Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Staying ahead on downside risk 145

In addition, several dynamic EVaR models have been developed in the recent
econometric literature and will be briefly reviewed in the next section. My contri-
bution is to suggest a simple asset allocation procedure that can be used to build
the optimal portfolio by minimizing risk as measured by the predicted EVaR.

6.2 Measuring downside risk: VaR and EVaR

6.2.1 Definition and properties

EVaR is introduced in Kuan, Yeh, and Hsu (2009). The concept of ω -expectile,
for 0 ω 1, can be found in Newey and Powell (1987). They defined μ ( ω ),
for any random variable Y with finite mean and cumulative distribution func-
tion F ( y ), as the solution of the equation:

μω

ω

ω

μω

μω

() ( ) ( ()) ()

()







EY y dFy

21

1

∞

∫

(6.1)

The ω -EVaR of Kuan et al. (2009) is just  μ ( ω ). Expression (6.1) can be
rearranged to give:

ωμω ωμω

μω

μω

( ( )) () ( ) ( ( ) ) ()

()

()

ydFy ydFy



∞

∞

∫∫^1

The differences that appear in the integrals are nonnegative. When ω  0.5,
the solution is simply μ ( ω )  E ( Y ), while as ω varies between zero and 0.5,
the expectile function μ ( ω ) describes the lower tail of the distribution of Y.
Intuitively, the quantities ω and 1  ω can be seen as asymmetric weights that
multiply the integrals of the deviations from μ ( ω ). If ω 0.5, then the weight
on the outcomes y that lie below μ ( ω ) (i.e., the weight on the integral on the
right hand side) dominates.
It can be shown that a solution exists and it is unique. Furthermore, the
expectiles of a linear transformation of Y , Y  aY  b can be easily found
since μ ( ω )  a μ ( ω )  b for any real numbers a and b.
Suppose that we have obtained a sample of n observations Y i. The sample
equivalent of the population expectile can be found by solving:

min

μ ω

ρμ()Yi

i

n



 1

∑

(6.2)

where

ρωω() |xIxx(<^0 )|

2