An application of Markowitz theorem on Tehran Stock Exchange

Article history: Received December 28, 2013 Accepted 24 March 2014 Available online March 29 2014 During the past 65 years, there have been tremendous efforts on portfolio selection problem. The standard Markowitz mean–variance model to portfolio selection includes tracing out an efficient frontier, a continuous curve demonstrating the tradeoff between return and risk. This frontier can be often detected via standard quadratic programming, categorized in convex optimization. Traditional Markowitz problem has been recently extended into a new form of mixed integer nonlinear problems by considering various constraints such as cardinality constraints, industry limitation, etc. This paper proposes a mixed integer nonlinear programming to determine optimal asset allocation on Tehran Stock Exchange. The results have indicated that a petrochemical firm named Farabi has gained 44% of the portfolio followed by a drug firm named Kosar Pharmacy gaining 28%. In addition, banking sector was the third winning firm where Eghtesad Novin bank gained nearly 10% of the portfolio. Minerals and mining firms were the next sector in our portfolio where Gol Gohar Iron Ore and Tehran Cement collected 0.73% and 0.57% of the portfolio, respectively. In our survey, auto industry gained only 0.26% of the portfolio, which belonged to Saipa group. © 2014 Growing Science Ltd. All rights reserved.


Introduction
For years, Markowitz theorem (Markowitz, 1952(Markowitz, , 1970) ) has been widely used to determine optimal investment strategies.The theory has been well studies under various conditions (Fabozzi et al., 2007).The standard Markowitz mean-variance model to portfolio selection includes tracing out an efficient frontier, a continuous curve demonstrating the tradeoff between return and risk.This frontier can be often detected via standard quadratic programming, categorized in convex optimization.Chang et al. (2000) considered the problem of locating the efficient frontier associated with the standard mean-variance portfolio optimization model.They extended the original model by considering cardinality constraints, which limited a portfolio to be limited to a specified number of assets, and to consider limits on the proportion of the portfolio held in a given asset.They also showed the differences arising in the shape of this efficient frontier when such constraints imposed and solved the resulted model using three heuristic algorithms based upon genetic algorithms, tabu search and simulated annealing for locating the cardinality constrained efficient frontier.Streichert et al. (2004) also solved the same portfolio optimization problem using evolutionary algorithms by considering the cardinality constrained.Maringer and Kellerer (2003) considered the same optimization of cardinality constrained portfolios with a hybrid local search algorithm.Soleimani et al. (2009) extended the problem by adding three options to the original model, which would lead Markowitz's model to a more practical one.They considered the minimum transaction lots, cardinality constraints and sector capitalization, which was proposed in this research for the first time as a constraint for Markowitz model.The explained that the new model could be formulated as an Np-Hard problem and they proposed a genetic algorithm to solve the resulted model.Branke et al. (2009) proposed to combined an active set algorithm optimized for portfolio selection into a multi-objective evolutionary algorithm (MOEA).The idea was to let the MOEA come up with some convex subsets of the set of all possible portfolios, solve a critical line algorithm for each subset, and then merge the partial solutions into a solution of the original non-convex problem.They showed that the resulting envelope-based MOEA substantially outperforms existing MOEAs.Anagnostopoulos and Mamanis (2010) considered the portfolio optimization model with three objectives and discrete variables.Skolpadungket et al. (2007) applied different techniques of multiobjective genetic algorithms to solve portfolio optimization by considering some realistic constraints, namely cardinality constraints, floor constraints and round-lot constraints.Fernández and Gómez (2007) considered the same portfolio selection using neural networks.

The proposed study
In this paper, we proposed an extended Markowitz model by considering different real-world limits on the original cardinality model including bound constraints, sector limitation, etc.

Variables and notations
X i : is the number of shares purchased from the share i, Z i : is the binary variable from the share i, if selected is equal to one and zero, otherwise Y j : j industry binary variables, if selected is equal to one, zero, otherwise W i : is weight of i share in portfolio g i : is the weight of j industry in portfolio i: stock index j: Industry Index Com: The fixed fee deals N: number of selected stocks σ: covariance between industry or stock P i : i free float shares F: The percentage of minimum commission rate of buying shares C i : the price of selective stock φ: volume of transactions P: Minimum number of free floating shares R: optimal return level r i : rate of expected return M: A big number IN: Total amount of investment S: Total number of industries B upperi : The maximum amount of investment in the share of the i B loweri : The minimum investment amount in the share of the i

Mathematical model
The mathematical model is formulated as follows, Z=min ∑ w w ∑ ∑ (1) subject to (2) (1, 2,..., ) (1, 2,... ) (1 ) , (1, 2,..., 1) The proposed model determines the amount of shares invested in each firm.In addition, parameters include monthly stock returns, monthly returns of covariance and industry are between returns of stock and industry and finally limitations include the budget, expected returns, volume of transactions, etc.The objective function minimizes the expected return by considering budget constrain.For more details, please see Chang et al. (2000), Branke et al. (2009) and Soleimani et al. (2009).The proposed model has been applied on monthly information gathered from Tehran Stock Exchange by considering Covariance between stock returns and mentioned industries, budget, investor optimum efficiency, free float stock, etc.The proposed model has been investigated in four different stages.
The first stage: Stocks returns of selected research companies were collected in three-year timeframe and Covariance of stock returns were calculated by Excel software.
The second stage: Returns of selected research industries were collected in three-year timeframe and covariance between industry returns was calculated by Excel software.
The third stage: Return of per share for a period of 3 years "36 months" has been calculated and its arithmetic mean has been used as the coefficients in limitations of the model.Accordingly, the return of each company has been calculated over a period of 3 years with taken into account.
The fourth stage: Based on information obtained, limitations of the model were defined including range of the asset, the minimum and the maximum choice of industry, transaction costs of buying shares, free float stock restrictions, etc.

The results
In this section, we present details of our findings on the implementation of the proposed model in four different scenarios.Table 1 shows details of our findings.

Table 1
The summary of the results of our survey As we can observe from the results of Table 1, during the time schedule of the study, drug industry has been the most attractive industry on Tehran Stock Exchange followed by petroleum industry, banking, non-metal as well as cements industry.We have performed sensitivity analysis on the proposed study under four different scenarios and Table 2 shows details of our findings.

Discussion and conclusion
In order to understand the behavior of the proposed model, we have applied it under four different stages.To implement this model in the first stage, collection of performance data was accomplished by RAH'AVARD NOVIN software.In the second stage, we have calculated the covariance between industry and stock.These values were used as the objective function coefficients of the decision variables.During the third stage, expected return measures of stock and industry, investment restrictions, the minimum and maximum choice of industry, transaction costs of buying stock and free float stock restrictions were calculated as constraints and parameters of the model.Then the values and parameters were used in the model.The results model has been coded in a commercial optimization software package and it was solved and its optimal solution obtained is as follows.
In our survey, a petrochemical firm named Farabi has gained 44% of the portfolio followed by a drug firm named Kosar Pharmacy gaining 28%.In addition, banking sector was the third winning firm where Eghtesad Novin bank gained nearly 10% of the portfolio.Minerals and mining firms were the next sector in our portfolio where Gol Gohar Iron Ore and Tehran Cement collected 0.73% and 0.57% of the portfolio, respectively.In our survey, auto industry gained only 0.26% of the portfolio, which belonged to Saipa group.We have discussed the results of the proposed model with some experts who were involved in Tehran Stock Exchange and they confirmed our survey result.

Table 2
The summary of portfolio optimization under four different scenarios