PORTFOLIO OPTIMIZATION USING THE MARKOWITZ MODEL:

CASE STUDY OF SELECTED COMPANIES IN GHANA 

by 

ALBERT K.M. COFIE

BSC (HONS) COMPUTER SCIENCE AND PHYSICS

FACULTY INTERN, ASHESI UNIVERSITY COLLEGE

OUTLINE

IntroductionReview of Available LiteratureProblem StatementObjectivesMethodResultsConclusionRecommendations

INTRODUCTION

•Investments play a vital role in any economy and can vary from small scale to large scale.

•Typically, an investor would have a collection of different assets (investments) in one place.

•This collection is often referred to as a “Portfolio”.

•An asset in a portfolio can represent a company’s stock (shares) that is traded on stock markets, government bonds, company bonds, Treasury bills, etc.

INTRODUCTION•Every asset is attributed with an expected return and an element of risk

•The expected return and the risk (variance or standard deviation) form an elementary aspect of a portfolio and are used as basis for selecting assets into a portfolio.

•The fundamental problem often faced by investors, which is known as the “Portfolio Selection problem”, is “how” to distribute an investment amount across a number of potential assets (investments).

REVIEW OF AVAILABLE LITERATURE

• 1952 – Markowitz, Harry: Portfolio Selection, • 1959 – Wolfe: Simplex method• 1984 – Perold• 1988 – Tayi and Leonard• 1990 – Dueck and Scheuer – Threshold Accepting

Algorithm• 1991 – Lai• 1992 – Dueck and Winker• 1993 – Speranza • 1995 – Kono & Suzuki• 1996 – Speranza

REVIEW OF AVAILABLE LITERATURE

• 1997 – Chunchachinda• 1997 – Borchers and Mitchel• 1999 – Kono and Wijanayake• 2000 – Winker• 2001 – Gilli and Kellezi• 2001 – Jobst et al• 2003 – Gaspero and Schaerf• 2005 – Konno and Yamamoto• 2007 – Bonami and Lejeune- probabilistic constraints

PROBLEM STATEMENT•Information regarding the risk level of companies and what proportions to invest in portfolios in order to spread the risks for some expected returns are not readily available to the public or prospective investors.

•Lack of knowledge of the risk levels may lead to ill-informed investments which may result in financial losses

OBJECTIVES

The main objectives are• To estimate the sensitivities(risk level) of six

selected companies trading on the Ghana Stock Exchange

• Formulate and solve the Markowitz Model by applying it to the Ghana Stock Exchange for these selected companies

METHOD

• A preliminary analysis was done by regression runs of the return of the companies against the market index

• Markowitz Model was formulated and solved using a quadratic programming add-in in MS Excel and the MS Excel Solver

METHOD

• Source of Data : Bank of Ghana• Type of Data:• 5 year historical, month by month data from

1998 to 2002 of six companies trading on the Ghana Stock Exchange

Contents of Data:• GSE All Share Index

METHOD

• Monthly beginning and closing stock prices of the six companies

• 91-day Treasury bill(also known as the Risk Free Rate

The six companies fall under four sectors of the economy and are:

Banking Sector• Ghana Commercial Bank-(GCB)• SG-SSB Bank-(SG-SSB)• Standard Chartered Bank-(SCB)

METHOD

Insurance Sector• Enterprise Insurance Company Limited

Real Estate• Home Finance Company

Oil and Gas• Total Ghana Limited

METHODInformation gleaned from data• Market Return

• Security Return

• Risk Free Rate

*100

m

Final Index Beginning IndexR

Beginning Index

Pr Pr

Pri

Final ice Beginning ice DividendsR

Beginning ice

100*12f

Annual RateR

METHOD

• FORMULATION OF MARKOWITZ MODEL• Consider a coordinate system of expected

return and standard deviation.• Slope subject to the constraint• • Stating expected return and std dev in general

form

P F

P

R R

1

1N

ii

X

1 1

1N N

F F i F i Fi i

R R X R X R

1

12

2 2

1 1 11

N

i i Fi

N N N

i i i j iji i i

j

X R R

X X X

METHOD

• Find partial derivatives and equate to zero• etc.

• Rewrite in the form• Differentiate using Chain and Product rule•

1

2

1 0

2. 0

d

dX

d

dX

12

2 2

1 1 1 1

(N N N N

i i F i i j iji i i j

j i

X R R X X X

32

2 2 2

1 1 1 1 11

1( 2 2

2

N N N N N

i i F i i i j ij k k j kji i i j ik

j j k

dX R R X X X X X

dX

METHOD

• Simplifying and re-arranging gives

• But is the Lagrange multiplier •

• This yields

21

2 2 1

1 1 1

( )0

N

i i F Ni

k k j kj k FN N Nij ki i i j ij

i i jj i

X R RX X R R

X X X

1

2 2

1 1 11

( )N

i i Fi

N N N

i i i j iji i j

j

X R R

X X X

2

1

0N

k k j kj k Fij k

X X R R

METHOD

• Multiplying ,• By extension

• Let • This gives or

2

1

0N

k k j kj k Fij k

X X R R

21 1 2 2 1 1... ... 0i i i i N N i N Ni i F

i

dX X X X X R R

dX

k kZ X2

1 1 2 2 1 1... ...i F i i i i N N i N NiR R Z Z Z Z Z

1

N

i F N Nii

R R Z

RESULTS

AVERAGE RETURNS AND STANDARD DEVIATIONS OF SELECTED COMPANIES

Invstmts 1 2 3 4 5 6 7

Name All-Share GCB SG-SSB SCB HFC EIC TOTAL

Return -1.98 -3.89 -2.18 -2.70 -2.76 -3.52 -2.29

Std Dev 8.1734 18.6741 13.2915 11.4043 10.3051 16.9908 14.2372

RESULTS

COVARIANCE MATRIX

All-Share GCB SG-SSB SCB HFC EIC TOTAL

All-Share 66.81 112.62 66.62 67.21 53.09 59.54 95.61

GCB 112.62 348.72 90.85 76.75 72.41 54.76 182.88

SG-SSB 66.62 90.85 176.66 50.62 40.47 231.63 82.77

SCB 67.21 76.75 50.62 130.06 67.97 63.82 89.54

HFC 53.09 72.41 40.47 67.97 106.20 59.51 91.54

EIC 59.54 54.76 231.63 63.82 59.51 288.69 55.43

TOTAL 95.61 182.88 82.77 89.54 91.54 55.43 202.70

RESULTS

• Preliminary Analysis: Regression Runs

=component of stock return that is independent of the market’s performance

The rate of return on the market index

A constant that measures the expected change in

given a change in

i i i mr a r

ir return on stock i

ia

mr i ir

mr

RESULTS

SUMMARY OF RESULTS OF REGRESSION RUNS

STOCK BETA

GCB 1.69 0.5154 0.48

SG-SSB -99.81 0.3831 0.62

HFC -80.85 0.4112 0.59

SCB -102.32 0.5377 0.46

EIC -90.64 0.1901 0.81

TOTAL -145.53 0.6901 0.31

2R 21 R

RESULTS

Setting up inputs to the Markowitz Model

Decision Variables

Fraction of portfolio to invest in industry

ir return on company imax (1)k imum risk factor

cov cov company ij ariance between i and j

( . company)N Portfolio size no of

jx

RESULTS

Objective

Markowitz Total Returns:

Constraints

Budget Constraint:

Maximum allowable risk:

1

*i ii

r x

1

1ii

x

1

*cov *j ji ii j J

kx x

N

RESULTS

SOLUTION TO THE MARKOWITZ MODEL

GCB SG-SSB SCB HFC EIC TOTAL GHA

0.01964 0.0893 0.1786 0.1071 0.0536 0.375

CONCLUSION

• A well diversified portfolio is one’s best bet for the growth of their investments

• GCB’s stock: very aggressive and sensitive and good for risk-loving investors

• Total Ghana Stock less risky hence Markowitz invested more in this stock, followed by GCB and the rest.

CONCLUSION

• The Markowitz Model could be solved for a series of expected returns, which could be plotted against standard deviation of returns to produce what is called an efficient frontier

RECOMMENDATIONS• Make continuous historical data accessible• Future research could extend the historical

period to ten or fifteen years• Increase the number of companies to involve

major sectors like oil and gas , agric, banking and finance and services sector

• Provide regular information on the efficient frontier of companies

This will provide periodic and relevant information to prospective local investors

RECOMMENDATION

• Government and policy-makers should include the study of finance and investment in the lower levels of the educational sector e.g. courses run by GSE should be extended to schools

• Companies must not be allowed to charge for data obtained for research and academic purposes

• “Don’t put all your eggs in one basket". Diversify.

THANK YOU!