International Journal of Accounting, Finance and Risk Management
Volume 2, Issue 1, January 2017, Pages: 21-30

Portfolio Optimization Using Matrix Approach: A Case of Some Stocks on the Ghana Stock Exchange

Abonongo John1, Anuwoje Ida Logubayom2, *, Ackora-Prah J.1

1College of Science, Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

2Faculty of Mathematical Sciences, Department of Statistics, University for Development Studies, Navrongo, Ghana

Email address:

(A. I. Logubayom)

*Corresponding author

To cite this article:

Abonongo John, Anuwoje Ida Logubayom, Ackora-Prah J. Portfolio Optimization Using Matrix Approach: A Case of Some Stocks on the Ghana Stock Exchange. International Journal of Accounting, Finance and Risk Management. Vol. 2, No. 1, 2017, pp. 21-30. doi: 10.11648/j.ijafrm.20170201.14

Received: September 11, 2016; Accepted: November 25, 2016; Published: February 13, 2017


Abstract: Analyzing risk has been a principal concern of actuarial and insurance professionals which plays a fundamental role in the theory of portfolio selection where the prime objective is to find a portfolio that maximizes expected return while reducing risk. Portfolio optimization has been applied to asset management and in building strategic asset allocation. The purpose of this paper is to construct optimal and efficient portfolios using the matrix approach. This paper used secondary data on 13 stocks (ETI, GCB, GOIL, TOTAL, FML, GGBL, CLYD, EGL, PZC, UNIL, TLW, AGA and BOPP) from the Ghana Stock Exchange (GSE) database comprising the monthly closing prices from the period 02/01/2004 to 16/01/2015. The results revealed that, all the portfolios were optimal and that portfolios 1, 2, 4, 5, 6, 9, 10, 11 and 12 with expected return 2.523, 2.593, 2.827, 3.642, 2.405, 2.812, 5.229, 3.559 and 5.928 respectively were efficient portfolios whereas portfolios 3, 7 and 8 with expected return 0.377, 0.699 and 0.152 respectively were inefficient portfolios with reference to the expected return of the global minimum variance portfolio (2.360). GGBL was seen as the stock with the highest allocation of wealth in most of the portfolios. Six out of the 12 portfolios had CLYD exhibiting the least asset allocation.

Keywords: Portfolio Optimization, Efficient Frontier, Mean-Variance, Matrix Approach


1. Introduction

After Markowitz ground-breaking work in portfolio selection Markowitz [10] portfolio optimization has been receiving greater attention from asset and liability managers, academics and risk managers. Most of the studies explain a portfolio optimization criterion such as mean-variance, conditional value-at-risk, value-at-risk, mean absolute deviation, stochastic dominance of first and second order among others.

The mean-variance is the traditional optimization approach introduced by Markowitz. But before Markowitz presented his approach, portfolio theory was a relevant area of research. However, the main focus of Bachelier and his successor was to improve performance. Markowitz focused on risk. He established volatility as a major risk measure in portfolio theory and showed how the risk can be reduced by diversification. He demonstrated how financial portfolios which have max expected return for a given risk level can be estimated.

In portfolio analysis, variance measures the volatility (risk) of an asset or group of assets, hence larger variance indicates greater risk and vice versa. When many assets are held together in a portfolio, assets decreasing in value are usually offset by portfolios asset increasing in value, hence minimizing risk. Also, the total variance of a portfolio is usually lower than a simple weighted average of the individual asset variances [5]. The return of any financial asset is described by a random variable, whose expected mean and variance are assumed to be reliably estimated from historical data. The expected mean and variance are interpreted as the compensation and the risk respectively. The portfolio optimization problem can be formulated as follows: given a set of assets, characterized by their returns and covariance, find the optimal weight of the asset such that the overall portfolio provides the lowest risk for a given overall return. This problem reduces to find the efficient frontier, which is the set of all achievable (attainable) portfolios that offers a higher return for a given risk level. When the number of assets in a portfolio becomes large, the total variance is actually derived from the covariance than from the variances of the assets [15].

The theory of portfolio optimization is generally associated with the classical mean-variance optimization framework of Markowitz [10]. The drawback of the mean-variance analysis is mainly related to its sensitivity to the estimation error of the means and covariance matrix estimation of the returns of the asset. Also, it is argued that estimates of the covariance matrix are more accurate than those of the expected returns ([12], [8]). Several studies concentrates on improving the performance of the global minimum-variance portfolio (GMVP), which provides the least possible portfolio risk and involves only the covariance matrix estimates.

The classical mean-variance framework depends on the perfect knowledge of the expected returns of the assets and their variance-covariance matrix. However, these returns are unobservable and unknown. The impossibility to obtain a sufficient number of data samples, instability of data, differing personal views of decision makers on the future returns [13] affect their estimation and has led to what [1] call estimation risk in portfolio selection. This estimation risk has shown to be the source of very erroneous decisions, for, as pointed in ([2], [6]), the composition of the optimal portfolio is very sensitive to the mean and the covariance matrix of the asset returns and agitation in the moments of the random returns can result in the difficulties in constructing different optimization.

[9], examined portfolio optimization with correlation matrix. The results showed how to perform portfolio optimizations using mean-correlation instead of mean-variance analysis and that the two alternatives set-up produced equivalent optimization weights if correlation-based number transformed back to mean-variance ones. Also, the analysis, presented strengthens the role of regression methods in portfolio analysis. [14], presented a simplified perspective of Markowitz contributions to Modern Portfolio Theory (MPT), foregoing in-depth presentation of the complex mathematical/statistical models typically associated with discussions of this theory and suggested efficient computer-based ‘short cuts’.

Also, ([3], [16], [7]) have studied the mean variance framework in a robust context, assuming that the expected return is stochastic. They characterize the parameters involved in the mean and the variance-covariance matrix with specific types of uncertainty, and built semi-definite or second-order cone programs.

On efficient and optimal portfolios, [4] stated that portfolios are efficient when they provide the maximum possible expected return for a certain risk level. When building efficient portfolio one need to assume that investors are risk-averse, meaning that they will choose the portfolio with the least risk. When faced with several portfolios with the same expected return, but with different risk levels. Also, a risk-averse investor will choose the portfolio with the highest return, when they have to choose from a set of portfolios with the same risk, but different expected returns. This indicates that efficient portfolios are located in the efficient frontier (minimum-variance frontier). An optimal portfolio is one that has the minimum risk for a given level of return and an efficient portfolio is one that has the maximum expected rerun for a given level of risk. Thus, all portfolios on the minimum-variance frontier are optimal, but only those in the upper portion-at above the global minimum-variance portfolio are efficient.

The purpose of this paper is to construct optimal and efficient portfolios using matrix approach. This will give investors an insight in diversification, asset management and risk management. It will also aid investors and academics on how to construct optimal and efficient portfolios using matrix approach.

2. Materials and Methods

2.1. Source of Data

This paper used secondary data of 13 stocks (ETI, GCB, GOIL, TOTAL, FML, GGBL, CLYD, EGL, PZC, UNIL, TLW, AGA and BOPP) from the Ghana Stock Exchange (GSE) database comprising the daily closing prices from the period 02/01/2004 to 16/01/2015.

2.2. Methods of Data Analysis

The daily index series were converted into compound returns given by;

(1)

where  is the continuous compound returns at time ,  is the current closing stock price index at time  and  is the previous closing stock price index. These returns were converted into monthly returns by assuming 365 days a year and averaging to get 30 days a month. This was then multiplied by the daily returns to obtain the monthly returns. The same method was employed in obtaining the monthly standard deviations by multiplying the square root of 30 by the daily standard deviations.

For an -asset portfolio problem with assets given by . Let  denote the return on asset  with a constant expected return model given by

(2)

(3)

Assuming that all wealth in the -asset is given by

(4)

Then, the portfolio return,  is given by

(5)

where  are the weights of the portfolio and  are the returns of the individual stocks.

From Equation 5, the expected return on the portfolio is given by

(6)

and the variance of the portfolio return given by

(7)

2.3. Portfolio Characteristics Using Matrix Approach

The asset returns and portfolio weights are given by an  column vector;

(8)

(9)

The probability distribution of  is the joint distribution of the elements of . In the constant expected model, all returns are jointly normally distributed and is characterized by the mean, variance and covariance of the returns. Applying matrix notation, the  vector of portfolio expected return is given by

(10)

and the  covariance matrix of returns;  is given by

(11)

(12)

Also, the condition that the portfolio weights sum to one (1) is given by

(13)

where  is an  vector with entries equal 1.

2.4. Estimating the Global Minimum Variance Portfolio

The global minimum-variance portfolio is simply the portfolio on the efficient frontier that has the least risk. It is given by

(14)

where  are the global minimum-variance portfolio weights for asset.

For an asset case, the constrained minimization problem is given by

(15)

Thus,

(16)

Also, the first order linear equation is given by

(17)

Therefore,

(18)

Equation (18) is of the form

(19)

where ,  and

Thus, solving for  from Equation (19), we get

(20)

where the elements of  are the portfolio weights  for the global minimum variance portfolio return,  and variance, .

2.5. Determining the Efficient Portfolios

For an n-asset case, the investment opportunity set in -space is explained by set of values whose shape depends on the covariance terms. Assuming that investors select portfolios that maximizes expected return subject to a target level of risk or minimize risk subject to a target expected return, the asset allocation problem can be streamlined by only concentrating on the set of efficient portfolios. These portfolios lie on the boundary of the investment opportunity set above the global minimum variance portfolio.

Following Markowitz [11], we assume that investors wish to find portfolios that have the best expected return-risk trade off. Firstly, investors wish to find portfolios that maximizes portfolio expected return for a given risk level as measured by portfolio variance or standard deviation. This constrained maximization problem to find an efficient portfolio is given by

(21)

(22)

Markowitz also showed that the investor’s problem of maximizing portfolio expected return subject to a target risk level has a correspondent dual representation in which the investor minimizes the risk of the portfolio subject to a given expected return. This dual problem is the constrained minimization problem which is given by

(23)

(24)

In this paper, the dual problem is considered due to computational convenience and that investors being more willing to specify target expected returns rather than risk.

In solving the constrained minimization problem in Equations (23 and 24), the Lagrangian function is employed and is given by

(25)

where  and  are the two constraints and  and  are the two Lagrangian multipliers.

The first order conditions for a minimum are given by the following linear equations

(26)

(27)

(28)

Also, representing the system of linear equations in matrix form, we get

(29)

Equation (29) is of the form

(30)

where ,  and

Solving for , we get

(31)

If  then portfolio,  is an efficient portfolio otherwise  is an inefficient portfolio. Also, all portfolios on the minimum variance frontier are optimal, but only those in the upper portion (at or above) the global minimum-variance portfolio are efficient.

3. Results and Discussion

Table 1 shows the descriptive statistics of the stocks. With much emphasis on the monthly returns and standard deviations, the results show that, the monthly expected return ranges from -0.547 to 5.928 with the highest return found in BOPP and the least return found in CLYD. All the stocks made gains (positive expected return) with the exception of CLYD which made a loss (negative expected return). The monthly standard deviation (risk) ranged from 8.498 to 35.547 with ETI (35.547) been the stock with the highest risk level compared with GGBL (8.498) which had the least risk level. Even though, the highest mean return was found in BOPP, its risk level was less than that of ETI. This implies that, ETI is much riskier than the rest of the stocks and that an investors need to reduce this risk by diversifying. For risk averse investors, it will be prudent to go in for GGBL since it had the least risk level compared to the rest of the stocks.

Table 1. Descriptive Statistics.

Stock Daily Expected Return Monthly Expected return Daily Std. Dev Monthly Std. Dev
ETI 0.083 2.523 6.463 35.547
GCB 0.085 2.593 1.926 10.594
GOIL 0.012 0.377 2.097 11.536
TOTAL 0.093 2.827 4.328 23.804
FML 0.120 3.642 2.157 11.864
GGBL 0.079 2.405 1.545 8.498
CLYD -0.018 -0.547 4.598 25.289
EGL 0.023 0.699 3.798 20.889
PZC 0.005 0.152 3.165 17.408
UNIL 0.093 2.812 1.882 10.352
TLW 0.172 5.229 5.270 28.985
AGA 0.117 3.557 3.346 18.403
BOPP 0.195 5.928 4.196 23.078

Table 2 shows the covariance matrix of the stocks. This provides a first-hand information on how the returns move together in a whole.

Table 2. Covariance Matrix of the Stocks.

Stock ETI GCB GOIL TOTAL FML GGBL CLYD EGL PZC UNIL TLW AGA BOPP
ETI 41.770 -2.013 -1.231 -2.284 1.047 0.207 0.362 4.116 -0.910 -1.550 0.355 -0.284 1.350
GCB -2.013 3.710 -0.047 1.615 -0.479 -0.188 0.027 -2.964 0.426 1.147 -0.095 0.852 -1.269
GOIL -1.231 -0.047 4.399 -0.166 -0.229 0.412 0.025 0.749 -0.042 -0.015 0.144 -0.015 0.111
TOTAL -2.284 1.615 -0.166 18.734 -0.646 -0.458 -1.379 -3.483 0.408 1.308 0.020 0.127 -0.174
FML 1.047 -0.479 -0.229 -0.646 4.653 0.799 1.455 1.732 -0.572 -0.545 -0.179 0.757 1.650
GGBL 0.207 -0.188 0.412 -0.458 0.799 2.387 0.446 2.006 0.128 -0.618 0.053 0.642 1.221
CLYD 0.362 0.027 0.025 -1.379 1.455 0.446 21.138 -0.144 0.394 -0.016 -0.425 -0.243 1.232
EGL 4.116 -2.964 0.749 -3.483 1.732 2.006 -0.144 14.427 -1.230 -2.382 0.174 -0.217 3.298
PZC -0.910 0.426 -0.042 0.408 -0.572 0.128 0.394 -1.230 10.017 1.229 -0.042 0.205 0.168
UNIL -1.550 1.147 -0.015 1.308 -0.545 -0.618 -0.016 -2.382 1.229 3.542 0.188 0.347 -1.635
TLW 0.355 -0.095 0.144 0.020 -0.179 0.053 -0.425 0.174 -0.042 0.188 27.777 -0.005 -0.504
AGA -0.284 0.852 -0.015 0.127 0.757 0.642 -0.243 -0.217 0.205 0.347 -0.005 11.199 -0.328
BOPP 1.350 -1.269 0.111 -0.174 1.650 1.221 1.232 3.298 0.168 -1.635 -0.504 -0.328 17.603

Figure 1 shows the monthly plot of risk-return of the stocks. The plot shows that, GOIL, PZC and EGL recorded low returns with higher risk levels compared with FML, AGA, BOPP and TLW which recorded higher returns with somewhat lower risk levels. ETI recorded the highest risk level of 35.547 with an expected return of 2.523. GGBL recorded the least risk with an expected return of 8.498. Since investors are only interested in forming optimal and efficient portfolios, CLYD was not considered since it made a loss. Also, the risk-return plot indicates that, equally weighted portfolio has higher expected return per the level of risk.

Figure 1. Monthly plot of the risk and return of the stocks.

Table 3, shows the estimates of the global minimum-variance portfolio. The results reveals that, the expected return on the portfolio called global minimum-variance is 2.360 and a risk level of 0.766. This means that, there is a 0.766 risk in investing in the minimum-variance portfolio that rewards 2.360. The global minimum-variance portfolio has portfolio weights (asset allocation) as follows;

.

In a vector form, the allocation of assets for the global minimum-variance portfolio is given by;

(32)

In other to achieve this return, an investor needs to allocate assets given Equation 32 for investing in the portfolio with the least risk.

Table 3. Global Minimum Variance Portfolio.

Stock Global Minimum Variance portfolio Weight Expected Return (μp,m) Portfolio Std. dev (σp,m)
ETI 0.014 2.360 0.766
GCB 0.166
GOIL 0.139
TOTAL 0.020
FML 0.150
GGBL 0.188
CLYD 0.014
EGL 0.024
PZC 0.061
UNIL 0.163
TLW 0.020
AGA 0.014
BOPP 0.026

Table 4, shows the efficient portfolio with the same expected return as a given stock. The results indicate that, when ETI, GCB, GOIL, TOTAL, FML, GGBL, EGL, PZC, UNIL, TLW, AGA and BOPP with expected returns 2.523, 2.593, 0.377, 2.827, 3.642, 2.405, 0.699, 0.152, 2.812, 5.229, 3.559 and 5.928 respectively, then in other to have a portfolio whose expected return will be the same as that of as any of the above assets, an investor need to bear a risk 0.771, 0.776, 1.288, 0.805, 1.019, 0.067, 1.157, 1.384, 0.802, 1.685, 0.990 and 2.017 for holding portfolio 1, 2,…,12 respectively which are all lesser than the individual risk associated with the assets. This indicates that, one reduces risk by diversifying in several assets that are uncorrelated. In portfolio 1, 2, …, 12, the highest proportion of asset allocation is found in GGBL (0.188), GGBL (0.189), GGBL (0.188), GOIL (0.382), GGBL (0.190), GGBL (0.194), GGBL(0.179), PZC (0.192), GGBL(0.190), FML (0.356), FML (0.236) and FML (0.406) respectively. GGBL is seen as the stock with the highest allocation of wealth in most of the portfolios. This is so because from Table 1, GGBL exhibited the least standard deviation (risk) and since investors are interested in minimizing risk given a target expected return, hence much wealth allocation in GGBL. The least allocation of wealth in portfolio 1, 2, …, 12 was CLYD (0.007), CLYD (0.004), BOPP (-0.102), CLYD (-0.005), CLYD (-0.040), CLYD (0.012), BOPP (-0.081), BOPP (-0.116), CLYD (-0.005), GOIL (-0.212), EGL (-0.177) and GOIL (-0.297) respectively. Six out of the 12 portfolios had CLYD exhibiting the least asset allocation. This is because even though from Table 1, CLYD had a higher risk level but no compensation for holding it since it made a loss. For an investor to adequately minimize risk in other to achieve the expected return in each portfolio, the proportion of wealth to be allocated to each asset in each portfolio is given by the following vectors;

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

The efficient portfolios were selected by taking into consideration the expected return of each portfolio. That is, any portfolio with expected return greater or equal the expected return of the global minimum-variance portfolio is considered efficient portfolio otherwise the portfolio is an inefficient one. From the results, portfolios 1, 2, 4, 5, 6, 9, 10, 11 and 12 with expected return 2.523, 2.593, 2.827, 3.642, 2.405, 2.812, 5.229, 3.559 and 5.928 respectively are considered efficient portfolios since their expected return each is greater than the expected return of the global minimum variance portfolio. This indicates that, these portfolios have maximum expected for the level risk estimated. Portfolios 3, 7 and 8 with expected return 0.377, 0.699 and 0.152 respectively are considered inefficient portfolios since their expected each is less than the expected return of the global minimum-variance portfolio.

Table 4. Efficient Portfolio with the same expected return as a given stock.

Portfolio No. Stock Stock Expected Return Weight Portfolio Expected Return (μp,o) Portfolio Std. dev (σp,o)
1 ETI* 2.523 0.014 2.523 0.771
  GCB 0.169
  GOIL 0.119
  TOTAL 0.021
  FML 0.162
  GGBL 0.188
  CLYD 0.007
  EGL 0.020
  PZC 0.052
  UNIL 0.169
  TLW 0.025
  AGA 0.017
  BOPP 0.037
  ETI 0.014
2 GCB* 2.593 0.170 2.593 0.776
  GOIL 0.111
  TOTAL 0.021
  FML 0.167
  GGBL 0.189
  CLYD 0.004
  EGL 0.017
  PZC 0.047
  UNIL 0.172
  TLW 0.027
  AGA 0.019
  BOPP 0.041
  ETI 0.119
  GCB 0.130
3 GOIL* 0.377 0.382 0.377 1.288
  TOTAL 0.011
  FML 0.008
  GGBL 0.178
  CLYD 0.098
  EGL 0.084
  PZC 0.179
  UNIL 0.084
  TLW -0.036
  AGA -0.028
  BOPP -0.102
  ETI 0.146
  GCB 0.174
  GOIL 0.082
4 TOTAL* 2.827 0.022 2.827 0.805
  FML 0.183
  GGBL 0.190
  CLYD -0.005
  EGL 0.010
  PZC 0.033
  UNIL 0.181
  TLW 0.034
  AGA 0.024
  BOPP 0.056
  ETI 0.155
  GCB 0.189
  GOIL -0.017
  TOTAL 0.026
5 FML* 3.642 0.242 3.642 1.019
  GGBL 0.194
  CLYD -0.040
  EGL -0.014
  PZC -0.015
  UNIL 0.213
  TLW 0.057
  AGA 0.041
  BOPP 0.109
  ETI 0.014
  GCB 0.167
  GOIL 0.134
  TOTAL 0.020
  FML 0.153
6 GGBL* 2.405 0.189 2.405 0.767
  CLYD 0.012
  EGL 0.023
  PZC 0.059
  UNIL 0.164
  TLW 0.022
  AGA 0.015
  BOPP 0.029
  ETI 0.012
  GCB 0.136
  GOIL 0.342
  TOTAL 0.013
  FML 0.031
  GGBL 0.179
  CLYD 0.084
7 EGL* 0.699 0.075 0.699 1.157
  PZC 0.160
  UNIL 0.097
  TLW -0.027
  AGA -0.021
  BOPP -0.081
  ETI 0.012
  GCB 0.126
  GOIL 0.409
  TOTAL 0.010
  FML -0.008
  GGBL 0.177
  CLYD 0.107
  EGL 0.091
8 PZC* 0.152 0.192 0.152 1.384
  UNIL 0.075
  TLW -0.043
  AGA -0.032
  BOPP -0.116
  ETI 0.015
  GCB 0.174
  GOIL 0.084
  TOTAL 0.022
  FML 0.182
  GGBL 0.190
  CLYD -0.005
  EGL 0.011
  PZC 0.034
9 UNIL* 2.812 0.180 2.812 0.802
  TLW 0.033
  AGA 0.023
  BOPP 0.055
  ETI 0.017
  GCB 0.218
  GOIL -0.212
  TOTAL 0.033
  FML 0.356
  GGBL 0.202
  CLYD -0.107
  EGL -0.062
  PZC -0.109
  UNIL 0.276
10 TLW* 5.229 0.102 5.229 1.685
  AGA 0.074
  BOPP 0.211
  ETI 0.015
  GCB 0.188
  GOIL -0.007
  TOTAL 0.025
  FML 0.236
  GGBL 0.194
  CLYD -0.036
  EGL -0.117
  PZC -0.010
  UNIL 0.210
  TLW 0.055
11 AGA* 3.559 0.039 3.559 0.990
  BOPP 0.103
  ETI 0.018
  GCB 0.231
  GOIL -0.297
  TOTAL 0.036
  FML 0.406
  GGBL 0.205
  CLYD -0.136
  EGL -0.083
  PZC -0.151
  UNIL 0.304
  TLW 0.122
  AGA 0.089
12 BOPP* 5.928 0.256 5.928 2.017

* Given stock

Figure 2, shows the efficient frontier of the portfolios under consideration. It can be seen that, all the portfolios are optimal since they are all on the minimum-variance frontier. Also, the efficient portfolios (ETI, GCB, TOTAL, FML, GGBL, CLYD, UNIL, TLW, AGA and BOPP) are located in the upper portion-at or above the global minimum-variance portfolio (Global minimum) whereas the inefficient portfolios (GOIL, PZC and EGL) are found beneath the Global minimum.

Figure 2. Efficient Frontier.

4. Conclusion

The purpose of this paper is to construct optimal portfolio using matrix approach. The results indicate that, all the portfolios are optimal and that portfolios 1, 2, 4, 5, 6, 9, 10, 11 and 12 with expected return 2.523, 2.593, 2.827, 3.642, 2.405, 2.812, 5.229, 3.559 and 5.928 respectively are efficient portfolios whereas portfolios 3, 7 and 8 with expected return 0.377, 0.699 and 0.152 respectively are inefficient portfolios with reference to the expected return of the global minimum variance portfolio (2.360). GGBL is seen as the stock with the highest allocation of wealth in most of the portfolios. Six out of the 12 portfolios had CLYD exhibiting the least asset allocation. It is therefore advisable for investors to consider investing in the efficient portfolios by taking into consideration the weight of each portfolio so as to minimize risk in other to get the desired return. Also, it is seen that investing in only one asset bares a higher risk than investing in several assets hence the need for investors to diversify their portfolios.


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