Stochastic portfolio optimization using efficiency

Management Decision Stochastic portfolio optimization using efficiency evaluation Paulo Rotela Junior Edson de Oliveira Pamplona Luiz Célio Souza Rocha Victor Eduardo de Mello Valerio Anderson Paulo Paiva

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Stochastic portfolio optimization using efficiency evaluation Paulo Rotela Junior, Edson de Oliveira Pamplona, Luiz Célio Souza Rocha, Victor Eduardo de Mello Valerio and Anderson Paulo Paiva Institute of Production Engineering and Management, Federal University of Itajuba, Itajuba, Brazil

Received 17 November 2014 Revised 7 May 2015 Accepted 1 July 2015

Abstract


Purpose – The purpose of this paper is to analyze portfolios chosen using an efficiency evaluation that considers risk and uncertainty and optimizes the allocation of invested capital using the Sharpe approach. Design/methodology/approach – The portfolios comprised shares on the Sao Paulo Stock Exchange. A chance-constrained data envelopment analysis stochastic optimization model was used, and return and variance were employed as input and output variables. Findings – The model was shown to be viable. It reduced the search space and considered data randomness. Originality/value – Three portfolios were proposed. The variation of the model’s risk criterion fulfilled the requirements of investors with different attitudes to risk. The model proposed can be used as a support tool for stock investment decisions. Keywords Financial management, Financial modelling, Stock markets, Assets management Paper type Research paper

1. Introduction Choosing portfolios involves allocating capital among a number of titles so that the investment generates the greatest return and minimizes risks, thus producing a returnto-risk ratio that is satisfactory for investors (Li et al., 2012). Fixed income funds and savings accounts have been losing their appeal. Investment options that are more profitable in the long term should attract investors over the coming years. Stock investments are becoming an alternative for diversification, as they can make a portfolio more profitable in the long term; this makes them a much better alternative than other financial instruments (Rotela et al., 2014). The Markowitz model, developed over 60 years ago (Markowitz, 1952), is a classic mean-variance approach that is still the main model used to allocate assets and manage portfolios. This model has led to new proposals (Tu and Zhou, 2011; Brodie et al., 2009; Ditraglia and Gerlach, 2013; Zopounidis et al., 2014). Charnes et al. (1978) developed data envelopment analysis (DEA), which is used to evaluate and compare organizational units that use many different inputs to produce outputs over a specified period of time (Kuo et al., 2010; Amirteimoori, 2011; Kao and Liu, 2014). This analysis has been amply discussed, and variations on it continue to be developed. Chen et al. (2013) and Silva et al. (2014) claim that fuzzy logic is already considered in some of them. Lertworasirikul et al. (2003) proposed a DEA model with Management Decision Vol. 53 No. 8, 2015 pp. 1698-1713 © Emerald Group Publishing Limited 0025-1747 DOI 10.1108/MD-11-2014-0644

The authors would like to thank FAPEMIG, CNPq, and CAPES for financial and research support.


fuzzy coefficients; Sengupta (1987) combined the DEA model with chance-constrained programming, as was proposed by Charnes and Cooper (1963). The main objective of this paper is to evaluate the efficiency of publicly traded stocks when there is risk and uncertainty. A chance-constrained data envelopment analysis (CCDEA) model, which includes stochastic variables, is used to propose portfolios that are more interesting to investors. The specific objectives of the study are to: •

introduce the CCDEA model;

•

use the CCDEA to reduce the search space while considering variable randomness and uncertainty;

•

analyze the efficiency of actions using different criteria and parameters; and

•

use the Sharpe (1963) approach to determine capital allocation in stocks that are considered efficient by the CCDEA.

2. DEA Measuring efficiency is important for all businesses and organization, as it allows them to compare their performance with that of their most important competitors. It also helps them develop a strategy for improving performance. Many efficiency measurement tools are available, such as conventional statistical analysis, non-parametric methods, and methods that use artificial intelligence. DEA can effectively measure the relative efficiency of decision-making units (DMUs) and has been widely used in many businesses and organizations, such as banks, schools, and hospitals (Bachiller, 2009; Kao et al., 2011; Liu et al., 2013; Kao, 2014). The DEA’s value is its ability to evaluate the relative efficiency or performance of DMUs within a group operating within a specific domain of application (Liu et al., 2013). These units are compared among themselves and are differentiated from one another according to the amount of resources (inputs) they consume and the goods (outputs) they produce (Cooper et al., 2007; Kao et al., 2011; Cook and Zhu, 2014). According to Silva et al. (2014) and Kao (2014), DEA allows benchmark DMUs to be identified from among the other DMUs. Bal et al. (2010) and Miranda et al. (2014) have shown that DEA stands out from the other types of quantitative modeling used for decision making and helps directors in many areas. Charnes et al. (1978) developed the first DEA model to evaluate the efficiency of public education systems. According to Cooper et al. (2007), input and output variables for each DMU should be chosen in order to represent the interests of the directors. Each input and output must have positive numerical values. Malekmohammadi et al. (2009) and Miranda et al. (2014) observe that the model known as DEA CCR, or the model with constant returns to scale, was proposed by Charnes et al. (1978). In linear form, as shown in the following equations, it is called the “multiplier model”: s X ur :yr0 (1) max w0 ¼ r¼1

Subject to: m X i¼1

vi xi0 ¼ 1

(2)

Stochastic portfolio optimization 1699

MD 53,8

s X

ur yr0

m X

r¼1


1700

vi xi0 p0

j ¼ 1; 2; ::::; n

(3)

i¼1

ur X 0;

r ¼ 1; 2; :::; s:

(4)

vi X 0;

i ¼ 1; 2; :::; m:

(5)

where j is the DMU index, j ¼ 1, …, n; r, the output index, r ¼ 1, …, s; i, the input index, i ¼ 1, …, m; yrj the rth output value for the jth DMU; xij the ith input value for the jth DMU; ur, the weight of the rth output; vi the weight of the ith output; w0 the relative efficiency of DMU0 (the DMU being evaluated); and yr0 and xi0, the technological coefficients of the matrix of input and output data, respectively (Miranda et al., 2014). When w0 ¼ 1, DMU0 (the DMU being analyzed) should be considered more efficient than the other units examined in the model. If w0 o 1, this DMU should be considered inefficient ( Jablonsky, 2012). 2.1 Chance-constrained DEA DEA is a non-parametric efficiency technique that is becoming popular for measuring financial performance. The classic models used more frequently in the literature are deterministic and fail to consider the random errors of inputs and outputs. According to Aigner et al. (1977) and Jin et al. (2014), generalized randomness in evaluation processes comes from data collection errors. Jin et al. (2014) also claim that the greatest contribution to stochastic DEA programming can be found in research developed by Sengupta (1987), who used chance-constrained programming, as was proposed by Charnes and Cooper (1963). Sengupta (1987) incorporated stochastic variables into the DEA formula and then transformed it into an equivalent deterministic model. Through this formula, stochastic DEA can be solved by any commercially available optimization software that deals with linear and quadratic programming (Sueyoshi, 2000). The stochastic DEA formula is obviously non-linear programming. Thus, in Equations (6) through (9), for the ith DMU, xî ¼ ðxî1 ; xî2 ; :::; xîa ÞT is the stochastic variable for the input vector, and yî ¼ ðyî1 ; yî2 ; :::; yîb ÞT is the stochastic variable for the output vector, and i ¼ i, …, n: max E ¼

b X

uq y^0q

(6)

q¼1

subject to: E

a X

! vp x^0p

¼1

(7)

p¼1

Pb Pr

uq yîq Pq¼1 p bi a p¼1 vp xîp

! X 1ai

i ¼ 1; 2; :::; n:

(8)


uq ; vp X 0:

(9)

Again, u1, …, ub, v1, …, va are weights that will be estimated by the model. The symbol uq represents the multiplier for the qth output, and vr represents the multiplier for the pth input. Pr is a probability, and superscript “^” indicates that xîp and yîq are random variables. For the restriction, the model formulates the segment that is less than or equal to βi, the level of efficiency expected for the ith DMU. According to Cooper et al. (1996) and Jin et al. (2014), the ith DMU has a variation of [0,1], which is defined as the future level of expected success. αi is a risk criterion that serves the decision maker’s purposes. 1-αi shows the chance of fulfilling the requirement and is considered a level of reliability ( Jin et al., 2014). Like βi, the risk criterion (αi) is a value measured in the interval between 0 and 1. In the model presented here, when αi ¼ 0, the ratio between input and output must become less than or equal to βi. When βi is defined to have a value of 1 and αi is defined as 0, the restrictions are defined as they are in the classic DEA model. To obtain a computationally viable model, the formula must be rewritten as proposed by Charnes and Cooper (1963). This proposal considers randomness. To do so, the stochastic variable xîp of each input can be expressed as xîp ¼ xip þ aip x; p has variance [1, b], and i has variance [1, n]; x ip is the expected value of xîp , and aip is the standard deviation. Stochastic variable yîq of each output can be expressed as yîq ¼ y iq þ biq x; q has variance [1, a], and i has variance [1, n]; y iq is the expected value of yîq , and biq is the standard deviation. Since the stochastic disturbance indicated that the areas come from data collection, it is natural to assume that the random variable ξ has a normal distribution (N(0, σ2)). After the model has been presented, its equivalent deterministic formula is acquired in order to make the model easier to solve. The objective function shown in Equation (6) can be rewritten as: E

b X

uq y^0q ¼

q¼1

b X

uq y 0q

(10)

q¼1

The restrictions shown in Equations (7) and (8), including the stochastic process, can be rewritten as in the following equations: ! ! a a X X _ vp x^0p vp x0p ¼ 1 (11) E p¼1

Pb Pr

uq yîq Pq¼1 pbi a p¼1 vp xîp

! ¼ Pr

p¼1 b X

uq yîq bi

q¼1

a X

! vp xîp p 0 X 1ai

p¼1

i ¼ 1; 2; :::; n: which can be rewritten in an equivalent form, as in the following equation: 0 Pb 1 Pa u y ^ b v x ^ p 0 E i q p ip iq i q¼1 p¼1 E i pffiffiffiffiffi Pr@ X pffiffiffiffiffiA X 1ai i ¼ 1; 2; :::; n: Vi Vi

(12)

(13)


MD 53,8

where Ei is the average of each random variable, and Vi, the variance of each random variable. This can be rewritten as in the following equations: Ei ¼

b X

_

uq yiq bi

q¼1

1702 Vi ¼

b X

a X

_

vp xip

(14)

p¼1

_

uq yiq bi

q¼1

a X

!2 _

vp xip

s2

(15)

p¼1


Pb pffiffiffiffiffi Pa Thus, for random variable q¼1 uq yîq bi p¼1 vp xîp p 0 E i = V i , the average distribution is 0, and the variance is 1. Equation (13) can be presented as in the following equation: E pffiffiffiffiffii X F1 ð1ai Þ i ¼ 1; 2; :::; n: Vi

(16)

An equivalent form is presented in the following equation: a X

b X vp bi x ip þ F1 ð1ai Þaip s uq y iq þ F1 ð1ai Þbiq s X 0i

p¼1

q¼1

i ¼ 1; 2; ::::; n

(17)

Finally, we come to the multiplier model. Here, Φ is a normal standard distribution function, and Φ1 is the inverse of the function. Thus, the original model can be reformulated as a linear programming model. Its equivalent is shown in the following equations: max

b X

uq :y 0q

(18)

vp x0p ¼ 1

(19)

q¼1

subject to: a X

_

p¼1 a X

b _ X _ vp bi xip þ F1 ð1ai Þaip s uq yiq þ F1 ð1ai Þbiq s X 0i

p¼1

q¼1

i ¼ 1; 2; ::::; n uq ; vp ; wr X 0:

(20) (21)

or even rewritten in the form of an envelope model, as in the following equations: miny

(22)

(23)

Stochastic portfolio optimization

(24)

1703

subject to: n X

_ _ li bi xip þ f1 ð1ai Þaip s p x0p

p ¼ 1; 2; ::::; a

i¼1 n X _ _ li yiq þ f1 ð1ai Þbiq s p y0q

q ¼ 1; 2; ::::; b

i¼1


li X 0

i ¼ 1; 2; :::; n:

(25)

The model of the multipliers (18)-(21) and the envelopes (22)-(25) amplify the DEA’s usefulness in financial decision making. In addition to the deterministic situation, efficiency can be measured when there are random variables. The expected levels of model success and reliability can be defined depending on the practical situation; it can deal with specific differences in each case. 3. Portfolio selection The basic theory of portfolio selection began with Markowitz (1952). In it, portfolio selection is based on a single-period investment model. The model proposed by Markowitz (1952), given by Equations (26) through (28), is operated using quadratic programming techniques. The objective is to optimize portfolios by considering the average, variance, and covariance of the expected returns of the stocks that make up the portfolio. These parameters are estimated from historical data based on a vector of averages and a variance-covariance matrix of the returns: min Z ¼

n X n X

xi :xj covij

(26)

i¼1 j¼1

subject to: j X

xi E ðriÞ ¼ E n

(27)

i¼1 j X

xi ¼ 1

(28)

i¼1

where xi and xj are the percentage of asset i and j in the optimized portfolio, E(ri), the expected return for asset i, from i ¼ 1 to j, and E*, the expected return of the portfolio. Markowitz (1952) suggested that portfolio selection should use probability estimates of future asset performance and an analysis of these estimates to determine an efficient set of portfolios and then choose the portfolios that best suit the preferences of the investor. According to Darolles and Gourieroux (2010), Sharpe (1963) extended the work of Markowitz (1952) by introducing a simplified model of the relationships between assets that offered evidence of costs. This model can be used for practical applications (Sharpe, 1963), as in the following equations: max Z ¼ l

N þ1 X i¼1

X i Ai

N þ1 X i¼1

X 2i Qi

(29)

MD 53,8

subject to Xi ⩾ 0 for each i from 1 to N: N X

X i Bi ¼ X n þ i

(30)

i¼1

1704

N X

Xi ¼ 1

(31)

i¼1


This formula shows why parameters An+1 and Qn+1 are used to describe variance and the future expected value of I and why this is called the “diagonal model.” The variance and covariance model, which is complete when N assets are considered, can be expressed as a matrix with non-zero values on the diagonal, including an (n+1)th asset, defined as has been shown (Sharpe, 1963). This drastically reduces the number of calculations needed to solve the portfolio analysis problem and allows the problem to be recognized directly in terms of the parameters of the diagonal model. According to Ben Abdelaziz et al. (2007), the mean-variance method proposed by Markowitz (1952) for portfolio selection has been fundamental for research and has served as a basis for the development of modern finance theory. However, researchers have developed more sophisticated models that use multi-period or dynamic extensions. The model presented below could provide another solution. 4. Materials and method According to Bertrand and Fransoo (2002) and Martins et al. (2014), this study can be classified as applied research with a descriptive empirical objective, as the aim is to create a model that adequately describes the causal relationships that can exist in practice, facilitating an understanding of real processes. As a quantitative approach to the problem, mathematical modeling is adopted. This study used the scientific foundation developed by Markowitz (1952) and Sharpe (1963) to present a method of selecting stock portfolios that considers stochastic variables. Chance-constrained DEA will be used to choose which assets are most efficient, as shown in Figure 1. Common (ON) and preferred (PN) shares of publically traded companies available on the Sao Paulo Stock Exchange (BOVESPA) and included in the Bovespa Index (Ibovespa) were chosen for the sample group. A sample of 59 companies was obtained. After the sample group was chosen, the input and output sets were determined using a CCDEA model efficiency analysis. Studies such as Powers and McMullen (2000) and Lopes et al. (2010) have used one- , two- , and three-year return and earnings per share (EPS) as output indicators and Definition of DMUs, inputs and outputs

Figure 1. Research flowchart

CCDEA

Efficient assets Bovespa

Application of Sharpe (1963) approach

Optimal solution

Analysis


β (60 months), price-earnings ratio (P/E), and volatility (36 months) as input sets for the DEA model. However, Rotela Junior et al. (2014) proposed replacing the indicators that comprise inputs with volatility in 12-month windows (i.e. one, two, and three years) and price-earnings ratio. This change of risk measurement indicator can provide information that will allow the DEA model to choose assets that are comparatively analyzed and found to be efficient. Attributes with advantages are treated as outputs and those with costs are treated as inputs in the model. The variables proposed were chosen following Powers and McMullen (2000), Lopes et al. (2010), and Rotela Junior et al. (2014). Return, or EPS, was chosen as the output of the model. The inputs were volatility, price-earnings ratio, and β. The Economática® computer program was used to collect historical data for the last 36 months for the chosen variables. For each DMU (stock), it was possible to calculate the average and the variance of each of the variables adopted for efficiency analysis. The data appear in Table I. The variables used may vary widely, as long as they represent the interests of the investors. The data were processed and subjected to various statistical tests using Minitab® software. For example, outliers were removed, and negative data were transformed by adding the value that turned the most negative value in the series positive without altering the efficiency analysis, as done in Cook and Zhu (2008), Rotela Junior et al. (2014), and Miranda et al. (2014). Because the independence of the data cannot be determined, the term σ2 presented in Equation (15) comes from the variance-covariance matrix shown in Table II. Only the covariance data that obtained statistical significance of less than 5 percent were presented. Software including Microsoft Excel® Solver, MaxDEA®, General Algebric Modeling® v23.6.5, and CPLEX v12.2.1 solver was used for CCDEA modeling. An efficiency level (βi) of 1 and risk criteria (αi) of 0.1, 0.3, 0.5, 0.7, and 0.9 were used to apply the proposed model. After the assets classified as efficient were selected for the risk criteria, the Sharp (1963) approach was used to determine the ideal allocation of capital among the assets of the portfolio. The difference among the expected returns of the portfolios provided by the capital asset pricing model (CAPM) proposed by Sharpe (1964) and the returns observed during the analysis period was calculated to identify abnormal returns. The average Special System for Settlement and Custody (Selic) rate for the period from August 2011 to July 2014 was applied for the return required for a risk-free (Rf) asset presented in the CAPM equation. This average is considered an indicator of the base interest rate of an economy with the risk of the Brazilian Federal Treasury. The behavior of Ibovespa from the beginning of 1995 to the end of the first half of 2014 was used as the return expected from the investment (Rm); this was calculated based on the average of the years observed. A value of 9.6 percent per year was used for the Selic average, and 15.30 percent per year was used for Rm. Finally, the term β was calculated for diversifiable risk. For each of the portfolios created, it was obtained from the average of the individual β of each asset multiplied by the percentage that was allocated for each paper in the portfolio. To ensure a thorough analysis, the Sharpe Ratio (SI) was also used. According to Zakamouline and Koekebakker (2009), this is regularly used to measure portfolio performance. Paschoarelli (2008) defines the index as a way to measure efficiency in order to take advantage of risk to generate returns. It establishes a relationship between excessive portfolio profitability relative to the Rf interest rate and the risk of the investment. The average Selic rate is used as the Rf interest rate, and the standard deviation of the portfolios’ monthly returns is used as base risk.


MD 53,8


1706

Table I. Input and output data of the model

Return DMU DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10 DMU11 DMU12 DMU13 DMU14 DMU15 DMU16 DMU17 DMU18 DMU19 DMU20 DMU21 DMU22 DMU23 DMU24 DMU25 DMU26 DMU27 DMU28 DMU29 DMU30 DMU31 DMU32 DMU33 DMU34 DMU35 DMU36 DMU37 DMU38 DMU39 DMU40 DMU41 DMU42 DMU43 DMU44 DMU45 DMU46 DMU47 DMU48 DMU49 DMU50

EPS

µ1

σ21

µ2

σ22

−1.579 2.438 0.273 0.516 0.335 1.194 0.406 −1.264 0.103 1.829 −4.406 1.497 1.033 0.665 0.085 3.319 1.416 0.150 −0.479 −0.026 0.620 −2.890 −1.436 1.414 0.040 3.078 −0.035 0.088 −2.820 −0.521 −0.427 −1.781 −0.293 0.093 0.232 0.771 −0.003 0.999 1.059 0.602 −3.024 −5.148 −1.258 0.204 −2.429 1.344 −4.515 −1.392 −0.820 1.380

4.126 2.478 3.684 5.257 4.436 2.174 1.084 0.903 2.128 1.801 5.128 3.100 4.329 2.569 7.536 1.946 3.900 1.828 3.186 8.637 2.155 10.645 10.256 1.403 0.821 10.232 4.563 12.807 12.983 3.337 3.015 10.482 17.131 1.254 1.658 4.438 2.775 1.439 2.669 3.190 11.110 10.287 2.904 5.629 7.283 1.297 2.843 1.584 1.255 1.877

0.232 0.615 0.557 2.204 3.020 2.746 2.746 2.669 4.346 1.220 −0.380 0.646 2.723 1.151 1.904 1.451 2.736 1.242 1.531 0.706 0.751 −1.872 1.340 0.624 0.831 0.548 1.080 −1.730 −0.547 0.978 1.402 −3.906 0.216 0.841 2.553 0.162 2.034 1.469 0.323 2.843 −1.523 −6.497 1.196 1.951 1.337 3.566 −0.520 1.947 1.947 0.015

0.028 0.003 0.000 1.121 4.493 0.021 0.021 5.551 0.408 0.166 0.543 0.017 0.506 0.037 0.003 0.050 7.352 0.091 0.091 0.015 0.011 22.030 3.125 0.163 0.047 0.060 0.021 0.984 2.745 0.049 0.158 1.248 0.072 0.006 0.083 0.037 0.536 0.091 0.002 0.080 0.463 4.285 0.067 0.007 0.460 0.290 1.006 0.260 0.260 0.013

Volatility µ3 σ23 7.699 5.693 6.947 6.448 5.026 5.906 5.638 7.172 8.004 5.863 12.889 4.345 8.197 6.547 9.581 5.860 5.211 4.786 7.290 8.283 5.151 10.898 11.767 7.539 5.001 8.497 7.879 8.599 15.454 8.461 8.503 15.473 11.421 6.219 6.651 11.915 6.558 5.732 6.944 7.542 16.508 13.944 14.171 6.536 11.758 6.527 13.079 9.842 8.467 6.052

0.770 2.862 0.771 2.529 0.797 1.840 1.967 1.728 4.939 0.469 6.394 0.877 8.185 2.804 3.165 1.560 0.439 0.660 10.184 1.693 3.328 16.350 7.196 0.542 4.876 2.275 5.407 0.707 11.636 1.394 1.143 2.427 17.679 0.583 0.571 11.660 3.390 2.094 2.274 2.366 22.877 21.748 6.398 1.576 6.442 3.013 5.109 4.673 2.249 0.751

P/E

β

µ4

σ24

µ5

26.122 23.564 20.846 11.904 7.286 10.388 10.813 12.859 5.723 34.524 −0.124 26.549 7.083 22.878 18.898 17.941 34.532 18.618 10.632 14.777 20.449 1.855 6.678 19.844 15.169 25.296 7.123 −11.513 −2.663 16.993 15.293 −3.157 39.230 9.578 10.916 25.865 12.088 22.191 38.163 22.631 −6.769 −1.773 8.375 22.779 5.175 25.388 −1.981 10.157 10.233 113.042

4.862 9.900 7.237 21.058 12.824 2.459 0.858 73.360 0.716 89.771 32.578 14.385 8.734 23.545 16.767 8.214 58.417 15.561 5.846 7.446 11.309 13.426 33.013 58.460 14.004 25.700 1.630 52.428 54.444 19.762 22.925 1.316 48.730 0.810 1.007 13.539 27.462 25.688 23.757 12.746 77.087 16.156 1.690 9.328 3.152 6.079 48.388 6.183 5.999 55.283

0.710 0.197 1.083 0.496 0.415 0.811 0.857 0.878 1.221 0.451 1.658 0.283 0.207 0.427 1.122 0.334 0.789 0.182 1.254 0.948 −0.029 0.573 0.643 0.351 0.184 0.801 1.221 1.039 1.978 1.174 1.196 1.888 1.212 0.915 0.968 1.333 0.450 0.703 1.018 0.994 1.016 1.621 1.730 0.704 0.687 0.588 1.653 1.133 1.094 0.454

σ25 0.078 0.005 0.014 0.025 0.004 0.004 0.013 0.009 0.006 0.005 0.023 0.018 0.007 0.007 0.008 0.007 0.023 0.008 0.099 0.032 0.012 0.042 0.193 0.109 0.027 0.074 0.037 0.065 0.022 0.016 0.007 0.130 0.050 0.018 0.019 0.039 0.021 0.019 0.021 0.017 0.102 0.107 0.012 0.011 0.023 0.033 0.019 0.036 0.020 0.000

(continued )

Return DMU


DMU51 DMU52 DMU53 DMU54 DMU55 DMU56 DMU57 DMU58 DMU59

EPS

µ1

σ21

µ2

σ22

−5.055 1.617 −1.769 1.115 0.976 1.379 1.358 1.923 −0.829

3.212 8.419 8.778 5.881 1.665 4.492 1.004 1.380 0.716

−0.103 2.499 0.863 1.058 3.952 0.644 2.263 1.891 3.437

0.317 0.206 1.060 0.010 0.261 0.028 0.019 0.078 8.406

Volatility µ3 σ23 13.851 7.134 11.295 6.534 4.852 7.354 4.270 4.885 6.279

10.405 2.353 10.090 0.369 1.124 7.052 0.397 0.968 1.397

P/E µ4 5.646 9.621 9.484 24.472 12.540 15.774 15.213 24.579 11.235

σ24 83.381 4.689 11.100 11.882 1.995 13.942 3.626 8.245 43.328

µ5

β

1.923 0.558 1.427 0.459 0.115 0.324 0.148 0.370 0.820

σ25 0.027 0.017 0.013 0.006 0.010 0.013 0.007 0.000 0.008

5. Analysis of results The efficiency results for the CCDEA model with different probability levels (1−αi) to meet the restrictions were obtained considering the determined risk criteria (αi). The results are shown in Table III. The alteration of the risk criterion leads to a different portfolio because the composition of the portfolio is given by the possible assets that have been classified as efficient. Table III shows that, as the number of risk criteria (αi) decreases, fewer assets are identified as efficient, indicating that risk aversion increases as the value of αi decreases, leading to the following classifications, following Sueyoshi (2000): αi o0.5 can be classified as conservative (risk averse); αi ¼ 0.5 can be classified as natural risk; and αi W0.5 can be classified as risky. It should be noted that, when the risk criterion was equal to 0.3 and 0.1, the probability of meeting the restrictions increased; no asset was classified as efficient for the sample being tested. Thus, none of the assets had efficiency equal to 1. For the assets considered efficient for the risk criteria adopted, the Sharpe (1963) approach was used to determine the ideal allocation of capital in the portfolio. The results shown above created three portfolios, with 10, 30, and 50 percent chances of meeting the restrictions. The participation levels of each asset in the portfolios are shown in columns 1-3 in Table III. Note that not all of the efficient assets are allocated when using the Sharpe (1963) approach, explaining the presence of zero values for participation in the columns mentioned above. Table IV shows the risk criteria (αi) adopted for each portfolio and the results obtained for return, standard deviation, and SI for each of the proposed portfolios. Portfolio 3 had a positive SI of 1.31. It was composed of only four assets. The analysis began with 59 assets taking part in Ibovespa. An efficiency evaluation was carried out considering a situation with risk and uncertainty using a risk criterion of 50 percent. This evaluation considered ten assets to be efficient. Then, the portfolio was optimized using the Sharpe (1963) method; as stated, only four assets were included in Portfolio 3. Portfolio 2 comprised six assets and had an SI of 1.24. As in Portfolio 3, there were 59 stocks at the beginning. A risk-analysis criterion of 70 percent was adopted, leading to 14 stocks being considered efficient. After optimization by the Sharpe (1963) model, six were chosen for the portfolio. Finally, Portfolio 1 evaluated the same 59 stocks but with a risk criterion of 10 percent. This produced 36 efficient assets, of which eight were selected through optimization by the Sharpe (1963) method. This produced a portfolio with an SI of 1.21.


Table I.

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1708

Table II. Variance-covariance matrix

DMU DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10 DMU11 DMU12 DMU13 DMU14 DMU15 DMU16 DMU17 DMU18 DMU19 DMU20 DMU21 DMU22 DMU23 DMU24 DMU25 DMU26 DMU27 DMU28 DMU29 DMU30 DMU31 DMU32 DMU33 DMU34 DMU35 DMU36 DMU37 DMU38 DMU39 DMU40 DMU41 DMU42 DMU43 DMU44 DMU45 DMU46 DMU47 DMU48 DMU49 DMU50

Return vs EPS

Volatility vs P/E

Volatility vs β

P/E×β

0.000 0.000 0.000 0.000 3.745 0.000 0.000 0.000 0.000 0.390 0.857 0.000 −1.298 0.000 0.000 0.000 0.000 0.325 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 −2.537 0.643 0.000 0.000 0.000 0.000 −0.214 0.000 0.000 1.661 0.000 0.000 0.162 1.410 0.000 0.000 0.000 0.000 0.083

0.000 −4.519 0.000 0.000 0.000 1.333 −0.664 9.957 0.000 0.000 −7.018 −2.553 −3.298 4.409 4.976 −1.481 0.000 1.258 5.441 1.679 0.000 −13.493 8.218 0.000 −3.310 0.000 0.000 0.000 −14.926 −1.825 0.000 0.000 0.000 0.000 0.000 0.000 −6.306 0.000 3.952 0.000 0.000 0.000 0.000 0.000 0.000 0.000 −5.970 0.000 0.000 0.000

−0.114 0.067 0.000 0.084 0.027 −0.039 −0.083 −0.099 0.087 0.019 0.000 0.000 0.000 −0.106 0.000 0.047 −0.060 0.025 0.706 0.000 0.167 0.312 0.678 0.089 0.342 0.188 0.349 0.125 0.000 0.000 0.000 0.000 −0.507 0.084 0.077 0.000 0.205 0.117 0.175 0.148 1.179 −1.094 0.000 0.000 0.000 −0.167 0.000 0.304 0.169 0.000

0.000 −0.126 −0.149 0.268 0.000 −0.058 0.000 −0.449 0.033 −0.339 0.000 0.000 0.000 −0.169 0.000 0.000 0.000 0.260 0.671 0.000 −0.355 −0.268 0.000 0.000 −0.363 −0.618 0.000 0.964 −0.485 0.000 0.000 0.172 0.000 0.000 0.000 0.000 −0.654 −0.249 0.000 −0.156 0.000 0.528 0.000 0.163 0.167 0.000 0.000 0.000 0.000 0.000

(continued )

DMU


DMU51 DMU52 DMU53 DMU54 DMU55 DMU56 DMU57 DMU58 DMU59

Return vs EPS

Volatility vs P/E

Volatility vs β

P/E×β

0.000 −0.937 0.000 0.000 0.496 0.000 0.000 0.000 0.000

0.000 −2.084 5.515 1.144 0.000 0.000 0.000 0.000 5.994

0.000 0.149 0.270 0.000 0.042 −0.235 −0.032 0.000 −0.036

0.000 −0.189 0.000 0.000 0.102 0.000 −0.082 0.000 −0.316

Table IV shows the expected return values (Re) of the portfolios. As mentioned, this can be calculated using the expected return of the market (Rm), return of a Rf investment, and the β-value of each portfolio, which are also listed in the table. The expected returns of the portfolios were very close, with values between 0.89 and 0.91 percent per month. The profitability values effectively obtained for the portfolios were 2.75 percent for Portfolio 3, 2.72 percent for Portfolio 2, and 2.68 percent per month for Portfolio 1, indicating that the returns for the three cases are abnormal when compared to the expected profitability obtained by the CAPM model. Finally, the accumulated return over the 36 months of the study was calculated for each portfolio. The accumulated return of Portfolio 3 was 165.49 percent, very different from the others. Portfolio 2 had a value of 153.01 percent, while Portfolio 1 had a value of 158.49 percent. Thus, Portfolio 2 had a better SI performance than Portfolio 1. 6. Conclusions This study has evaluated the efficiency of stock available on the Sao Paulo Stock Exchange by considering uncertainty and risk and using CCDEA, a stochastic DEA model. The purpose was not to avoid the approaches developed by Markowitz (1952) or Sharpe (1963) but to stochastically use different variables that would allow a smaller sample of efficient stocks to be found by considering the randomness of the variables comprising the model. The proposed model can thus be seen as a tool for stock investment decision making. The stocks chosen were evaluated according to a set of input and output variables, rather than just return and risk, revealing their relative efficiency. This model can aid established portfolio optimization theories. Varying the CCDEA model’s risk criterion allows the needs of investors with different attitudes to risk (such as conservative investors and those more open to risk) to be met. The efficient stocks then underwent the Sharpe (1963) procedure to determine the ideal allocation of capital in each of the assets in the portfolios. Three groups of efficient assets were created by altering the chances of the CCDEA model restrictions. The groups were then run through the optimization model to propose investment capital allocation, resulting in three portfolios. The CCDEA model was shown to be viable and to describe the analysis units (DMUs) well. Portfolio 3 stood out from the others when the performance measured by SI was taken into account. Another benefit is the reduced number of stocks comprising the three portfolios, which, along with the maintenance of risk control, can reduce the costs of rebalancing portfolios, providing investors with indirect earnings.

Stochastic portfolio optimization 1709 Table II.

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1710

Table III. Efficiency and allocation of assets in the portfolio

(1−αi) Probability 10% Portfolio 1 (%) 30% Portfolio 2 (%) 50% Portfolio 3 (%) 70% 90% DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10 DMU11 DMU12 DMU13 DMU14 DMU15 DMU16 DMU17 DMU18 DMU19 DMU20 DMU21 DMU22 DMU23 DMU24 DMU25 DMU26 DMU27 DMU28 DMU29 DMU30 DMU31 DMU32 DMU33 DMU34 DMU35 DMU36 DMU37 DMU38 DMU39 DMU40 DMU41 DMU42 DMU43 DMU44 DMU45 DMU46 DMU47 DMU48 DMU49 DMU50 DMU51

0.553 1.350 0.852 1.215 1.468 1.432 1.255 0.895 1.530 1.069 0.996 1.443 1.643 0.913 0.878 1.363 1.190 1.040 1.086 0.994 1.318 0.998 1.088 1.039 1.011 0.969 1.386 2.432 1.514 0.842 0.909 2.120 0.503 1.258 1.184 0.797 1.110 1.021 0.850 0.866 2.529 0.370 1.019 0.818 0.954 0.935 1.151 0.800 1.023 1.008 0.696

– 0.209 – 0 0 0 0 – 0 0.117 – 0.014 0 – – 0.493 0.002 0 0 – 0 – 0 0.005 0 – 0 0 0 – – 0 – 0 0 – 0 0 – – 0 – 0 – – – 0 – 0 0 –

0.473 1.132 0.700 0.953 1.194 1.122 1.030 0.817 1.117 0.904 0.751 1.162 1.229 0.788 0.665 1.148 0.958 0.865 0.814 0.723 1.137 0.642 0.856 0.889 0.875 0.863 0.907 1.612 0.793 0.646 0.686 0.664 0.439 0.944 0.943 0.591 0.886 0.822 0.685 0.725 1.084 0.137 0.652 0.706 0.889 0.817 0.811 0.746 0.750 0.843 0.589

– 0.228 – – 0 0 0 – 0 – – 0.031 0 – – 0.517 – – – – 0 – – – – – – 0 – – – – – – – – – – – – 0 – – – – – – – – – –

0.431 1.000 0.616 0.809 1.000 0.947 0.872 0.739 1.000 0.807 0.613 1.000 1.000 0.703 0.566 1.000 0.854 0.812 0.677 0.604 1.000 0.583 0.782 0.800 0.785 0.755 0.714 1.000 0 0.602 0.537 0.572 0.417 0.392 0.788 0.778 0.494 0.770 0.732 0.590 0.630 0.796 0.089 0.588 0.629 0.813 0.731 0.638 0.692 0.701 0.748 0.514

– 0.264 – – 0 – – – 0 – – 0.059 0 – – 0.563 – – – – 0 – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

0.403 0.888 0.519 0.709 0.861 0.768 0.713 0.633 0.820 0.713 0.486 0.845 0.936 0.625 0.499 0.909 0.754 0.718 0.560 0.511 0.873 0.490 0.673 0.719 0.695 0.670 0.568 0.735 0.465 0.455 0.484 0.299 0.349 0.637 0.645 0.451 0.691 0.647 0.505 0.539 0.608 0.065 0.486 0.544 0.693 0.644 0.500 0.582 0.587 0.651 0.416

0.371 0.745 0.407 0.583 0.725 0.587 0.542 0.531 0.626 0.570 0.360 0.638 0.794 0.525 0.421 0.785 0.546 0.600 0.431 0.351 0.735 0.384 0.545 0.606 0.586 0.569 0.436 0.506 0.336 0.378 0.401 0.210 0.293 0.475 0.518 0.323 0.599 0.544 0.403 0.469 0.434 0.045 0.380 0.477 0.576 0.571 0.371 0.469 0.473 0.507 0.319

(continued )

(1−αi) Probability 10% Portfolio 1 (%) 30% Portfolio 2 (%) 50% Portfolio 3 (%) 70% 90%


DMU52 DMU53 DMU54 DMU55 DMU56 DMU57 DMU58 DMU59

1.551 0.863 0.939 1.303 1.173 1.425 1.341 1.046

Risk criterion (αi) (%) Portfolio Beta (β) Expected return (Re) (%) SD (%) Return (%) Sharpe index Number of assets Accumulated return (%)

0.013 – – 0 0 0 0.147 0

1.127 0.630 0.808 0.011 0.947 1.160 1.092 0.967

0.018 – – – – 0.027 0.176 –

0.939 0.585 0.724 1.000 0.833 1.000 0.941 0.855

– – – 0

– 0.114 – –

0.794 0.486 0.644 0.942 0.748 0.914 0.826 0.718

0.666 0.386 0.548 0.860 0.634 0.749 0.657 0.603

Portfolio 1

Portfolio 2

Portfolio 3

90 0.316 0.90 1.46 2.68 1.21 8 158.49

70 0.333 0.91 1.45 2.72 1.24 6 153.01

50 0.273 0.89 1.41 2.75 1.31 4 165.49

On its own, the Sharpe (1963) approach can recommend a great many assets for a portfolio, as well as participation that is impossible in practice, making it difficult to use. Finally, the stochastic approach to the DEA model can assist in portfolio development. When combined with the Sharpe (1963) approach, this has several advantages: efficiency evaluation when there is uncertainty or risk; the consideration of data randomness; more stochastic variables in the analysis, improving traditional deterministic models; the use of different variables chosen by the analyst for comparing different DMUs; and the selection of fewer assets for portfolios, reflecting its viability. Future studies could compare different variables for use in the CCDEA model and employ more indicators. Moreover, a long-term analysis could be carried out, as well as a comparison with the fuzzy DEA model and other models designed to insert uncertainty into DEA models. Finally, stochastic optimization models could be used to determine the ideal allocation of capital in portfolios. References Aigner, D., Lovell, A. and Schmidt, P. (1977), “Formulation and estimation of Stochastic Frontier production function models”, Journal Economics, Vol. 6 No. 1, pp. 21-37. Amirteirmoori, A. (2011), “A DEA two-stage decision processes with shared resources”, Central European Journal of Operations Research, Vol. 21 No. 1, pp. 141-151. Bachiller, P. (2009), “Effect of ownership on efficiency in Spanish companies”, Management Decision , Vol. 47 No. 2, pp. 289-307. Bal, H., Orkcu, H. and Çelebioglu, S. (2010), “Improving the discrimination power and weights dispersion in the data envelopment analysis”, Computer & Industrial Engineering, Vol. 37 No. 1, pp. 99-107. Ben Abdelaziz, F., Aouni, B. and El Fayedh, R. (2007), “Multi-objective stochastic programming for portfolio selection”, European Journal of Operational Research, Vol. 177 No. 3, pp. 1811-1823.

Stochastic portfolio optimization 1711 Table III.

Table IV. Results for the portfolios by risk criterion

MD 53,8

Bertrand, J. and Fransoo, J. (2002), “Operations management research methodologies using quantitative modeling”, International Journal of Operations & Production Management, Vol. 22 No. 2, pp. 241-264. Brodie, J., Daubechies, I., Mol, C., Giannone, D. and Loris, I. (2009), “Sparce and stable Markowitz portfolios”, Proceedings of the National Academy of Sciences of the United States of America, Vol. 106 No. 30, pp. 12267-12272.

1712

Charnes, A. and Cooper, W. (1963), “Deterministic equivalents for optimizing and satisfying under chance constraints”, Management Science, Vol. 11 No. 1, pp. 18-39. Charnes, A., Cooper, W. and Rhodes, E. (1978), “Measuring the efficiency of decision-making units”, European Journal of Operational Research, Vol. 2 No. 6, pp. 429-444.


Chen, Y.-C., Chiu, Y.-H., Huang, C.-W. and Tu, C.-H. (2013), “The analysis of bank business performance and market risk – applying fuzzy DEA”, Economic Modelling, Vol. 32 No. 3, pp. 225-232. Cook, W. and Zhu, J. (2008), Data Envelopment Analysis: Modelling Operational Processes and Measuring Productivity, Create Space Independent Publishing Platform, Worcester. Cook, W. and Zhu, J. (2014), Data Envelopment Analysis – A Handbook on the Modeling of Internal Structures and Networks, Springer Science+Business, New York, NY. Cooper, W., Huang, Z. and Li, S. (1996), “Satisficing DEA models under chance constraints”, Annals Operations Research, Vol. 66 No. 4, pp. 279-295. Cooper, W., Seiford, L. and Tone, K. (2007), Data Envelopment Analysis: A Comprehensive Text with Models, Application, References and DEA-Solver Software, 2nd ed., Springer Science +Business, New York, NY. Darolles, S. and Gourieroux, C. (2010), “Conditionally fitted Sharpe performance with an application to hedge fund rating”, Journal of Banking & Finance, Vol. 34, pp. 578-593. Ditraglia, F. and Gerlach, J. (2013), “Portfolio selection: an extreme value approach”, Journal of Banking & Finance, Vol. 37 No. 2, pp. 305-323. Jablonsky, J. (2012), “Multicriteria approaches for ranking of efficient units in DEA models”, Central European Journal of Operations Research, Vol. 20 No. 3, pp. 435-449. Jin, J., Zhou, D. and Zhou, P. (2014), “Measuring environmental performance with stochastic environmental DEA: the case of APEC economies”, Economic Modelling, Vol. 38 No. 3, pp. 80-86. Kao, C. (2014), “Efficiency decomposition for general multi-stage systems in data envelopment analysis”, European Journal of Operational Research, Vol. 232 No. 1, pp. 117-124. Kao, C. and Liu, S. (2014), “Multi-period efficiency measurement in data envelopment analysis: the case of Taiwanese commercial banks”, Omega, Vol. 47 No. 6, pp. 90-98. Kao, L.-J., Lu, C.-J. and Chiu, C.-C. (2011), “Efficiency measuremente using independent component analysis and data envelopment analysis”, European Journal of Operational Research, Vol. 210, pp. 310-317. Kuo, L., Huang, S. and Wu, Y. (2010), “Operational efficiency integrating the evaluation of environmental investment: the case of Japan”, Management Decision, Vol. 48 No. 10, pp. 1596-1616. Lertworasirikul, S., Fang, S., Joines, J. and Nuttle, H. (2003), “Fuzzy data envelopment analysis (DEA): a possibility approach”, Fuzzy Sets and Systems, Vol. 139 No. 2, pp. 379-394. Li, X., Shou, B. and Qin, Z. (2012), “An expected regret minimization portfolio selection model”, European Journal of Operational Research, Vol. 218 No. 2, pp. 484-492. Liu, J.S., Lu, L.Y., Lu, W.-M. and Lin, B.J. (2013), “A survey of DEA applications”, Omega, Vol. 41 No. 5, pp. 893-902.


Lopes, A.L., Carneiro, M. and Schneider, A. (2010), “Markowitz na otimização de carteiras selecionadas por data envelopment analysis – DEA”, Revista Gestão e Sociedade, Vol. 4 No. 9, pp. 640-656. Malekmohammadi, N., Lotfi, F. and Jaafar, A. (2009), “Data envelopment scenario analysis with imprecise data”, Central European Journal of Operations Research, Vol. 19 No. 1, pp. 65-79. Markowitz, H. (1952), “Portfolio selection”, Journal of Finance, Vol. 7 No. 1, pp. 77-91. Martins, R., Mello, C.H. and Turrioni, J.B. (2014), Guia para elaboração de Monografia e TCC em Engenharia de Produção, Atlas, São Paulo. Miranda, R., Montevechi, J.A., Silva, A.F. and Marins, F.A. (2014), “A new approach to reducing search space and increasing efficiency in simulation optimization problems via the FuzzyDEA-BCC”, Mathematical Problems in Engineering, Vol. 2014, pp. 1-15, Article ID 450367, doi: 10.1155/2014/450367. Paschoarelli, R. (2008), Como Ganhar Dinheiro No Mercado Financeiro, 2nd ed., Editora Elsevier, São Paulo. Powers, J. and Mcmullen, P. (2000), “Using data envelopment analysis to select efficient large market cap securities”, Journal of Business and Management, Vol. 7 No. 2, pp. 31-42. Rotela, P. Junior, Pamplona, E.O. and Salomon, F. (2014), “Otimização de portfólios: Análise de eficiência”, RAE – Revista de Administração de Empresas, Vol. 54 No. 4, pp. 405-413. Sengupta, J. (1987), “Data envelopment analysis for efficiency measurement in the stochastic case”, Computer and Operational Research, Vol. 14 No. 2, pp. 117-129. Sharpe, W.F. (1963), “A simplified model for portfolio analysis”, Management Science, Vol. 9 No. 2, pp. 277-293. Sharpe, W.F. (1964), “Capital assets prices: a theory of market equilibrium under conditions of risk”, Journal of Finance, Vol. 19 No. 3, pp. 425-442. Silva, A.F., Marins, F.A. and Santos, M.V. (2014), “Programação por metas, análise por envoltória de dados e teoria fuzzy na avaliação da eficiência sob incerteza: Aplicação em minifábricas do segmento de autopeças”, Gestão e Produção, Vol. 21 No. 3, pp. 543-554. Sueyoshi, T. (2000), “Stochastic DEA for restructure strategy: an application to a Japanese petroleum company”, Journal of Financial Economics, Vol. 99 No. 1, pp. 385-398. Tu, J. and Zhou, G. (2011), “Markowitz meets talmud: a combination of sophisticated and naive diversification strategies”, Journal of Financial Economics, Vol. 99 No. 1, pp. 204-215. Zakamouline, V. and Koekebakker, S. (2009), “Portfolio performance evaluation with generalized Sharpe ratios: beyond the mean and variance”, Journal of Banking & Finance, Vol. 33 No. 7, pp. 1242-1254. Zopounidis, C., Doumpos, M. and Fabozzi, F. (2014), “Preface to the special issue: 60 years following Harry Markowitz’s contributions in portfolio theory and operations research”, European Journal of Operational Research, Vol. 234 No. 2, pp. 343-345.

Corresponding author Paulo Rotela Junior can be contacted at: [email protected]

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