Portfolio Optimization of Some Stocks on the Ghana Stock Exchange Using the Markowitz Mean-Variance Approach

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Journal of Financial Risk Management, 2019, 8, 29-41
http://www.scirp.org/journal/jfrm
ISSN Online: 2167-9541
ISSN Print: 2167-9533
DOI:
10.4236/jfrm.2019.81003 Mar. 22, 2019 29
Journal of Financial Risk Management
Portfolio Optimization of Some Stocks on the
Ghana Stock Exchange Using the Markowitz
Mean-Variance Approach
Anuwoje Ida Logubayom, Togborlo Annani Victor
Department of Statistics, Faculty of Mathematical Sciences, University for Development Studies, Tamale, Ghana
Abstract
An investment portfolio is a collection of financial assets consisting of in-
vestment tools such as stocks, bonds, and
bank deposits, among others, which
are held by a person or a group of persons.
Constructing a portfolio with
standardized optimization remains a myth in Ghana and henc
e this study
displayed how the Markowitz model can be applied on the Ghana Stock Ex-
change and also unraveled the most efficient portfolio among selected stocks
to the relief of the investor. Historical monthly data of the stock returns from
2011 to 2016 was used for the study. The study revealed that, GCB Bank li-
mited had the best average returns (returns of 4.2%) with a risk of 13.1% fol-
lowed by CAL (returns of 3.5%) and 11.7% risk. UGL had the lowest risk (risk
of 6.8%) and lowest average returns of 2.1%. A risk lover may go in for GCB
and CAL while an investor who is completely risk averse can opt for UGL
since it comes with the lowest risk. A two-way combination of t
he portfolios
also concluded that, the most efficient portfolio is the combination of GCB
and CAL and recommended that a risk tolerant investor can invest all his as-
sets in GCB while a risk averse investor can invest 39.21% of his assets in
GCB and 60.79% in CAL. In terms of expected returns, a combination of CAL
and GCB bank limited gives the highest returns of about 3.9% with a risk of
10.6%, followed by the combination of TOTAL and GCB with expected re-
turns of about 3.40% and a high risk of 12.3%. The re
latively high expected
return of the combination of TOTAL and GCB could not reflect in the Sharpe
ratio because of the high level of risk which implies that the portfolio cannot
compensate much for this high level of risk. Also, CAL and GCB achieving
the highest Sharpe ratio shows that, this portfolio is expected to offer the best
compensation for the risk taken by an investor and therefore the most effi-
cient portfolio for investor. The lowest risk (which is what the risk averse in-
vestor is interested in) was achieved from the combination of HFC and UGL
How to cite this paper:
Logubayom, A. I., &
Victor
, T.A. (2019). Portfolio Optimiza
tion
of Some Stocks on the Ghana Stock E
x-
change Using
the Markowitz Mean-Variance
A
pproach.
Journal of Financial Risk Ma
n-
ag
ement, 8,
29-41.
https://doi.org/10.4236/jfrm.2019.81003
Received:
February 11, 2019
Accepted:
March 19, 2019
Published:
March 22, 2019
Copyright © 201
9 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access

A. I. Logubayom, T. A. Victor
DOI:
10.4236/jfrm.2019.81003 30
Journal of Financial Risk Management
which is 5.2% with a Sharpe ratio of 6.7% and a covariance of −0.00051.
Keywords
Portfolio Optimization, Mean-Variance, Variance-Covariance Matrix
1. Introduction
Investment activity is essential to the promotion of economic well-being; it is
one of the most important economic activities that individuals, businesses and
governments undertake. The commitment of resources in anticipation that an
affirmative rate of return will be achieved is known as investment (Mensah,
2008). Major considerations when investing include what to invest in, how much
to invest and the level of risk an investor is prepared to bear in order to achieve
his/her investment goal. A portfolio is simply a collection of financial assets in-
volving investment tools such as bonds, foreign exchange, stocks, gold, as-
set-backed securities, real estate certificates and bank deposits which are held
simultaneously by one person or a group of persons.
Risk is the probability of the losses one incurred on portfolio investment
whiles the return is the profit or benefit one derives from portfolio investment.
Investment is the net worth on long term financial assets such as bonds, shares
and mutual funds. Investment risk is most properly understood when it is ex-
pressed in statistical terms that consider the entire range of an investment’s
possible returns. Markowitz states that, the expected return (mean) and variance
and standard deviation (risk) of a portfolio are the whole criteria for portfolio
selection and construction. These parameters can be used as a possible maxim
for how investors need to act. It is interesting to note that, the whole model is
based on an economic fact of “Expected Utility”. The concept of utility here is
based on the fact that different investors have different investment goals and can
be satisfied in different ways.
The theory of portfolio optimization is generally associated with the classical
mean-variance optimization framework of Markowitz (Markowitz, 1952). 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 accu-
rate than those of the expected returns (Merton, 1980; Jagannathan & Ma, 2003).
Several studies concentrates on improving the performance of the global mini-
mum-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 as-
sets and their variance-covariance matrix. However, these returns are unobserv-
able and unknown. The impossibility to obtain a sufficient number of data sam-
ples, instability of data, and differing personal views of decision makers on the

A. I. Logubayom, T. A. Victor
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Journal of Financial Risk Management
future returns affect their estimation and have led to what call estimation risk in
portfolio selection (Mulvey & Erkan, 2003; Bawa, Brown, & Klein, 1979). This
estimation risk has shown to be the source of very erroneous decisions; as
pointed in (Ceria & Stubbs, 2006; Cornuejols & Tutuncu, 2007), 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 re-
sult in the difficulties in constructing different optimization.
Consequently, every investor seeks to maximize their utility (satisfaction) by
maximizing expected return and minimizing risk (variance). The Markowitz
model could be summarized as follows; Calculate the expected return rates for
each stock to be included in the portfolio; Calculate the variance or standard
deviation (risk) for each stock to be included in the portfolio; Calculate the
co-variance or correlation coefficients for all stocks, treating them as pairs
(Fabozzi, 1999).
2. Methods of Data Analysis
2.1. Data Collection Methods
Secondary data on the monthly returns of selected companies on the Ghana
Stock Exchange was collected from the Ghana Stock Exchange database ranging
from 2011 to 2016.
2.2. Mathematics of the Markowitz Mean-Variance Model
The Markowitz model involves some mathematics, which makes it possible to
construct stock portfolio with different combinations where short sale and lend-
ing or borrowing might be allowed or not. The Markowitz model is all about
maximizing return, and minimizing risk, but simultaneously. We should be able
reach a single portfolio of risky assets with the least possible risk that is preferred
to all other portfolio with the same level of return. Our optimal portfolio will be
somewhere on the ray connecting risk free investment to our risky portfolio and
where the ray becomes tangent to our set of risky portfolios. This point has the
highest possible slope. In mathematics, optimization refers to the selection of a
best element, with regard to certain conditions, from a set of possible alterna-
tives.
2.3. Different Approaches for Portfolio Optimization
Even in Mean-Variance framework, there are different approaches when it
comes to searching for optimum portfolio of risky asset. In the following lines
we will introduce three different approaches, based either on desired expected
return and risk or on finding the portfolio with highest reward-to-risk ratio.
Optimum Portfolio for Particular Rate of Return
When investor wants to construct a portfolio which yields a particular rate of
return and simultaneously minimize the portfolio risk, he is facing a linear pro-
gramming problem in a form:

A. I. Logubayom, T. A. Victor
DOI:
10.4236/jfrm.2019.81003 32
Journal of Financial Risk Management
Min
T
WWΣ
ST.
( )
T
01
pi
r W W We
µ
≥Λ ≥Λ =
where
e
is column vector of ones, i.e.
[ ]
T
1,1, ,1e=
, and
μ
is desired rate of re-
turn.
Optimum Portfolio for Particular Risk Rate
Another approach for finding optimum portfolio of risky assets is to set a de-
sired level of portfolio risk and find a combination of securities, which maximize
expected return. This process can be once again formulated in terms of linear
programming problem, as:
Max
( )
p
rW
ST.
TT
01
i
W W W We
σ
Σ ≤Λ ≥Λ =
where
σ
is level of portfolio variance, which should not be exceeded.
Similarly, to the first approach, we will arrive to efficient frontier by changing
the value of
σ
. Moreover, it has been proved by (Palmquist, Pavlo, & Uryasev,
2002) that, by these optimization approaches we will arrive at the same results.
Thus, we obtain the identical efficient frontier in both cases.
Optimum Portfolio Dependent on Risk Aversion Parameter
There is still a third way how to arrive to the efficient frontier. We can con-
sider a problem proposed by (Sharpe, 1994).
Min
( )
TT
*Er W W W
λ
− +Σ
ST.
0
i
W≥
T
1We=
where
λ
is risk aversion parameter. By varying
λ
one can arrive to the same effi-
cient frontier as in two previous cases.
2.4. Assumptions of the Markowitz Mean-Variance Model
The Markowitz model has the following assumptions:
1) That an investor is apprehensive with return distribution over a single pe-
riod.
2) Investors try to maximize the expected return of total wealth.
3) All investors are risk-averse, i.e. they will simply take a higher risk if they
are rewarded for higher expected return.
4) Investors based their investment judgments on the expected return and
risk.
5) All markets are perfectly effective.
6) Investments are also by a single period. This means that, investors make
their portfolio decisions at the start of a period and then wait until the close of
the period when the rate of return on their portfolio is realized. Also the investor
cannot make any intermediate changes in the composition of his portfolio; and
finally the investor makes his choice with the aim of maximizing expected utility

A. I. Logubayom, T. A. Victor
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Journal of Financial Risk Management
of wealth at the end of the period (final wealth).
The Markowitz approach is often described as a mean-variance method since;
it simply takes those two parameters, mean return and variance of return into
consideration to characterize the investor’s portfolio. The expected return of the
portfolio is quantified by the mean return, while the risk of the portfolio is
measured by the variance. The variance facilitates simple modeling and also is a
good measure of risk under the supposition that returns are normally distri-
buted. The concept established by Markowitz is also centered on maximizing the
expected utility of the investor’s terminal fortune. The utility function is defined
according to the expected return and the standard deviation of the wealth.
2.5. The Sharpe’s Ratio
This ratio is a measurement for risk-adjusted returns and was developed by Wil-
liam F. Sharpe.
There is a risk and return characteristics of the portfolio that will change in a
non-linear fashion as the weighting of the component assets change. The Sharpe
ratio characterizes how well the return of an asset compensates the investor for
the risk taken. The higher the returns mean better investment option.
risk premium
systematic risk
p
rR
Sp p
δ
−
= =
where
rp
is the average returns of portfolio
p
,
R
is the risk-free rate of returns,
p
δ
is the standard deviation (risk) of portfolio
p.
2.6. Risk and Return
The expected returns of portfolio
p
=
( ) ( )
T
p
r W WEr=
.
w
denotes the vector of the portfolio weights and the
( )
Er
denotes the vec-
tor of the expected returns of the portfolio instruments.
Variance
( )
T
pw w W
= Σ
where
w
denotes the vector of portfolio weights and
Σ
denotes the va-
riance-covariance matrix
Risk (standard deviation) =
( )
T
p
W WW
ρ
= Σ
where
ρ
denotes the risk of the portfolio
.
3. Results and Discussion
Statistical Estimation for Each Asset
Table 1 presents the six-year period of mean return and standard deviation, the
skewness and the kurtosis of the individual assets. Skewness is used in statistics
to describe asymmetry from the normal distribution in a set of statistical data.

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Table 1. Risk, return, skewness and kurtosis of assets.
STOCK
ST. DEV
RETURNS
SKEWNESS
KURTOSIS
CAL
0.117
0.035
0.857
1.033
EBG
0.088
0.02
1.182
1.629
GCB
0.131
0.043
1.667
5.18
SCB
0.152
0.016
−2.586
18.308
HFC
0.084
0.016
1.908
6.542
UGL
0.069
0.022
0.067
2.495
GGL
0.093
0.019
−1.023
4.412
FMGL
0.137
0.013
−2.827
18.561
PZC
0.167
0.003
−0.98
12.901
TOTAL
0.166
0.025
−2.301
15.405
Skewness can come in the form of negative skewness or positive skewness, de-
pending on whether data points are skewed to the left and negative, or to the
right and positive of the data average. For a positively skewed data, the mean
and the median of the set are both greater than the mode and in most cases, the
mean is greater than the median. Thus average higher returns will be obtained
for such an investment. Conversely, when data are skewed to the left, the mean
and the median are both less than the mode and the mean possibly less than the
median. A skewed data indicate deviation from normality. By understanding
which way data is skewed, an investor can better estimate whether a given future
data point will be more or less than the mean. Many casual equity investors look
at the chart of a stock’s price and try to make investments in companies that are
positively skew, which in the equity markets is a stock price that is greatly
skewed positively with possible higher average returns. From the results, the po-
tentially great fortune of GCB can be seen with its positive skewness of 1.667 as
it is the second highest behind HFC (with 1.908 skewness) indicating possible
higher gains. SCB, GGL, FMGL, PZC and TOTAL yielded negative skewness
which indicates losses while the rest of the stocks had positive skewness indicat-
ing gains. Kurtosis is a measure of the likelihood that an event occurring is ex-
treme in relation to a given distribution and is often referred to as the volatility.
The higher the kurtosis coefficient is above the normal level (which is 3), the
more volatile the future return (thus either extremely large or extremely small).
CAL, EBG and UGL had kurtosis lesser than 3 (they are platykurtic) while the
others had kurtosis greater than 3 (are leptokurtic). SCH, FMGL, PZC and
TOTAL had kurtosis of 18.308, 18.561, 12.901 and 15.405 respectively which are
far above the normal level (3) and imply a high likelihood of their future returns
being either extremely large or extremely small.
By investing solely in one of the ten assets, it is not possible to achieve a return
more desirable than the asset itself. An investor would prefer a risk/return rela-
tionship yielding a high return associated with low risk; however this is not

A. I. Logubayom, T. A. Victor
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possible due to the fact that high return almost always comes with high risk. By
constructing a portfolio of the assets, the author strives to reach a return more
desirable than any of the assets themselves, a combined portfolio with a mini-
mized risk and a more desirable return.
From the ten companies selected, we considered two-asset portfolio. This
yielded 45 possible combinations out of which the most efficient portfolio had to
be determined. To do this, equal weights were assigned to each combination to
determine the Sharpe Ratio of each of the combinations. The Sharpe ratio tells
how well the return of assets compensates for the risk taken and hence the high-
er the ratio, the better the portfolio.
The highest Sharpe Ratio obtained would belong to the best combination as it
represents the best combination with the best excess return over the risk of each
portfolio. Table 2 gives the 45 asset combinations with their returns, risk and
Sharpe Ratios. From Table 2, the highest Sharpe ratio is ascertained from the
combination of CAL and GCB. This portfolio yielded a Sharpe ratio of about
21.6 with a risk of 10.6% and a covariance of 0.007089. This positive covariance
implies that both assets move in the same direction. Thus, whenever there is an
increase (or a decrease) in the returns of CAL, the returns of GCB also expe-
rience an increase (or decrease). CAL and GCB achieving the highest Sharpe ra-
tio implies that, it is the portfolio which is expected to offer the best compensa-
tion for the risk taken by an investor and therefore the most efficient portfolio
for a rational investor to opt for. In terms of expected returns, the combination
of CAL and GCB had the highest of about 3.9% with a risk of 10.6% followed by
the combination of TOTAL and GCB with expected returns of about 3.40% and
a high risk of 12.3%. The relatively high expected return of the combination of
TOTAL and GCB could not reflect in is Sharpe ratio because of the high level of
risk which implies that the portfolio cannot compensate much for this high level
of risk. Also, the lowest risk (which is what the risk averse investor is interested
in) was achieved from the combination of HFC and UGL which is 5.2% with a
Sharpe ratio of 6.7% and a covariance of −0.00051. The negative covariance im-
plies that the two assets move in opposite directions. Thus, when one is expe-
riencing an increase in returns, the other is experiencing a decrease in returns.
However, the covariance only shows directional relationship between two assets
but cannot show the strength of the relationship. It is also sensitive to high vola-
tility (volatile assets include those with returns that are farther from the mean)
and hence correlation coefficients is a better measure.
The correlation coefficients show the strength of the relationships between the
companies or it measures the extent to which there is a linear relationship be-
tween the two assets. The correlation between two assets may be negative (cor-
relation,
10
ρ
−≤ <
), positive (correlation,
01
ρ
<≤
), no correlation (
0
ρ
=
).
There is a perfect positive/negative if correlation coefficient of 1 or −1 respec-
tively. Whenever there exists a perfectly positive correlation between two assets,
there is no need to diversify because it does not reduce the unsystematic risk
whereas a perfectly negative correlation (a correlation coefficient of −1) between

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Table 2. Optimum portfolio combinations.
ASSET
COMBINATIONS
PORTFOLIO
RETURNS
PORTFOLIO
COVARIANCE
PORTFOLIO
RISK
PORTFOLIO
SHARPE RATIO
CAL-EBG
0.027
0.000
0.074
0.159
CAL-GCB
0.039
0.007
0.106
0.216
CAL-SCB
0.025
−0.001
0.094
0.101
CAL-HFC
0.025
0.000
0.072
0.136
CAL-UGL
0.028
0.001
0.070
0.179
CAL-GGL
0.027
0.004
0.086
0.131
CAL-FMGL
0.024
−0.003
0.082
0.096
CAL-PZC
0.019
0.001
0.105
0.030
CAL-TOTAL
0.030
0.003
0.109
0.130
EBG-GCB
0.032
0.000
0.080
0.198
EBG-SCB
0.018
0.003
0.096
0.025
EBG-HFC
0.018
0.002
0.067
0.041
EBG-UGL
0.021
0.001
0.059
0.093
EBG-GGL
0.020
0.001
0.066
0.064
EBG-FMGL
0.016
0.000
0.081
0.010
EBG-PZC
0.012
0.001
0.096
−0.041
EBG-TOTAL
0.023
0.001
0.097
0.073
GCB-SCB
0.029
0.000
0.101
0.134
GCB-HFC
0.030
0.000
0.078
0.179
GCB-UGL
0.032
0.003
0.082
0.202
GCB-GGL
0.031
0.004
0.092
0.167
GCB-FMGL
0.028
0.005
0.106
0.113
GCB-PZC
0.023
0.003
0.114
0.064
GCB-TOTAL
0.034
0.008
0.123
0.149
SCB-HFC
0.016
0.001
0.089
0.005
SCB-UGL
0.019
0.001
0.087
0.036
SCB-GGL
0.018
0.000
0.089
0.022
SCB-FMGL
0.014
0.003
0.108
−0.013
SCB-PZC
0.009
0.001
0.114
−0.055
SCB-TOTAL
0.020
0.000
0.112
0.043
HFC-UGL
0.019
−0.001
0.052
0.068
HFC-GGL
0.018
0.001
0.067
0.034
HFC-FMGL
0.015
0.002
0.087
−0.013
HFC-PZC
0.010
0.001
0.095
−0.062
HFC-TOTAL
0.021
0.001
0.096
0.054

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Continued
UGL-GGL
0.021
0.001
0.062
0.081
UGL-FMGL
0.017
0.002
0.081
0.020
UGL-PZC
0.012
0.003
0.098
−0.032
UGL-TOTAL
0.024
0.002
0.096
0.082
GGL-FMGL
0.016
0.003
0.091
0.004
GGL-PZC
0.011
0.004
0.106
−0.042
GGL-TOTAL
0.022
0.003
0.102
0.065
FMGL-PZC
0.008
0.003
0.115
−0.068
FMGL-TOTAL
0.019
−0.001
0.104
0.031
PZC-TOTAL
0.014
0.002
0.122
−0.012
Table 3. The correlation matrix of the stocks.
CORRELATION MATRIX
Cal
EBG
GCB
SCB
HFC
UGL
GGL
FMGL
PZC
TOTAL
Cal
1
0.032
0.462
−0.046
0.011
0.082
0.351
−0.162
0.074
0.027
EBG
0.032
1
0.027
0.237
0.209
0.117
0.072
0.005
0.035
0.079
GCB
0.462
0.027
1
0.020
−0.010
0.285
0.341
0.262
0.145
−0.145
SCB
−0.046
0.237
0.020
1
0.055
0.125
−0.003
0.124
0.021
−0.043
HFC
0.011
0.209
−0.010
0.055
1
−0.089
0.142
0.192
0.046
−0.186
UGL
0.082
0.117
0.285
0.125
−0.089
1
0.147
0.165
0.236
0.083
GGL
0.351
0.072
0.341
−0.003
0.142
0.147
1
0.231
0.266
−0.057
FMGL
−0.162
0.005
0.262
0.124
0.192
0.165
0.231
1
0.129
−0.171
PZC
0.074
0.035
0.145
0.021
0.046
0.236
0.266
0.129
1
−0.079
TOTAL
0.027
0.079
−0.145
−0.043
−0.186
0.083
−0.057
−0.171
−0.079
1
assets implies that a certain combination of these assets can reduce the unsyste-
matic risk to zero (Sharpe, 1994). From Table 3, the compounded portfolio has
an overall low correlation such that apart from the correlation of an asset by it-
self which certainly is perfectly correlated, the highest correlation coefficient,
0.462, occurred between CAL and GCB. A compounded portfolio with an over-
all low correlation is crucial for investors who aim to diversify in order to elimi-
nate unsystematic risk (Sharpe, 1994). The correlation coefficient between CAL
and GCB (0.462) implies that an increase (or decrease) in one of the two will in-
crease (or decrease) in the returns of the other. The correlation coefficient be-
tween FMGL and PZC (−0.171) implies that, when the returns of one of them
increase, the returns of the other decreases and the reverse is true. There exists a
very positive weak correlation (0.005) between FGL and EBG which implies that
an increase (or decrease) in the returns of one will only result in a smaller in-
crease (or decrease) in the other. SCB and GGL also had a very weak negative

A. I. Logubayom, T. A. Victor
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Journal of Financial Risk Management
correlation (−0.003) between each other implying that an increase in the returns
of one will only result in a smaller decrease in the other and the reverse is true.
Having found the optimal portfolio assuming equal weights, this study applied
Excel solver to the optimal weights to maximize the returns, minimizing the risk
and maximizing the Sharpe Ratio.
From Table 4, the expected returns when trying to maximize returns was
achieved with 0.00% weight to CAL and 100% weight to GCB with the corres-
ponding returns being 4.27% and 13.1% risk. In trying to minimize risk howev-
er, the returns and risk expected were 3.77% and 10.5% respectively with weights
given as 60.79% for Cal Bank and 39.21% for GCB. By maximizing the Sharpe
ratios, the weights attained were 36.21% for CAL and 63.79% for GCB with ex-
pected return and risk given as 3.98% and 10.9% respectively. The weights at the
various columns tells what percentage of an investor’s asset should be allocated
to the company in question at the optimized level when minimizing risk, max-
imizing returns and when maximizing the Sharpe ratio. Therefore an investor
who will invest in the efficient portfolio (CAL and GCB) and wanting to endure
the minimal risk (a risk averse person) of 10.5% should invest 60.79% of his as-
sets in CAL and 39.21% in GCB and he will achieve a 3.77% returns with a
Sharpe ratio of 21.64 monthly. An investor in the efficient portfolio who wants
to ascertain the maximum return of 4.27% monthly regardless of the risk (a risk
tolerant person) should invest all (100% of) his assets in CAL but should expect
a 13.13% risk with a Sharpe ratio of 20.61 monthly. Also an investor in the effi-
cient portfolio who wishes to have a maximum Sharpe ratio of 21.92% should
allocate 36.21% of his assets to CAL and 63.79% to GCB and should expect a risk
of 10.99% and a 3.98% returns monthly which implies an expected annual return
and risk to be 47.7% and 38.1% respectively (thus, annual return = 0.0398 × 12 =
0.477, annual risk = 0.1099 ×
12
= 0.381).
4. Conclusion and Recommendations
In conclusion, the researcher has been able to estimate the expected returns,
standard deviation (risk), skewness and kurtosis of the selected stocks. Based on
the Markowitz Mean-Variance analysis, 45 optimal portfolios were generated to
Table 4. Optimal portfolio.
OPTIMAL PORTFOLIO
ASSETS
EQUAL WEIGHTS
MAX RETURNS
MIN RISK
MAX SR
CAL
0.5000
0.0000
0.6079
0.3621
GCB
0.5000
1.0000
0.3921
0.6379
TOTAL WEIGHTS
1.0000
1.000
1.0000
1.0000
RETURNS
0.0386
0.0427
0.0377
0.0398
STD. DEV
0.1061
0.1313
0.1052
0.1099
SR
0.2164
0.2061
0.2099
0.2192

A. I. Logubayom, T. A. Victor
DOI:
10.4236/jfrm.2019.81003 39
Journal of Financial Risk Management
satisfy the various investor attitudes to risk. A further analysis using excel solver
revealed how the risk of the efficient portfolio could be minimized, returns
maximized and Sharpe ratios maximized by adjusting the weights of the indi-
vidual stocks. These results contribute significantly to the existing knowledge on
the Ghana Stock Exchange since an average investor can create optimal portfo-
lios to guide him/her in investment decision making.
The study reveals that, stocks in GCB has the highest expected returns of
4.27% with a 13.13% risk whiles UGL has the least expected monthly risk of 6.9%
with 2.2% expected monthly returns. It implies that GCB has an annual expected
return of 51.24% with 45.48% risk while UGL has the least annual risk of
23.905% and 26.4% returns. An uninformed risk averse investor will opt for
UGL which has the least expected annual risk of 23.9% and expect 26.4% annual
returns. The ideal thing to do as an investor is to diversify in order to reduce
unsystematic risk. This study put together a two-asset portfolio which yielded 45
portfolio combinations and it was revealed that the combination of GCB and
CAL yielded the highest Sharpe ratio of 0.2164 with monthly expected returns
and risk of 3.86% and 10.61% respectively. The highest Sharpe ratio implies that
it is the portfolio which compensates best for the risk taken by the investor and
hence the most efficient portfolio. If an investor invests in this portfolio and al-
locates equal proportions of his assets to the two stocks, he will still achieve a
high expected annual return of 46.32% with the expected annual risk reduced
drastically to 36.75% which emphasizes the importance of diversification. If the
risk averse investor is informed and he/she opts for the optimal portfolio with
least risk (which is the combination of HFC and UGL) instead of singling out
UGL for investment, he/she can expect a monthly return and risk of 1.92% and
5.2% respectively with a Sharpe ratio of 6.7 which means that he/she should ex-
pect 23.04% annual expected returns with the expected annual risk reduced to
18.0% through diversification. Again the study reveals, that risk is minimized
when the weights allocated to the individual stocks that make up the portfolio
are adjusted by reducing the weight attached to the riskier stock. When the effi-
cient portfolio was optimized using excel solver (from Table 4), it revealed that
the monthly expected risk could be reduced to 10.5% when the investor allocates
60.79% of his assets to CAL and 39.21% to GCB but still achieved a monthly
3.77% returns which implies a minimized expected annual risk of 36.4% yielding
45.2% expected annual return. Therefore a risk averse investor who is informed
to invest in the efficient portfolio should allocate 60.7% of his assets CAL and
39.2% to GCB and enjoy a 45.2% annual expected returns with a minimal 36.4%
risk which offers a far better prospects to the investor compared to naively in-
vesting in only UGL or even in the least risk optimal portfolio (HFC and UGL).
It is therefore recommended that, the most efficient portfolio is the combina-
tion of CAL and GCB and therefore investors should opt for it. A risk tolerant
investor should allocate all his/her assets to GCB while a risk averse investor can
allocate 60.79% of his/her assets to CAL and 39.21% to GCB. A risk neutral in-

A. I. Logubayom, T. A. Victor
DOI:
10.4236/jfrm.2019.81003 40
Journal of Financial Risk Management
vestor should allocate 36.21% and 63.79% for his/her assets to CAL and GCB
respectively or invest in any of the optimal portfolios constructed by this study.
Since excel can be used to construct more than 45 optimal portfolios, it is
recommended that investors can develop portfolios with larger number of stocks
to help them take informed investment decisions.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this pa-
per.
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A. I. Logubayom, T. A. Victor
DOI:
10.4236/jfrm.2019.81003 41
Journal of Financial Risk Management
Appendix
CAL CAL Bank LTD
EBG Ecobank Ghana LTD
EIC Enterprice Insurance Company
ETI Ecobank Transnational Incorporated
GCB Ghana Commercial Bank
SIC State Insurance Company
HFC HFC bank LTD
SCB Standard Chartered Bank LTD
TBL The Trust Bank LTD
UT Unique Trust Bank LTD
MPT Modern Portfolio Theory
UNIL Unilever Company Limited
FML Fan Milk Ghana LTD
PZC PZ Cussons Ghana
GGL Goil Ghana LTD
UGL Uniliver Ghana LTD
TOTAL Total Petroleum Ghana