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Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 957893, 14 pages

http://dx.doi.org/10.1155/2015/957893

## Model to Estimate Monthly Time Horizons for Application of DEA in Selection of Stock Portfolio and for Maintenance of the Selected Portfolio

Institute of Production Engineering and Management, Federal University of Itajubá, 37500-903 Itajubá, MG, Brazil

Received 24 March 2015; Revised 10 June 2015; Accepted 25 June 2015

Academic Editor: Yan-Jun Liu

Copyright © 2015 José Claudio Isaias et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the selecting of stock portfolios, one type of analysis that has shown good results is Data Envelopment Analysis (DEA). It, however, has been shown to have gaps regarding its estimates of monthly time horizons of data collection for the selection of stock portfolios and of monthly time horizons for the maintenance of a selected portfolio. To better estimate these horizons, this study proposes a model of mathematical programming binary of minimization of square errors. This model is the paper’s main contribution. The model’s results are validated by simulating the estimated annual return indexes of a portfolio that uses both horizons estimated and of other portfolios that do not use these horizons. The simulation shows that portfolios with both horizons estimated have higher indexes, on average 6.99% per year. The hypothesis tests confirm the statistically significant superiority of the results of the proposed mathematical model’s indexes. The model’s indexes are also compared with portfolios that use just one of the horizons estimated; here the indexes of the dual-horizon portfolios outperform the single-horizon portfolios, though with a decrease in percentage of statistically significant superiority.

#### 1. Introduction

The investing in stocks requires taking into account the expected returns. To minimize the probability of loss, one should select stocks based on a prediction of their likely future earnings. The stock market does permit investors to obtain satisfactory returns, though the market can often be hostile and deliver sometimes severe losses and this return is accompanied by a certain level of risk [1–3].

The problem with capital losses in a stock market is the oscillating nature of a stock’s price trajectory. That is, investors are able to occasionally obtain satisfactory returns, over time, even with little knowledge. At other times, however, investors are unable to achieve their target returns even when possessing a high degree of knowledge about the stock or its market sector [4, 5].

To make lucrative investments in the stock market, one can benefit from the support of a properly structured tool. One such tool is Data Envelopment Analysis (DEA), which has a record of producing good results in the selection of stock portfolios [2, 4, 5].

According to the research that forms the basis of Institute Scientific Information (ISI) Web of Knowledge, no mathematical models, methods, or procedures have been found to create monthly time horizon estimates. Such a tool would be used to collect data to be used in applying a DEA model to select a stock portfolio and to maintain a stock portfolio so selected.

Hence, this study’s general objective is to better estimate monthly time horizons. Such an estimate could take two forms: simultaneous and individual. The study aims to first propose a mathematical programming model to estimate three monthly time horizons. Second, the model will estimate, simultaneously, both the monthly time horizons. Third, it will estimate, individually, monthly time horizons, collecting data to be used in the application of a DEA model as it selects a stock portfolio. And fourth the model will estimate, also individually, the monthly time horizons in the maintenance of a stock portfolio so selected.

#### 2. Data Envelopment Analysis

Data Envelopment Analysis (DEA) is a set of mathematical programs that verify efficiency, comparing production units according to their proximity to an efficiency frontier. DEA, first proposed by Charnes et al. in 1978 [6], consists of a technique used to calculate the efficiency of decision maker units (DMUs). DMUs must be components of homogeneous production and it uses equal set of inputs and set of products. The differences between the DMUs consist only in the intensity and magnitude of utilization of inputs and of production of outputs [7].

In 1973, The International Journal of Management Science published the first study on partial productivity measure or efficiency. The article analyzed the productivity of manufacturing units in the USA. The study found problems in analyzing productivity indicators that considered an output type per an input type [8].

According to Charnes et al. [8, 9], in modeling the original DEA problem, the optimal point occurs determining the vector values of the input and output weights, represented by and . By linearizing the original model, the authors gave rise to the CCR Primal model, which is oriented to inputs according to (1)–(4). It is this model that has been adopted in the current research.

In the set of equations is the efficiency of DMU in analysis; , are the weights of inputs, , and outputs, , respectively; and are the inputs and outputs of the DMUs and are the inputs and outputs of the DMU in analysis [10, 11]. ConsiderIn the system of equations, (1) maximizes the efficiency of DMU under analysis. Equation (2) constrains the results of the virtual input of DMU under analysis to 1, meaning that it defines the consumption of DMU as a reference for all the others. Equation (3) constrains the difference between virtual outputs and virtual inputs to zero at the maximum. So as to impose the constraint, the efficiency results are equal, at their maximum, to 1. Equation (4) relates to the nonnegativity of vectors and [10, 11].

Data Envelopment Analysis encompasses two classical models, the CCR and the BCC. The CCR, named after Charnes, Cooper, and Rhodes in 1978, is also known as the constant return scale (CRS). This model estimated, using multiple inputs and multiple outputs, the technical efficiency of schools. One characteristic of the model is that it admits only positive data in its applications. The BCC model, named after Banker, Charnes, and Cooper in 1984, is also known as the variable return scale (VRS). This model differs mainly from the CCR in its utilization of the return variable scale [11–13].

#### 3. Gaps of DEA Applications in Stocks

DEA has yielded good results in picking stock portfolios, although it is known to have weaknesses. These weaknesses concern its capacity to estimate the monthly time horizon’s extension for data collection and to estimate the monthly time horizon’s extension of maintenance of the selected portfolio.

According to research in the Institute Scientific Information (ISI) Web of Knowledge, Lamb and Tee [4] came up with a new theory. This theory utilizes the DEA in constructing an index of risk-returns for investment funds. To identify an appropriate form of returns to scale, the authors explored the so-called production possibility set of investment funds. They proposed that measures of risk and return can be combined in a justifiable manner. The theory indicates how to deal with downside risks and identify appropriate sets of measures to avoid them. The research identifies the problem as a result of not treating the diversification in investment funds. Finally, the authors proposed an iterative process to address the problem, a process that in practice proves its efficiency at simulating the utilization.

Pätäri et al. [14] examined the applicability of DEA in using multiple criteria to select stock portfolios. The stock portfolios formed in their study were obtained using three variants of DEA models: the pure CCR, the CCR with super efficiency, and the CCR with cross-efficiency. The results of this study showed that the DEA approach is really able to add value to the selection of stock portfolios. The study compared the superior performance of the main portfolios with both a portfolio of comparable funds from the market in question and the stock market average. In both comparisons, over all the performance measures used, the superiority of the main portfolios was statistically significant. The paper concluded that the DEA is particularly useful for multicriteria applications in the stock market. Therefore DEA holds interesting implications for the practice of portfolio management.

A new methodology for selecting the variables of inputs and outputs for DEA approaches was presented by Edirisinghe and Zhang [1]. Their paper proposed an iterative model of optimization constructed in such a way that it is executed in two stages. The first stage addresses the benefit of utilizing expert information to determine a performance evaluation. The second stage maximizes the correlation with the metric of return of DMUs under analysis. The methodology was applied in fundamental analyses of publicly traded US companies. The aim was to determine a single fundamental indicator based on the proposed DEA model. More than 800 companies from all major sectors of the US stock market were used in the empirical evaluation of the proposed model. The simulation of results of the model’s application in companies proved the methodology’s superiority as a decision tool for investment.

According to the research that forms the basis of Institute Scientific Information (ISI) Web of Knowledge, no mathematical models, methods, or procedures have been found to generate monthly time horizon estimates. The purpose of such an approach would be to collect data to be used in applying a DEA model to select a stock portfolio and to maintain a stock portfolio so selected. The lack of such an approach is a result of the fact that when the aforementioned researchers applied some DEA model to the stock market, they left arbitrarily the time horizons of the collection variables and the maintaining of the portfolio. Or they simply utilized third-party extensions.

#### 4. Conception of the Proposed Mathematical Model

The step-by-step procedure followed in this chapter is intended to detail the design of this paper’s model. As the paper’s main contribution, this model estimates monthly time horizons of data collection to select a stock portfolio and monthly time horizons to maintain a portfolio so selected. The paper’s research problem is determining how to estimate these time horizons.

The model, consisting of (5) through (17), comprises two sets of equations. The first set, (5)–(12), is made up of equations reformulated from the DEA CCR primal model, oriented to inputs so as to adapt to paper data. This is because the DEA CCR primal model oriented to the inputs is selected to be applied by research. The second set, (13)–(17), is binary programming and minimization of square errors, and it is made up of original equations to solve the research problems presented in Section 3.

In this section we takes into consideration a number of factors. These include the selection of the DEA model used and the redefinition of its terms, the matrices of DEA efficiency that are planned to be generated, the formulations for minimizing square errors, the formulations for minimizing square errors, the matrices that aid in the solution in the minimizing square errors which are defined, and finally the proposed mathematical model for the phase of minimization of square errors.

The model’s purpose is to estimate the monthly time horizons that were obtained by research and that, when used together, are most representative of one another. Such time horizons would offer greater maximizing tendencies of the indices of return of a portfolio that utilizes them. After all, the portfolio is formed under the constraint of a boundary that accords with the results of the DEA efficiency adopted by the model. Furthermore, the only output of this DEA model is the index of stock returns. Consequently, the greatest representability between the monthly time horizons utilized will increase the efficiency of the cutting line regarding the return generated. This cutting line is described in Section 6.2.

##### 4.1. DEA Model Utilized

The DEA model applied in this research is the CCR Oriented Inputs. This model, when it is utilized in the stock market for the formation of portfolios, generally has superior results to other models [14]. Also this research conducted several tests to verify among the classical DEA models and their variants the one that best represented the oscillations of stock returns. The CCR Oriented Inputs had the best results.

According to Charnes, Cooper, and Rhodes, DEA models should be oriented either to inputs or to outputs. The orientation of the CCR model in this study is to inputs. The objective of this orientation is that the model maintains the input level and maximizes the only output—the returns of the stocks studied. Thus, the stocks with higher indices of return also have higher levels of efficiency [9, 15].

###### 4.1.1. Redefinitions of the DEA Model Terms

The model utilized for this research, the CCR Oriented Inputs, requires a redefinition of their terms. The purpose of redefining is clear when the model uses stock data related to the past and related to the future of the collection’s eight sections. This is in line with sections of collection adopted in other researches.

When the DEA modeling uses only the historical past of the eight time frames from the collection, the CCR Model Oriented Inputs described in Section 2 will have to replace the terms found in the literature review. What follows are redefinitions of those terms:(a)The term is renamed and represents the efficiency of the stocks in analysis of each application of the model. The result set generates the matrix which is described in Section 4.1.2.(b)The terms are the data relating to inputs . The 3 variables of the set are market value, price for sales, and price per equity value. The terms are the data relating to outputs . They are only the return variables of each stock.(c)The terms and are the weights of the inputs and the output . The variables are the same as those cited in (b) item for sets and .(d)The terms and are related to inputs and outputs of stock in analyses. The variables are the same as those cited in (b) item for sets and .(e)The set is . The set ranges up to 35 because this is the number of stocks that were used in the study that applies the model proposed by (5) through (17).Thus, the model is redefined by (5)–(8) to measure efficiency based on only the historical past of the collection’s eight sections: Now, when the modeling uses future data in relation to the collection’s eight sections, the model will have to replace the terms of the review with the following redefinition:(a)The term is renamed and represents the efficiency of the stocks in analyzing each application of the model. The result set generates the matrix , which is described in Section 4.1.2.(b)The terms are the data relating to inputs . The 3 variables of set are also market value, price for sales, and price per equity value. The terms are the data relating to outputs . They are also only the return variables of each stock.(c)The terms and are the weights of the inputs and the outputs . The variables are the same as those cited in (b) item for sets and .(d)The terms and are related to inputs and outputs of stock in analyses. The variables are the same as those cited in (b) item for sets and .(e)Set . The set ranges up to 35 because this is the number of stocks that were used in the study applying the model proposed by (5) through (17).In this way the model is redefined by (9)–(12) to measure efficiency using only future data from the collection’s eight sections:

###### 4.1.2. Matrices of Efficiency Generated by DEA Model

The DEA model calculates the results of efficiency starting from data adjusted in the last of the collection’s sections. The model’s objective is to assemble a set of matrices starting from the efficiency of each stock of the set in all the tested monthly time horizons .

The number of efficiency matrices is equal to the number of the collection’s sections, where . The eight sections are equivalent to monthly closing of BM&F Bovespa stock exchange from September 2011 to April 2012. These are the exact points in time from which the data will be collected. Monthly time horizons from the past will be used to collect data for portfolio selection; sometimes future monthly time horizons will be used to collect data for stock portfolio maintenance.

The number of monthly time horizons from the past is represented by the set of tested horizons. The amount of these monthly time horizons is fourteen, where . This dimension sets the number of columns for each of the eight arrays.

The number of tested stocks that define the vertical dimension of the matrices is 35, where .

This yields a set of eight matrices of efficiency from the results of the stocks’ history:Similarly, when applying the DEA model to find results of efficiency using future data or to maintain portfolios initiated in sections or in the exact collection points, the set of matrices is equal to .

The other dimensions of the set of efficiency matrices are the number of sets, , and the number of future monthly horizons, or portfolio maintenance, in relation to the size of the sections . The first dimension defines the number of rows and the second dimension defines the number of columns.

In this case, the die assembly is the generated of stock efficiency results from sections. This uses future data in applying the adopted DEA model. So the set of matrices can be defined as

##### 4.2. Formulations for Minimization of Squares Errors

Presented below are definitions of matrices that aid in the solution of the problem and the proposed mathematical model both in phase minimization of errors squares. These matrices and equations consider the phase minimization of error squares.

###### 4.2.1. Matrices of Assistance to Solution

At this point it is necessary to define which matrices facilitate the solution. The first two are binary matrix lines and .

Matrix aids in the estimation of the monthly time horizon of data collection for the formation of the portfolio. In this—as occurred in the matrix of efficiency of results from the past —each column represents a monthly time horizon from the past with the eight sections, where .

The matrix aids in the estimation of the monthly time horizon of maintenance of the portfolio selected. In this—as in the matrix of efficiency of future —each column represents a monthly time horizon of maintenance portfolio, where .

Thus, the two matrices areThese matrices’ binary lines are defined so as to be used in constraints. This is because the binary matrices’ lines have the same number of columns as the number of monthly time horizons tested, when associating each of its columns to a monthly time horizon and restricting the sum of the columns of each matrix to a value equal to 1. Thus, in a model for minimizing square errors, it is possible to determine the monthly time horizons optimum.

The following describes the constraints that were included in the model of minimization of error squares that was elaborated starting from binary matrices’ rows:After determining that the constraints defined by (13) and (14) are still necessary, the two matrices are defined regarding the tested monthly time horizons in this study.

When the matrix refers to monthly time horizons of data collection to form the portfolio, it is represented by and it has the same dimension as the matrix binary row .

When the matrix refers to monthly time horizons of maintenance of the selected portfolio, it is represented by and has the same dimension as the matrix binary row . These matrices areWith matrices and defined, they may be equated with the two scalars and to find the result. This is done first for the results estimated for the monthly time horizon of data collection to create the portfolio. It is done next for the results estimated for the monthly time horizon for maintenance of the selected portfolio. To do this, just insert the following equations in the model of minimization of square errors:

###### 4.2.2. Model Proposed for Minimizing of Square Errors

The estimation of the monthly time horizons for this search is done by looking for the best representability between efficiency results starting from the past, represented by matrix , with the results of the efficiency of future, represented by matrix .

Based on concepts from the method to minimization of errors squares, “which is well known for being the main concept in studies using variations of Regression Analysis,” [16] the model can be proposed to estimate the monthly time horizon searches.

The proposal of the model of minimization of error squares is described starting from the definitions of the matrices of efficiency calculated from sections of the collection to the past, , of the matrices of efficiency calculated from the sections of the collection to the future, , and of the binary matrices lines and . With these matrices, it is possible to calculate the scalars and .

The result is then inserted into the model by objective function to minimize square errors described by (17). In the model, (17) minimizes the square of the product of binary matrices and . This product still multiplies the errors between the results of efficiency of matrices with the efficiency results of matrices .

The intention is to identify the columns—one from the past and one from the future—that best represent each other. Equations (13) and (14) constrain the sums of the binary matrices’ rows and to extract a value equal to 1. Thus, as these binary matrices rows vary in sets 0 and 1, only one column of and one column of will be equal to 1.

Consequently, the column of matrix that will be equal to 1 represents the monthly time horizon estimated from the data collection of the past to the portfolio selection. The column of matrix that will be equal to 1 defines the monthly time horizon estimated to maintain the selected portfolio: It is important to be clear that (15) and (16) did not constrain the model but are calculated only for scalars and .

#### 5. Collection and Treatment of Data

This chapter explains how the collecting and treatment of data was planned in this study. Such data make up the inputs and outputs utilized in the initial applications of the equations of the proposed mathematical model in Section 4.

##### 5.1. Decision Maker Units

The object of study was a set of stocks traded on the São Paulo exchange, the BM&F Bovespa. As of November 2014, the stock exchange offered over 500 investment options in stocks. To utilize the data of all these stocks would make an analysis of this research unfeasible due to the extensive calculations that would need to be executed. The solution is to utilize, as first filters, the so-called Market Indices.

The most important of the Brazilian stock markets is the stock exchange of São Paulo—the Bovespa Index. Among the indexes, this one represents the average of prices and the profile of the negotiations of the cash market for stocks with high degrees of liquidity [17].

The current study intends to scrutinize the behavior of stocks traded on the BM&F Bovespa. The Bovespa Index represents well the BM&F Bovespa. Thus the first filter in the selection of stocks here was that the stocks will be components of the BM&F Bovespa Index portfolio—the Ibovespa.

The research is defined to simulate an investment starting in April 2013. In this scenario, the fictitious investor intends to build a portfolio of BM&F Bovespa and to keep it for a maximum of one year before reassessing the market. Thus, the last BM&F Bovespa Index portfolio before the investment simulation period is of the first quarter of 2013.

The stocks that this study looks at, consisting of stocks from the Bovespa Index in the first quarter of 2013, must have data available for the period of application of the proposed mathematical model, from January of 2009 to April of 2013, relating to fundamental indices utilized.

For the period of applying this model, it was observed that, of the 69 stocks comprising the Bovespa Index in the first quarter of 2013, only 42 were available in the period outlined by the fundamental indices. The third constraint was that the number of stocks to be studied must be at least 80% of the 42 stocks that are available on the fundamental indices used.

Thus, we randomly selected 33 stocks in the code: BISA3, BRFS3, BRKM5, CESP6, CMIG4, CPLE6, CSNA3, CYRE3, DASA3, DTEX3, ELET3, ELET6, ELPL4, EMBR3, GGBR4, ITSA4, LAME4, LIGT3, LREN3, MRFG3, MRVE3, OIBR3, OIBR4, PCAR4, PETR3, RENT3, RSID3, SUZB5, TIMP3, USIM3, VAGR3, VALE3, and VIVT4.

##### 5.2. Delimiting of Scale

To obtain the correct monthly time horizons for this research, there needed to be an imposition of a scale of inputs and outputs homogeneous in their maximum and minimum limits. This is before applying the DEA model.

The need to impose these limits is that tests show that when in a table for application of the model DEA, CCR Oriented Inputs exist only for DMUs with low values of inputs and outputs; the other table exists only for DMUs with high values of inputs and outputs, even if in each of the tables there is a DMU with the same values of inputs and outputs. The result is that these DMUs have different efficiency values. The divergence of values occurs due to differences in the limitation of scales generated by the DEA model used for each data set.

Tests also show that the problem can be prevented using, in each table of input and output, the values of two fictitious DMUs. Around the period of application of the proposed mathematical model, a monthly return index value that was the lowest among them all and also a monthly return index value that was the largest among them all were estimated.

Thus, the first fictitious DMU will have results of the output return equal to or lower than those found during the entire modeling period, and the second fictitious DMU will have results of the output return equal to or greater than those found during the entire modeling period. In both fictional DMUs, the three inputs used had to have average values, which in this case were utilized values equal to 1. Thus, each table in the application of the model had 33 preselected stocks, increased by two more fictitious stocks from the delimitation of scale.

Between the data from monthly returns encountered during the whole period of application of the proposed mathematical model, it was observed that the lowest value was −19.66% and the highest value was 10.25%.

##### 5.3. Tested Horizons and Sections of Collection

The first boundary of this research is in relation to the monthly time horizons of data collection of the past from points or sections of that collection. In this case, the maximum monthly time horizon is, according to surveys, 32 months or the approximate average between two and three years. Within this maximum period, the relation of four months of the Bovespa Index portfolio is maintained. So in the initial stage eight horizons are tested, starting at 4 months and ending at 32 months.

The second boundary relates to the monthly time horizons of data collection of the future from points or sections of that collection, which represent the time horizons of maintaining the stock portfolio. In this case, the study uses the typical maximum monthly time horizon, according to research, which is 12 months. Within this maximum period, again the relationship of four months to the Bovespa Index portfolio is maintained. Hence, three horizons are tested in an initial phase, beginning at 4 months and ending at 12 months.

After applying the model to estimate the local monthly time horizons according to the preliminary scope, the study searches for, within the limits of this scope, the global optimal time horizons.

In practice, the horizons of data collection for the portfolio selection tested were, in months, 4, 8, 12, 16, 20, 24, 25, 26, 27, 28, 29, 30, 31, and 32. The horizons for the maintenance portfolio that had already been tested were, in months, 4, 8, 9, 10, 11, and 12.

This is because the results of the first application of the model proposed identified 28 months as the optimal monthly time horizon for collecting data for portfolio selection and 12 months as the optimum monthly time horizon of maintenance of the selected portfolio. The global optimum within the boundaries of the search scope is presented in Section 6.1.

The third definition of search time relates to the collection sections. These sections are exact points in time from which the data were collected. Sometimes the section is from the past, representing the variation of monthly time horizons for portfolio selection; sometimes the section is for the future, representing the variation monthly time horizons of portfolio maintenance. In total there are eight sections. The first section occurs at the close of the stock exchange in September 2011, and the last section occurs at the close of April 2012 of the BM&F Bovespa exchange. In this range, each monthly closing corresponds to a collection section.

The main constraint for the extensions of the two monthly time horizons of the first two delimitations, as well as the section number of the collection, was not to use 2008 data, because of the distortion from that year’s economic crisis.

The fourth boundary for the times of research concerns the simulation period of the portfolio that used the time horizons estimated by the proposed mathematical model. The planning was that, 12 months after the last point of data collection, the simulation of the stock portfolio would be initiated with the time horizons estimated. This was chosen because the major monthly time horizon of maintenance portfolio tested was also 12 months. In this way, the period of application of the proposed mathematical model to estimate the time horizons and the period of simulation of the results of the portfolio with the estimated horizons were completely independent, giving more credibility to the study. The simulation period took place from the beginning of May 2013 to the end of June 2014, a total of 14 months.

##### 5.4. Data Utilized and Their Definitions

The research presupposes that an investor at the end of April 2013 intends to build a portfolio of stocks from the BM&F Bovespa and to keep it at least one year, before assessing the need to change it. It is assumed then that the investor has access to the historical index of stocks on the BM&F Bovespa.

The fictitious investor intends to obtain a set of data to measure the efficiency of stocks solely to obtain the return index. In this way, the only data of output utilized by the DEA model is the monthly return of each stock.

Regarding other dates to be used by the DEA model as inputs, Nagano et al. [18] argued that “analyzing fundamental indices applying regression techniques to see which have the most power to explain the return of stocks BM&F Bovespa, its concluded among them those indexes the which show strong relationship with return are: Profit per price, market value, price per equity value, liquidity, beta and price per sales.”

However, modular correlation coefficients ranging between 0.10 and 0.30 are able to explain a response variable, while values lower than 0.10 cannot [19]. Thus the study analyzed which fundamental indices had modular correlation coefficients with the monthly stock returns. The study then studied which was greater than or equal to 0.20, which is the average of the variation of correlation that, according to Moore [19], is the least bit representative.

“The stock market is dynamic and their trends constantly change in short periods” [3]. To minimize the possibility of errors by changes in trends, data was used up to 32 months before situating the fictitious investor at the end of April of 2013. The extension of 32 months was chosen also to represent the maximum time extension tested in the survey to collect data on the portfolio formation.

Thus, the correlation calculations were executed with data from September 2010 to April 2013. That is, for this period we calculated the correlation between each of the fundamental indices selected by Nagano et al. [18], for the returns of all stocks studied here. Subsequently, a total average for the index was obtained.

The values that correlated between the indexes were as follows: profit per price = −0.17, market value = 0.26, price per equity value = 0.26, liquidity = −0.07, beta = 0.01, and price per sales = 0.25. In this way, the fundamental indices selected as inputs for the DEA model were the following: market value, price per equity value, and price for sale.

##### 5.5. Inputs Adjustments and Parameters

The fundamental indices that were used as inputs in this research need to be adjusted to improve their representability in relation to the stock returns studied. This adjustment is calculated as the monthly percentage variation that the indices fell or rose in one unit.

For example, if in a certain section of data collection we wanted to calculate the adjusted input to be used, then we assigned this input the value of 1 plus the percentage monthly variation for the horizon of data collection for the portfolio selection at a distance of months. The calculation must agree that the value of crude input after collecting the month section is and that the same input to previous months has the value of . So to calculate the percentage of monthly variation , this index number should solveTo calculate the interference in one unit of the monthly percentage variation , the formula to use isTo calculate the result of a particular section for the future of that same section, representing the direction of maintaining the portfolio, the variable of the section must be and of the future at months should be .

The justification for the adjustment is that the fundamental indices change significantly in size from one stock to another, which would make it an unfair or incorrect use of modeling in its own index number. If we want to see the maximum efficiency of stocks at producing a return, this way seeks to minimize any alteration in the results of efficiency caused by alterations in the inputs.

The database used in this research is from the* Economática* Software. The collection parameters for each input index number are as follows:(a)Market value = enterprise value at the monthly close, in the country’s currency, to three decimal places.(b)Price per equity value = index value at the consolidated monthly close, in the country’s currency, to three decimal places.(c)Price for sales = index value at the consolidated monthly close (based on the demonstrations of monthly returns for the last year), in the country’s currency, to three decimal places.

##### 5.6. Output Adjustments and Parameters

If for a certain section of data collection we want to calculate the adjusted value of output return, according to the percentage monthly variation for a horizon of data collection for a portfolio selection, it is done in extension months. Here the value of the index number of the return for the section’s month is , and the same number index at previous months is . The value of the adjusted output to be used by the DEA model can then be obtained byWhen the adjusted output from the percentage monthly variation initiated in the future section needs to be calculated, the index number in the section is and in the future months is .

The first portion of the equation is multiplied by 100 so as to alter the percentage values obtained for integer values with three decimals. This is mainly to maximize the interference of variations of return on the results of efficiency.

The value of 2 × 19,657 is necessary due to the constraint of the DEA model CCR Oriented Inputs being unable to use negative data. As the period of application of the model, the lowest value of is −19.657; then, as recommended in the literature, this value is summed twice in the adjustment of the data to eliminate the negative.

The equation is squared to maximize the differences of stock return indexes. The motivation is to make the DEA model be able to absorb the minimum difference in stock return indexes and reflect this difference in the efficiency results.

The collection parameters of this index are as follows: value of return on monthly closing calculated based on the previous month, in the original currency of the country (adjusted for dividends), with integer values obtained by multiplying the percentage by 100 and to three decimal places.

This index was collected in monthly percentage values. Before applying the calculation formula of (20) to the data, all stocks were attributed, in December 2008, an index number equal to 1, that is, one month before the period of application of the model. Subsequently, this index number was adjusted for all stocks and in all months. The index number represents the respective value of the monthly return for each stock.

##### 5.7. Negative Inputs or Outputs Adjustments

The DEA CCR model oriented to inputs must treat those variables presenting negative data. As this is the model used in the research, it should verify the possible presence of negative data on inputs and outputs.

The formulas used in (19) and (20) for the adjustment of inputs and outputs were designed to eliminate negative data.

Equation (19), which is utilized for treatment of inputs, also has a structure that eliminates negative data.

An alternative to eliminating negative data in models DEA to each data of particular variable must be added a constant. This constant must have a positive value (+)greater than or equal to lowest data negative (−)that was found in the variable data set [20].

This is the procedure adopted for the elimination of negative data by (20) when adding up 2 × 19.657.

#### 6. Results and Discussion

This chapter presents the results used to validate the proposed mathematical model. First, the extension is estimated in months for the time horizon of utilizing the DEA model in selecting stock portfolios. At the same time, the extension is estimated in months for the time horizon of maintaining the selected portfolio.

We validate the model by simulating the estimated annual indexes of return, both for the portfolio that utilizes the two horizons found and the others that do not.

To validate the proposed mathematical model for hypotheses testing, we compare the estimated annual indexes of returns. Here the study verifies the possible existence of statistically significant differences between the indexes of the portfolio formed using the two horizons estimated and the indexes of those not utilizing them.

It is important to clarify that the estimation period of the monthly time horizons is independent of the simulation period of the indices of return of portfolios. This is to highlight the conceit that the simulation was executed by a fictitious investor situated at a point in time after obtaining the horizons, with no knowledge of the stock market’s future.

The period during which the model was applied to estimate the time horizon was from January of 2009 to April of 2013. Hence, the simulation period of the portfolio with the determined monthly time horizons, as well as all the other portfolios formed for comparison, begins in early May of 2013 and finishes at the end of June 2014.

##### 6.1. Results of Minimization of Squares Error

At this point, to assist calculations Microsoft* Excel* software can be used. In the software the matrices efficiency results and the matrices efficiency results must be mounted. This is done for each of the eight sections in the collection.

This way, it is possible to execute the modeling of this stage to obtain the horizons sought. This is done according to the objective function given by (17), constraints by (13) and (14), and scalars by (15) and (16) of the system proposed in Section 4.

After applying the model using* Excel* Solver so that the model’s equations are applied to the minimization of square error over the matrices and , we obtain the first response of scalar .

The value of the scalar is obtained by multiplying the matrix , of the time horizons of data collection, by results of the binary matrix line of variables of decision.

Table 1 provides the matrix of options for the horizons of data collection for the formation of portfolio , the binary matrix of decision variables , the results of the scalars , and also the result of the objective function of the model in the phase of minimization of errors square.