PORTFOLIO OPTIMIZATION USING THE MARKOWITZ MODEL CASE STUDY

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 • Security Return • Risk Free Rate

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

METHOD • Find partial derivatives and equate to zero • etc. • Rewrite in the form • Differentiate using Chain and Product rule •

METHOD • Simplifying and re-arranging gives • But • • This yields is the Lagrange multiplier

METHOD • Multiplying , • By extension • Let • This gives or

RESULTS AVERAGE RETURNS AND STANDARD DEVIATIONS OF SELECTED COMPANIES Invstmts 1 Name 2 3 4 5 6 7 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

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

RESULTS Setting up inputs to the Markowitz Model Decision Variables Fraction of portfolio to invest in industry

RESULTS Objective Markowitz Total Returns: Constraints Budget Constraint: Maximum allowable risk:

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.

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