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

Portfolio optimization with “ Threshold Accepting ” : a practical guide 205

stable and thus preferable, working with ratios practically never caused problems
in our studies. See also Section 9.5.3.
Many special cases for such ratios have been proposed in the literature, the
best known certainly being the Sharpe ratio ( Sharpe, 1966 ). For this ratio,
reward is the mean portfolio return and risk is the standard deviation of port-
folio returns, so it corresponds closely to mean – variance optimization. Rachev,
Fabozzi, and Menn (2005) give an overview of various alternative ratios that
have been proposed in the academic literature; further possible specifications
come from financial advisors, in particular from the hedge fund and commod-
ity trading advisor field ( Bacon, 2008 , Chapter 4).
In general, these ratios can be decomposed into “ building blocks ” ; we will
discuss some examples now.

Partial moments

For any portfolio return r t , the equation

rrtdrrtd rrdt

desired return upside

 max( , )^00 max( , )

ddownside

(9.2)

always holds for any desired-return threshold r d ( Scherer, 2004 ). Partial
moments are a convenient way to distinguish between returns above and below
r d , i.e., the “ upside ” and “ downside ” terms in Equation (9.2), and thus to cap-
ture potential asymmetry around this threshold. For a sample of portfolio
returns r[]rr 12 ...rn
S
with nS observations, the partial moments Pγ()^ ()rd
can be estimated as:

P

S

γ

 γ

()r ( )

n

ddrr

rrd

1

∑

(9.3a)

P

S

γ

 γ

()r ( )

n

ddrr

rrd

1

∑

(9.3b)

The superscripts  and  indicate the tail (i.e., upside and downside).
Partial moments take two more parameters: an exponent γ and the threshold
r d. The expression “ r r d ” indicates that we sum only over those returns that
are greater than r d.

Conditional moments

Conditional moments can be estimated by:

Cγ γ



() 

#{ }

r ()

rr

d rr

d

d

rrd

1

∑

(9.4a)