Transcript PORTFOLIO OPTIMIZATION USING THE MARKOWITZ MODEL: …
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
Introduction Review of Available Literature Problem Statement Objectives Method Results Conclusion Recommendations
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
METHOD
• Information gleaned from data Market Return
R m
• Security Return
R i
Final
Pr Pr
Beginning
Pr
ice
*100 • Risk Free Rate
R f
100 *12
METHOD
• • • • • FORMULATION OF MARKOWITZ MODEL Consider a coordinate system of expected return and standard deviation.
Slope subject to the constraint
R F
1
R F
i N
1
R P X i
R F
P
R F
i N
1
X R i F
i N
1
X i
Stating expected return and std dev in general 1 form
i N
1
i N
1
X i
R i X i
2
i
2
R F
i N
1
N i j
1 1
X X i j
ij
1 2
METHOD
• • • • Find partial derivatives and equate to zero
d
1
dX
1 0 etc.
2.
d
dX
2 0
N
i
1
i i
F
N
i
1
X i
2 2
i N
N
1
j j i
1 Differentiate using Chain and Product rule
X X i
j ij
1 2 •
d
dX k
i N
1
i i
R F
2
i N
1
X i
2
i
2
i N
1
j N
1 1
j X X i j
ij
3 2 2
X k
k
2 2
i N
1
X j
kj
METHOD
• Simplifying and re-arranging gives • • •
i N
1
i N
1
X i i
2
X N
1
i i
2 2
i N
This yields 1
i i
2
i N
1
X i
i
j
1 (
N i
R i N R
1
F X i j j
)
X X
N
k
1 1
j R F k
2
X X
ij i
)
N
i
1
X X j
k
k
2
ij j
kj
N
i
1
R k X
j
kj R F
R k
0
R F
0
METHOD
• • • • Multiplying ,
X
By extension
d
dX i
1 1
i
2 2
i k
2
N
i
1
X
i i
2 Let
Z k
This gives
X k i
F Z
1 1
i
Z
2 2
i kj
R k
R F
0
X N
1
N
1
i
N Ni
i R F
0
Z
i i
2
Z N
1
Z
N Ni
or
R i
R F
i N
1
Z N
Ni
RESULTS
AVERAGE RETURNS AND STANDARD DEVIATIONS OF SELECTED COMPANIES
Invstmts Name Return Std Dev 1 2 All-Share GCB -1.98
8.1734
-3.89
18.6741
3 SG-SSB -2.18
13.2915
4 SCB -2.70
11.4043
5 HFC -2.76
10.3051
6 EIC -3.52
16.9908
7 TOTAL -2.29
14.2372
RESULTS
All-Share All-Share 66.81
GCB SG-SSB SCB 112.62
66.62
67.21
HFC EIC TOTAL 53.09
59.54
95.61
GCB 112.62
348.72
90.85
76.75
72.41
54.76
182.88
COVARIANCE MATRIX
SG-SSB SCB HFC 66.62
90.85
176.66
50.62
67.21
76.75
50.62
130.06
53.09
72.41
40.47
67.97
40.47
231.63
82.77
67.97
63.82
89.54
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
r i
i
r i m i r
a i
=component of stock return that is independent of the market’s performance
r m
i
The rate of return on the market index A constant that measures the expected change in
r i
given a change in
r m
RESULTS
SUMMARY OF RESULTS OF REGRESSION RUNS
STOCK BETA
R
2 GCB 1.69
0.5154
SG-SSB HFC SCB EIC TOTAL -99.81
-80.85
-102.32
-90.64
-145.53
0.3831
0.4112
0.5377
0.1901
0.6901
0.48
1
R
2 0.62
0.59
0.46
0.81
0.31
RESULTS
Setting up inputs to the Markowitz Model
r i
k
max cov
ij
cov (1)
N
( . company)
j x
Decision Variables Fraction of portfolio to invest in industry
RESULTS
Objective Markowitz Total Returns: Constraints Budget Constraint: Maximum allowable risk:
i
1
r i
*
x i
i
1
x i
1
i
1
x j
*cov *
ji x i
N k
GCB 0.01964
RESULTS
SOLUTION TO THE MARKOWITZ MODEL
SG-SSB SCB HFC EIC 0.0893
0.1786
0.1071
0.0536
TOTAL GHA 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.