Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm

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Abstract

Heuristic algorithms strengthen researchers to solve more complex and combinatorial problems in a reasonable time. Markowitz’s Mean-Variance portfolio selection model is one of those aforesaid problems. Actually, Markowitz’s model is a nonlinear (quadratic) programming problem which has been solved by a variety of heuristic and non-heuristic techniques. In this paper a portfolio selection model which is based on Markowitz’s portfolio selection problem including three of the most important limitations is considered. The results can lead Markowitz’s model to a more practical one. Minimum transaction lots, cardinality constraints (both of which have been presented before in other researches) and market (sector) capitalization (which is proposed in this research for the first time as a constraint for Markowitz model), are considered in extended model. No study has ever proposed and solved this expanded model. To solve this mixed-integer nonlinear programming (NP-Hard), a corresponding genetic algorithm (GA) is utilized. Computational study is performed in two main parts; first, verifying and validating proposed GA and second, studying the applicability of presented model using large scale problems.

Introduction

Markowitz’s modern portfolio theory has made a new paradigm of portfolio selecting for investors in order to form a portfolio with the highest expected return at a given level of risk tolerance (the lowest level of risk tolerance at a given expected return) (Markowitz, 1952, Markowitz, 1959, Oh et al., 2006). Lots of efforts have been performed by experts in order to solve and expand Markowitz’s model. These attempts, regarding the limitations of a factual market, have tried to make his model more practical.

In 1956, Markowitz represented the critical line method to solve his quadratic model (Markowitz, 1956). Wolfe tried to solve Markowitz’s model by Simplex algorithm (Wolfe, 1959). Konno’s new definition of risk in his mean absolute deviation (MAD) model, has been led several investigations. Interestingly, Konno’s model can be solved by linear methods like Simplex, (Konno, 1990, Konno and Yamazaki, 1991). Again, Markowitz, himself, studied more complex objective functions, based on the notions of semi-variance (Markowitz, Todd, Xu, & Yamane, 1993).

In 1993, Speranza introduced a more general model with a weighted risk function (Speranza, 1993) and also proposed a mixed-integer programming, considering realistic characteristics in portfolio selection, such as minimum transaction lots and maximum number of securities. His researches have been always based on MAD model of Konno not on Markowitz’s (Speranza, 1996). Other investigations added some other constraints; Yoshimoto considered multi period portfolio selection with transaction costs based on Markowitz’s (Yoshimoto, 1996). Speranza and Mansini’s research, regarded transaction costs with and without minimum transaction lots but again not based on Markowitz’s model (Mansini, & Speranza, 1997). In 2001, Konno proposed an algorithm for his portfolio optimization problems regarding transaction costs and minimum transaction lots (Konno, & Wijayanayake, 2001).

In the field of model solving, Arnone presented a Genetic Algorithm for an unconstrained portfolio optimization problem (Arnone, Loraschi, & Tettamanzi, 1993), but the first use of genetic algorithm for Markowitz’s model (without any extra constraints) was done by Shoaf (Shoaf, & Foster, 1996). Rolland utilized Tabu search (TS) to solve Markowitz’s (Rolland, 1997). Later, to corroborate the necessity and desirability of heuristic algorithms, Mansini and Speranza proved that the portfolio selection problem with minimum transaction lots is an NP-complete problem. Subsequently, they proposed three heuristic algorithms to figure out the MAD model of Konno (Mansini, & Speranza, 1999). Afterwards, they (with Kellerer) extended their model in order to take fixed transaction costs into account (Kellerer, Mansini, & Speranza, 1999).

At the first years of 20th century, some studies threw more light on the capabilities of heuristic algorithms in portfolio selection problems; Xia, Orito, Streichert, Fieldsend and Lai in showing the excellent performance of GA, Gilli in utilizing Simulated Annealing (SA), and Chang’s group experimented a variety of meta heuristics, including SA, TS and GA for portfolio selection model without trading and turnover constraints. Although Chang mentioned that GA is best able to approximate the Unconstrained Efficient Frontier (UEF) but he could not found any individual heuristic performed better than the other ones (Chang et al., 2000, Fieldsend et al., 2004, Gilli and Këllezi, 2000, Lai et al., 2006, Orito et al., 2003, Streichert et al., 2003, Xia et al., 2000). Schaerf first improved Chang’s article, thereafter, in another research he proposed a Tabu Search algorithm in order to solve Markowitz’s model with cardinality of the portfolio (Schaerf, 2002a, Schaerf, 2002b). Lin considered the multi objective genetic algorithm for portfolio selection problem (Lin, Wang, & Yan, 2001). Crama utilized a Simulated Annealing to solve Markowitz’s model with cardinality and Turn Over constraints (Lin et al., 2001).

Oh proposed a new portfolio selection algorithm based on portfolio beta and regarding sector capitalization, he profited genetic algorithm as his solver (Oh, Kim, & Min, 2005). Stein exploited Meta heuristics for Markowitz’s model with cardinality and Buy-in thresholds (Stein, Branke, & Schmeck, 2005). Finally, in 2007, Markowitz’s model and heuristic algorithms are still taken into considerations; Lin and Liu (2007) considered Markowitz’s model with minimum transaction lots and they presented three other models. They also utilized GA to solve their proposed models. Fernández and Gómez (2007) paid attention to Markowitz’s model with cardinality and bounding constraints. They utilized neural networks as their solver.

On the other side, genetic algorithms, roughly speaking, are one of the most popular heuristic optimization techniques were originally developed by Holland, 1975, Holland, 1992. Subsequently, the GA was applied to optimization problems in biology, engineering, and operations research. In ninety decade, different aspects of GA were expanded; Hartl researched on convergence of GA (Hartl, 1990), Radcliffe and Bean studied on cross over operator (Bean, 1992, Radcliffe, 1991, Radcliffe, 1992), Miller and Goldberg (1995) researched on selection strategies, and finally, Vose and Koza (Koza by developing Genetic Programming), investigated on whole concept of basic GA of Holland (Koza, 1994, Vose, 1991). More information about GAs and also Evolutionary Algorithms can be found in Reeves, Mitchell and Baeck’s researches (Baeck et al., 1997, Mitchell, 1996, Reeves, 1993).

In this paper, a portfolio selection model which is based on Markowitz’s portfolio selection problem including three of the most important limitations is considered. The results can lead Markowitz’s model to a more practical one. Minimum transaction lots, cardinality constraints (both of which have been presented before) and market (sector) capitalization (which is proposed in this research for the first time as a constraint for Markowitz model), are taken into account in presented model. In particular, in Section 2 the expanded model is represented and described. In Section 3 the characteristics of proposed GA is explained. Empirical experiments which research the performance of proposed model and also the GA are discussed in Section 4. Finally, in Section 5 conclusions are remarked and possible future researches are provided.

Section snippets

The proposed model

In this section the model will be described. It has been mentioned that the proposed model is based on Markowitz’s mean–variance portfolio selection model which doesn’t consider the situation of a real market as cardinality constraint and minimum transaction lots. In order to explain the model let:

    N

    be the total number of securities available

    ri

    be the expected return of security “i” (i = 1,  , N)

    σi

    be the covariance between securities “i” and “j” (i = 1,  , N; j = 1,  , N)

    R

    be the minimum desired expected

Utilizing genetic algorithm

Genetic Algorithms which were originally developed by Holland in 1975, is one of the most popular heuristic optimization techniques (Holland, 1975, Holland, 1992). GAs originate from the famous Darwin’s principal of natural selection and the well-known phrase; “survival of the fittest”. Actually, it can explore the solution space by using the concept taken from natural genetics and evolutionary theory. In Genetic Algorithms, an initial population which consists of several feasible solutions

Empirical experiments

The empirical experiments are organized in four parts; first the parameters of the proposed GA are set, second to verify the logical reactions of proposed GA, data are retouched intentionally and the results are analyzed, third in order to validate proposed GA (which is coded by MATLAB 7), it is compared with model results in LINGO which can generate global optima for small problems (but because of LINGO’s limitation, this comparison couldn’t be performed in large scale data) and finally, the

Conclusions and future researches

In this paper a portfolio selection model based on Markowitz’s mean–variance portfolio selection model is proposed. The proposed model covered minimum transaction lots, cardinality constraints and also regards market (sector) capitalization. A Genetic Algorithm which is coded by MATLAB 7 software is produced to solve the proposed model which belongs to mixed-integer nonlinear programming (NP-Hard). The empirical studies first compare the proposed GA and the model results in LINGO; in addition

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