Technology and Investment, 2013, 4, 73-77 Published Online February 2013 (http://www.SciRP.org/journal/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 Email: [email protected] 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 2. Portfolio selection models based on In general, portfolio theory deals with the selection of an Omega function appropriate mix of assets in a portfolio in order to meet Omega function is measure which incorporates all the predetermined properties. Various mathematical models, distributional, characteristics of a return series. The which measure the portfolio performance measurement measure is a function of the returns leveled and requires can be used to support the decision making process of no parametric assumption on the distribution. Precisely, selection of the portfolio assets. The aim is to determine it considers the returns below and above a specific loss the allocation of the available resources in the selected threshold and provides a ratio of total probability group of assets that results in maximization of portfolio weighted losses and gains that fully describes the risk performance. Follow this idea, various measurements of reward properties of the distribution [8]: performance can be used. The performance measurement b techniques are e.g. Sharpe ratio, Treynor ratio, Jensen's − (1( ))dxxF alpha, Information Ratio, Sortino ratio, Omega function =Ω ∫MAR MAR)( MAR and the Sharpe Omega ratio ([1], [2], [3], [4], [5], [6], [7], )( dxxF [8]). The paper deals with Omega function, which com- ∫a putability is difficult due to substandard structures of performance level and therefore the use of standard tech- where: niques seems to be relatively complicated. The alterna- MAR denotes the return level regarded as a loss threshold, tive computing procedures includes variety of different (a, b ) denotes yields range, approaches. Nowadays a lot of research attention is fo- F(x) the cumulative distribution function of asset returns. cused on evolutionary algorithms. The paper presents the In the next part we formulate the problem of portfolio algorithm of differential evolution that is able to deal selection based on omega function performance measure. with nonlinear objective function with success (enables As it is mentioned, the Omega function involves consid- quick achievement of suboptimal solutions) and thus eration of all the information contained in the time series demonstrates suitability of evolutionary algorithms for of returns. The aim of portfolio selection problem is to financial modeling. Above mentioned approach is ana- maximize the level of Omega performance measure, lyzed on assets included in the Down Jones Industrial 1 where the variables w1, w2,…wd (where d represents the Index and its historical data published from July 1st number of assets) represent the weights of each asset in 2001 to April 1st 2012 on weekly basis were used. the portfolio. Corresponding problem can be formulated as follows [9]: 1 http://finance.yahoo.com/ (2012) Copyright © 2013 SciRes. TI 74 P. Juraj ET AL. T CREATE test vector; T − ∑ max (wrt MAR,0) IF (EVALUATION of test vector) > Ω=t =1 max (w ) T (EVALUATION of current selected indi- T vidual) THEN (SUBSTITUDE the selected ∑ max (MAR − wrt ,0) t =1 individual with the test vector) ; Subject to: ENDIF weT = 1 ENDFOR ENDWHILE w ≥ 0 EVALUATE process of calculating; Where: END T represents the number of periods, r represents returns vector of portfolio assets in the pe- t Evolutionary algorithms differ from more traditional riod t = 1,2,…T, optimization techniques in such a way that they involve a w denotes a vector of variables w , w ,…w representing 1 2 d search from a "population" of individuals, not from a the weight of each asset in the portfolio. single one. Each individual represents one candidate so- lution for the given problem that is represented by para- The computational complexity of the presented problem meters of individual. Associated with each individual is arises from its non-standard structure. Therefore, evolu- also the fitness, which represents the relevant value of tionary algorithms seem to be a suitable alternative to objective function. A population can be viewed as np.d standard techniques, due to its ability to achieve the sub- matrix (np – number of individuals in the population, d – optimal solutions in relatively short time. The differential number of parameters of individual). Every step involves evolution is one of the popular and well known tech- a competitive selection that carried out poor solutions. niques. Consider the problem of portfolio selection it is possible to summarize the steps of the algorithm as follows: 3. Differential Evolution Setting of the control parameters. Differential evolution is controlled by a special set of parameters. Recom- Differential evolution (introduced by Price and Storn mended values for the parameters are usually derived [10]) belongs to the class of evolutionary techniques, empirically from experiments ([15], [16], [17]): comprise a large number of nontraditional computing d – dimensionality. Number of parameters of individual techniques whose common characteristic is that they are is equal to number of assets. inspired by the observation of the nature processes (ge- np – population size. Number of individuals in popula- netic algorithms, ant colony optimization, differential tion. recommended setting is 5d to 30d, respectively evolution, etc.). Nowadays evolutionary algorithms are 100d, in case the optimized function is multimodal ([15], considered to be effective tools that can be used to search [16]). for solutions of optimization problems (napr. [11], [12], g – generations. Represent the maximum number of ite- [13], [14]). The big advantage over traditional methods is ration (g is also stopping criterion). that they are designed to find global extremes (with built-in stochastic component) and that their use does not cr – crossover constant, cr∈ 0,1 . The value of cr was require a priori knowledge of optimized function (con- set on the base of experiments. vexity, differential etc.), and in that way they work well f – mutation constant, f ∈ 0,1 . The value of f was set on to solve continuous non-linear problems, where is hard to the base of experiments. use traditional mathematic methods. The principle of ()0 basic version of differential evolution can be described Initialization. The population P was randomly initia- by following pseudocode: lized at the beginning of evolutionary process according to the rule: BEGIN rnd 0,1 ()()00= = = SETTING of control parameters; P()i = wij, d i1,2,... np jd1,2,... ()0 INITALIZATION of population; ∑ wij, EVALUATION of each individual; j=1 WHILE (STOPPING CRITERION is not satisfied) that ensure that the total weights of portfolio is equal DO to one. Each individual is then evaluated with the fitness FOR (each individual of the population) DO (given by function Ω(w)). (REPRODUCTIVE CYCLE): CREATE differential vector; CREATE trial vector; Copyright © 2013 SciRes. TI 75 P. Juraj ET AL. The test of stopping condition. In its canonical So that process continues in each generation for all indi- form, the only stopping criterion is to reach the viduals. The result is a new generation with the same number of individuals. maximal number of iterations (represent by parameter g). Reproductive cycle. This cycle comprise the crossing and Evaluation. The whole process of reproduction continues mutation to create individuals for the next generation. until the last (users specified) number of generations is For each individual wig, i =1, 2,...np, from the population reached. The value of the best individual from each gen- another three different individuals are chosen (vectors r1, eration is reflected to history vector, which shows the r2, r3). The difference of the first two vectors (r1 and r2) progression of an evolutionary process. gives the differential vector, which is multiplied by mu- tation constant f and added to vector r3. Thus, we get trial vector v. Formally: 4. Empirical Results tt t t v j=+−w r3, j fw.() rj1, w r 2, j jd= 1,2,... , tg= 1,2,... The portfolio analyze was based on index Dow Jones After the mutation process comes the formation Industrial, which is one of the major market indexes, as well as one of the most popular indicators of the U.S. of a new individual, which is also called test vector wtest market. It is a stock market index, and one of several so that one element after another is selected from the indices created by Wall Street Journal editor and Dow currently selected individual wig and from the trial vec- Jones & Company co-founder Charles Dow. It was tor v and for every pair is generated a random number founded on May 26, 1896. It is an index that shows how 0,1 30 large publicly owned companies based in the United from the interval , which is compared with the States have traded during a standard trading session in crossing constant cr. If the generated random number is the stock market. (so called Large-Cap companies – less than or equal to cr, to the relevant position of wtest companies with market capitalization above 10 mild comes the element of trial vector v, otherwise of USD).