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

Portfolio optimization with “ Threshold Accepting ” : a practical guide 209

9.2.2 The problem summarized

Our optimization problem can be summarized as follows: let x  []xx 12 ...xnA^
be the holdings of the individual assets, and J be the set of assets in the
portfolio, i.e.:

J {|ixi ≠^0 }

The objective function Φ will be a ratio of reward and risk. If we just want
to minimize some risk function, we replace the reward by a constant; for solely
maximizing reward, we set the risk to a constant. The following table gives
some examples.

Reward Risk

Constant

CQ 1 ()q

Minimize Expected Shortfall for q th quantile

Constant

Q 0 Minimize maximum loss

P^1

()r

d P 2

()r

d

Upside potential ratio ( Sortino, van der Meer, &

Plantinga, 1999 )

P^1

()r

d P 1

()r

d

Omega ( Keating & Shadwick, 2002 ) for threshold r d

1

n

r

S

∑

(^) D
max
Calmar ratio ( Young, 1991 )
CQγ()p
CQδ()q
Rachev generalized ratio for exponents γ and δ
( Biglova, Ortobelli, Rachev, & Stoyanov, 2004 )
Now the problem, including constraints, can be written as:
min
#{ }
inf sup
inf sup
x
xxxjjj j
KK
Φ
 ∈

for J
J
where xinfj and xjsup are minimum and maximum holding sizes, respectively,
for those assets included in the portfolio (i.e., those in J ). K (^) inf and K (^) sup are car-
dinality constraints that set a minimum and maximum number of assets in J.
We do not include minimum return constraints. Note that we will always mini-
mize, which is not restrictive since it is equivalent to maximizing  Φ.