A portfolio optimization model with three objectives and discrete variables

ARTICLE IN PRESS Computers & Operations Research 37 (2010) 1285–1297

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A portfolio optimization model with three objectives and discrete variables K.P. Anagnostopoulos, G. Mamanis  Democritus University of Thrace, Greece

a r t i c l e in fo

abstract

Available online 13 October 2009

We formulate the portfolio selection as a tri-objective optimization problem so as to find tradeoffs between risk, return and the number of securities in the portfolio. Furthermore, quantity and class constraints are introduced into the model in order to limit the proportion of the portfolio invested in assets with common characteristics and to avoid very small holdings. Since the proposed portfolio selection model involves mixed integer decision variables and multiple objectives finding the exact efficient frontier may be very hard. Nevertheless, finding a good approximation of the efficient surface which provides the investor with a diverse set of portfolios capturing all possible tradeoffs between the objectives within limited computational time is usually acceptable. We experiment with the current state of the art evolutionary multiobjective optimization techniques, namely the Non-dominated Sorting Genetic Algorithm II (NSGA-II), Pareto Envelope-based Selection Algorithm (PESA) and Strength Pareto Evolutionary Algorithm 2 (SPEA2), for solving the mixed-integer multiobjective optimization problem and provide a performance comparison among them using metrics proposed by the community. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Evolutionary multiobjective optimization Class constraints Multiobjective portfolio selection NSGA-II PESA Quantity constraints SPEA2

1. Introduction The portfolio selection problem, which involves computing the proportion of the initial budget that should be allocated in the available assets, is at the core of the field of financial management. A fundamental answer to this problem was given by Markowitz who proposed the mean–variance model which laid the basis of modern portfolio theory [1,2]. In Markowitz’s approach the problem is formulated as an optimization problem involving two criteria: the reward of a portfolio, which is measured by the mean and should be maximized, and the risk of the portfolio (measured by the variance of return) that should be minimized. In the presence of two criteria there is not a single optimal solution (portfolio), but a set of optimal portfolios, the so-called efficient portfolios, which tradeoff between risk and return. Since the mean–variance theory of Markowitz, an enormous amount of papers have been published extending or modifying the basic model in three directions: (i) the simplification of the type and amount of input data; (ii) the introduction of alternative measures of risk; and (iii) the incorporation of additional criteria and/or constraints. In this study we concentrate on the third direction of incorporating additional criteria and/or constraints in portfolio modeling.

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E-mail addresses: [email protected] (K.P. Anagnostopoulos), [email protected], [email protected] (G. Mamanis). 0305-0548/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2009.09.009

Recently, researchers have recognized the usefulness of incorporating additional criteria beyond variance and return into the portfolio selection model [3–9]. These studies rather than focusing on the ‘‘standard’’ investor, they consider the so-called suitable-portfolio investor [3]. This type of investor is also concerned with the number of assets in the portfolio, the maximum amount invested in any asset, the social responsibility, the amount invested in R&D and so forth. Having more than two objectives we are not searching for an efficient line anymore, rather we are searching for an efficient surface in a high dimensional space. Finding the exact efficient surface for multiobjective optimization problems is a very difficult task especially when at least one of the objectives possesses discrete or nonsmooth characteristics. Steuer et al. [9] point out that a discrete picture of the true efficient surface can be found when the additional objectives and/or constraints are linear. They notice also, that they are able to obtain the exact efficient surface for three objective problems with one quadratic and two or three linear objective functions. However, if at least one more objective is nonlinear, alternative optimization techniques are required. In this paper, we formulate the portfolio selection as a triobjective optimization problem so as to find tradeoffs between risk, return and the number of securities included in the portfolio. We also impose restrictions on the proportions of the portfolio invested in assets and in group of assets in order to avoid very small holdings and excessively investing in assets with common characteristics. At this point, one might well wonder why to modify the number of assets in the portfolio as an objective and not as a constraint. The reason is that, in multiple criteria

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optimization, as pointed out in [3], ‘‘we distinguish an objective from a constraint when it is not easy to fix a right-hand side value for the constraint without knowing the levels of the other objectives’’. In the particular case it is very difficult for the decision maker to know on beforehand the optimal number of securities that should be included in his/her portfolio without examining all the tradeoffs between risk, return and the cardinality of the portfolio. In any case, an investor might lose important solutions with large tradeoff between the objectives when he/she is forced to fix the number of assets in the portfolio on beforehand. Furthermore, the rational in multiobjective optimization, as well as in the standard approach of Markowitz, is to first solve for the efficient surface (or at least a discrete approximation of it) and then to select a point from this surface that better fits the investors needs. Since the function that measures the number of assets in the portfolio possesses discrete and non-smooth aspects, approximate optimization techniques may be the only way to attack such problems. For this reason we have experimented with the current state-of-the-art multiobjective evolutionary algorithms (MOEAs) [10]. MOEAs are especially designed evolutionary algorithms (EAs) for multiobjective optimization problems and are able in most cases to find a good approximation of the true efficient surface. There exist many studies applying metaheuristics or other strategies in the portfolio problem. Most of them consider a single-objective optimization problem, usually by minimizing the variance of the portfolio while satisfying a lower return bound and a maximum number of assets in the portfolio. One of the first attempts was made by Dueck and Winker [11], which use a local search technique, called threshold accepting, in a portfolio selection problem with semi-variance as a risk measure. Chang et al. [12] have applied three heuristic algorithms based upon genetic algorithms, tabu search and simulated annealing, and they show that the best results are obtained when optimal solutions from all the algorithms are selected. Various algorithms have also been proposed for solving the constrained portfolio optimization problem: a branch-and-bound algorithm combined with heuristics [13]; hill climbing, simulated annealing and tabu search [14]; a simulated annealing algorithm applied to an extended version of the model with trading and turnover constraints [15]; an hybrid local search algorithm which combines principles of simulated annealing and evolutionary strategies [16]; a threshold accepting heuristic for minimizing value-at-risk and expected shortfall [17]. Apart from those techniques there are a few papers that apply evolutionary multiobjective optimization techniques or other multiobjective metaheuristics to tackle the multiobjective portfolio optimization problem. These applications are: a MOEA to solve a tri-objective quadratic programming problem with two sources of risk [18]; greedy search, simulated annealing and ant colony optimization [19]; NSGA-II, PESA and SPEA2 for solving the standard portfolio optimization problem [20]; MOEAs with local search for feasible solutions [21]; an hybrid multiobjective optimization approach combining evolutionary computation with linear programming for solving a tri-objective portfolio optimization problem with one return and two risk measures [22]; a MOEA with an ordered based representation [23]. A recent survey of applications of MOEAs in the portfolio problem can be found in Tapia and Coello [24]. According to our knowledge there is one more study in the literature that concerns with the mean–variance-number of assets in the portfolio problem [25]. In this study a MOEA is applied to the problem in order to find a discrete approximation of the efficient surface in a single run. In addition to this attempt, our study (i) evaluates the current state of the art algorithms in the field of evolutionary multiobjective optimization using measures

widely accepted by the community, (ii) solves larger scale instances with 200 and 300 securities, (iii) add practical constraints and proposes a data structure for chromosomal representation for handling the additional constraints imposed to the model. A comparison of this chromosomal representation and the usual representation used in evolutionary portfolio optimization literature together with the constrained domination operator proposed by Deb et al. [26] is done to evaluate its effectiveness. The rest of the paper is organized as follows. In Section 2, after a short introduction to multiobjective optimization terminology, the tri-objective portfolio optimization problem considered in this study is described. Section 3 presents the three multiobjective evolutionary algorithms and how they were implemented in this problem. Section 4 is devoted to numerical results, and some concluding remarks are presented in Section 5.

2. Multiobjective portfolio optimization 2.1. Multiobjective optimization A multiobjective optimization problem is formulated as follows:   optimize FðxÞ ¼ f1 ðxÞ; . . . ; fk ðxÞ s:t: x A X where x ¼ ðx1 ; . . . ; xn Þ is the vector of decision variables and X is the set of feasible solutions. The objective function vector F(x) which contains the values of k objectives maps the feasible set X into the set F (the feasible region in the objective space) which represents all possible values of the objective functions. The objective functions may all be maximized, minimized or be in a mixed form. The usual process in multiobjective optimization is to find all non-dominated or Pareto optimal solutions of the problem, i.e. every solution which we cannot improve one objective function without deteriorating another. Let I1 (I2) be a set containing the indexes of the objective functions that should be minimized (maximized). We say that a solution x A X dominates x A XðX  XÞ iff 8i A I1 ; fi ðx Þ rfi ðxÞ4(i A I1 : fi ðx Þ ofi ðxÞ and 8i A I2 ; fi ðx Þ Zfi ðxÞ4(i A I2 : fi ðx Þ 4fi ðxÞ The Pareto optimal set (or non-dominated set) is constituted by all solutions that are not dominated by any other in the feasible set. P ¼ fx A Xj) x0 A X: x0  xg The image of the Pareto optimal solutions in the objective space defines the Pareto front (or non-dominated front). PF ¼ fu ¼ FðxÞjx A Pg 2.2. A tri-objective portfolio optimization problem The standard portfolio optimization problem assumes a single investment period and n assets available for investment. The investor should determine the proportion weights of the initial budget which will be allocated in the available assets. These weights are denoted as a vector of decision variables x ¼ ðx1 ; . . . ; xn Þ and constitute the portfolio. Following the pioneering work of Markowitz the portfolio optimization problem is modeled as a two-objective optimization problem. The goal in portfolio selection is to find a feasible portfolio x that simultaneously maximize return and minimize risk. However, since the two criteria are conflicting such a

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portfolio does not exist. Contrarily, between the expected return and risk there is a tradeoff that is offered by a number of investment alternatives. The only way to improve portfolio’s return is to increase the risk level that the decision maker is disposed to accept. These tradeoff portfolios are usually called efficient portfolios and their mapping in mean–variance space gives the efficient or non-dominated frontier. The essence of portfolio optimization is to search for all feasible portfolios that offer the best tradeoff between risk and return. The mean– variance bi-objective portfolio optimization problem is formulated as follows: min

rðxÞ ¼

n X n X

xi xj sij

i¼1j¼1

max

mðxÞ ¼

n X

xi mi

i¼1

s:t:

xAX

where the m’s, and s’s symbolize the expected returns, covariances and variances of the securities which are the inputs of the portfolio optimization problem. The feasible set of portfolios X is P defined by the budget constraint ð i xi ¼ 1Þ, which requires that all the available capital is invested, and the non-negativity constraints that implies that no short sales are allowed ði:e: x Z0Þ. However, the portfolio selection is usually formulated as a single objective optimization problem. The most popular approach for the mean–variance optimization problem is the one that tries to minimize variance while return is constrained to have a lower level. min s:t:

rðxÞ mðxÞ Zd; x A X

With the above formulation, solving for different return levels of d A ½dmin ; dmax  the efficient set of portfolios can be found. The above formulation has the advantage that it is reduced to a scalar optimization problem (a quadratic optimization problem) and can be solved very efficiently if X is convex. Solving the above problem for the efficient frontier several effective quadratic programming algorithms exist as well as the critical line method of Markowitz which computes the true efficient frontier by identifying corner points and then computes all efficient portfolios via convex combination of corner points. An alternative formulation of the problem is the weighted sum approach. In this approach a parameter l is used which combines P P the two objectives into a scalar ðmin l ni¼ 1 nj¼ 1 xi xj sij  Pn ð1  lÞ i ¼ 1 xi mi Þ. By varying the parameter value l and solving a sequence of quadratic programming problems (for each l) the efficient portfolios from the minimum variance portfolio (l =1) to the maximum return portfolio (l = 0) can be found. However, when additional objectives are imposed into the model, solving for the efficient surface with these methods becomes problematic since the required number of single optimization problems that should be solved is increased with the number of objectives. Furthermore, a non-uniform sample of the true non-dominated frontier may be returned by the algorithm. In this study, beyond risk and return we wish to consider an additional objective which minimizes the number of assets in a portfolio. Besides the additional difficulty of the introduction of a third objective there is another one which is imposed by the nonsmoothness of the additional objective function. These difficulties have led us to experiment with the state of the art multiobjective evolutionary algorithms in order to find a good approximation of the true efficient surface. The third objective function is incorporated by counting the number of non-negative weights in a portfolio. The number of

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assets in the portfolio should be minimized. min cardðxÞ ¼

n X

1xi 4 0

i¼1

We will refer to the tri-objective model with only budget and nonnegativity constraints as the unconstrained problem and its solution as the unconstrained efficient frontier. Quantity constraints can be added to the model using a binary variable, which is equal to 1 if the asset i =1,y,n is held in the portfolio and 0 otherwise. Introducing finite upper and lower bounds for the stock weight, quantity constraints are represented by the following inequality: li di r xi r ui di ; i ¼ 1; . . . ; n Moreover class constraints can be added by letting Gm , m= 1, y, M, be M mutually exclusive sets of assets and Lm and Um be the lower and upper proportion limit for class m. The constraints can be defined as X xi rUm ; m ¼ 1; . . . ; M Lm r i A Gm

Introducing a third objective into the portfolio optimization model the efficient frontier becomes a surface in the threedimensional space and finding the exact efficient surface is very difficult if not impossible. However, a discrete approximation of the efficient surface which provides adequate information of the tradeoffs between the objectives is usually acceptable. Solving a multiobjective optimization problem requires a preference relation to be denoted. The usual relation that is used in multiple criteria optimization is the dominance relation defined in Section 2.1. In the particular problem at hand we say that a portfolio y dominates another portfolio x if m(y) Z m(x), r(y)r r(x) and card(y)rcard(x) with at least one strict inequality. The set of all portfolios that their images in the objective space are non-dominated comprises the efficient set.

3. Multiobjective evolutionary algorithms Evolutionary algorithms (EAs) are powerful stochastic search techniques that mimic the Darwinian principles of natural selection (survival of the fittest) and are well suited for solving optimization problems with difficult search landscapes (e.g., large solution spaces, multimodal search spaces, constraints, nonlinear and non-differentiable functions, multiple objectives). The last ability of EAs—due to their population-based nature—to handle problems having multiple objective functions has given rise to the field of evolutionary multiobjective optimization (EMO) which refers to the use of EAs to solve complex multiobjective optimization problems (e.g., many objectives, very large search spaces, noise, disjoint non-dominated curves, etc.). The algorithms designed for this purpose are usually identified under the rubric multiobjective evolutionary algorithms (MOEAs) and they differ from their single-objective counterparts mainly in the way selection is performed. MOEAs use a non-dominated ranking and selection to guide the population towards the Pareto front, and diversity preserving techniques to avoid convergence to a single point on the front. The main advantage of MOEAs is that they generate reasonably good approximations of the nondominated frontier in a single run and within limited computational time. Since the first approach by Schaffer [27] several MOEAs have been proposed [10,28,29]; for example the Multiobjective Genetic Algorithm (MOGA), the Niched-Pareto Genetic Algorithm (NPGA) and the Non-dominated Sorting Genetic Algorithm (NSGA). These

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early attempts have been criticized, mainly because of the lack of elitism, a technique that plays crucial role in the performance of MOEAs. Elitism is usually incorporated to MOEAs by using an external population (archive) to retain the non-dominated individuals found along the evolutionary process or the use of a (m + l) selection strategy in which parents compete with their children and those which are non-dominated are selected for the following generation. Most cited elitist MOEAs are the Strength Pareto Evolutionary Algorithm (SPEA, SPEA2) [30,31], PAES, PESA and NSGA-II [29]. In this study we use three of the most cited MOEAs, the Nondominated Sorting Genetic Algorithm II (NSGA-II) [26], the Strength Pareto Evolutionary Algorithm 2 (SPEA2) and the Pareto Envelope-based Selection Algorithm (PESA) [32] in order to explore the multiobjective portfolio optimization problem domain for the best tradeoff portfolios hopefully to provide a good approximation of the (unknown) efficient set.

3.1. MOEAs description In this section we will describe the basic operators and structure for the three algorithms considered in this study. All algorithms are described through the general framework outlined in Laumans et al. [33] with slight modifications. This allows us to identify more easily similarities and differences of the tested algorithms. All algorithms fit the following general framework and apply in a different manner the operators. One difference is that PESA applies the evaluate operator after truncation in the archive of the next generation (Fig. 1). All MOEAs employ a normal population of individuals B together with an archive A in order to ensure the preservation of good non-dominated solutions. At first, the archive A0 is set to the empty set and the population B0 (of size Npop) to a random sample of the solution space through the initialize operator. At each generation, and while a stopping criterion is not satisfied, the evaluate operator assigns fitness to individuals from both the archive and the normal population. This fitness assignment operator must comply with the fundamental goal of MOEAs which is the generation of non-dominated solutions as spread as possible in the efficient frontier. Next, the archive is updated by the best individuals of the composite population A [ B. This is performed by using the update operator and the truncate operator if the number of solutions exceeds the user-specified maximum archive size Narc. The sample and vary operators specify the particular selection and reproduction scheme and are the same as in traditional evolutionary algorithms. At the last stage the best solutions from the archive and the final offspring population is returned by the algorithm. The main difference among the algorithms lies in their fitness assignment technique (evaluate operator). All MOEAs fitness assignment scheme give first priority to non-dominance and

t=0 (A0, B0) = initialize() while (termination = false) do evaluate(At, Bt) At+1 = truncate(update(At, Bt)) Bt+1 = vary(sample(At+1)) t = t +1 end while return (update(At, Bt) Fig. 1. Structure of a MOEA.

second priority to diversity. However, they achieve this by using three different dominance-based ranking and diversity preserving techniques [10, p. 79]. NSGA-II uses the dominance depth method for sorting the individuals according to the dominance relation and a crowding technique to preserve diversity. The dominance depth method identifies at which frontier an individual is located. First, the nondominated individuals are identified so that to constitute the first non-dominated frontier and are assigned a rank 1. Next, these individuals are ignored and the second frontier of non-dominated individuals is identified given the rank 2. The process continues until all individuals in the population are classified. The crowding mechanism is applied to discriminate between individuals with identical ranks and has the goal to preserve diversity in population. The technique is applied separately in each layer and works as follows. The population is sorted according to each objective function value, and the extreme solutions are assigned a large distance value so that they are always selected. The remaining solutions are assigned a distance value equal to the absolute normalized difference in the function values of two adjacent solutions. SPEA2 does this in a different manner. It utilizes a finer grained mixed strategy to emphasize non-dominated individuals based on the dominance rank and dominance count method, and a clustering technique to preserve diversity. First, each individual is assigned a strength value equals to the number of individuals that dominates (dominance count method). Thereafter, the fitness of a solution is simply the sum of the strengths of its dominators. Thus, all non-dominated solutions have a zero fitness value. The density information is incorporated by adding to the fitness of each individual a value that is equal to the inverse of the k-th smallest Euclidean distance (measured in objective space) plus two. PESA uses a niching approach to preserve diversity by forming an implicit hyper-grid which divides the objective space into hyper-boxes. The fitness of an individual is the number of other solutions in the archive that resides in the same box. This process can be seen as a secondary information since it is applied only to non-dominated individuals (see update operator below). Again, minimization of fitness is assumed. Update operator is very important for MOEAs since manipulates the elitism intensity in the process. We will describe update and truncate operators together as they are closely related. NSGA-II update strategy returns the best Narc individuals from the union of the archive and the population based on the rank value and the crowding distance. Individuals with the lower rank have the priority to survive. If a number of solutions that have the same rank does not all fit the archive, then the less crowded individuals from the particular rank are selected to enter the archive. On the other hand, SPEA2 returns all non-dominated individuals from the combined set. There are two possibilities with this process. Either the individuals returned by the update operator exceed the maximum archive size Narc or they are not enough. In the latter case the best dominated solutions according to their fitness values are selected. In the former case, a truncate operator is applied that recursively deletes surplus solutions based on the nearest neighbor Euclidean distance. At each stage, if there is more than one solution with the same minimum distance, the decision is done considering the second nearest neighbor and so forth. In PESA the archive update and truncate operator are performed iteratively. The newly generated solutions Bt are incorporated into the archive one by one. A candidate enters the archive when it is non-dominated within Bt, or it is not dominated by any current member of the archive A. If the addition of a solution renders the archive over-full then a current solution with the maximum fitness value is deleted.

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Sample and vary operators are the same as in usual evolutionary algorithms. We use identical schemes for all tested algorithms in order to ensure a fair comparison. For selecting the parents we use binary tournament selection for all algorithms. Variation operators are described in Section 3.3. 3.2. MOEAs performance indicators Comparing the quality of different MOEAs as well as different algorithm configurations is an important issue. Several quality metrics have been developed which are able (i) to measure each qualitative characteristic of an approximation set (the output of a MOEA) separately, i.e. the closeness to the true Pareto front, the uniformity and spread of the solutions along the Pareto front [34]; and (ii) to identify whether one approximation set is better than another based on certain set preference relations [35]. In this study we have used the e-indicator and the hypervolume metric proposed in [35,30]. The e-indicator is able to detect whether one approximation set is better than another and defines how close the set is to a reference set. As a reference set, usually the true or the best known efficient frontier is used. The e-indicator metric gives the minimum value, a reference set must be multiplied or added in order to be dominated by the approximation set. If the approximation set matches exactly the reference set then it takes the identical value of one (or zero in the additive version). Low values for this metric reveals that the approximation set are very close to the reference set. Since, in this study we do not know the true non-dominated surface, as reference set we have used the non-dominated solutions from the combined set produced by all algorithms and all replicates for a particular problem instance. Hypervolume metric is also compliant with the set preference relations and is able to measure how well the algorithms perform in identifying solutions along the full extent of the Pareto surface [35]. It simply calculates the volume of the objective space dominated by the solutions produced from a particular algorithm bounded by some reference point. Thus, higher values are preferable. Furthermore, in order to allow the objectives to contribute approximately equally we have used the following linear normalization technique [36]. Each objective function value i was transformed according to the following equation: fi0 ¼

fi  fimin max fi  fimin

where fimin and fimax are the minimum and maximum values, that the ith objective can take. The reference point required to compute the hypervolume metric was zref = {1, 0, 1} normalized. In some cases, the two indicators may return opposite preference orderings for a pair of approximation sets. For such a result the two sets should be incomparable [36]. 3.3. MOEAs implementation When evolutionary algorithms are implemented in real world optimization problems, several issues are crucial for their performance and should be determined. The basic components are the solution representation (coding, chromosomal data structure), the input parameters of the algorithm such as the population size, the maximum archive size, the crossover and mutation probabilities, and the termination condition. The solution representation (chromosome data structure) is probably the most important future of an evolutionary algorithm and a MOEA in particular since it determines the search space and if it is chosen wisely could lead in improving performance. For

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example, for the portfolio optimization problem Streichert et al. [21] proposed a hybrid representation, where an additional binary string is included to reflect the existence of the assets in the portfolio resulting in better algorithm performance. 3.3.1. MOEAs chromosomal representation and encoding Initially, we have chosen the same hybrid representation with Streichert et al. [21], but the results were poor when class constraints were added into the model as we will see later. In the hybrid solution representation two vectors are used for defining a portfolio, a binary vector specifies whether a particular asset participates in the portfolio, and a real-valued vector used to compute the proportions of the budget invested in the assets. Thus we have

D ¼ fd1 ; . . . ; dn g; di 2 f0; 1g; i ¼ 1; . . . ; n W ¼ fw1 ; . . . wn g; 0 r wi r 1; i ¼ 1; . . . ; n In order to find the real portfolio x associated with the above encoding we proceed as follows: first, the weights of assets that are not elements of the portfolio are vanished ði:e: wi ¼ 0; if di ¼ 0Þ. Thereafter the remaining weights are normalized in order to satisfy the budget constraint. Thus the real proportion xi P is calculated by xi ¼ wi = w for every i= 1,y,n. With this encoding and decoding process we account only for budget and non-negativity constraints. The portfolio calculated may violate quantity and/or class constraints. To account with these inequality constraints we have used a technique introduced by Deb et al. [26]. The usual dominance relation defined in Section 2.2 is replaced by the constrained dominance relation to bias the search towards feasible non-dominated regions. A portfolio x1 is said to constrained dominates a portfolio x2 if any of the following conditions is true: (1) Portfolio x1 is feasible and portfolio x2 is not. (2) Both portfolios are infeasible, but portfolio x1 has a smaller overall constraint violation. (3) Both portfolios are feasible and x1 dominates x2. However, with this encoding and constraint-handling technique we have obtained poor results regarding a part of the efficient surface that lies close to the boundary of feasible, and infeasible regions (in particular the surface composed by portfolios with the highest expected return). This problem is probably due to the ineffective exploitation of infeasible solutions. The archive of individuals converges very quickly (approximately in 7–8 generations for NSGA-II and SPEA2; and PESA does not exploit any infeasible solution at all once it archives a feasible one, and again this happens in early generations) for the particular problem, thus making solutions close to the boundary of feasible and infeasible regions hard to find. The necessity to maintain diversity in population has been noticed by numerous researches [37]. We think that this is especially true for constrained multiobjective optimization problems. We want also to point out that this ineffectiveness of the constraint handling technique is deteriorated due to interactions among the weights when they are normalized to find the real portfolio x. For this reason, we have utilized a problem-specific chromosome representation and a repair mechanism, which are able to produce only feasible solutions. In order to handle class constraints, we introduce a new vector of real values in the above mentioned chromosome representation to facilitate with the proportion of the initial budget that should be invested in each class. C ¼ fc1 ; . . . ; cM g; 0 r cm r1; m ¼ 1; . . . ; M

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The class weights c are associated with each group of assets m and they are normalized to find the real proportion invested in assets belonging to the particular class using the following formula: cm rcpðmÞ ¼ Lm þ PM

j ¼ 1 cj



1

M  X Lj ;

m ¼ 1; . . . ; M

ð1Þ

j1

where rcp(m) is the real class proportion invested in class m. Next, the real proportion of each class is shared in the corresponding assets of the particular class. The proportion associated with each asset in the portfolio is calculated by the following equation:     X wd rcp classðiÞ  xi ¼ li di þ P i i lj dj ; i ¼ 1; :::; n ð2Þ wj dj jAG j A GclassðiÞ

classðiÞ

class(i) returns the group that the asset i belongs. With the above encoding and decoding process just described we are able to obtain a feasible portfolio in most of the times. However, we may obtain an infeasible portfolio. If this happens we use a simple repair technique which is described below. One way to get an infeasible solution is when the portfolio does not contain an asset from a specific class which must have a positive proportion weight in the portfolio (i.e. Li 40 for the particular class i). When this happens the rcp of the particular class does not share and the resulting solution violates budget as well as one class constraint. We fix this infeasible solution by inserting a random asset from the specific class or classes into the portfolio (i.e. by changing a zero value to 1 in the D string). Another way the above formulation results in an infeasible portfolio is when the parenthesis in (2) is negative. That is when the rcp for a particular class is not enough to be shared in the corresponding assets so that to satisfy lower quantity constraints for these assets. For example, suppose that the rcp for a class is 0.05 and there are six or more assets from the particular class in the portfolio, then this class proportion is not enough if we suppose that these assets requires at least 0.01, thus resulting in a 0.06 minimum class proportion. Again, we repair this infeasible solution by simply deleting surplus assets belonging to the particular class (e.g. by setting a value from 1 to 0 in D) starting from the one with the smallest weight (in W vector). In this way the portfolio x satisfies budget, lower quantity and lower class constraints. To account with upper quantity and class bounds, we propose the following repair algorithm outlined in [12]. If a real class proportion rcp(m) violates its upper limit Um, then the real class proportion for the particular class m takes this value, and the remaining class values are normalized again according to equation described above. This process is continued until all upper limits are satisfied. The same process is used for the real asset weights. 3.3.2. MOEAs variation operators For reproducing the offspring population, we have used the uniform crossover operator in each string of the chromosome. In uniform crossover two selected individuals generates a single child and its value for each array is selected with equal probability from one or another parent. The children were considered also for mutation. We have used different mutation probabilities for each vector. In real-valued vectors the Gaussian random mutation was applied with standard deviation 0.05, while in the binary string bit flip mutation in a randomly defined position was applied.

4. Empirical results In this section we report the experimental results we have obtained on randomly generated data that were constructed

utilizing the method described in [38]. The procedure requires the expected value and standard deviation of variances and covariances to be given, and then generates randomly a covariance matrix with these characteristics. For providing such values we have calculated the expected value and standard deviation of the required parameters from the British FTSE 100 data set as taken from the publicly available OR-Library retained by Beasley (http:// people.brunel.ac.uk/  mastjjb/jeb/orlib/portinfo.html). Two different data sets containing n = 200 and 300 securities have been constructed. Numerous problem instances with and without additional constraints have been solved in order to test algorithms’ robustness. All algorithms have been implemented in C++ and run on a personal computer Core 2 Duo at 2.1 GHz.

4.1. Parameter setting Before the algorithms were compared some tuning was done to find good parameter values that make them to perform well. The analysis in this section is referred to the smallest problem instance (i.e., n= 200). For identifying the best population and archive size, the following configurations were tested for each algorithm: Npop = Narc = 200, 400, 500 and Npop =50, Narc = 500. The last configuration with 1:10 ratio between population and archive size was used to identify if we could obtain some gain on the performance. This ratio was suggested in some studies, for example in [32]. For each configuration the algorithms were left to run until 300,000 solutions were generated. The performance of each algorithm was analyzed in different number of generations in order to have a picture of the whole evolution of MOEAs. In this stage of experiments for crossover and mutation probabilities we have used values that are most commonly used with MOEAs in prior literature. Crossover probability was set at 0.9, Gaussian mutation probability at 1.0 and bit flip mutation probability at 1/n for all algorithms. For measuring the quality of each configuration the hypervolume metric was used. For normalizing the objective values the minimum and maximum bounds for each objective function for this instance are set at fimin ¼ f0:000141; 0:00217; 1g and fimax ¼ f0:001337; 0:00762; 125g. In Fig. 2 the mean values over five runs for each algorithm and configuration are provided. We observe that as the archive size increases the hypervolume metric increases but with lower rate. However, the computational time increases with the archive size as well. All algorithms converge to their maximum metric value when approximately 80,000 solutions are generated. Based on these experiments we have chosen a population and archive size of 500 individuals for all algorithms, and we have performed a number of experiments to find good parameter values for crossover and mutation probabilities. We have fixed four crossover rates from 0.7 to 1.0 with step 0.1, and 3 bit flip mutation probabilities (0.001, 0.005, 0.01), thus having 12 different configurations for each algorithm. Each configuration was run five times yielding a total of 180 trial runs. We have not parameterized mutation in real valued vectors. We have used a Gaussian mutation probability of 1.0 as suggested in [21]. A small amount of testing with different values has shown no improvement on the results. Table 1 contains the average values over five runs for each configuration and MOEA. The best combinations among the 12 configurations for each algorithm are noted with bold font. Fortunately, there are several combinations that generate good values very close to the best one. In NSGA-II two values which are very close have identified. For this algorithm we have chosen a crossover probability of 0.9 since with this rate we have obtained more reliable results across mutation. PESA requires an additional

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PESA

0.8

0.8

0.75

0.75

0.7 0.65

Npop = Narc = 200 Npop = Narc = 400 Npop = Narc = 500 Npop = 50, Narc = 500

0.6 0.55

Hypervolume

Hypervolume

NSGA−II

1291

0.7 0.65

Npop = Narc = 200 Npop = Narc = 400 Npop = Narc = 500 Npop = 50, Narc = 500

0.6 0.55

0

0.5

1 1.5 2 2.5 3 x 105 Generated Solutions

0

0.5

1 1.5 2 2.5 3 x 105 Generated Solutions

SPEA2 0.8

Hypervolume

0.75 0.7 0.65

Npop = Narc = 200 Npop = Narc = 400 Npop = Narc = 500 Npop = 50, Narc = 500

0.6 0.55 0

0.5

1 1.5 2 2.5 3 Generated Solutions x 105

Fig. 2. Population and archive size testing.

Table 1 Average values over five runs for each configuration and MOEA. 0.005

0.01

0.001

1 0.9 0.8 0.7

NSGA-II 0.791397 0.793578 0.792723 0.792917

0.793097 0.793848 0.792869 0.792216

0.783050 0.793602 0.793857 0.793422

1 0.9 0.8 0.7

PESA 0.781893 0.794925 0.795154 0.795629

0.778742 0.794647 0.795524 0.796129

0.737547 0.793439 0.795266 0.795822

1 0.9 0.8 0.7

SPEA2 0.796936 0.798957 0.799250 0.798568

0.798057 0.798954 0.798956 0.798439

0.796103 0.798718 0.799006 0.798890

parameter to be set, which is the number of divisions of the search space in each axis. After trial and error we have set that value equal to 50.

4.2. Unconstrained multiobjective portfolio optimization problem This section describes the computational results obtained when no class or quantity constraints were used on the

securities weights. In Figs. 3 and 4 the best efficient surface from the combined set of all replicates for each algorithm and for both instances are represented. The figures show for each cardinality level the mean–variance efficient frontier. We can see that as the number of assets in the portfolio increases the variance decreases but the expected return of the portfolio decreases as well. Furthermore, for a fixed level of expected return, there are various portfolios with varying variance but generally portfolios with smaller variance contain more securities and this is certainly a tradeoff since investors prefer to have small portfolios. By examining the above surfaces decision makers can therefore find portfolios that suit to their preferences the best. The current practice in literature to handle the number of assets in the portfolio is to fix a cardinality level K or a range of levels Kmin through Kmax and to solve for the mean–variance efficient frontier. This corresponds to a particular efficient line in our graph or the mean–variance efficient portfolios from the combined lines between Kmin and Kmax. To find such a surface (like Figs. 3 and 4) by fixing the cardinality constraint one could solve 30 two-objective optimization problems (one optimization problem for each K) or 500 or more nonlinear optimization problems with different values of return and K. As for algorithms’ performance we can visually see that PESA and SPEA2 achieve good diversity characteristics which are better than NSGA-II. In order to have a sense and to provide some evidence for the effectiveness of the proposed techniques we plot the mean–variance efficient frontier extracting from the above surfaces for each algorithm.

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NSGA−II 10−3

6 4 2 0 40

x 8 7 6 5 4 3 2 30

Expected Return

Expected Return

x 8

PESA

10−3

30 20 Ca rdin alit 10 y

1.5

1 0

0.5

0

20 rdin 10 alit y

Ca

−3

0 e x1

nc Varia

0.5

0 0

1

1.5 −3

10 e x

nc Varia

Expected Return

SPEA2 x 10−3 8 7 6 5 4 3 2 30 20 Ca rdin

10 alit y

1

0.5

ce arian

0 0

1.5 −3

x 10

V

Fig. 3. Unconstrained efficient surface for the problem with n =200 securities.

PESA

Expected Return

0.01 0.008 0.006 0.004 0.002 0 40

30 Car 20 10 din ality

0 0

0.01 0.008 0.006 0.004 0.002 0 30

1.5 1 −3 0.5 0 1 e x c n ia Var

20

Car

10 ality

din

0 0

SPEA2 Expected Return

Expected Return

NSGA−II

0.01 0.008 0.006 0.004 0.002 0 30 20 Car din 10 ality

0

0

1.5 1 −3 0.5 x 10 e c n Varia

Fig. 4. Unconstrained efficient surface for the problem with n =300 securities.

1.5 1 −3 0.5 0 1 e x c n ia Var

ARTICLE IN PRESS K.P. Anagnostopoulos, G. Mamanis / Computers & Operations Research 37 (2010) 1285–1297

Fig. 5. Mean–variance efficient frontier for the problem with n= 200 securities.

Fig. 6. Mean–variance efficient frontier for the problem with n= 300 securities.

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0.35

0.8

NSGA−II PESA SPEA2

0.3 0.25

0.7

NSGA−II PESA SPEA2

0.65

ε−Indicator

Hypervolume

0.75

0.2 0.15 0.1

0.6

0.05 0

0.55 0

0.5

1 1.5 2 2.5 3 x 105 Generated Solutions

0

0.5

1 1.5 2 2.5 3 x 105 Generated Solutions

0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35

NSGA−II PESA SPEA2

0

0.5

1 1.5 2 2.5 3 Generated Solutions x 105

ε−Indicator

Hypervolume

Fig. 7. Performance indicators for the problem with n= 200 securities.

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

NSGA−II PESA SPEA2

0

0.5

1 1.5 2 2.5 3 Generated Solutions x 105

Fig. 8. Performance indicators for the problem with n= 300 securities.

Table 2 Average runtime (in s) required by each algorithm.

NSGA-II PESA SPEA2

Unconstrained problem

Constrained problem

n= 200

n= 300

n= 200

n= 300

2254.2 604.9 4720.8

3091.3 766.1 5563.9

12100.9 2632.4 23480.6

16813.4 3115.7 26640.4

Figs. 5 and 6 show that all algorithms have generated a large number of portfolios very close to the true Markowitz’s efficient frontier (UEF) obtained using an exact method. For the n =200 securities problem instance, the approximation error is 0.0622% for NSGA-II, 0.0693% for PESA and 0.0596% for SPEA2. For the problem with n = 300 securities approximation errors are 0.0426%, 0.0492% and 0.0436%, respectively, revealing that the generated portfolios are very close to the optimal ones. Approximation error was estimated as follows: For each solution xa generated by a MOEA, the percent error was calculated as: Z ¼ ðJFðxa Þ  Fðxr ÞJ2 =JFðxr ÞJ2 Þ100%, where xr is the solution from the true efficient frontier that gives the smallest Euclidean distance to xa and FðxÞ ¼ ½rðxÞ; mðxÞT . The approximation error is the average of all percent errors. Consequently, solving the three-objective portfolio optimization problem with MOEAs generalizes Markowitz model since it provides the investor with additional portfolios which are not mean–variance efficient but have less securities in the portfolio.

The investor can examine the tradeoffs among expected return, variance and the number of securities in the portfolio to select the one that best suits his/her needs. Figs. 5 and 6 give some evidence for the performance of each algorithm to provide an approximation of the tradeoffs between expected return, variance and the cardinality of the portfolio. Furthermore, we have visually seen that SPEA2 and PESA achieve better diversity characteristics than NSGA-II. Figs. 7 and 8 illustrate for each MOEA the average values over 10 simulation runs for both the hypervolume and e-indicator metrics in different number of generations. The results confirm that NSGA-II is the worst algorithm for this problem and show that SPEA2 has the best performance since it generates better values for both metrics. The results, however, are very close especially for SPEA2 and PESA. In addition, we observe that all algorithms have similar convergence behavior for both metrics since they all come close to their maximum metrics values in about 160 generations (i.e., 80,000 generated solutions). With respect to runtime, form Table 2 it is seen that PESA is the fastest approach while NSGA-II comes in the second and SPEA2 in the third place, respectively. However, it must be noted that a more efficient implementation of SPEA2 truncate operator could improve its performance.

4.3. Constrained multiobjective portfolio optimization problem For both instances we have classified the securities into 10 classes (i.e., M= 10) with identical number of assets in each class. The lower proportion allowed for investment in each class was set

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PESA

x 10−3 7

x 10−3 Expected Return

Expected Return

NSGA−II

6 5 4 3 2 30

25 20 Ca rdin alit 15 y

1295

7 6 5 4 3 2 30

4

2

10 0

nce

Varia

6

25 rdin 20 alit y

Ca

−4

x 10

15

2 nce Varia

10 0

6

4 −4 x 10

SPEA2 x 10−3 Expected Return

7 6 5 4 3 2 40 30 rdin 20 alit y

Ca

10 0

2

4 −4 x 10

ce

n Varia

6

Fig. 9. Constrained efficient surface for the problem with n =200 securities: (a) NSGA-II; (b) PESA; and (c) SPEA2.

x 10−3 8 7 6 5 4 3 2 40 30 Ca 20 rdin alit y

PESA

Expected Return

Expected Return

NSGA−II

2 nce Varia

10 0

6

4

−4

x 10

x 10−3 8 7 6 5 4 3 2 40 30 Ca rdin

20 y

alit

10 0

2 nce Varia

4

6

SPEA2

Expected Return

x 10−3 8 7 6 5 4 3 2 40 30 20 rdin alit y

Ca

10 0

2 nce a V ria

4

−4

x 10

6 −4

x 10

Fig. 10. Constrained efficient surface for the problem with n= 300 securities: (a) NSGA-II; (b) PESA; and (c) SPEA2.

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0.76

0.72 NSGA−II PESA SPEA2

0.7 0.68 0.66 0.64 0.62 0

2

4

ε−Indicator

Hypervolume

0.74

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

6 8 10 12 14 16 18 Generated Solutions x 105

NSGA−II PESA SPEA2

0

2

4

6 8 10 12 14 16 18 Generated Solutions x 105

Fig. 11. Performance indicators for the problem with n= 200 securities.

0.7

0.16

0.68

0.14

0.66

0.12

NSGA−II PESA SPEA2

0.64 0.62 0.6

ε−Indicator

Hypervolume

0.18

NSGA−II PESA SPEA2

0.1 0.08 0.06 0.04

0.58

0.02

0.56

0 0

2

4

6 8 10 12 14 16 18 Generated Solutions x 105

0

2

4 6 8 10 12 14 16 18 Generated Solutions x 105

Fig. 12. Performance indicators for the problem with n =300 securities.

at 5% (i.e., Li = 0.05, i= 1,y,M) while the minimum share of each security was fixed at 1% (i.e., li = 0.01, i= 1,y,n). For normalizing the objective values, the minimum and maximum bounds for each objective function for n=200 are fimin ¼ f0:000141; 0:00217; 9g and fimax ¼ f0:00067; 0:007; 125g; and for n=300 are fimin ¼ f0:000125; 0:00218; 9g and fimax ¼ f0:00055; 0:008; 150g. Figs. 9 and 10 show the efficient surfaces for the constrained multiobjective portfolio optimization problem. We see that PESA and SPEA2 have good diversity characteristics well capturing the tradeoffs among the three objectives, and the smallest portfolio is consisting of 10 assets. Since a lower proportion weight for each class has been imposed, at least one asset from each class should be selected in the portfolio. These surfaces capture the same tradeoffs between variance, expected return and number of assets in the portfolio; however, the portfolios are better constructed due to the restrictions imposed on the amount invested in each class and security, leading in more practical and well diversified portfolios. Again for the constrained problem we can visually see that SPEA2 and PESA achieve better diversity characteristics than NSGA-II, and the quantitative performance comparison among the algorithms confirms this conclusion (Figs. 11 and 12). From the computational comparison we notice that while NSGA-II converges faster than the other two for both metrics it fails to provide better results at the end of the run. Again for this type of problem SPEA2 is the best technique since it wins in both metrics. The average runtime for each algorithm and for both instances is given in Table 2. Experiments for the problem with n =200 securities were performed using the hybrid representation combined with the constraint domination operator proposed in [26]. This combination was integrated into SPEA2 search process, and the best value for the hypervolume metric over five simulation runs was 0.725. As Fig. 11 indicates, this result is worst than the performance

achieved by every algorithm with the problem specific chromosome representation used for handling the additional constraints.

5. Conclusion In this paper a tri-objective portfolio optimization problem with quantity and class constraints has been proposed. The first two objectives were the variance and the expected return of the portfolio as commonly utilized in most portfolio selection problems, and the third one measures the number of assets held in the portfolio and should be minimized. We have experimented with NSGA-II, PESA and SPEA2 for finding an approximation of the best possible tradeoffs between return, risk and the cardinality of the portfolio. Visual comparisons have shown that MOEAs generate surfaces with good diversity characteristics in general. The computational comparison has demonstrated that SPEA2 is the best algorithm for both the constrained and unconstrained multiobjective portfolio optimization problem, and PESA comes at the second place while being the fastest technique. The computational analysis confirms that the algorithms provide a good approximation of the return–risk frontier when solving the triobjective problem. Thus, solving the tri-objective problem with MOEAs generalizes mean–variance approach by providing the investor with additional portfolios which are not mean–variance efficient but have fewer securities in the portfolio. References [1] Markowitz HM. Portfolio selection. Journal of Finance 1952;7:77–91. [2] Markowitz HM. Portfolio selection, efficient diversification of investments. Cambridge MA, Oxford UK: Blackwell; 1990.

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