown that EVaR satisfies all the properties required by the definition of Artzner et al.
(1999) , including monotonicity and subadditivity.
Table 6.1 α corresponding to ω when ω -EVaR and α -VaR are equal
ω Normal t (30) t (5) t (3)
1% 4.3% 4.0% 3.0% 2.4%
5% 12.6% 12.3% 10.0% 8.5%
10% 19.5% 19.0% 16.6% 14.5%