Technology and Investment, 2013, 4, 73-77
Published Online Febr uary 20 http://www.SciRP.org/journal/ti)
Copyright © 20 SciRes. TI
Portfolio Selection by Maximizing Omega Function using
Differential Evolution
PEKÁR Juraj, BREZINA Ivan, ČIČKOVÁ Zuzana, REIFF Marian
Department of Operations Research and Econometrics, University of Economics, Bratislava, Slovakia
Received 2012
ABSTRACT
Paper presents alternative solution seeking approach for portfolio selection problem with Omega function performance
measure which allows determining capital allocation over the number of assets. Omega function computability is diffi-
cult due to substandard structures and therefore the use of standard techniques seems to be relatively complicated. Dif-
ferential evolution from the group of evolutionary algorithms was selected as an alternative computing procedure. Al-
ternative approach is analyzed on the Down Jones Industrial Index data. Presented approach enables to determine good
real-time solution and the quality of results is comparable with results obtained by professional software.
Keywords: differential evolution, Omega function, problem of portfolio selection
1. Introduction
In general, portfolio theory deals with the selection of an
appropriate mix of assets in a portfolio in order to meet
predetermined properties. Various mathematical models,
which measure the portfolio performance measurement
can be used to support the decision making process of
selection of the portfolio assets. The aim is to determine
the allocation of the available resources in the selected
group of assets that results in maximization of portfolio
performance. Follow this idea, various measurements of
performance can be used. The performance measurement
techniques are e.g. Sharpe ratio, Treynor ratio, Jensen's
alpha, Information Ratio, Sortino ratio, Omega function
and the Sharpe Omega ratio ([1], [2], [3], [4], [5], [6], [7],
[8]). The paper deals wi th Omega function, which com-
putability is difficult due to substandard structures of
performance level and therefore the use of standard tech-
niques seems to be relatively complicated. The alterna-
tive computing procedures includes variety of different
approaches. Nowadays a lot of research attention is fo-
cused on evolutionary algorithms. The paper presents the
algorithm of differential evolution that is able to deal
with nonlinear objective function with success (e na b l es
quick a c hieve ment of suboptimal solutions) and thus
demonstrates suitability of evolutionary algorithms for
financial modeling. Above mentioned approach is ana-
lyzed on assets inc lude d in the Down Jones Industrial
Inde x and its historical data1
1
publ is hed from July 1st
2001 to April 1st 2012 on weekly basis were used.
http://finance. yahoo. co m/ (2012)
2. Portfolio selection models based on
Omega function
Omega function is measure which incorporates all the
distributional, characteristics of a return series. The
measure is a function of the returns leveled and requires
no parametric assumption on the distribution. Precisely,
it considers the returns below and above a specific loss
threshold and provides a ratio of total probability
weighted losses and gains that fully describes the risk
reward properties of the distribution [8]:
where:
MAR denotes the return level regarded as a loss threshold,
(a, b ) denotes yields range,
F(x) the cumulative distribution fu nction of asset returns.
In the next part we formulate the problem of portfolio
selection based on omega function performance measure.
As it is mentioned, the Omega function involves consid-
eration of all the information contained in the time series
of returns. The aim of portfolio selection problem is to
maximize the level of Omega performance measure,
where the variables w1, w2,…wd (where d represents the
number of assets) represent the weights of each asset in
the portfolio. Corresponding problem can be formulated
as follows [9]:
∫
∫−
=Ω MAR
a
b
MAR
dxxF
dxxF
MAR
)(
))(1(
)(
13
13 (

P. Juraj ET AL.
TI
( )
( )
1
1
max ,0
max( )
max ,0
T
t
t
T
t
t
MAR
MAR
=
=
−
Ω=
−
∑
∑
T
T
wr
w
wr
Subject to:
1
0
=
≥
T
we
w
Wher e :
T represents the number of periods,
rt represents returns vector of portfolio assets in the pe-
riod t = 1,2,…T,
w denotes a vector of variables w1, w2,…wd representing
the weight of each asset in the portfolio.
The computationa l c omplexity of the presented problem
arises from its non-standard structure. Therefore, evolu-
tionary algorithms seem to be a suitable alternative to
standard techniques, due to its ability to achieve the sub-
optimal solutions in relatively short time. The d iffere ntial
evolution is one of the popular and well known tech-
niques.
3. Differential Evolution
Differential evolution (introduced by Price and Storn
[10]) belongs to the class of evolutionary techniques,
comprise a large number of nontraditional computing
techniques whose common characteristic is that they are
inspired by the observation of the nature processes (ge-
netic algorithms, ant colony optimization, differential
evolution, etc.). Nowadays evolutionary algorithms are
considered to be effective tools that can be used to search
for solutions of optimization problems (napr. [11], [12],
[13], [14]). The big advantage over traditional methods is
that they are designed to find global extremes (with
built-in stochastic component) and that their use does not
require a priori knowledge of optimized function (con-
vexity, differential etc.), and in that way they work well
to solve continuous non-linear problems, where is hard to
use traditional mathematic methods. The principle of
basic version of differential evolution can be described
by following pseudocode:
BEGIN
SETTING of control parameters;
INITALIZATION of population;
EVALUATION of each individual;
WHILE (STOPPING CRITERION is not satisfied)
DO
FOR (each individual of the population) DO
(REPRODUCTIVE CYCLE):
CREATE differential vector;
CREATE trial vector;
CREATE test vector;
IF (EVALUATION of test vector) >
(EVALUATION of current selected indi-
vidual ) THEN (SUBSTITUD E the selected
individual with the test vector) ;
ENDIF
ENDFOR
ENDWHILE
EVALUATE process of calculating;
END
Evolutionary algorithms differ from more traditional
optimization techniques in such a way that they involve a
search from a "population" of individuals, not from a
single one. Each individual represents one candidate so-
lution for the given problem that is represented by para-
meters of individual. Associated with each individual is
also the fitness, which represents the relevant value of
objective function. A population can be viewed as np.d
matrix (np – number of individuals in the population, d –
number of parameters of individual). Every step involves
a competitive selection that carried out poor solutions.
Consider the problem of portfolio selection it is possible
to summarize the steps of the algorithm as follows:
Setting of the control parameters. Differential evolution
is controlled by a special set of parameters. Recom-
mended values for the parameters are usually derived
empirically from experiments ([15], [16], [17]):
d – dimensio na lity. Number of parameters of individual
is equal to number of asse ts.
np – population size. Number of individuals in popula-
tion. recommended setting is 5d to 30d, respectively
100d, in case the optimized function is multimodal ([15],
[16]).
g – generations. Represent the maximum number of ite-
ration (g is also stopping criterion).
cr – crossover constant, cr
0,1∈
. The value of cr was
set on the base of experiments.
f – mutation constant, f
0,1∈ . The value of f was set on
the base of experiments.
Initializatio n. The population
( )
0
P
was randomly initia-
lized at the beginning of evolutionary process according
to the rule:
( )
( )( )
( )
00
,0
,
1
0,1
P =
i
ij d
ij
j
rnd
w
w
=
=∑
1,2,...i np= 1,2, ...jd=
that ensure that the total weights of portfolio is equal
to one. Each individual is then evaluated with the fitne ss
(given by function Ω(w)).
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P. Juraj ET AL.
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The test of stopping condition. In its canonical
form, the only stopping criterion is to reach the
maximal number of iterations (represent by parameter g).
Reproductive cycle. This cycle comprise the crossing and
mutation to create individuals for the next generation.
For each individual wig, i =1, 2,...np, from the population
another three different individuals are chosen (vectors r1,
r2, r3). The difference of the first two vectors (r1 and r2)
gives the differential vector, which is multiplied by mu-
tation constant f and added to vector r3. Thus, we get
trial vector v. Formally:
( )
3,1, 2,
.
ttt t
jr jrjrj
wfw w=+−v
1,2, ...jd=
,
1,2,...tg=
After the mutation process comes the formation
of a new individual, which is also called test vector wtest
so that one element after another is selected from the
currently selected individual wig and from the trial vec-
tor v and for every pair is generated a random number
from the interval
0,1
, which is compared with the
crossing constant cr. If the generated random number is
less than or equal to cr, to the relevant position of wtest
comes the element of trial vector v, otherwise of
current selected individual wig . Formal ly :
( )
3 12
, if 0,1
, otherwise
j
g gg
r jrjr jj
test
g
ij
wf wwrandcrjk
w
w
+−≤ ∨=

=


where
1,2,...i np=
, 1,2, ...jd= ,
{ }
1,2, ...kd∈
, for
{ }
,,1, 2,...np∈r1 r2r3 ,
i≠≠≠r1 r2r3
To ensure the feasibility of solution we use the following
rule: if
0
test
j
w<
, then
0,1
test
j
w rnd=
and
( ) ,
1
P = w
i
test
j
test test
ij d
test
j
j
w
w
=
=
∑
1,2,...i np=
1,2, ...jd=
.
where k is a random index, which always ensures a
change of at least one parameter in the test vector. The
value of the objective function for the test vector is com-
pared to the value of objective function of the current
selected individual and to the next generation is selected
the vector with the better objective value.
test
1cos cos
, ak ()()
, otherwise
test g
gt ti
ig
i
ff
+
≤

=


w ww
ww
So that process continues in each generation for all indi-
viduals. The result is a new generation with the same
number of individuals.
Evaluation. The whole process of reproduction continues
until the last (users specified) number of generations is
reached. The value of the best individual from each gen-
eration is reflected to history vector, which shows the
progression of an evolutionary process.
4. Empirical Results
The portfolio analyze was based on index Dow Jones
Industrial, which is one of the major market indexes, as
well as one of the most popular indicators of the U.S.
market. It is a stock market index, and one of several
indices created by Wall Street Journal editor and Dow
Jones & Company co-founder Charles Dow. It was
founded on May 26, 1896. It is an index that shows how
30 large publicly owned companies based in the United
States have traded during a standard trading session in
the stock market. (so called Large-Cap companies –
companies with market capitalization above 10 mild
USD). Data2
Company Name
are processed weekly for the period July
1st 2001 to April 1st 2012. A total of 559 data were ana-
lyzed.
Table 1 : Company overview DJI.
Ticker
4M Co. MMM
Alcoa Inc. AA
American Express Co. AXP
AT&T Inc. T
Bank of America Corp. BAC
Boeing Co. BA
Caterpillar Inc. CAT
Chevron Corp. CVX
Cisco Systems Inc. CSCO
Coca-Cola Co. KO
DuPo nt DD
Exxon Mobil Corp. XOM
General Electric Co. GE
Hewlett-Packard HP Q
Home Depot Inc. HD
IBM IBM
Intel INT C
Johnson & Johnson JNJ
JPMorgan Chase & Co. JPM
Kraft Foods Inc. Cl A KFT
McDonald's Corp. MCD
2 http://finance.yahoo .co m/ (2012)
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Merck & Co. Inc. M RK
Microsoft Corp. MSFT
Pfizer Inc. PFE
Procter & Gamble Co. PG
Travelers Cos. Inc. TRV
United Technologies Corp. UTX
Verizon Communications Inc. VZ
Wal-Mart Stores Inc. WMT
Walt Disney Co. DIS
The input parameter of threshold (MAR) was set to 0.055.
A disadvantage of algorithm of differential evolution, as
well as of other evolutionary approaches, is that it has a
dependence on the control parameter setting. Due to this
fact, our effort was to determine effective settings of the
parameters f and cr. The tests were done on above men-
tioned data, with the simultaneous use of the set np = 300
a g = 500. The tested values of parameters f and cr were
from the interval
0,1 as sequence of levels 0.1, 0.2, 0.3,
0.4, 0.5, 0.6, 0.7, 0.8, 0.9. The interval limits were not
considered during testing (purely deterministic and pure-
ly stochastic nature of the algorithm). For each combina-
tion of pairs, five exp er imen ts were conducted. The
average value of Omega functions for each combination
of pairs is shown in Figure 1. The control parameters
were set on the base of the article [18], which describes
the possibility of setting the parameters with the help of
some statistical methods e.g. Kr uskal-Wallis test, Bart-
lett's test, Cochran-Hartley’s test.
Figure 1
The algorithms were implemented in MATLAB 7.1. Two
functions were created: Differential evolution adapted for
solving portfolio selection problem and the function for
calculation of objective function (Omega function) value.
All the experiments were run on PC INTEL(R) Core(TM)
2 CPU, E8500 @ 3.16 GHz, 3.25 GB RAM under Win-
dows XP. The best result was the value 1.230054969 of
O mega function.
Based on the testing parameters problem was ten times
re-solved (Table 2) wi th parameter settin gs f = 0.1,
cr = 0.2, np = 3000 a g = 2000, and best obtained value
of the Omega function was 1.230055034. Convergence
of the solution can be seen in Figure 2. The rapid con-
vergence till 200 generations is evidently seen from the
Figure 2. Values obtained afte r 200 generations are close
enou gh to the final solution.
Based on model results, recommended allocation is to
inves t assets in McDonald's companies at the rate
82.230 %, Caterpillar Inc. at the rate 13.998 % and IBM
3.772 %. According to result achieved, it is not recom-
mended to invest to other companies since values of
weights (variables) are equal to 0 %.
Figure 2
Table 2 : Solutions values
Value of Omega function
Solution 1 1.23005496457
Solution 2 1.23005497790
Solution 3 1.23005498325
Solution 4 1.23005495551
Solution 5 1.23005496369
Solution 6 1.23005498844
Solution 7 1.23005498579
Solution 8 1.23005499256
Solution 9 1.23005504187
Solution 10 1.23005500989
MAX 1.23005504187
MIN 1.23005495551
Avera ge 1.23005498635
Mentioned problem was also solved using the Risk Solv-
er Platform V.12.0, with the result equal to 1.23004199,
which is a smaller value compared to the best value
computed of Omega function (the difference is
0.000013). The relevance of presented approach is dem-
onstrated also by the fact even lower values of control
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P. Juraj ET AL.
TI
parameters (np = 300 and g = 500) provided the solution
on the level 1.230054969. Based on showed resul ts, it
can be stated the suitability of presented approach, which
enables to determine the good real time solution.
5. Conclusion
The portfolio selection problem is one of the basic prob-
lems of allocating capital over the number of assets.
Fro m differ ent sets of performance measurement tools to
assist us with our portfolio evaluations, authors chose
portfolio performance measure - Omega function, which
computability is difficult due to substandard structures
and therefore the use of standard techniques seems to be
relatively complicated. Based on it, we use one of the
evolutionary algorithms that allow solving various types
of optimization problems (differential evolution). Pre-
sented approach enables to determine good real-time
solution. The quality of results is comparable with results
obtained by professional software.
6. Acknowledgements
This paper is supported by the Grant Agency of Slovak
Republic – VEGA, grant no. 1/0104/12 „Modeling
supply chain pricing policy in a competitive
environ ment“.
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