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

206 Optimizing Optimization

Cγ−()=−∑ γ

#{ }

r ()

rr

d rr

d

d

rrd

1

< <

(9.4b)

where again  and  indicate the tail, and “ # { r r d } ” is a counter for the
number of returns higher than r d.
Conditional and partial moments are closely related. For a fixed threshold r d ,
the lower partial moment of order γ equals the lower tail’s conditional moment
of the same order times the lower partial moment of order 0. That is,

PCP

PCP

γγ

γγ









() () ()

() () ()

rrr

rrr

ddd

ddd

0

(^0)
The partial moment of order 0 is simply the probability of obtaining a return
beyond r d. Still, for a given r d , both conditional and partial moments convey
different information, since both the probability and the conditional moment
need to be estimated from the data to obtain a partial moment. In other words,
the conditional moment measures the magnitude of returns around r d , while
the partial moment also takes into account the probability of such returns.
Both partial and conditional moments, in our definitions, are centered
around r d. In the finance literature, r d is often regarded as exogenously fixed,
for instance at the risk-free rate or at some minimal acceptable return. In this
case, we can directly work with Equations (9.3) and (9.4). An alternative way
to set r d , often chosen in risk management applications, is to equate r d to some
quantile of the return distribution. This is the usual convention for conditional
moments like Expected Shortfall, as they can then be compared with the corre-
sponding VaR values. But now we must not optimize with equations like (9.4)
anymore: Fixing a quantile says nothing about the location of the return dis-
tribution that our optimization algorithm selects, and we may end up with a
dominated distribution (see Figure 9.2 ).
The simplest remedy is not to center around r d , and modify Equation (9.4) to:
Cγ γ

()
#{ }
r
rr
d r
d rrd
1
∑
(9.5a)
Cγ γ

()
#{ }
r
rr
d r
d rrd
1
∑
(9.5b)
But now without centering, we have no more guaranty for the sign of r.
Hence, for γ  1, we replace r by max( r , 0) in Equation (9.5a), and by min( r , 0)
in Equation (9.5b).