Bi-objective Portfolio Optimization Using a Customized Hybrid NSGA-II Procedure

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1 Bi-objective Portfolio Optimization Using a Customized Hybrid NSGA-II Procedure Kalyanmoy Deb 1, Ralph E. Steuer 2, Rajat Tewari 3, and Rahul Tewari 4 1 Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, PIN , India, deb@iitk.ac.in, Also Department of Business Technology, Aalto University, School of Economics, Helsinki, Finland 2 Department Banking and Finance, Terry College of Business, University of Georgia, Athens, Georgia , USA rsteuer@uga.edu 3 Benaras Hindu University, Benaras, India rajat.tewari.min08@itbhu.ac.in 4 Deutsche Bank Group, Mumbai, India rtewari.iitk@gmail.com Abstract. Bi-objective portfolio optimization for minimizing risk and maximizing expected return has received considerable attention using evolutionary algorithms. Although the problem is a quadratic programming (QP) problem, the practicalities of investment often make the decision variables discontinuous and introduce other complexities. In such circumstances, usual QP solution methodologies can not always find acceptable solutions. In this paper, we modify a bi-objective evolutionary algorithm (NSGA-II) to develop a customized hybrid NSGA-II procedure for handling situations that are non-conventional for classical QP approaches. By considering large-scale problems, we demonstrate how evolutionary algorithms enable the proposed procedure to find fronts, or portions of fronts, that can be difficult for other methods to obtain. 1 Introduction Portfolio optimization inherently involves conflicting criteria. Among possible objectives, minimizing risk and maximizing expected return (also known as the mean-variance model of Markowitz [1]) are the two objectives that have received the most attention. The decision variables in these problems are the proportions of initial capital to be allocated to the different available securities. Such biobjective problems give rise to fronts of trade-off solutions which must be found to investigate the risk-return relationships in a problem. In addition to the two objectives, these problems possess constraints [2]. The calculation of expected return is a linear function of the decision variables and the calculation of portfolio risk involves a quadratic function of the decision variables. Thus, the overall problem, in its simplest form, is a bi-objective quadratic programming (QP) problem, and when all constraints are linear and all variables are continuous, such problems can be solved exactly using the QP solvers [2, 3]. However, in practice, there can be conditions which make QP solvers difficult to apply. For instance, a portfolio with very small investments in one or more securities is likely not to be of interest on managerial grounds alone. This then

2 2 creates a need for decision variables that either take on a value of zero or a nonzero value corresponding to at least a minimum investment amount. Moreover, users may only be interested in portfolios that involve limited numbers of securities. Also, there can be governmental restrictions contributing to the complexity of the process, thus only adding to the difficulties of handling portfolio problems by classical means. Evolutionary multi-objective optimization (EMO) has, over the years, been found to be useful in solving different optimization problems. EMO has also been used to solve different portfolio optimization problems [2, 4 7]. Chang et al. [8] used genetic algorithms (GAs), tabu search, and simulated annealing on a portfolio optimization problem with a given cardinality on the number of assets. Other approaches including simulated annealing [9], differential evolution [10], and local search based memetic algorithms [11, 12] have also been attempted. Here, we suggest and develop a customized hybrid NSGA-II procedure which is particularly designed to handle the non-smooth conditions alluded to above. The initialization procedure and the recombination and mutation operators are all customized so that the proposed procedure starts with a feasible population and always creates only feasible solutions. Conflicting objectives are handled using the elitist non-dominated sorting GA or NSGA-II [13]. To make the obtained solutions close to optimal solutions, the NSGA-II solutions are clustered into small groups and then a local search procedure is commenced from each solution until no further improvements are possible. The remainder of the paper is organized as follows. Section 2 discusses portfolio optimization in greater detail. Thereafter, the customized hybrid NSGA-II procedure for solving portfolio optimization problems is described in Section 3. The local search component of the overall approach is described next. Section 4 presents results obtained by the proposed procedure. Concluding remarks constitute Section 5. 2 Practical Portfolio Optimization In a portfolio problem with an asset universe of n securities, let x i (i = 1, 2,..., n) designate the proportion of initial capital to be allocated to security i. There are typically two goals maximize portfolio expected return and minimize portfolio risk [1]. To achieve the first goal, one might be tempted to pursue the securities with the highest individual expected returns r i, but these are often the riskiest securities. In its most basic form, this results in the following problem: Minimize f 1 (x) = n n i=1 j=1 x iσ ij x j, Maximize f 2 (x) = n i=1 r ix i, subject to n i=1 x i = 1, x i 0. (1) The first objective is portfolio risk as computed from a given n n risk matrix [σ ij ]. The second objective is portfolio expected return as computed from a weighted sum of the individual security expected returns. The first constraint

3 3 ensures the investment of all funds. The second constraint ensures the nonnegativity of each investment. It is clear that the objectives are conflicting. Thus, the solution to the above is the set of all of the problem s Pareto-optimal solutions as this is precisely the set of all of the problem s contenders for optimality. One of the ways to solve for the front of (1) is to convert the problem into a number of single-objective problems, the popular way being to convert the expected return objective to a constraint as in the e-constraint formulation: Minimize f 1 (x) = n n i=1 j=1 x iσ ij x j, subject to n i=1 r ix i R, n i=1 x i = 1, x i 0. With the above a QP problem, the idea is to solve it repetitively for many different values of R. If the number of securities n is more than about 50, it can be expected that almost any solution of (2) will contain many securities at the zero level. That is, it is expected that for many i, x i = 0. It is also expected that for at least some securities, x i will be a very small number. However, to have a practical portfolio, very small investments in any security may not be desired and are to be avoided. Thus, there is the practicality that for any portfolio to be of interest, there is to be a lower limit on any non-zero investment. That is, either x i = 0 (meaning no investment in the i-th security) or x i α (meaning that there is a minimum non-zero investment amount for the i-th security). Additionally, there may also an upper bound β, limiting the maximum proportion of investment to any security. Unfortunately, the solution of Equation 2 for any given R does not in general carry with it the guarantee that all such requirements will be satisfied. In addition to the above, there a second practicality which we now discuss and it is that any portfolio on the Pareto-optimal front of (1) may contain any number of non-zero investments. Over this, a user may wish to exert control. To generate practical portfolios, a user may be interested in specifying a fixed number of investments or a range in the number of investments. This constraint is also known as the cardinality constraint and has been addressed by other researchers as well [3, 12]. Taking both practicalities into account, we have the following bi-objective optimization problem: (2) Minimize f 1 (x) = n n i=1 j=1 x iσ ij x j, Maximize f 2 (x) = n i=1 r ix i, subject to (1st constraint) n i=1 x i = 1, (2nd constraint) x i = 0 or α x i β, (3rd constraint) d min d(x) d max, where α > 0 and d(x) is given as follows: n { 1, if xi > 0, d(x) = 0, if x i = 0. i=1 (3) (4)

4 4 2.1 Difficulties with Classical Methods Classical QP solvers face difficulties in the presence of discontinuities and other complexities. While the 1st constraint is standard, the 2nd constraint requires an or operation. While x i = 0 or x i = α is allowed, values between the two are not. This introduces discontinuities in the search space. The 3rd constraint involves a parameter d which is defined by a discontinuous function of the decision variables, given in Equation 4. It is the presence of the 2nd and 3rd constraints that makes the application of standard methodologies difficult. In the following section, we discuss the GA based methodology of this paper for dealing with constraints like these. 3 Customized Hybrid NSGA-II Procedure Generic procedures for handling constraints may not be efficient in handling the constraints of this problem. Here, we suggest a customized hybrid NSGA- II procedure for handling the specific constraints of the portfolio optimization problem. The 1st constraint ensures that all variables added together become one. Most evolutionary based portfolio optimization methodologies suggest the use of random keys or a dummy variable (x i ) and repair the variable vector as x i x i / n i=1 x i to ensure satisfaction of this constraint. In the presence of the 1st constraint alone, this strategy of repairing a solution is a good approach, but when other constraints are present, the approach may not be adequate. The 2nd constraint introduces an or operation between two constraints representing two disjointed feasible regions. One approach suggested in the literature is to use an additional Boolean variable ρ i [0, 1] for each decision variable x i, as in [3]: αρ i x i βρ i. When ρ i = 0, the variable x i = 0 (meaning no investment in the i-th security). But when ρ i = 1, the variable x i takes a value in the range [α, β]. This is an excellent fix-up for the or constraint, but it introduces additional variables into the optimization problem. The 3rd constraint involves a discontinuous function of decision variables. If Boolean variables are used to take care of the 2nd constraint, the cardinality constraint can be replaced by the following: d min n ρ i d max. i=1 The presence of equality and inequality constraints would make it difficult for a generic penalty function approach to find a feasible solution and solve the problem. Here, we suggest a different approach which does not add any new variables, but instead considers all constraints in its coding and evaluation procedures. First, a customized initialization procedure is used to ensure creation of feasible solutions. Thereafter, we suggest recombination and mutation operators which also ensure the creation of feasible solutions.

5 5 3.1 Customized Initialization Procedure To create a feasible solution, we randomly select an integer value for d from the interval [d min, d max ]. Then we create a portfolio which has exactly d non-zero Σx i < 1 securities. That is, we randomly select d δ of the n variables and assign zeros to the Σx i > 1 other (n d) variables. Since the non-zero x i values must lie within [α, β], we randomly create a number in the range of α each of the non-zero variables. However, the random numbers for the non-zero variables α α+ 1 Raw Value (x_i) β 2 β may not sum to one, thereby violat- ing the first constraint. To make the solution Fig. 1. The mapping function to sat- feasible, we repair the solution as isfy the first constraint. follows. We map the non-zero variables to another set of non-zero variables in the range [α, β] in such a way that they sum to one. The mapping function is shown in Figure 1. If n i=1 x i = 1, we have a feasible solution, else we follow the procedure described below. 1. If n i=1 x i 1, the decision vector x is modified as follows: For each i, if x i is within [α, α + 1 ], set x i = α. For each i, if x i is within [β 2, β], set x i = β. For each i, if x i is within [α + 1, β 2 ], x i is kept the same. 2. Compute X = n i=1 x i. If X is still not equal to one, then stretch the mapping line by moving its mid-point by an amount δ, as follows: If X < 1, move the mid-point upwards by a distance δ in a direction perpendicular to the line (as shown in Figure 1). This process is continued until n i=1 x i is arbitrarily close to one. If X > 1, move the mid-point downwards by a distance δ in a direction perpendicular to the line. This process is continued until n i=1 x i is arbitrarily close to one. The parameters 1 and 2 allow us to map near-boundary values to their boundary values. However, if this is unwarranted, both of these parameters can be chosen to be zero, particularly in the cases when all x i values are either within [α, α+ 1 ] or within [β 2, β]. The parameter δ may be chosen to be a small value and it helps to iterate the variable values so that the 1st constraint is satisfied within a small margin. A small value of δ will require a large number of iterations Mapped value β Σ x i = 1

6 6 to satisfy the first constraint but the difference between n i=1 x i and one will be small and vice versa. It remains as an interesting study to establish a relationship between δ and the corresponding error in satisfying the equality constraint, but here, in all of our simulations, we have arbitrarily chosen the following values: 1 = 0, 2 = and δ = After this repair mechanism, the 1st constraint is expected to be satisfied within a small tolerance. The 2nd and 3rd constraints are also satisfied in the process. Thus, every solution created by the above process is expected to be feasible. 3.2 Customized Recombination Operator Once a feasible initial population is created, it is to be evaluated and then recombination and mutation operators are to be applied. Here, we suggest a recombination operator which always produces feasible offspring solutions by recombining two feasible parent solutions. Two solutions are picked at random from the population as parents. Let the number of non-zero securities in the parents be d 1 and d 2, respectively. We illustrate the recombination procedure through the following two parent solutions having n = 10: Parent1: a 1 0 a 3 0 a a 8 0 a 10 Parent2: b 1 0 b 3 b 4 b 5 0 b b 10 It is clear that d 1 = 5 and d 2 = 6 in the above. The following steps are taken to create a child solution. 1. An integer d c is randomly selected from the range [d 1, d 2 ], say, for the above parents, d c = The child solution inherits a zero investment for a particular security if both parents have zero investments in that security. For the above parents, this happens for three (n 0 ) securities: i = 2, 6 and 9. Thus, the partial child solution is as follows: Child1: The securities which have non-zero investments in both parents will inherit a non-zero investment value, but the exact amount will be determined by a real-parameter recombination operator (bounded form of SBX [14]) applied to the parent values. This operator ensures that values are not created outside the range [α, β]. We discuss this procedure a little later. For the above parents, the child solution then takes on the form: Child1: c 1 0 c 3 c c 10 Note that the number of common non-zero investments (n c ) between the two parent solutions is such that n c [0, min(d 1, d 2 )]. In this case, n c = 4.

7 7 4. The number of slots that remain to be filled with non-zero values is w = d c n c. Since, d c [d 1, d 2 ], w is always greater than or equal to zero. From the remaining (n n 0 n c ) securities, we choose w securities at random and take the non-zero value from the parent to which it belongs. For the above example parents, w = 5 4 = 1 and there are (10 3 4) or 3 remaining places to choose the w = 1 security from. The remaining securities are assigned a value zero. Say, we choose the seventh security to have a non-zero value. Since the non-zero value occurs in parent 2, the child inherits b 7. Thus, the child solution looks like the following: Child1: c 1 0 c 3 0 c 5 0 b c The above child solution is guaranteed to satisfy both 2nd and 3rd constraints, but needs to satisfy the 1st constraint. We use the procedure described in Subsection 3.1 to repair the solution to a feasible one. We also create a second child from the same pair of parents using the same procedure. Due to the creation of the random integer d c and other operations involving random assignments, the second child is expected to be different from the first child. However, both child solutions are guaranteed to be feasible. The bounded form of the SBX operator is described here for two non-zero investment proportions for a particular security (say, a 1 and b 1, taken from parents 1 and 2, respectively). The operator requires a user-defined parameter η c : 1. Choose a random number u i within [0, 1]. 2. Calculate γ qi using the equation: γ qi = ( (κu i ) 1 ηc+1, if ui 1 κ, ηc+1 ) κu i, otherwise. (5) For a 1 b 1, κ = 2 ζ (ηc+1) and ζ is calculated as follows: ζ = min[(a 1 α), (β b 1 )]/(b 1 a 1 ). For a 1 > b 1, the role of a 1 and b 1 can be interchanged and the above equation can be used. 3. Then, compute the child from the following equation: c 1 = 0.5[(1 + γ qi )a 1 + (1 γ qi )b 1 ]. (6) The above procedure allows a zero probability of creating any child solutions outside the prescribed range [α, β] for any two solutions a 1 and b 1 within the same range. 3.3 Customized Mutation Operator Mutation operators perturb the variable values of a single population member. We use two mutation operators.

8 8 Mutation 1 In this operation, a non-zero security value is perturbed in its neighborhood by using the polynomial mutation operator [15]. For a particular value a 1, the following procedure is used with a user-defined parameter η m : 1. Create a random number u within [0, 1]. 2. Calculate the parameter µ as follows: { 1 (2u) ηm+1 1, if u 0.5, µ = 1 ηm+1 1 [2(1 u)], otherwise. (7) 3. Calculate the mutated value, as follows: a 1 = a 1 + µ min[(a 1 α), (β a 1 )]. The above procedure ensures that a 1 lies within [α, β] and values close to a 1 are preferred more than values away from a 1. The above procedure is applied to each non-zero security with a probability p m. Since values are changed by this mutation operator, the repair mechanism of Subsection 3.1 is applied to the mutated solution. Mutation 2 The above mutation procedure does not convert a zero value to a non-zero value. But, by the second mutation operator, we change z zero securities to non-zero securities, and to keep the number of investments d the same, we also change z non-zero securities to zero securities. For this purpose, a zero and a non-zero security are chosen at random from a solution and their values are swapped. The same procedure is re-applied to another (z 1) pairs of zero and non-zero securities. To illustrate, we set z = 2 and apply the operator on the following solution: a 1 0 a 3 0 a a 8 0 a 10 Mutation 2 operator is applied between the first and fourth securities and between the ninth and tenth securities as follows: 0 0 a 3 a 1 a a 8 a 10 0 Since no new non-zero values are created in this process, the 1st constraint is always be satisfied and this mutation operator always preserves the feasibility of a solution. In all our simulations, we have chosen z = 2, although other values of z may also be tried. Thus, the proposed procedure starts with a set of feasible solutions. Thereafter, the creation of new solutions by recombination and mutation operators always ensures their feasibility. These procedures help NSGA-II to converge close to the Pareto-optimal front trading off the two conflicting objectives quickly.

9 9 3.4 Clustering NSGA-II is expected to find a set of solutions representing the trade-off frontier. The NSGA-II procedure requires a population size (N) proportional to the size of the problem (number of securities) for a proper working of its operators. However, this can cause NSGA-II to find more portfolios than a user may desire. To reduce the number of obtained trade-off portfolios, we apply a k-mean clustering procedure to the obtained set of solutions and pick only H (< N) well-distributed portfolios from the obtained NSGA-II solutions. For all our problems, we use H = 30 to get an idea of the trade-off frontier, however in practice other numbers of solutions can be employed. 3.5 Local Search To obtain near-pareto-optimal solutions, we also employ a local search procedure from all H clustered solutions. We use a single-objective GA with the abovediscussed operators for this purpose. From a seed (clustered) solution z, first an initial population is created by mutating the seed solution N ls times (where N ls is the population size for the local search). Both mutation operators discussed above are used for this purpose. The mutation probability p m and parameter η m are kept the same as before. Since the mutated solutions are repaired, each population member is guaranteed to be feasible. The objective function used here is the following achievement scalarizing function [16]: F (x) = 2 min j=1 f j (x) z j fj min f max j j=1 f j (x) z j fj min f max j. (8) Here, f 2 (x) is considered to be the negative of the expected return function (second objective) given in Equation 3. The parameters fj min and fj max are the minimum and maximum objective values of all obtained NSGA-II solutions. In the local search GA, the above function is minimized. The search continues until no significant improvement in consecutive population-best solutions is obtained. 4 Results We are now ready to present some results using our proposed customized hybrid NSGA-II procedure. For simulation, we consider a data-set (containing an r expected return vector and a [σ ij ] risk matrix) for a 1,000-security problem. The data-set was obtained from the random portfolio-selection problem generator specified in [17]. Figure 2 shows the risk-return trade-off frontier first obtained by solving the e-constraint problem (Equation 2) many times via QP for different values of R. Solutions that have low risk (near A) also have low expected returns. Solutions that have high risk (near C) also have a high expected returns. Although this is a known fact, the discovery of the trade-off frontier provides the actual manner in which the trade-offs take place. For example, for the risk matrix and return vector chosen for this case study there seem to be good tradeoffs near B, and a most preferred portfolio may well be selected from the region

10 B C A Fig. 2. Obtained trade-off front of the original problem using the quadratic programming (QP) technique. marked. This is because to choose a solution near A over a solution from B, one has to sacrifice a large amount of expected return to get only small reduction in risk. Similarly, to choose a solution C over a solution from B, one has to take on a large amount of extra risk to get a only small increase in expected return. However, since not all trade-off frontiers are so nicely formed and who is to know from where along a trade-off frontier a given decision maker will make a final selection, it is not our intention to become involved in decision making at this point. Rather, our goal here is to develop comprehensive sets of trade-off frontier portfolios in problems involving practicalities so that no region possibly containing an optimal portfolio will be missed in the analysis. For this case study, we analyzed all solutions and observed that out of the 1,000 securities, only 88 ever took on non-zero investments. We extract for these securities their r-vector and [σ ij ] risk matrix information and confine our further studies to only these n = 88 securities. For the NSGA-II procedure, we selected a population size of N = 500, a maximum number of 3,000 generations, a recombination probability of p c = 0.9, a SBX index of η c = 10, a mutation probability of p m =, and a polynomial mutation index of η m = 50. For the local search procedure, we selected N = 200 and the ran until 500 generations were reached. All other GA parameters were kept the same as in the NSGA-II simulations. It is important to mention here that since NSGA- II considers all constraints given in Equation 3 and QP considered only the first constraint, the NSGA-II solutions are not expected to be better than QP solutions. But NSGA-II solutions are expected to be non-dominated to those QP solutions that satisfy all constraints given in Equation Portfolio Optimization with No Size Constraint As the 1st constraint is always present, we start by imposing the 2nd constraint with α = and β = Of the 483 QP obtained portfolios characterizing

11 11 the front of the problem without any discontinuities, 453 satisfy the bounds of the 2nd constraint. Due to the use of β = 0.04, at least 25 securities, of course, are in every portfolio. However, in our algorithm presented, we do not specify any restriction on the number of securities. Figure 3 shows 294 trade-off solutions obtained by the customized NSGA-II procedure alone. The QP solutions that satisfy the α = and β = 0.04 bounds are also shown in the figure. The figure clearly shows that the proposed procedure is able to find a good set of distributed solutions close to the points obtained by the QP procedure. High risk solutions are somewhat worse than those obtained by QP. But we show shortly that the hybrid method in which a local search is applied on the NSGA-II solutions is a better overall strategy Proposed method Baseline Fig. 3. Obtained trade-off front for α = and β = 0.04 and with no restriction on d by a typical NSGA-II run. Fig. 4. Attainment curve of the 21 runs for this problem for the case α = and β = 0.04 and with no restriction on d. Genetic algorithms are stochastic optimization procedures and their performance is shown to depend somewhat on the chosen initial population. In order to demonstrate the robustness of the proposed customized NSGA-II procedure over variations in the initial population, we created 21 different random initial populations and ran the proposed procedure independently from each one. The obtained solutions are collected together and 0%, 50% and 100% attainment curves [18] are plotted in Figure 4. The 0% attainment curve indicates the trade-off boundary which dominates all obtained trade-off solutions. The 100% attainment curve indicates the trade-off boundary that is dominated by all obtained trade-off solutions collectively. The 50% attainment curve is that trade-off boundary that dominates 50% of the obtained trade-off solutions. The three curves obtained for the 21 non-dominated fronts are so close to each other that they cannot be well distinguished visually. This means that all 21 independent runs produce almost identical frontiers thereby indicating the reliability of the proposed customized NSGA-II procedure. From the obtained non-dominated front of all 21 runs, we now apply the clustering algorithm and choose only 30 well-distinguished portfolios covering the entire front. These solutions are then local searched. The complete method

12 12 is called the customized hybrid NSGA-II procedure. The obtained solutions are shown in Figure 5 by circles. The solutions are close to the QP solutions, but Proposed method Baseline number of securities solution with lowest return solution with a little higher return solution with a high return solution with highest return grades of securities Fig. 5. A set of 30 trade-off portfolios obtained using clustering and local search procedure for the case with α = 0.005, β = 0.04, and no restriction on d. Fig. 6. Investment pattern for four solutions in each of the five grades of diminishing return. importantly a well-distributed set of portfolios is created by our proposed procedure. In order to show the investment pattern of different trade-off portfolios, we choose two extreme solutions and two intermediate solutions from the obtained set of 30 portfolios. Based on the return values r i, all securities are ordered from the highest return value to the lowest. They are then divided into five grades of 18 securities each, except for the fifth grade which has the lowest 16 securities. For each of the four solutions, the number of non-zero securities in each grade is counted and plotted in Figure 6. It can be seen that the highest return solution allocates the maximum number of grade 1 securities in order to maximize the overall return. The solution with the lowest return (or lowest risk) invests more into grade 5 securities (having lowest return). The intermediate solutions invest more into intermediate grades to make a good compromise between return and risk. 4.2 Portfolio Optimization for a Fixed d Now we impose the 3rd constraint. Initial population members are guaranteed to satisfy the 3rd constraint and every recombination and mutation operation also guarantees maintaining the 3rd constraint. First, we consider d min = d max = 28, so that portfolios with only 28 securities are desired. The variable bounds of α = and β = 0.04 are enforced. The QP generated frontier now has only 24 portfolios which satisfy these constraints. They are shown in Figure 7. There are a total of 168 portfolios found by a typical run of the proposed procedure. They are shown in the figure as well. It is clear from the figure that the proposed procedure is able to find widely distributed sets of solutions. In order to investigate the robustness of the customized NSGA- II procedure alone, the three attainment curves are drawn for the 21 different

13 Proposed method Baseline Fig. 7. Obtained trade-off front for d = 28 and with α = and β = 0.04 by a typical NSGA-II run. Fig. 8. Attainment surface of the 21 runs for the problem with d = 28 and α = and β = runs in Figure 8, as before. The closeness of these curves to one another signifies that the proposed procedure is able to find almost identical trade-off frontiers on any run. Finally, all solutions from 21 runs are collected together and non-dominated solutions are clustered into 30 widely separated solutions. They are local searched and shown in Figure 9. In this case, it is clear from the figure that the proposed Proposed method Baseline 6 Proposed method Fig. 9. A set of 30 trade-off portfolios obtained using clustering and local search procedure for the case with d = 28, α = 0.005, and β = Fig different portfolios found for d = 50, α = 0.005, and β = 0.04 obtained using the proposed procedure. Base-line QP solution set did not have any solution with d = 50. procedure is able to find much wider and better distributed sets of portfolios than classical QP alone. We consider another case in which exactly d = 50 securities are desired in each portfolio. The QP solution set did not have any such portfolio. When we apply the customized hybrid NSGA-II procedure, we obtain the front with 30 portfolios shown in Figure 10. These results amply demonstrate the working of the proposed procedure when both a discontinuous variable bound and a discrete number of securities are desired.

14 Portfolio Optimization for a Given Range of d Next, we consider the case in which a user wishes the portfolios to vary within a prescribed range [d min, d max ]. Customized operators to handle such variables have been described before. We consider d min = 30 and d max = 45 and the variable bounds α = and β = The distribution of solutions obtained after local search is shown in Figure 11. There are 261 QP obtained solutions that satisfy these constraints. The closeness of the obtained solutions with the QP solutions is clear from the figure Proposed method Baseline Number of Invested Securities Fig. 11. A set of 30 portfolios obtained for d [30, 45], α = 0.005, and β = Fig. 12. Number of invested securities (d) versus accumulated risk. In order to understand the investment pattern, we consider 50 clustered solutions from the set of all 21 runs and plot the number of non-zero securities in each solution with the corresponding risk value (objective f 1 (x)) in Figure 12. It is interesting to note that low risk portfolios (also with low returns) can be obtained with a wide variety of investments, almost uniformly from 30 to 44 securities. However, high risk (therefore, high return) portfolios only require users to invest in smaller numbers of securities. 5 Conclusions In this paper, we have suggested a customized hybrid NSGA-II procedure for handling a bi-objective portfolio optimization problem having practicalities. First, the proportion of investment in a security can either be zero (meaning no investment) or be between specified minimum and maximum bounds. Second, the user may have a preference for a certain number, or a range in the number, of non-zero securities. Such requirements make the resulting optimization problem difficult to solve via classical techniques. The hallmark of our study is that our procedure uses a repair mechanism to make all generated solutions feasible. By taking a large-sized problem, we have systematically demonstrated the efficacy of the proposed procedure over different user-defined requirements. The reliability of the proposed customized hybrid NSGA-II procedure is demonstrated by simulating the algorithm from 21 different initial populations. The accuracy of the proposed procedure is enhanced by coupling the method with

15 a clustering and a local search procedure. The results are compared with those obtained with the quadratic programming method. The proposed methodology involves a number of parameters ( 1, 2, δ, z). A parametric study is now needed to find suitable values of these parameters. A computational complexity study of the proposed repair mechanism is also needed to establish a relationship between δ and extent of violation of the equality (1st) constraint. The methodology suggested here is now ready to be applied to more complex portfolio optimization problems. The QP method, when applicable, is generally the best approach to pursue. But practice is full of complexities which can limit the use of standard QP methods. The flexibility of genetic algorithms demonstrated here provides us with confidence about their applicability to reallife portfolio optimization problems. This study should encourage further use of QP and GA approaches together in a probably more computationally efficient manner to make the overall portfolio optimization problem more pragmatic. 15