MODERN PORTFOLIO THEORY TOOLS A METHODOLOGICAL DESIGN AND APPLICATION

MODERN PORTFOLIO THEORY TOOLS A METHODOLOGICAL DESIGN AND APPLICATION Su Han Wang A research report submtted to the Faculty of Engneerng and the Bult Envronment, of the Unversty of the Wtwatersrand, Johannesburg, n partal fulflment of the requrements for the degree of Master of Scence n Engneerng. Johannesburg, 2008

DECLARATION I declare that ths research report s my own, unaded work. It s submtted n partal fulflment of the requrements for the degree of Master of Scence n Engneerng n the Unversty of Wtwatersrand, Johannesburg. It has not been submtted before for any degree of examnaton n any other Unversty. Su Han Wang day of (year)

ABSTRACT A passve nvestment management model was developed va a crtcal lterature revew of portfolo methodologes. Ths model was developed based on the fundamental models orgnated by both Markowtz and Sharpe. The passve model was automated va the development of a computer programme that can be used to generate the requred outputs as suggested by Markowtz and Sharpe. For ths computer programme MATLAB s chosen and the model s logc s desgned and valdated. The demonstraton of the desgned programme usng securtes traded s performed on Johannesburg Securtes Exchange. The selected portfolo has been sub-categorsed nto sx components wth a total of twenty- seven shares. The shares were grouped nto dfferent components due to the nvestors preferences and nvestment tme horzon. The results demonstrate that a test portfolo outperforms a rsk- free money market nstrument (the government R194 bond), but not the All Share Index for the perod under consderaton. Ths desgn concludes the reason for ths s due n part to the use of the error term from Sharpe s sngle ndex model. An nvestor followng the framework proposed by ths desgn may use ths to determne the rsk- return relatonshp for selected portfolos, and hopefully, a real return.

To my famly, for ther support v

ACKNOWLEDGEMENTS I would lke to thank the followng people whom have helped me durng varous stages of ths report. My frst supervsor, Dr. Harold Campbell, for hs nvaluable gudance, nsghts and tme. To Prof. Snaddon, for hs consstent support and hs wllngness to take over the supervson of ths project after Dr. Campbell left Unversty of the Wtwatersrand. To Ms. Bernadette Sunjka, for her consstent gudance, nsghts and her enthusasm when she was apponted to take over the supervson of ths desgn. To my fr, Randall Paton, for hs patence and assstance n wrtng the MATLAB codes. To Joanne Hobbs, for the detaled foundaton she lad n her honours project. To Thomas Tengen, for hs nsghts n the further mprovements of the mathematcal models. To Mchael Boer, Megan Chatterton, Peter Langeveldt, Mchael Mll, Martn Perold and Po-Hsang Wang for all the proof readng they have done and ther support. v

TABLE OF CONTENTS DECLARATION... ABSTRACT... ACKNOWLEDGEMENTS...v TABLE OF CONTENTS...v LIST OF FIGURES...x LIST OF TABLES...x LIST OF SYMBOLS...x NOMENCLATURE.xv Chapter 1 Introducton...1 1.1 Background...1 1.2 Motvaton...2 1.3 Scope of Desgn...3 1.4 Lmtatons of Desgn...4 1.5 Statement of Task...4 1.6 Methodology...6 Chapter 2 Development of a Passve Management Model Va a Crtcal Lterature Revew of Portfolo Methodologes...8 2.1 Introducton...8 2.2 Modern Portfolo Theory...9 2.3 Fnancal Engneerng...11 2.4 Actve and Passve Management...12 2.4.1 Actve Management...12 2.4.2 Passve Management...13 2.5 Portfolo Constructon...14 2.5.1 Securty Valuaton...15 2.5.2 Asset Allocaton...19 2.5.3 Portfolo Optmsaton...21 2.5.4 Performance Measurement...21 v

2.6 Development of The Model...21 2.6.1 Markowtz s Mean-Varance Framework...22 2.6.2 Sharpe s Sngle Index Model...25 2.6.3 Effcent Market Hypothess...34 2.7 Next Steps...38 Chapter 3 Development of Computer Programme...39 3.1 Desgn Requrement Specfcatons...39 3.1.1 The Objectves...39 3.1.2 Needs Analyss...39 3.1.2.1 Desgn Overvew...39 3.1.2.2 Desgn Requrement Specfcaton...40 3.2 Software Selecton...41 3.2.1 Introducton...41 3.2.2 Types of Statstcal Packages...41 3.2.3 Decson Process...45 3.3 Code Wrtten for Computer Programme...46 3.3.1 Introducton...46 3.3.2 Detaled Computer Programme Logc...46 3.3.3 Fnal Computer Programme...50 3.3.4 Testng of Computer Programme...54 Chapter 4 Selecton of Test Portfolo...57 4.1 Choce of Consttuents n Test Portfolo...57 4.1.1 Portfolo Selecton...57 4.2 Choce of Index...66 Chapter 5 Desgn Outcomes...67 5.1 Introducton...67 5.2 The Data...67 5.3 Results wth Dscusson...67 5.3.1 Results of Components...69 5.3.2 Results of Overall Test Portfolo...96 5.4 Summary...105 v

Chapter 6 Conclusons & Further Work...107 6.1 Conclusons...107 6.2 Drectons for Further Work...110 Chapter 7 References & Bblography...112 References...112 Bblography...119 Appces..121 Appx A: MATLAB Code for Analysng Components of the Test Portfolo Wth Error Terms...122 Appx B: MATLAB Code for Analysng Components of the Test Portfolo Wthout Error Terms...134 Appx C: Instructons for Runnng MATLAB Codes...149 Appx D: MATLAB Code for Valdatng The Computer Programmes...161 Appx E: Valdaton Results...164 Appx F: Sample Sze of Test Portfolo... 168 Appx G: Ratonale for Shares Inclusons n the Test Portfolo...170 Appx H: Ordnary Shares Lsted Based on Market Captalzaton...174 Appx I: Dvds & Weghtngs Used for Beta Calculatons...188 v

LIST OF FIGURES Fgure 1.1: Proposed Methodology...6 Fgure 2.1: Structure of Lterature Revew & Model Development...9 Fgure 2.2: Asset Allocaton Approaches...19 Fgure 2.3: Markowtz's Mean-Varance Framework...23 Fgure 2.4: Sharpe's Sngle Index Model (Part I)...27 Fgure 2.5: Sharpe Sngle Index Model (Part II)...28 Fgure 2.6: Process Flow Dagram of the Model...37 Fgure 3.1: Types of Statstcal Packages Consdered...42 Fgure 3.2: Order of Dscusson...47 Fgure 3.3: Requred Outputs...47 Fgure 3.4: Inputs Parameters Used...48 Fgure 3.5: Process Flow Dagram for Beta Calculaton...49 Fgure 3.6: Process Flow Dagram for Alpha Calculaton...50 Fgure 3.7: Overall Process Flow Dagram for MATLAB code Includng Error Terms..52 Fgure 3.8: Overall Process Flow Dagram for MATLAB code Excludng Error Terms.53 Fgure 3.9: Steps for Valdaton...54 Fgure 4.1: Structure of Test Portfolo...60 Fgure 4.2: All Share Economc Group Breakdown...61 Fgure 5.1: Structure of Dscusson for Desgn Outcomes...68 Fgure 5.2: Weghted Average Beta for Balanced Component over Test Perod...71 Fgure 5.3: Weghted Average Alpha for Balanced Component over Test Perod...72 Fgure 5.4: Returns Excludng Errors for Balanced Component over Test Perod...73 Fgure 5.5: Returns Includng Errors for Balanced Component over Test Perod...74 Fgure 5.6: Weghted Average Beta for Conservatve Component over Test Perod...76 Fgure 5.7: Weghted Average Alpha for Conservatve Component over Test Perod...77 Fgure 5.8: Returns Excludng Errors for Conservatve Component over Test Perod...78 Fgure 5.9: Returns Includng Errors for Conservatve Component over Test Perod...79 Fgure 5.10: Weghted Average Beta for Core Alternatve Component over Test Perod80 x

Fgure 5.11: Weghted Average Alpha for Core Alternatve Component over Test Perod...81 Fgure 5.12: Returns Excludng Errors for Core Alternatve Component over Test Perod...82 Fgure 5.13: Returns Includng Errors for Core Alternatve Component over Test Perod...83 Fgure 5.14: Weghted Average Beta for Core Component over Test Perod...84 Fgure 5.15: Weghted Average Alpha for Core Component over Test Perod...86 Fgure 5.16: Returns Excludng Errors for Core Component over Test Perod...86 Fgure 5.17: Returns Includng Errors for Core Component over Test Perod...87 Fgure 5.18: Weghted Average Beta for Md-Term Component over Test Perod...89 Fgure 5.19: Weghted Average Alpha for Md-Term Component over Test Perod...90 Fgure 5.20: Returns Excludng Errors for Md-Term Component over Test Perod...91 Fgure 5.21: Returns Includng Errors for Md-Term Component over Test Perod...92 Fgure 5.22: Weghted Average Beta for Small Caps Component over Test Perod...93 Fgure 5.23: Weghted Average Alpha for Small Caps Component over Test Perod...94 Fgure 5.24: Returns Excludng Errors for Small CapsComponent over Test Perod...95 Fgure 5.25: Returns Includng Errors for Small Caps Component over Test Perod...95 Fgure 5.26: Daly Comparson of Expected Returns Excludng Errors of Test Portfolo over Test Perod...97 Fgure 5.27: Daly Comparson of Expected Returns Includng Errors of Test Portfolo over Test Perod...99 Fgure 5.28: Repo Rate Changes over Test Perod...101 Fgure 5.29: Exchange Rate over Test Perod... 102 Fgure 5.30: Average Returns Excludng Errors Comparsons Over Test Perod...104 Fgure 5.31: Average Returns Includng Errors Comparsons Over Test Perod...105 x

LIST OF TABLES Table 3.1: Crtera for Desgn Requrements...41 Table 3.2: Decson Matrx of Concepts...46 Table 4.1: Securtes Categores for Portfolo Sub-Dvson...62 Table 4.2: Securtes Included n Test Portfolo, Includng Sector Dvson...64 Table 4.3: Investment Composton...66 Table 5.1: Summarsed Results for Balanced Component...75 Table 5.2: Summarsed Results for Conservatve Component...79 Table 5.3: Summarsed Results for Core Alternatve Component...83 Table 5.4: Summarsed Results for Core Component...88 Table 5.5: Summarsed Results for Md-Term Component...92 Table 5.6: Summarsed Results for Small Caps Component...96 Table E1: Outcomes from Valdatng Computer Programme...165 Table E2: Outcomes from Manual Calculatons...166 Table E3: Errors Comparson Between Table E1 and Table E2...167 Table F1: Calcualton of Sample Sze n Terms of Confdence Intervals...169 Table G1: Ratonale for Shares Inclusons...170 Table H1: Ordnary Shares Lsted Based on Market Captalzaton... 174 Table I1: Dvds & Weghtngs for Balanced Portfolo...188 Table I2: Dvds & Weghtngs for Conservatve Portfolo...188 Table I3: Dvds & Weghtngs for Core Alterantve Portfolo... 189 Table I4: Dvds & Weghtngs for Core Portfolo...189 Table I5: Dvds & Weghtngs for Md-Term Portfolo...190 Table I6: Dvds & Weghtngs for Small Caps Portfolo...190 x

LIST OF SYMBOLS,t BA Alpha of securty, at tme t Alpha estmate, by regresson analyss, n ths desgn OLS, of the ndvdual securty Alpha calculated by applyng adjusted beta value usng Vascek s technque ML Alpha calculated by applyng adjusted beta value usng Merrll Lynch s adjustment Beta estmate by regresson analyss, n ths desgn OLS, of the ndvdual securty Beta of securty, at tme t,t Beta of securty, j at tme t j,t BA ML Average of the betas of all stocks n the portfolo Adjusted beta value usng Vascek s technque Adjusted beta value usng Merrll Lynch s adjustment D Dvd of securty,, at tme t,t e Random error assocated wth securty at tme t,t nr Nomnal nterest rate I Returns n j th securty n th component, j m N P 0 P,t Number of compoundng perods Sample sze Intal prce of an ndvdual securty,.e. the ntal reference pont Prce of an ndvdual securty,, at tme t r Effectve nterest rate R Return of an ndvdual securty,, at tme t,t R Sample mean of ndvdual securty, x

R Sample mean of ndvdual securty, at tme t,t R Return on market at tme t M,t R Sample mean of market M R Sample mean of market at tme t M,t R Return of n th subportfolo or component n,p R Overall return of test portfolo OP, j,t Correlaton coeffcent between R and 2 Varance of securty 2 Varance of market at tme t M,t 2 Varance of the beta estmate P R j at tme t Cross- sectonal standard devaton of all beta estmate n the portfolo Standard devaton of ndvdual securty,, at tme t,t Standard devaton of ndvdual securty, j at tme t j,t, j,t Covarance between R return on asset and R j return on asset j at tme t w w j Weght assocated wth securty Weght assocated wth securty j w Weght assocated wth n th securty n n th subportfolo or component nn x j Amount of nvestment n j th securty n th subportfolo or component Investment fracton assocated wth th subportfolo or component x

NOMENCLATURE AFB Alexander Forbes Lmted AGL Anglo Amercan plc ALSI FTSE/ JSE Afrca All Share Index AMS Anglo Platnum Ltd. ASA ABSA Group Ltd. BA Bayesan Adjustments BAW Barloworld Lmted BCX Busness Connexon Group Lmted BDEO Bdvest Call Opton BVT The Bdvest Group Ltd. CLH Cty Lodge DRS Desgn Requrement Specfcaton DST Dstell Group Lmted EMH Effcent Market Hypothess ERP ERP.com Holdngs Ltd. FBR Famous Brand Lmted FSR FrstRand Lmted IPL Imperal Holdngs Ltd. JSE Johannesburg Securtes Exchange LBT Lberty Internatonal plc ML Merrll Lynch Adjustment MPT Modern Portfolo Theory MTN MTN Group Ltd. MUR Murray & Roberts Holdngs Lmted OLS Ordnary Least Squares PIK Pck n Pay Stores Lmted PPC Pretora Portland Cement Company Ltd. REM Remgro Lmted xv

RLO SAB SBK SHP TBS VNF WHL Reunert Lmted SABMller plc Standard Bank Group Ltd. Shoprte Holdngs Ltd. Tger Brands Lmted VenFn Ltd. Woolworths Holdngs Ltd. xv

Chapter 1 Introducton 1.1 Background South Afrca s a country regarded as a developng and emergng market (Internatonal Marketng Councl of South Afrca, 2007 and L, 2007), where there s potental for growth, thus, ts bullsh economc phase wll contnue for the very near future (L, 2007). The mmedate entry to a country s economy s through ts securtes market, n ths case, the JSE Securtes Exchange (hereforth known as JSE) (JSE, 2007). The JSE Securtes Exchange South Afrca was prevously known as the Johannesburg Stock Exchange. JSE s South Afrca s only securty exchange and t s also ranked as Afrcan s largest securty exchange. The JSE has operated as a tradng ground for fnancal products for nearly 12 decades. Therefore the JSE s a valuable money market nstrument n South Afrca s economc landscape (JSE, 2007). The JSE s not as heavly traded as many other exchange markets, for example: New York, Chcago and London. The effcency of the JSE s an ssue of mportance to South Afrcan nvestors. Durng the last three decades numerous studes have addressed ths ssue and concluded that the market effcency for JSE s sem-strong 1 (Correa et al., 2003). A securtes exchange may be a far reflecton of an economy. Many nvestors consder enterng the securty market to gan a better access to the overall market. Therefore, some may thnk beatng or outperformng the market s not a dffcult task n an emergng 1 Sem-strong asserts that securty prces adjust rapdly to the release of all new publc nformaton, thus the securty prces fully reflect all publc nformaton. Ths s dscussed n more detals n Chapter 2. 1

market. However, the consstent out-performance of benchmark postons 2 s rare (Hobbs, 2001). The rarty of out-performng the market gves rse to the two broad classes of market vews as well as the asset nvestment management approach. When an nvestor analyses a market, he or she ts to take one of the two vews namely contraran 3 or smart money 4 vews (Malkel, 1999 and Schweser Kaplan Fnancal, 2006b). Once an nvestor has commtted to one of these tradng vews, the management approach may be decded. The approach that an nvestor can adopt s ether the actve or the passve management approach. For the actve management approach, the nvestors need to research the market thoroughly and know when they are to sell or to buy; whereas for the passve approach, an nvestor mostly practces the buy-and-hold strategy. Passve management s favoured by rsk-averse nvestors, where the key to proftablty les wth portfolo selecton and asset allocaton. The allocaton between actve and passve management approaches deps on sklls, and rather subjectvely, personal preferences (Sorensen et al., 1998). 1.2 Motvaton South Afrca s GDP (PPP) 5 per capta ncome s $13300, ths s lower than the developed economes of USA wth $44000, Japan $33100, UK $31800 and France $31300 n 2006 6. When ctzens save, ther funds may not be suffcent to hre fnancal advsors and managers 7 due to the hgh servce costs nvolved. Nevertheless, these prvate nvestors 2 In ths desgn, benchmark poston refers to the ndex chosen,.e. ALSI. 3 Contrarans argue that the majorty of the market s generally wrong; hence they do the opposte to what the majorty of nvestors are dong. (Schweser Kaplan Fnancal, 2006b: p.170) 4 Smart nvestors know what they are dong, so nvestors better follow them whle there s stll tme. (Schweser Kaplan Fnancal, 2006b: p.170) 5 Gross domestc product at purchasng power party, where purchasng power party (PPP) s a theory that states the exchange rates between currences are n equlbrum when ther purchasng power s the same n each of the two countres. 6 These fgures, were lsted by CIA World Factbook, and were taken drectly from http://en.wkpeda.org/wk/lst_of_countres_by_gdp_%28ppp%29_per_capta. 7 Ths s referrng to the general publc and does not nclude the eltes of the socety. 2

may seek sutable nvestment opportuntes on the JSE for ther funds. By nvestng accurately and cautously, these nvestors can avod the reducng purchasng power of money due to the nterest rate, nflaton and tax. An ncrease n nterest rate leads to the ncreased nterest costs for the busnesses, hence busnesses rase the prces of ther goods. As a drect consequence, ths leads to the reducng purchasng power of money as for the same amount of money, customers can now buy less than what they could pror to the nterest ncrease. The desgn proposed n ths document attempts to provde a framework whch these nvestors can use to make better nvestment decsons. Some questons that an nvestor may ask when conductng the nvestgaton related to ths desgn are the followng: What are the aspects that one should consder when constructng an nvestment portfolo? How may one determne the optmal splt between asset classes wthn the portfolo? How would one determne a reasonable rate of returns on the portfolo? Ths desgn attempts to address these pertnent questons, hence prvate nvestors wll gan understandng and knowledge n ths feld. As a result, an nvestor can make sound decsons on nvestments based upon modern theory. 1.3 Scope of Desgn The objectve of ths desgn s to develop a passve portfolo management model usng both Markowtz s mean-varance framework and Sharpe s sngle ndex model that may be easly used by a prvate nvestor through ts automaton va a computer program. The market for the automated models s prvate nvestors or the potental prvate nvestors on the JSE. To acheve ths objectve, the desgn s approached n two stages. Frstly, a model for passve portfolo management usng Modern Portfolo Theory (hereforth known as MPT) tools s developed va a crtcal lterature revew. Secondly, a computer 3

programme s developed. The computer programme s the valdaton vehcle for the model developed. In the frst stage, the model valdaton s completed through an exstng test portfolo. The test portfolo s then passed through the computer programme, where a set of results are generated. The reasons for securty selecton as well as the outcomes are dscussed. The specfc outcomes are the returns of portfolo. These wll be compared to the rsk-free money market nstrument,.e. a government bond, n the chapters to come n ths document. 1.4 Lmtatons of Desgn A lmtaton of ths desgn s that the model developed s lmted to MPT related tools, the valdaton conducted for the computer programme was usng lmted sectors on the JSE, ths s seen as a lmtaton snce the lmted sectors do not gve a holstc vew of JSE, short-sellng of securtes has not been dscussed n ths desgn report, and R-squared statstcs have been left out of ths desgn report, as ths desgn focuses on the desgn of the methodology. 1.5 Statement of Task Ths desgn ams to: develop a model for passve portfolo management usng MPT tools va a crtcal lterature revew, and develop a computer programme where the model s valdated through the use of a test portfolo. One of the elements that the computer programme wll be evaluated on s ts user-frlness (ths s defned n Desgn Requrement Specfcaton). 4

1.6 Methodology In Chapter 2, a crtcal lterature revew s dscussed. Through ths dscusson, a model for passve portfolo management s developed. In Chapter 3, the development of computer programme s developed. Ths dscusson had been dvded nto three stages, namely desgn requrement specfcaton, software selecton and the code wrtten for computer programme. Each of the three stages are dscussed below: Frst stage, desgn requrement specfcaton of a computer programme s ntroduced, where the crtera and constrants of the computer programme are tabulated and dscussed. The computer programme s desgned based on the model for passve portfolo management. Second stage, the computer packages consdered for the computer programme s dscussed. The dscusson ncludes the advantages and dsadvantages of each of the packages. Based on ths, an evaluaton matrx s drawn, and a fnal decson s reached on the package selecton. Thrd stage, the detaled desgn logc s dscussed, where the procedures on the formaton of the computer programme s descrbed. Ths stage concludes wth the valdaton of the model. In Chapter 4, the applcaton of the valdated automated model s necessary. Therefore, the test portfolo and the benchmarks are selected. The reasons for these selectons are ntroduced. In Chapter 5, the outcomes acheved by applyng the automated model to the test portfolo are analysed and dscussed n detal. In Chapter 6, conclusons and fndngs of ths desgn are revsted and summarsed. The proposed methodology s graphcally represented n Fgure 1.1 below. 5

Crtcal Lterature Revew where model s developed Development of Computer Programme Desgn Requrement Specfcaton Software Selecton Computer Programme Code Development Test Portfolo Selecton Model Valdaton va use of a test portfolo Analyss of the Test Portfolo Conclusons Fgure 1.1: Proposed Methodology In summary, the fundamental elements of software development project management methodology have been employed. Thus, n the forthcomng chapters of ths report, the crtcal lterature revews are dscussed, n partcular, the Markowtz s mean-varance model and Sharpe s sngle ndex model are dscussed crtcally n the lterature revew. The MPT model forms the requrement for the development of the computer programme. A test portfolo s chosen for the valdaton of the automated model, and the outcomes are 6

dscussed. Lastly, the major conclusons reached from the analyss are dscussed, and a dscusson of possble mplcatons for further work. 7

Chapter 2 Development of a Passve Management Model Va a Crtcal Lterature Revew of Portfolo Methodologes 2.1 Introducton In ths chapter, the lterature that forms the foundatons and technques of MPT s crtcally revewed. The structure of the revew s represented n Fgure 2.1. The revew begns wth the broad concept of fnancal engneerng, narrowng down the concept to the specfc management approaches that are currently beng employed n the ndustry, such as actve and passve management 8. The prmary focus of ths revew s on the passve management approach ncludng the foundatons and the technques assocated wth t. The motvaton for usng the passve management approach wll be dscussed later. A revew of a general portfolo constructon method whch forms the base of the model desgn methodology s then undertaken followed by an analyss of the applcaton of Markowtz s mean-varance framework and Sharpe s sngle ndex model. Ths chapter concludes wth the presentaton of passve management MPT model whch s the prmary objectve of ths desgn. 8 Actve management approach refers to the use of human element n managng a portfolo. Passve management refers to an nvestment strategy whch mrrors ndex composton and doesn t attempt to beat the market, (Hobbs, 2001). 8

Modern Portfolo Theory Fnancal Engneerng Management Approaches Market Vews Actve Management Approach Passve Management Approach Contraran Vew: Do opposte to what majorty nvestors are dong Smart Money Vew: Do what the smart nvestors are dong Markowtz s Mean- Varance Framework Portfolo Constructon Methodology Sharpe s Sngle Index Model Model Fgure 2.1: Structure of Lterature Revew & Model Development 2.2 Modern Portfolo Theory MPT s an overall nvestment strategy that seeks to construct an optmal portfolo by consderng the relatonshp between rsk and return (Correa et al., 2003). Ths theory s generally perceved as a body of models that descrbes how nvestors may balance rsk and reward n constructng nvestment portfolos. (Holton, 2004: p. 21). MPT s otherwse known as portfolo management theory (Relly, 1989). The man ndcators used n MPT are the alpha and the beta of nvestment (Hobbs, 2001). Beta s a measurement of volatlty of an asset or a portfolo relatve to a selected benchmark, usually a market ndex. A beta of 1.0 ndcates that the magntude and drecton of movements of returns for an asset or a portfolo are the same as those of the benchmark. A beta value greater than 1.0 ndcates the hgher volatlty, and a beta value 9

less than 1.0 ndcates the lesser volatlty when measured aganst the benchmark (Yao et al., 2002). Alpha calculates the dfference between what the portfolo actually earned and what t was expected to earn gven ts level of systematc 9 rsk, beta value. A postve alpha ndcates return of the asset or the portfolo exceeds the general market expectaton. A negatve alpha ndcates return of the asset or the portfolo falls short of the general market expectaton (Yao et al., 2002). Although the growth of MPT has been both normatve and theoretcal, there are some general ssues assocated wth MPT (Compass Fnancal Planner Pty Ltd., 2007), as follows: 1) Volatlty s a measure of rsk n a hstorcal perod. One reles heavly on hstorcal data when attemptng to predct the future. It can also be understood as a measure of uncertanty that quantfes how much a seres of nvestment returns vares around ts mean or average. Mathematcally, volatlty s represented by standard devaton (Yao et al., 2002). Uncertanty s assocated wth randomness and one of the best ways to deal wth randomness s the use of non-parametrc models, namely neural networks (Harvey et al., 2000). Non-parametrc refers to nterpretaton whch does not dep on the data fllng any parameterzed dstrbutons (Wnston, 2004). A neural network s a set of nodes, whch can be categorsed nto three components, namely the unts, neurons and processng elements. A neural network s usually appled to pattern recognton, content addressable recall and approxmate, common sense reasonng (Campbell, 2007). 2) One should not put too much fath n an effcent portfolo performng at all well f world markets become unstable for a lttle whle (Harvey et al, 2000). A study done by Merrll Lynch n 1979 showed that a typcal dversfed nvestment portfolo elmnates so much of the specfc rsk, that roughly 90 percent of all the 9 Systematc rsk refers to the rsks that cannot be dversfed away, such that they are nherent n the system. 10

portfolo rsk s market rsk, therefore f market s unstable, an nvestor should not be dsapponted f the portfolo s not performng (Derby Fnancal Group, 2008). Further to the ssues that are assocated wth MPT, the mplementatons of ths theory have also been lmted. The three major reasons for the lmted mplementaton of MPT are (Elton et al., 1976: p. 1341): 1) The dffculty n estmatng and dentfyng the type of data necessary for correlaton matrces. 2) The tme and expenses needed for generatng effcent portfolos that s the costs assocated wth solvng a quadratc programmng problem. The nput data requrements are volumnous for portfolos of a practcal sze (Renwck, 1969). 3) The dffculty n educatng portfolo managers to express the rsk-return trade-off n terms of covarances, returns and standard devatons (Renwck, 1969). The lterature suggests that the development of MPT has led to the development of the feld of fnancal engneerng. 2.3 Fnancal Engneerng Fnancal engneerng s a relatvely new dscplne; t orgnated n the late 1980s when the feld of fnance was changng (Fnancal Engneerng News, 2006). Ths s one of the new dscplnes whch emerged from MPT. Fnancal engneerng s the art 10 of rsk management where fnancal opportuntes are exploted through complex fnancal formulatons. Ths s supported by the followng: Topper (2005: p. 3) asserts that (t)he art of fnancal engneerng s to customze rsk. Fnancal engneerng s based on certan assumptons regardng the statstcal behavour 10 The word art refers to the methods or the technques used. 11

of equtes (securtes), exchange rates and nterest rates. In MPT, customzng rsk refers to managng a measurement of uncertantes of expected returns (Yao et al., 2002). Addtonally, Jack Marshall, as cted n the Fnancal Engneerng News (2006), suggests that (f)nancal engneerng nvolves the development and creatve applcaton of fnancal theory and fnancal nstruments (securtes) to structure solutons to complex fnancal problems and to explot fnancal opportuntes. Through ths dscplne, one would be able to reach sound decsons regardng savngs, nvestng, borrowng, lng and managng rsk (Fnancal Engneerng News, 2006). One of the core objectves of fnancal engneerng s to manage rsk; therefore the actve and passve management approaches need to be understood, as each refers to a dfferent method of portfolo rsk management. 2.4 Actve and Passve Management To gan a better understandng of these management approaches, ths report proceeds to dscuss both actve and passve management approaches n more detal. 2.4.1 Actve Management Ths management approach refers to the actve frequent tradng of securtes. It s an attempt to outperform the market as measured by a partcular ndex (Sharpe, 2006 and Frank Russell Company, 2006). An actve portfolo manager uses research fndngs and market forecasts to purchase securtes that he beleves wll outperform varous benchmarks; when he feels the value of the nvestment s at ts peak, he wll sell the securtes (Hobbs, 2001). 12

Ths approach s assocated wth the constant rebalancng of asset classes wthn a portfolo (Evanson Asset Management, 2006). Rebalancng s referrng to the process of resettng a portfolo at a predetermned nterval back to a default asset allocaton (Compass Fnancal Planner Pty Ltd., 2007). Rebalancng can also mean adjustng the weght of each asset n the portfolo or droppng certan assets from the portfolo (Yao et al, 2002). The core beneft of an actve nvestment strategy s the potental for hgher returns. The greatest drawbacks are the hgh operatng expenses (Hobbs, 2001 and Evanson Asset Management, 2006). 2.4.2 Passve Management Passve management s commonly known as ndexng. It s an nvestment approach based on nvestng n dentcal securtes, n smlar proportons as those n an ndex (Sharpe, 2006 and Evanson Asset Management, 2006). Passve managers generally beleve t s dffcult to outperform the market, thus strateges such as purchasng, holdng and adjustng a selecton of securtes are used to replcate the performance of a gven ndex (Hobbs, 2001). The benefts of a passve management strategy are the lower operatng expenses and acton-free requrements from nvestors (Hobbs, 2001 and Frank Russell Company, 2006). Passvely managed portfolos seek to provde only the market returns, hence ndex performance dctates portfolo performance (Mesrow Fnancal Holdngs Inc., 2006). In lght of passve management, acton-free means that on average the same performance can be acheved by smply buyng the entre asset class or a representatve sample (as the chosen benchmark) wthout usng ether securty selecton or market tmng (Hultstorm, 2007). 13

Passve portfolo management s desgned to be stable and to match the long term performance of one segment of the captal market. It has dstnct sectoral and asset emphass depng on the nvestors atttude toward rsk and the economc envronment (Rudd, 1980). Whle the understandng of both management approaches allow rsk assocated wth portfolo to be optmsed (Ln et al., 2004), the model focuses on passve management, the buy- and- hold strategy. Cheng et al. (1971: p. 11) have explaned ths choce, (t)he buy- and- hold strategy under effcent markets s an optmal strategy snce t mnmzes transacton costs. The reason for ths choce s that the foundatons of MPT form part of the orgn for passve management approach (Hobbs, 2001). The foundaton of MPT les n Markowtz s and Sharpe s work, both of whch were developed n the 1950s and 60s (Hobbs, 2001). The prmary reason for these choces of models was that these models have rekndled nterest n normatve (modern) portfolo theory (Frankfurter, 1990); ths s renforced by wnnng the 1990 Nobel Prze n economcs (Njavro et al., 2000). Pror to the theoretcal dscusson of Markowtz s mean-varance framework and Sharpe s sngle ndex model, n secton 2.6.1 and secton 2.6.2 respectvely, t s mportant to understand the methodologcal framework, that s, portfolo constructon through whch these models are appled as set out n secton 2.5. 2.5 Portfolo Constructon The applcatons of MPT are outlned as follows accordng to Hagn (1979): securty valuaton, asset allocaton, portfolo optmsaton, and performance measurement. 14

Each of the four steps s dscussed below. 2.5.1 Securty Valuaton Ths s the frst step n developng a portfolo. At ths ntal stage, one needs to be able to select securtes wth the potental for sustanable growth (Malkel, 2003). Value nvestng refers to the determnaton or dentfcaton of a frm s ntrnsc value 11 (Buffet et al., 2002 and Bernsten, 1992). Value nvestng s an nvestment paradgm that generally nvolves the dentfcaton and buyng under-prced securtes (Graham et al., 1962). The ntrnsc value can be estmated by the usng two of the most commonly used technques, namely the fundamental and the techncal analyses, dscussed below. 1. Fundamental Analyss Fundamental analyss s a tool that fnancal analysts use to determne a frm s value through ts fnancal data and operatons. The vew s echoed by Malkel (1999: p. 127), who asserts that (f)undamental analyss s the technque of applyng the tenets of the frm-foundaton theory to selecton of ndvdual stocks (securtes). Ths analyss can be used to determne a securty s proper value. The suggested determnants are (Malkel, 1999): expected growth rate, expected dvd payout, and degree of rsk. Ths choce of determnants s echoed by Graham et al. (1962). These three determnants are usually predcted usng a frm s hstorcal fnancal data. As a result, sets of ratos are generated. A rato expresses the relatonshp between one quantty and another, thus 11 The underlyng far value of a stock based on ts future earnngs potental. 15

through rato analyss one would be able to tell how a frm s dong, what ts fnancal condtons are and what ts weaknesses are (Fenberg, 2005). Ratos are often used by analysts to make predctons regardng the future, hence the factors whch affect these ratos should also be consdered. The usefulness of the ratos s depent upon the analyst s sklful applcaton and nterpretaton of them (Correa et al., 2003). Ratos often used for the fnancal analyss are (Fenberg, 2005): Return On Equty (ROE) Debt/ Equty Rato Prce Earnng Rato (P/E) Earnngs Per Share Dvd Per Share Dvd Yeld Ths report wll, thus, use ratos, to determne a frm s fnancal poston. These ratos are usually gven n a frm s fnancal statements. Fundamental analyss consders the varables that are drectly related to the company tself, rather than the overall state of the market. Techncal analyss, on the other hand, consders the overall market drectly and complements the fundamental analyss. 2. Techncal Analyss Techncal analyss s usually understood as the makng and nterpretng of securty charts. From these securty charts, the past (both movements of common securty prces and the volume of tradng) wll be studed for an ndcaton of the lkely drecton of future change. Ths s supported by Ryan (1978: p. 116), who says, (t)echncal, or chart, analyss s the term appled to the work of a partcular school of stock (securty)-market analysts whose theores of stock (securty) prce movements rely heavly on the use and nterpretaton of varous types of charts or graphs. 16

The key prncples of techncal analyss are as follow (Standard Bank Group, 2006): everythng s dscounted and reflected n market prces, prces move n trs and trs persst, and market acton s repettve. Ths report uses ths stance as proposed by Standard Bank Group. Techncal analyss prncples are based on the market movements, where t s assumed that the movements are repettve and all nformaton s reflected n the market prces. 3. Combnaton of Fundamental & Techncal Analyses Instead of usng ether fundamental or techncal analyss alone n order to analyse a frm, t s recommed to use the combnaton of both together for frms analyss. One of the most sensble procedures for selectng the securtes whch are attractve for purchase can be summarzed by the followng three rules (Malkel, 1999). The followng rules also concde wth Buffet s methodology (Buffet et al., 2002). Rule 1: Buy only companes that are expected to have above-average earnngs growth for fve or more years. (Malkel, 1999: pp. 141-142) The sngle most mportant element contrbutng to the success of most securty nvestments s an extraordnary long-run earnngs growth rate. The contnued, repeated performance s more mpressve than a sngle occurrence (Graham et al., 1962). Ths refers to the sustanablty of the frm. Therefore, the securty whch has been performng consstently n the past s more lkely to be purchased. Ths s usually done by examnng the tr for prce earnng (hereforth know as P/E) rato. P/E rato represents a valuaton rato of a company s current share prce compared to ts per-share earnng. In general, a hgh P/E rato suggests that nvestors can expect hgher earnngs growth n the future compared to companes wth a lower P/E. 17

Rule 2: Never pay more for a stock (securty) than ts frm foundaton of value. (Malkel, 1999: pp. 142-143) Ths rule can be summarsed as never payng more for a securty than ts ntrnsc value. Ths renforces Buffet s approach of ntrnsc value nvestments (Buffet et al., 2002). Ths valuaton process usually conssts of the followng basc components (Graham et al., 1962): expected future earnngs, expected future dvds, captalzaton rates of dvds and earnngs, and asset values It should be noted that these four components nclude a number of elements that are both quanttatve and qualtatve n nature. Chef among these are the past and expected rates of proftablty, stablty and growth; the abltes of the management va corporate governance concept (Graham et al., 1962). A rough estmaton of a frm s ntrnsc value s usually calculated by ts Return on Investment (ROI) rato. Rule 3: Look for stocks (securtes) whose stores of antcpated growth are of the knd on whch nvestors can buld castles n the ar. (Malkel, 1999: pp. 143-144) Ths rule refers to the possblty of future news beng released by the frm whch wll affect the securty s prce. Ths can be demonstrated wth use of Economc Value Added (henceforth known as EVA). EVA s a fnancal measure that attempts to capture a creaton of shareholder wealth over tme (Correa et al., 2003). Thus, EVA s a relevant performance measure for ths rule. EVA s calculated by takng a frm s proft after tax then subtracts the rate of the cost of the captal multpled by the average total assets less the average non-nterest bearng current labltes (Fenberg, 2005). 18

2.5.2 Asset Allocaton Portfolo theory ams to optmse the relatonshp between rsk and reward for an nvestment, and ths optmsaton s reached through dversfcaton of assets. Asset allocaton s the dvson of nvestments among asset categores, that s (a)asset allocaton s an nvestment portfolo technque that ams to balance rsk and create dversfcaton by dvdng assets among major categores such as cash, bond, stocks (securty), real estate and dervatves. (Investopeda Inc., 2003). Asset allocaton wth effcent dversfcaton s the heart of portfolo theory (Jacquer et al., 2001). Asset allocaton s a major determnant of return and rsks, as well as the nvestment performance (Elton et al., 2000 and Derby Fnancal Group, 2008). The process of asset allocaton ncludes one or all of the followng approaches, and they are dsplayed n Fgure 2.2 below: Asset Allocaton Strategc Tactcal Dynamc Fgure 2.2: Asset Allocaton Approaches Strategc asset allocaton refers to the use of hstorcal data n an attempt to understand how the asset has performed and predct ts future performance. Tactcal asset allocaton uses perod assumptons regardng performance and characterstcs of the asset and/ or the economy. Dynamc asset allocaton s depent upon the changes n nvestors crcumstances (Derby Fnancal Group, 2008). 19

Furthermore, there are two attrbutes that need to be consdered under asset allocaton (Gallant, 2005): a) Fnancal stuaton and nvestment goals Items consdered are the age of the nvestors, the amount of captal avalable and the possble future needs and nvestment purposes. Based on dfferent fnancal goals set, an nvestor chooses dfferent securtes. For example, f an nvestor s rsk- seekng and the nvestment perod s short-term, then dervatves would be a better opton than cash and bonds. b) Personalty and rsk tolerance One should decde, whether one s wllng to encounter more rsks n exchange for hgher potental returns. An nvestor needs to decde on what level of rsk he or she wants to take n order to receve a hgher return. Thus for a rsk-seekng 12 nvestor, an aggressve portfolo can be formed and hgher returns can be the outcome. Asset allocaton s depent on the two attrbutes mentoned above. An nvestor s fnancal poston, nvestment goals and personal rsk tolerances would affect the asset classes chosen. The most famlar rule of thumb for asset allocaton are (Campbell, 2002): Aggressve nvestors should hold stocks (securty), conservatve nvestors should hold bonds. Long-term nvestors can afford to take more stock market rsk than short-term nvestors. That s dfferent types of nvestors and tme horzons set for nvestments would affect the asset classes chosen. For example: for a conservatve nvestor 13, he/ she would seek to mantan the purchasng power of hs/ her money. Ths s usually done by holdng the rsk- free securty, namely the bonds. Alternatvely, for a rsk-seekng nvestor, he/ she would seek to obtan a hgher return; therefore he/ she would consder securtes n hs/ her nvestment portfolos. 12 Rsk- seekng refers to aggressve. These terms wll be used nterchangeably throughout ths report. 13 Conservatve refers to rsk- averse. These terms wll be used nterchangeably throughout ths report. 20

2.5.3 Portfolo Optmsaton Portfolo optmsaton refers to a group of assets whch have been grouped together to ether maxmse the returns for a gven level of rsk or to mnmse the rsk for a gven expected return (Cuthbertson et al., 2004 and WebFnance Inc., 2007a). The goal of portfolo optmsaton s to maxmze the nvestor s expected utlty by takng nto account all relevant nformaton (Sharpe, 2006). Expected utlty refers to the total satsfacton receved or experenced. 2.5.4 Performance Measurement Performance can be defned as the outcomes of nvestment actvtes over a gven perod of tme (Sharpe, 2006). The most common performance or dmenson of a portfolo would be ts return,.e. ts proftablty. More mportantly, an nvestor should also consder sustanablty for future returns, e. whether the future returns can be mantaned ndefntely. Future returns are depent on the sustanablty of a frm and ts ntrnsc value. To examne portfolo performance, Markowtz s and Sharpe s models are used as the bass for data analyss. Markowtz s framework forms the foundaton for MPT. Sharpe s model elaborates on applcatons of Markowtz s framework. 2.6 Development of The Model The model has been developed by usng both Markowtz s mean- varance framework and Sharpe s sngle ndex model. Each of the pertnent models are dscussed n more detals below. 21

2.6.1 Markowtz s Mean-Varance Framework Markowtz s (1952) mean-varance framework forms a bass for hs portfolo selecton model. Ths s a tool for quantfyng the rsk-return trade-off of dfferent assets (Lynu, 2002), and t leads to mnmum varance portfolos (Luenberger, 1998). The pertnent statement s supported by the nvestors who attempted to mnmze portfolo varances at any gven level of expected returns (Fsher et al., 1997). Markowtz s mean-varance framework has had many fnancal applcatons n macroeconomcs and monetary theory (Tobn, 1981). Markowtz mean-varance framework s, however, usually appled n portfolo selecton, where t nvolves the estmaton of means, varances and covarances of the parameters chosen. Ths s supported by Barry (1974: p. 515), who says, (t)he use of mean-varance analyss n portfolo selecton nvolves the estmaton of means, varances, and covarances for the returns of all securtes under consderaton. Markowtz s model s dscussed through a drect adaptaton from Elton et al. (2003). Ths s ntroduced n Fgure 2.3 below. Therefore, the necessary nput data for Markowtz s model are the hstorcal estmates of (Hagn, 1979): 1. Expected returns for each securty Markowtz (1959) suggests that the expected returns for each securty can be calculated by: R,t P P D,t 0,t... (2.1) P 0 West (2005) places emphass on equaton (2.1) regardng ts smplcty n determnng the expected returns of a fnancal securty. 22

23 Fgure 2.3: Markowtz's Mean-Varance Framework Suppose an nvestor has a portfolo wth n assets, the th of whch delvers a sngle perod return R wth mean µ and a varance 2 σ. Suppose further, that the weght assgned to asset n the portfolo s w. Then the sngle perod return on the portfolo s: n w R R 1 The expected return on the portfolo s then: n 1 n 1 n 1 w R E E R w R E w R E R E The varance of the portfolo s j j n j n j j n j n j j j n j n n j j j j n n σ w w R σ R, R covar w w R σ µ R µ R w E w R σ µ R w µ R w E R σ µ R w E R σ µ R E R σ 1 1 2 1 1 2 1 1 2 1 1 2 2 1 2 2 2 Where, j σ s the covarance between R the return on asset and j R the return on asset j. σ

2. Standard devaton for each securty The sample standard devaton has been used as an estmator of the populaton standard devaton (Mason et al, 1990). It s represented by equaton (2.2).,t N R,t R,t 1 N 1.. (2.2) 2 Where R,t N 1 R N,t, the mean of an ndvdual securty, s calculated as the sum of ts returns by ts sample sze (Sharpe, 1970). 3. Correlaton coeffcent between each possble par of securtes for the securtes under consderaton Ths s defned as the covarance between two random parameters dvded by the product of ther standard devatons, and represented by equaton (2.3) (Ryan, 1978). 2,j,t,t j,t m.t,j,t..... (2.3),t j,t,t j,t The correlaton coeffcent s bound n the range between -1.0 and +1.0, whch corresponds to perfect negatve and postve correlaton respectvely (Ryan, 1978). The covarance between two varables s expressed n equaton (2.4)., j,t N 1, j1 R R R R,t,t N 1 j,t j,t.. (2.4) 24

Further to the above, Markowtz s model can be formulated as the followng: Assume that there are N assets. The mean (or expected) returns are R 1, R 2,, the covarances are, j, t R N and for, j = 1, 2,, N. A portfolo s defned by a set of N weghts w, = 1, 2,, N, that sum to 1. To fnd a mnmum- varance portfolo, the mean value s fxed at some arbtrary value R. Thus the problem can be formulated as follows (Adapted from Cuthbertson et al., 2004): Mnmze 1 2 N, j1 w w j, j,t Subject to N 1 w R R N 1 w 1 1 There s no partcular sgnfcant reason for the constant value n the above 2 formulaton, ts presence just make the algebra neater (Cuthbertson et al., 2004: p. 143), ths can be nterpreted as makng the mathematcs easer to understand and follow. An dentcal model was proposed by Luenberger (1998). Markowtz s model provdes the foundaton for sngle-perod nvestment theory. Sngleperod refers to a partcular perod as defned by the nvestor, that s an nterval of tme characterzed by a sngle occurrence of an nvestment decson. Ths model explctly addresses the trade-off between the expected rate of return and the varance of the return n a portfolo (Luenberger, 1998). 2.6.2 Sharpe s Sngle Index Model Sharpe shows that the ndex model can smplfy the portfolo constructon problem as proposed by Markowtz (Jacquer et al., 2001). The smplfcaton was acheved by 25

ntroducng assumptons. Ths s shown by Ryan (1978: p. 90), who says that ()ndex models owe ther orgn to a semnal paper by Sharpe whch ntroduced a smple but farreachng modfcaton to the basc Markowtz framework. Sharpe added an addtonal assumpton that observed covarance between the returns on ndvdual securtes s attrbutable to the common depence of securty yelds upon a sngle common external force a market ndex Even though assumptons were ntroduced n ths model, these wll not affect the qualty of results generated as the sngle ndex model, developed to smplfy the nputs to portfolo analyss and thought to lose nformaton due to smplfcaton nvolved, actually does a better job of forecastng than the full set of hstorcal data. (Elton et al., 2003: p. 147) The sngle ndex model (Sharpe, 1964) s mplemented when one tres to estmate a correlaton matrx, conduct effcent market tests or equlbrum tests (Elton et al., 2003). Ths s a smplfed approach to portfolo formulaton. Sharpe s sngle model s dscussed by a drect adaptaton from Elton et al., (2003). Ths s descrbed n the Fgure 2.4 and Fgure 2.5. 26

Basc Equaton R R e for all stocks (securtes) = 1 n M By Constructon Mean of e = E( e ) = 0 for all stocks (securtes) = 1 n By Assumpton 1. The ndex s unrelated to unque return: E[ e ( R M R M )] = 0 for all stocks (securtes) = 1 n 2. Securtes are only related through ther common response to the market: E[ e ej ] = 0 for all pars of stocks (securtes) = 1 n and j = 1 n but j By Defnton 1. Varance of e = E( e ) 2 = 2. Varance of R M = E(R 2 σ e 2 M R M ) 2 M The expected return, varance and covarance for Sngle Index Model are: 1. The mean return, R R M 2. The varance of a securty s return, 2 2 2 M 3. The covarance of return between securtes and j, 2 e j j 2 M The expected return on a securty s E(R ) E[ R M e ] E( ) E( R M ) E(e ) α and By lnearty of expectatons, snce value of e s zero by constructon, thus, β are constants and snce the expected E(R ) R M The varance of return on a securty s gven by: σ 2 E(R R ) 2 Fgure 2.4: Sharpe's Sngle Index Model (Part I) 27

28 Fgure 2.5: Sharpe Sngle Index Model (Part II) 2 M M 2 M M 2 2 2 M M 2 2 M M 2 ) E(e R R E e 2 R R E e R R E ] R e R E[ Snce by assumpton E[ e ( M M R R )] = 0, thus, 2 e 2 M 2 2 2 2 M M 2 2 ) E(e R E R The covarance between any two securtes can be wrtten as j j j R R R R E σ Substtutng for j R, R, R and j R yelds, j M M j M M j 2 M M j j j M M j M M j M j j j M j j M M j E e e R R E e R R E e R R E e R R e R R E R e R R e R E Snce the last three terms are zero, by assumptons. Therefore: 2 M j j Where by regresson analyss, the beta and alpha values can be calculated as follows: N 1 t 2 Mt Mt N 1 t Mt Mt t t 2 M M R R R R R R Mt t R R

The nput data requrements for performng a portfolo analyss usng Sharpe s sngle ndex model are the hstorcal estmates of (Hagn, 1979): expected return for each securty, expected return of the market (n ths report, the market refers to the ndex chosen), standard devaton for each securty, standard devaton for the market, and correlaton coeffcents between each securty and the market. The pertnent hstorcal estmates have been establshed by applyng and adaptng the equatons (2.1) to (2.4). The basc equaton for Sharpe s sngle ndex model s represented by equaton (2.5). Ths s also the basc equaton for a lnear regresson model (Raftery et al., 1997). R,t R e.... (2.5),t,t M,t,t for all stocks (securtes) = 1 N From equaton (2.5), R, t s represented as a lnear functon of M, t R and e, t. Ths vew s supported by Cuthbertson (2004: p.179), who ndcated that a return on any securty R,t can be adequately represented as a lnear functon of a sngle (economc) varable (parameter) R M, t where e, t s a random error term. The parameters represented n equaton (2.5), are, t, known as alpha,, t, as beta and e,t a random error term. The nterpretatons of these constants are that alpha represents the extent to whch a securty s msprced (Tucker et al., 1994: p. 577), and beta s a measure of systematc rsk of a securty or portfolo, (Tucker et al., 1994: p. 577). 29

These values can be estmated by regresson analyss. Beta and alpha can be represented mathematcally by equatons (2.6) and (2.7) respectvely (Elton et al., 2003: pp. 140-141).,t,t,M,t 2 M,t N t1,t R R R R M,t,t N R M,t R M,t t1,t M,t 2 M,t..... (2.6) R R..... (2.7) Beta represents the senstvty of an ndvdual share to changes n the market. The market has a beta of one. Indvdual securtes wll thus have betas reflectng ther relatve senstvtes to the market beta of one (Correa et al., 2003). Alternatvely, beta can be explaned by the slope of a securty lne n the Captal Asset Prcng Model (CAPM) (Correa et al., 2003). When the beta value s less than 1, ths suggests a lower gradant slope, e. a flatter slope and a low rate of change between the prce of securtes and the market ndex, as a result, lower volatlty. Furthermore, the parameter beta s also one of the performance measures of ths model. It can be nterpreted as the senstvty of a securty s return to an underlyng factor. (Tucker et al., 1994: p. 577) The calculated beta value, usng equaton (2.6), s also known as ordnary least square (hereforth known as OLS) beta. OLS betas are adjusted n an attempt to mprove predctve ablty of the betas on securtes and portfolos (Elton et al., 2003), snce ndvdual securtes betas have a regresson tency towards grand mean of all the securtes on the exchange. The adjustments are dscussed n more detals later. Alpha represents the dfference between a portfolo s returns and ts expected returns gven ts rsk level as measured by ts beta. It gves an ndcaton of the extent to whch a securty s msprced. Based on equaton (2.7), from a mathematcal perspectve, t s reasonable to deduce that alpha s nversely related to beta. The error s also estmated by usng the regresson model. The followng descrbes the formulaton of the parameters for the regresson model. 30

Let the sample subset of returns on the market ndex have n elements. Denote ths as { M, R ( 1) M,..., R (2) M ( n ) R }. Let y be the (n by 1) vector of returns on share, the response parameter (n s the same for each of the securtes n the test portfolo, for more detals please refer to Chapter 5 Desgn Outcome The Data). Let X equal to the (n by 2) matrx of predctor parameters (Adapted from Hobbs, 2001: p.16): 1 R M (1).. X....... (2.8).. 1 R M ( n ) s the vector of unknown regresson coeffcents:... (2.9) e s the vector of error terms: e (1). e....... (2.10). e ( n) So that unknown. e ( t values are random varables, the parameters of whose dstrbuton are ) The regresson model s gven by (Hobbs: 2001, p. 16): y X e...... (2.11) 31

The least squares estmator X' X y f X X s non-sngular. 1. (2.12) For the purpose of ths desgn, the vectors y and X are known. These values have been calculated usng the raw daly prce data collected. The least square estmator s then establshed usng equaton (2.12). The error vector s calculated by changng the subject of formula n equaton (2.11). The equaton (2.11) becomes: e y X. When the errors are establshed, the values obtaned are substtuted nto equaton (2.5), to calculate R. There are two adjustments whch are made to the OLS beta values; these are Bayesan and Merrll Lynch s adjustments. 1. Bayesan Adjustment Vascek s technque s an applcaton of Bayesan adjustment (hereforth known as BA) (Bradfeld, 2003). BA presents the method of adjustng a securty s beta based on the degree of uncertanty nstead of assumng all securtes move by the same amount toward the average (Elton et al., 2003). The BA equaton s shown n equaton (2.13) (Bradfeld, 2003), where the adjusted beta value s equal to the sum of both the product of a weght factor wth the OLS beta estmate and the product of 1 less the weght factor wth the average of the betas of all the securtes n the portfolo. BA,... (2.13) BA BA, 1 32

The weght factor n equaton (2.13) s calculated usng equaton (2.14), shown below, (Bradfeld, 2003: p. 50): 2 P BA,...... (2.14) 2 2 P Ths technque s relevant to South Afrcan s envronment, snce Cadz Fnancal Strategsts use t to determne beta values on JSE (Profle Group (Pty) Ltd., 2006a). 2. Merrll Lynch s Adjustment The motvaton for Merrll Lynch s (known as ML hereafter) adjustment on OLS beta estmates s the observaton that, on average, the beta coeffcent of securtes seems to regress toward 1 over tme (Elton et al, 2003: p. 144). Jarnecc et al. (1997: p. 7) suggest the statstcal explanaton for ths s that when beta s estmated over a partcular sample perod, an unknown samplng error of estmated beta s sustaned. The greater the dfference between the estmated beta and 1, the greater the chance that a large estmaton error has occurred; when the same beta s estmated n a subsequent sample perod, the new estmate would be closer to 1. Beta s adjusted by takng the sample beta estmate, OLS n ths desgn, multplyng ths value by two-thrds then plus a thrd (Jarnecc et al., 1997). The equaton s shown n equaton (2.15). The sgnfcance of constant, 1, from equaton (2.15) has been descrbed above. 2 1 ML.1... (2.15) 3 3 Furthermore, from Sharpe s sngle ndex model, alpha, can be determned by applyng equaton (2.7). The assocaton between the two relevant adjustments and Alpha s also determned usng equaton (2.7); the results wll be dfferent due to the dfferent beta outcomes. Beta, β, can also be estmated dynamcally by the use of Kalman Flterng. 33

A Kalman flter, also known as lnear quadratc estmaton, s a set of mathematcal equatons that provde an effcent computatonal means to estmate the state of a process (Welch et al., 2001). The Kalman flter s appled to estmate the state of a system from measurements whch contan random errors. Ths technque s usually used n control theory and control systems engneerng (Welch et al., 2004). Ths technque also has applcatons n fnance (Wells, 1996). It s often used for the dynamc estmaton of beta values (Bradfeld, 2003). Ths s done by the two dstnctve phases n Kalman flterng, that s, predct and update. The predct phase uses the estmate from the prevous tme state to produce an estmate for the current tme state. In the update phase, the measurement nformaton at the current tme s used to refne ths predcton n order to arrve at a new, hopefully more accurate estmate, for current tme (Welch et al., 2001). Ths report has chosen to model beta usng a regresson model. The adjustments that were done to the OLS beta, are BA and ML (Profle Group (Pty) Ltd., 2006a). Kalman flterng s not used due to the dynamc nature of ths tool. The models that are applcable to MPT have been dscussed above. The examnatons of the envronment of the nvestment, namely the forms of the market, are ntroduced below. 2.6.3 Effcent Market Hypothess An effcent market s assumed for the concept of passve management approach (Hobbs, 2001). The Effcent market hypothess (EMH) s the set of arguments leadng to the asserton that market prces fully reflect avalable nformaton. (Tucker et al., 1994: p.580) EMH s a set of mplcatons that are assocated wth each dfferent form of the market. There are three forms of the EMH: 34

1. Weak Form The weak form of the EMH assumes that current securty prces fully reflect all securty market nformaton, ncludng the hstorcal sequence of prces, prce changes, tradng volume and any other market nformaton such as odd lot transactons (Relly, 1989, Correra et al., 2003 and Cuthbertson et al., 2004). Therefore, techncal analyss s of no use when attemptng to outperform the market; t s merely an approach that s used n the hope of predctng future trs (Hobbs, 2001). Yet, ths form of the EMH suggests that future securty prces cannot be predcted by the use of hstorcal prces, ths means that future cannot be predcted by usng hstorcal data, that further suggests that whatever happened n the past s unlkely to happen n the future, thus stock prces behave accordng to a random walk (Malkel, 1999). 2. Sem-Strong Form The sem-strong form of the EMH asserts that securty prces adjust rapdly to the release of all new publc nformaton; thus securty prces fully reflect all publc nformaton (Relly, 1989, Correra et al., 2003 and Cuthbertson et al., 2004). Thus, fundamental analyss s of no use n outperformng the market, nstead t s used n the hope of dentfyng new nformaton (Hobbs, 2001 and Correra et al., 2003). 3. Strong Form The strong-form of the EMH conts that securty prces fully reflect all nformaton, whether t mght be publc or prvate (Relly, 1989, Correra et al., 2003 and Cuthbertson et al., 2004). In other words, not even nsder nformaton can be used n the quest to outperform the market. The tools derved n ths report may perform dfferently n dfferent market envronments. 35

From the above, the theores and methodologes for the model have been revewed and developed. The model s graphcally represented n Fgure 2.6 and summarsed as follows: 1. calculate returns of securtes, usng equaton (2.1), 2. calculate the averages of securtes and the chosen ndex, 3. estmate the error terms from Sharpe s sngle ndex model, usng equatons (2.8) to (2.12), 4. calculate the varances of securtes, usng equaton (2.2), 5. calculate the covarances between securtes, usng equaton (2.4), 6. estmate OLS beta values by regresson model, usng equaton (2.6), 7. perform adjustments to OLS beta, the adjustments done were: a. Bayesan adjustment, usng equaton (2.13), b. Merrll Lynch adjustment, usng equaton (2.15), 8. estmate the alpha values usng equaton (2.7), and 9. calculate the expected returns usng equaton (2.5). 36

Avalable Data Inputs, P,t, P 0 & D, t for securtes Avalable Data Inputs for the chosen ndex Calculate returns on securtes and the chosen ndex usng equaton (2.1) and equaton (2.1) wthout the dvds term respectvely Calculate the averages of securtes and the chosen ndex Calculate the varances of the securtes & the chosen ndex usng equaton (2.2) Calculate the covarances between securtes and between the securtes and the chosen ndex, usng equaton (2.4) Estmate the error terms from Sharpe s sngle ndex model, usng equaton (2.8) to (2.12) Estmate OLS beta values by regresson model, usng equaton (2.6) Perform adjustments to OLS beta values Bayesan Adjustment usng equaton (2.13) Merrll Lynch s Adjustment usng equaton (2.15) Estmate the alpha values usng equaton (2.7) Calculate the expected returns usng equaton (2.5) Fgure 2.6: Process Flow Dagram of the Model 37

The model s subject to the followng assumptons and lmtatons: Investors behavour plays a sgnfcant role n nvestment returns (Frdson, 2007). Investors are assumed to behave ratonally, for example: a. Investors consder each nvestment alternatve as represented by a probablty dstrbuton of expected returns over some holdng perod. b. For a gven level of rsk, nvestors prefer hgher returns to lower returns. Smlarly, for a gven level of expected returns, nvestors prefer lower to hgher rsks. Investors base ther decsons solely on expected returns and rsk, so ther utlty curves are a functon of expected return and varance (or standard devaton) of returns only. There s assumed to be a perfectly effcent nvestment market, whch suggests zero tradng costs, et cetera. Investment decsons are based only on the rsk-return preferences of nvestors. Ths model wll also gve an effcent fronter. The nvestor has a quadratc utlty functon, but ths s not always possble. Securty movements are related to the changes n the overall market. Ths model also assumes that the expected value of a resdual s zero and there s no correlaton between the market returns and resduals (Kam, 2006). The resduals of assets are uncorrelated. Ths suggests that any assocaton between the returns of assets s attrbutable only to the common market movement (Kam, 2006). 2.7 Next Steps To satsfy the objectves of: valdty of the model and user- frly utlsaton of the model The model s automated va a computer program. 38

Chapter 3 Development of Computer Programme In ths chapter, the development of the computer programme s dvded nto three stages, namely desgn requrement specfcatons, software selecton and code wrtten for the computer programme. Each of the stages are dscussed below. 3.1 Desgn Requrement Specfcatons In ths secton, the objectves of ths computer programme are dscussed. Ths leads to a needs analyss where a desgn requrement specfcaton (hereforth known as DRS) s developed. The DRS conssts of a lst of requrements, crtera and constrants assocated wth the computer programme. 3.1.1 The Objectves The motvaton for creatng ths computer programme has been dscussed n secton 1.2, and the desgn objectves have been made apparent. The objectve s acheved by completng the followng tasks: develop a model for passve portfolo management usng MPT tools va a crtcal lterature revew as dscussed n Chapter 2, and based on the above, develop an automated model va a computer programme that shall perform the relevant calculatons as descrbed n the crtcal lterature vew. 3.1.2 Needs Analyss 3.1.2.1 Desgn Overvew The computer programme desgned s nted to be used by prvate nvestors. The level of computer competency needed s mnmal. Mnmal refers to the basc sklls n Mcrosoft Offce packages, n partcular, the Excel package. 39

3.1.2.2 Desgn Requrement Specfcaton As a drect consequence of the above, the requrements, constrants and crtera of the computer programme are dscussed below. Functonal Requrement The computer programme developed needs to demonstrate the automaton of the model as dscussed n Chapter 2. The computer programme follows the approach as proposed n Fgure 2.6. Constrants The constrants wth regards to ths desgn of the computer programme were: lmted tme, lmted fnancal resources, therefore some of the more advanced statstcal packages were not consdered, and lack of experence n wrtng a computer programme n all computer languages. Crtera The crtera form the gudelnes to whch the computer programme needs to adhere. Furthermore, the crtera consdered need to be classfed as ether demand (hereforth known as D) or hgh wsh (hereforth known as HW). D refers to the crteron that s the must- have and hgh wsh refers to the crteron that s nce to have. The crtera consdered for ths computer programme have been tabulated n Table 3.1. 40

Table 3.1: Crtera for Desgn Requrements Desgn Requrements The model to be bult based on the crtcal lterature revew The outcomes of the model need to be specfed The computer package should be easy to learn The computer package used should be relatvely nexpensve wthout compromsng the accuracy of calculatons All data resultng from the model should be satsfactory for recordng and analysng Model should process data speedly Model should be clearly defned and structured n a logcal manner Crtera D D D HW HW D D 3.2 Software Selecton 3.2.1 Introducton In ths secton, the processes followed to acheve the fnal software selecton are dscussed. The secton starts wth the ntroducton of the types of statstcal packages, namely Mcrosoft Excel, MATLAB and SAS, that were consdered for the computer programme. Each package s ntroduced, followed by ther respectve applcatons, advantages and dsadvantages. Ths secton concludes wth the decson matrx used for software selecton. 3.2.2 Types of Statstcal Packages As mentoned under the needs analyss, n secton 3.1.2, the way to acheve the objectves that were set for ths desgn s to buld a model through the use of statstcal packages. The types of statstcal packages consdered for ths desgn s shown n Fgure 3.1 below. 41

Types of Statstcal Packages Mcrosoft Excel MATrx LABoratory (MATLAB) Statstcal Analyss System (SAS) Fgure 3.1: Types of Statstcal Packages Consdered 3.2.2.1 Mcrosoft Excel Mcrosoft Excel (full name Mcrosoft Offce Excel) s a spreadsheet 14 applcaton wrtten and dstrbuted by Mcrosoft. It features calculaton, graphng tools, pvot tables and a macro programmng language called Vsual Basc for Applcaton (henceforth known as VBA) (Mcrosoft Corporaton, 2003 and Wkmeda Foundaton Inc., 2007a). There are varous add-on applcatons avalable that can conduct more n-depth analyss, examples of whch are Analyss ToolPak and Solver Add-In. Some strengths and weaknesses of Mcrosoft Excel are descrbed below: Strengths It s user- frly, very easy to learn. It can mport, organse and explore data sets (Mcrosoft Corporaton, 2007). Ths mples that Excel has strong analytcal functonalty. As a result, professonallookng graphs can be created. Ablty to graphcally compare results from a model and observatons (Carleton College, 2007). 14 A spreadsheet s a grd of nformaton, often fnancal nformaton, (Wkmeda Foundaton Inc., 2007b). 42

Smart documents. These are documents that are programmed to ext the functonalty of a workbook by dynamcally respondng to the context of ones actons. For example, the documents can be connected to a database that automatcally flls n some of the requred nformaton (Mcrosoft Corporaton, 2003). Weaknesses Mcrosoft Excel was bult based on floatng pont calculaton. As a drect consequence, ts statstcal accuracy has been crtczed, snce t lacks certan statstcal tools (Wkmeda Foundaton Inc., 2007a). It s effectve at certan tasks and not others (Wkmeda Foundaton Inc., 2007b). Excel s effectve at analytcal functons, such as generatng graphcs, but not effectve n mathematcal modellng. It s loosely structured. Therefore t s easy for someone to ntroduce an error, ether accdentally or ntentonally. An example of ths s that there s a lack of revson control. It s dffcult to determne who changed what and when. Ths can cause problems wth regulatory complance, among other thngs (Wkmeda Foundaton Inc., 2007b). 3.2.2.2 MATLAB MATLAB s the abbrevaton for MATrx LABoratory. It s a hgh performance language for techncal computng. It can ntegrate vsualsaton, computaton and programmng n an easy-to-use envronment, where problems and solutons are expressed n famlar mathematcal notaton. Some applcatons of ths programme are maths & computaton, data acquston, data analyss, graphcs applcaton, modellng, smulaton and statstcal analyss (The MathWorks Inc., 2006 and Wkmeda Foundaton Inc., 2007c). Some strengths and weaknesses of MATLAB are descrbed below: 43

Strengths It s relatvely easy to learn (Northeastern Unversty: College of Computer and Informaton Scence, 2003). MATLAB code s optmsed to be relatvely quck when performng matrx operatons. It s an nteractve system whose basc elements don t requre dmensonng. Therefore, ths package s more robust than Excel, allowng complcated techncal problems to be solved (The MathWorks Inc., 2006 and Northeastern Unversty: College of Computer and Informaton Scence, 2003). There are varous toolboxes (add-on applcatons for specfc solutons n a feld) that can be accessed easly (The MathWorks Inc., 2006). Although the package s prmarly procedural, MATLAB does have some object orentated elements (Wkmeda Foundaton Inc., 2007c). Weaknesses MATLAB s an nterpreted language, makng t, for most part, slower than a compled language such as C++ (Northeastern Unversty: College of Computer and Informaton Scence, 2003). It s desgned for scentfc computaton; therefore t s not a general purpose programmng language and not sutable for some thngs. (Northeastern Unversty: College of Computer and Informaton Scence, 2003). An example s that MATLAB doesn t support references, whch makes t dffcult to mplement certan data structures (Wkmeda Foundaton Inc., 2007c). Ths pont can also be dentfed as a characterstc of ths package. 3.2.2.3 SAS SAS (orgnally known as Statstcal Analyss System) s an ntegrated system of software products. Some applcatons of ths software are statstcal & mathematcal analyss, operatons research & project management, busness plannng, forecastng & decson supports, report wrtng and graphcs. Some strengths and weaknesses of SAS are descrbed below: 44

Strengths Beng one of the most powerful data mnng technologes, there s a huge user base for ths software (Yates, 2006). It can handle large data sets (Mtchell, 2007). It can perform the vast majorty of statstcal analyses. Weaknesses Relatvely hard to learn (Yates, 2006 and Wkmeda Foundaton Inc., 2007d) for a person wth lmted programmng experence. One of the reasons s that the syntax t uses s unlke that of any other programmng language. Doesn t have sophstcated graphcal functons (Mtchell, 2007 and Wkmeda Foundaton Inc., 2007d). The graphcs generated by SAS are not as clear and structured as those produced by Excel. Costs, especally when compared to ts open source compettors such as R- squared statstcs. It s an open source statstcal package that can be downloaded free of charge. 3.2.3 Decson Process Based on the DRS, dscussed n secton 3.1, the most mportant factors 15 that affect the choce of statstcal packages used, as dentfed from Table 3.1, are: the processng speed of the package, the cost to obtan the lcence of the package and the ease of learnng the package. By combnng DRS and the strengths & weaknesses of each of the packages consdered, ths gves rse to Table 3.2 below, where each of the packages have been benchmarked aganst each other. From Table 3.2, each of the three factors have been assgned dfferent weghtng factor, based on the DRS. Also, the score of 5 refers to the package beng consdered as the best 15 The term factor has later on become category n Table 3.2. 45

n the category and 0 beng the least desrable n the category. The choce of scores was chosen to show the dfferentaton between the choces. Therefore, the scores were made to demonstrate a decson matrx. Table 3.2: Decson Matrx of Concepts Weghtng Maxmum Mnmum Mcrosoft MATLAB SAS Factor Score Score Excel Speed 0.5 10 0 0 3 5 Cost 0.2 10 0 5 3 0 Ease to 0.3 10 0 5 3 0 Learn TOTAL 2.5 3 2.5 Therefore, the package wth the hghest score from Table 3.2, MATLAB was chosen as the fnal package that s to be used for ths desgn project. Wth ths decson, a complete programme for the dscusson n Chapter 2 needs to be undertaken, followng the other desgn requrements n Table 3.1. 3.3 Code Wrtten for Computer Programme 3.3.1 Introducton In ths secton, the detaled model logc s dscussed, whch ncludes the codng of the computer programme. 3.3.2 Detaled Computer Programme Logc The computer programme (hereforth known as model) logc has been segmented nto three stages, namely nputs, computer programme and outputs. In ths secton, the detals assocated wth each of the stages are descrbed. The order of dscusson s outputs, nputs 46

and computer programme, as shown n Fgure 3.2 below. The ratonale, for ths order of dscusson, s that t s mportant to keep n mnd the set objectves of ths desgn, followed by examnng the nputs that are avalable and can be used to establsh the objectve. Fnally, the computer programme s wrtten to convert the avalable nputs nto the proposed outputs. Outputs Inputs Computer Programme Fgure 3.2: Order of Dscusson 3.3.2.1 Outputs The proposed method to acheve ths relatonshp requres the followng output parameters, as seen n Fgure 3.3 below. Requred Outputs Alpha (α) Beta (β) Expected Returns (R) Fgure 3.3: Requred Outputs The detaled calculatons of the pertnent parameters wll be covered n secton 3.3.2.3. 47

3.3.2.2 Inputs The nput parameters that are needed to calculate beta, alpha and the expected returns of the portfolo are the followng, whch are also graphcally presented n Fgure 3.4: daly closng share prces for each of the securtes n the portfolo, weght 16 assgned to each securty, dvds of each securty over a partcular tme frame, and daly closng value of All Share Index (also known as ALSI). Inputs Daly Closng Share Prces for Securtes (P,t ) Weght Assgned to Each Securty (w ) Dvds of Securtes (D,t ) Daly Closng Values for ALSI (P M,t ) Fgure 3.4: Inputs Parameters Used 3.3.2.3 Computer Programme Ths computer programme serves as a tool that s necessary for the converson from nputs to outputs. The nputs are fed nto the model n one of two ways. Frstly, communcaton was set up between the nput n raw data form n Excel as extracted from the source and MATLAB software. Alternatvely, a user-nterface was created to allow the user to enter the requred nformaton. As dscussed above, the requred outputs are beta, alpha and expected return of a portfolo. In ths secton, the flow process dagrams for each of the requred outputs are dscussed separately before they are combned n the overall computer programme s flow process dagram. 16 Weght, n ths case, refers to the nvestment composton that s assgned to the securty. 48

Beta Calculaton Beta s calculated by usng the proposed nputs and applyng them to the equatons that were ntroduced n Chapter 2. The flow chart s shown below, Fgure 3.5. P,t D,t P M,t Calculate R,t usng equaton (2.1) Calculate R M,t usng equaton (2.1), exclude D M,t term Calculate R, t by takng the averages of R,t Calculate R M, t by takng the averages of R M,t Calculate, t by substtutng above nformaton nto equaton (2.6) Adjustments done on, t,t becomes Calculate BA usng equaton (2.13) Calculate ML usng equaton (2.15) Fgure 3.5: Process Flow Dagram for Beta Calculaton 49

Alpha Calculaton Alpha s now calculated by applyng equaton (2.7). The nput parameters needed for equaton (2.7) have been calculated above under beta calculaton, shown n Fgure 3.5. The process of calculatng alpha has been represented graphcally n Fgure 3.6 below. The values for R, t and M, t R are calculated from the above beta calculaton BA ML Substtute the above nformaton nto equaton (2.7), then 3 cases of alpha values are generated α BA ML Fgure 3.6: Process Flow Dagram for Alpha Calculaton Expected Returns Calculaton Expected returns are calculated by applyng equaton (2.5). All of the parameters from equaton (2.5) can be calculated by applyng equatons from sectons 2.6. These parameters nclude beta, alpha and the error terms. 3.3.3 Fnal Computer Programme From above, the detals of the error terms from Sharpe s sngle ndex model have been dscussed n Chapter 2. The ntroducton of error calculatons was done n secton 2.6.2. 50

As a consequence, two sets of MATLAB codes have been wrtten, one to nclude the error term from the sngle ndex model (Appx A MATLAB Code for Analysng Components of the Test Portfolo Wth Error Terms, p. 122) and the other to exclude t (Appx B MATLAB Code for Analysng Components of the Test Portfolo Wthout Error Terms, p. 134). The nstructons for runnng the MATLAB codes are set out n Appx C Instructons for Runnng MATLAB Code (p. 149). A set of codes to exclude error terms s wrtten for the generc analyss. Ths code calculates the parameters, n solaton 17, for an nvestor. If an nvestor wants to examne the parameters n relaton to the general economc envronment, t s necessary to nclude the error terms. By ncludng the error, an nvestor would gan a more holstc vew of hs/her nvestment n relaton to that of an economc envronment. Hence, a separate set of codes are wrtten for ths reason. Comparsons are made between the results. Process flow dagrams have been drawn for the cases where the error terms are ncluded and excluded. These are shown below n Fgure 3.7 and Fgure 3.8 respectvely. 17 Isolaton refers to a closed system. In ths research, t means to examne shares wthout consderng the general economc envronment. 51

Defned Inputs n secton 3.3.5.2 Set up communcaton wth chosen document Create user nterface, by enterng the values needed Save these nputs for processng n the wrtten codes Intalse the processng of MATLAB codes Calculate the returns nclude dvds where possble (R,t & R M,t ) Error terms estmaton Calculate the averages, R M,t R, t and Calculate varances Calculate covarances Establsh standard devatons Beta calculatons & ts adjustments (refer to Fgure 3.5 for more detals) Alpha calculatons wth each of 3 cases of beta (refer to Fgure 3.6 for more detals) Expected returns, nclude error terms Outcomes wrtten to selected workbook Fgure 3.7: Overall Flow Process Dagram for MATLAB code Includng Error Terms 52

Defned Inputs n secton 3.3.5.2 Set up communcaton wth chosen document Create user nterface, by enterng the values needed Save these nputs for processng n the wrtten codes Intalse the processng of MATLAB codes Calculate the returns nclude dvds where possble (R,t & R M,t ) Calculate the averages, R M,t R, t and Calculate varances Calculate covarances Establsh standard devatons Beta calculatons & ts adjustments (refer to Fgure 3.5 for more detals) Alpha calculatons wth each of 3 cases of beta (refer to Fgure 3.6 for more detals) Expected returns, exclude error terms Statstcal analyss done on expected returns Outcomes wrtten to selected workbook Fgure 3.8: Overall Flow Process Dagram for MATLAB code Excludng Error Terms 53

3.3.4 Testng of Computer Programme Testng (whch can also be nterpreted as valdaton) s a process that conssts of four dstnct steps, namely software, hardware, method and system sutablty valdatons. Ths s represented below, n Fgure 3.9 (Waters Corporaton, 2007): Valdaton Software Hardware Method System Sutablty Fgure 3.9: Steps for Valdaton The testng of ths computer programme s demonstrated through the use of an example as descrbed below. The gven data s as follows: Observaton P 1 P M 1 12 50 2 13 54 3 10 48 4 9 47 5 20 70 6 7 20 7 4 15 8 22 40 9 15 35 10 23 37 54

To ensure that the analytcal system s valdated, a valdatng computer programme has been wrtten (Appx D MATLAB Code for Valdatng The Computer Programmes, p. 161). In ths report, the analytcal system refers to the computer programme wrtten. The valdatng computer programme wrtten s smlar to the fnal programmes found n Appx A and Appx B. The fnal computer programmes wrtten have been broken down nto smaller parts for ease of valdaton. The valdatng computer programme can be run by carryng on the steps (5) and (6) as descrbed n Appx C as well as selectng an output fle to whch the results are wrtten. The valdatng programme conssts of the followng parts: calculaton of the returns for ndvdual share and the ndex, calculaton of the arthmetc averages for ndvdual share and the ndex, calculaton of the varance, calculaton of the covarance, calculaton of the OLS beta and OLS alpha, adjustments of the beta by usng Bayesan and Merrll Lynch adjustments, and calculaton of the adjusted alpha values. The results from ths valdaton demonstraton are found n Appx E Valdaton Results, p. 164. The valdatng computer programme and the results can be found on the CD provded. Valdaton ensures that the model meets ts nted requrements n terms of the method employed and results obtaned. The valdatng computer programme s a reasonable model as the outcomes have matched the manual calculatons wth sutable precson. Thus the valdaton results, the error comparsons between the results obtaned by the valdatng computer programme and the manual calculatons are neglgble. It s evdent that the procedures followed n ths report are vald, snce the errors are neglgble. The valdatng computer programme was then modfed to gve rse to the fnal computer 55

programme. The fnal computer programme s n a generalsed format and s able to ncorporate more data than the valdatng computer programme. 56

Chapter 4 Selecton of Test Portfolo 4.1 Choce of Consttuents n Test Portfolo The theoretcal prelmnares and desgn model logc have been establshed n the lterature survey and development of the computer programme respectvely. The next phase s to nvestgate the reasons for the consttuents n the test portfolo. Ths secton dscusses the structure of the test portfolo. 4.1.1 Portfolo Selecton Ths s an ex-ante 18 concept (Fr et al., 1965) and the process of selectng a portfolo can be dvded nto two stages. The frst stage begns wth observaton and experences and s wth a belef regardng the future performances of the avalable securtes. The second stage starts wth the relevant future performance belef and s wth portfolo choce (Markowtz, 1952). In portfolo selecton, there are four areas that one usually looks at (Cohen et al., 1987), namely the macroeconomc factors, nvestors profle, fundamental and techncal analyses.. Macroeconomc factors: these refer to factors that can affect the entre economy (Muradzkwa et al., 2004). An nvestor should ask and obtan answers to the followng questons n order to consder the relevant factors for the portfolo selecton (Cohen et al., 1987): What s the state of busness or the economy? Is t a favourable tme to nvest? Where are we n the busness cycle? Is a boom lkely to top out shortly? Is a recesson near at hand? 18 Ex- ante means before, frst or pror to. 57

What s the state of the market? Are we n the early stages of a bull market? Has the low pont of a bear market been reached? What ndustres are lkely to grow most rapdly? Are there any specal factors that favour a partcular ndustry? Whch companes wthn the ndustry are lkely to do best? Whch companes are to be avoded because of poor prospects? These pertnent questons are assocated wth macro-economc factors of the economy. By takng these factors nto consderaton, a better understandng of the economy s ganed and more nformed decsons are made regardng the portfolo selecton. Once the macroeconomc factors have been dentfed, one would decde upon the techncal vews that are gong to be followed,.e. whether t would be a contraran or a smart money vew... v. Investors profle: An nvestor s rsk tolerance and nvestment goals play an mportant part n portfolo selecton. These attrbutes have been dscussed n secton 2.5.2. Fundamental analyss: Ths refers to examnaton of a frm s fnancal data and operatons whle gnorng the overall state of the market. Ths analyss s often referred to as rato analyss. The ratos of nterest n portfolo selecton are generally earnngs per share, prce earnng and return on nvestment. These have been dscussed n secton 2.5.1. Techncal analyss: Ths refers to nvestment decson-makng by the use of charts. Ths gves a reasonable ndcaton of the market and the drecton t s headng; these have been explaned n the dscusson of secton 2.5.1. Fundamental and techncal analyses are mportant n estmatng the ntrnsc value of a frm. From the former, an nvestor would be able to decde upon the frm s potental. 58

From the latter, an nvestor would be able to dentfy the possble trs of the frm n the future based on the chart patterns. In the test portfolo, the macroeconomc factor of partcular nterest s the FIFA Soccer World Cup. On the 15 th May 2004, t was decded that South Afrca would be the host country for the 2010 Soccer World Cup (Wkmeda Foundaton Inc., 2004). Ths mmedately suggests the followng: a) New stadums need to be constructed, whle exstng ones need to be upgraded. b) Government needs to mprove the current publc transport nfrastructure. c) Specal measures need to be taken to ensure the safety and securty of toursts. The general consensus from a revew of the lterature regardng the 2010 Soccer World Cup s that an nvestor should pay specal attenton to the followng sectors: a) Basc materals b) Consumer goods and servces: these would contrbute towards toursm. c) Telecommuncatons d) Industral e) Fnancals Brnson et al. (1995) gve a set of gudelnes for desgnng a portfolo, whch nvolves at least four steps:. Determne what asset classes or sectors are to be ncluded and excluded from the portfolo. Ths supplements the concept of asset allocaton, dscussed under secton 2.5.2.. Decde on the tme horzon of the portfolo, whether t would be a short-, medum- or long- term nvestment; and on the weghts assocated wth each of the asset classes. 59

. v. From a strategc perspectve, an nvestor should rebalance the portfolo annually to capture excess returns from short-term fluctuatons (n captal gan) n asset classes. These fluctuatons may be due n part to economc condtons. Select ndvdual securtes wthn an asset class, whch would acheve superor returns relatve to the rest of that partcular class. These are usually referred to as blue-chp or growth securtes. The structure of the test portfolo wll take nto account the sector breakdown as t appeared on JSE as well as the securtes categorsatons. Ths s represented graphcally below, Fgure 4.1. Major Sectors Breakdown on JSE n Fgure 4.2 Securtes Categorsaton for Portfolo Sub-dvson n Table 4.1 Securtes Included n Test Portfolo, Includng Sector Dvson n Table 4.2 Fgure 4.1: Structure of Test Portfolo It s relevant to know whch of the major sectors these shares fall under, therefore the major sector dvson of the ALSI s shown n Fgure 4.2. There are Roman numeral superscrpts present wth each of the major sector dvsons n Fgure 4.2. The purpose of superscrpts s to cross-reference between the major sector dvson and the securty n the test portfolo. Ths wll be evdent n the sectons to follow. 60

Ol and Gas Basc Materals Industrals All Share Economc Group Consumer Goods v Health Care v Consumer Servces v Telecommuncatons v Fnancals v Technology x Fgure 4.2: All Share Economc Group Breakdown The next procedure s to determne the number of shares to be ncluded n an nvestment portfolo. As Sharpe (1995: p. 85) states, (t)he number of securtes n a portfolo provdes a farly crude measure of dversfcaton. Ths means many securtes must be ncluded n a portfolo n order to acheve dversfcaton. The overall test portfolo used n ths research ncludes a total of 27 shares (Appx F Sample Sze of Test Portfolo, p. 168). Ths s a reasonable number of securtes, snce a well-dversfed stock (securty) portfolo must nclude at least 30 stocks (securtes) for a borrowng nvestor (Statman, 1987: p. 362). Therefore the benefts of dversfcaton are experenced n the test portfolo, and rsk reductons are evdent. 61

Securtes ncluded n ths portfolo are mert frms. Mert frms refers to companes wth sold fundamentals. Ths s mostly emphassed by ther presence n the headlne ndces such as the FTSE/JSE Afrca Top 40 Index and Top 100 Securtes n FTSE/JSE Afrca All Share Index. Each of the frms s a leader n ts partcular ndustry. The test portfolo s dvded nto sx components as dsplayed n Table 4.1. Ths dvson s due to dfferent nvestment tme horzons, market captalsatons and selecton crtera. The securtes categores shown n Table 4.1 are dscussed below (Standard Bank, 2007). Table 4.1: Securtes Categores for Portfolo Sub-Dvson Balanced Conservatves Core Alternatves Core Md-Term Small Caps Commodty Blue- chp Blue- chp Blue- chp Blue- chp Small Caps Cyclcal Income Value Commodty Cyclcal Growth Growth Value Value Commodty securtes are the frms whose securty prce s depent on a value of commodty such as gold or ol. An example of these securtes s Anglo Platnum plc. Cyclcal securtes fortunes are ted closely wth the economcal cycle. South Afrca s currently preparng for FIFA Soccer World Cup 2010. There s nfrastructure whch needs to be bult, therefore cement and constructon frms were chosen. These are Pretora Portland Cement (PPC) and Murray & Robert (MUR). Growth securtes are the frms who have consstently produced above-average growth n revenue and profts for many years and look lkely to contnue n the future, such as Anglo Platnum plc. These are the securtes that are supported by Buffet, who beleves n the sustanablty of frms (Buffet et al., 2002). 62

The securtes of proftable companes that are sellng at a reasonable prce compared to ther ntrnsc value are the value securtes. Examples are Woolworths Holdngs Ltd. (WHL) and Shoprte Holdngs Ltd (SHP). Income securtes are those securtes whose securty prces may be unexctng but wll contnue to pay out generous dvds and as a result yeld very good returns to nvestors. Blue-chp securtes are the most stable ones, as they are large, fnancally sold frms that have been around for years and ther securtes are held by both professonal and prvate nvestors. Examples are Standard Bank Group (SBK) and Anglo Platnum plc (AMS). Smaller Caps securtes: There s always a possblty of nvestng early on n a frm that may become a growth securty or blue chp of tomorrow. The categores of securtes can overlap due to the nature of the securty. An example s AMS whch s a blue-chp frm and a commodty-based frm wth strong sustanable growth due to the current needs for platnum. Hence AMS can be categorsed as bluechp, commodty and growth securty smultaneously. Usually, when a frm can be placed nto more than one category, the frm s a good securty recommaton to an nvestor. 63

Table 4.2: Securtes Included n Test Portfolo, Includng Sector Dvson Balanced Conservatves Core Alternatves Core Md-Term Small Caps ANGLOPLAT ABSA v ALEXFBS v ANGLO BARLOWORLD BCX x CITYLDG v BIDVEST FIRSTRAND v BARLOWORLD FIRSTRAND v BDE MTN v IMPERIAL SAB PLC v LIB-INT v M &R HLD DISTELL v PPC REUNERT STANBANK v PICK N PAY v MTN v ERP.COM x SHOPRITE v VENFIN v TIGER BRANDS v REMGRO PPC FAMBRANDS v WOOLIES v REUNERT SAB PLC v SHOPRITE v STANBANK v TIGER BRANDS v WOOLIES v In Table 4.2, the securtes under each category are shown. Also, the numercal superscrpts assocated wth each securtes, are referrng to the correspondng sectors n Fgure 4.2. Through ths, the securtes are pared wth ther respectve sectors. In Table 4.2, the categores of securtes chosen for each of the sx components are dsplayed. In summary, the consttuents of the test portfolo form part of the headlne ndces. It s observed that the securtes chosen are fnancally sold and ther dversfcatons are evdent. Ths s supported by the ratos calculated (Profle Group (Pty) Ltd., 2006b), the nvestments made n other frms as well as the cross-lstng structures of some frms. Therefore ther mert s recognsed. An n-depth dscusson on reasons for each securty s ncluson s avalable, (Refer to Appx G Ratonale for Shares Inclusons n the Test Portfolo, p. 170). Furthermore, the choce of ths test portfolo was supported by Korner (2005). 64

The reasons for the choce of securtes have been dscussed. Next, the model formulaton and ts composton wll be consdered. The generc formulaton of the test portfolo s as follows: I j 1 N and 1 j N R R 1,P w11i11 w12i12... w1ni1n 2,P w 21I21 w22i 22... w2ni2n : : R R n,p OP w I w I... w I...... (4.1) 1 n1 n1 1,P 2 n2 n2 2,P n nn nn R R... R.. (4.2) n,p n 1...... (4.3) 1 : > 0 In ths desgn, N goes up to 6. The returns calculated usng equatons (4.1) and (4.2) form the effectve nterest rate. A converson needs to be conducted to convert the effectve nterest rate nto the nomnal nterest rate format. The reason for ths converson s that the yeld of the rsk-free nterest money market nstrument, the government R194 bond, s gven n nomnal form, compounded sem-annually. Equaton (4.4) s used for ths converson: nr m r 1 1 m 1... (4.4) In Table 4.3, the nvestment composton s dsplayed; the percentages nvested are based on the monetary value nvested n each component. 65

Table 4.3: Investment Composton Component Name Amount Invested Percentage Invested Balanced R 15 000 18.75% Conservatves R 10 000 12.50% Core Alternatves R 10 000 12.50% Core R 15 000 18.75% Md- Term R 20 000 25.00% Small Caps R 10 000 12.50% R 80 000 100.00% 4.2 Choce of Index The choce of ndex determnes how much the portfolo return s correlated wth the market (Hobbs, 2001: p.21). The benchmark chosen s the FTSE/JSE Afrca All Share Index, snce t represents 99% of the full market captal value of all ordnary securtes lsted on the JSE that are elgble for ncluson n the ndex (JSE, 2007). The All Share Index s domnated by the frms n the resource sector whch s the nature of the domestc economc envronment. The consttuents chosen for the test portfolo are the headlnes ndces consttuents; ths emphasses the mert of these frms. The frms chosen also account for more than a thrd of the equty market captalsaton, (Appx H Ordnary Shares Lsted Based on Market Captalsaton, p. 174). Ths renforces the vew that the sample chosen s a good representaton of the market as a whole. Ths mples that the benefts of dversfcatons have been experenced and rsk reductons become evdent. 66

Chapter 5 Desgn Outcomes 5.1 Introducton In ths chapter, the results obtaned by applyng the computer programme, as outlned n Chapter 3, are dscussed. These dscussons are based on the models formulated n the crtcal lterature revew n Chapter 2. 5.2 The Data Daly data from 1 st September 2005 to 31 st January 2007 was used to perform analyses. The test perod began on 1 st September 2005 because the test portfolo was only actve as of that date, and the test perod s on 31 st January 2007 as the government bond R194 had been redeemed around that tme. The choce of usng daly data was made snce there was lmted monthly and yearly data avalable over ths test perod. Also over ths perod, the market dsplayed a bullsh state. Ths s shown n the ncreasng tr of the All Share Index. The data was sparse for one partcular share n the test portfolo, namely VenFn Ltd., snce t was de-lsted from the JSE equty market on 1 st March 2006. The de-lstng of VenFn was because of ts acquston by Vodafone. (VenFn Group, 2006: p.10) VenFn was kept n the portfolo to provde the holstc vew of the component over the chosen test perod. 5.3 Results wth Dscusson Each of the shares, makng up the components (also known as subportfolos) whch made up the test portfolo, was ndvdually regressed aganst the FTSE/JSE Afrca All Share 67

Index. The raw data of each component was processed through sets of MATLAB code. The MATLAB codes were wrtten based on the sngle ndex model. The process flow dagram of ths computer programme has been dscussed n Chapter 3. Results may be found under the Fnal Results folder on the dsk provded. The folder has further been categorsed nto two sectons, one beng the results wthout error terms and the other beng wth error terms. In the next sectons, these outcomes are revewed, accordng to dfferent components, and the overall portfolo outcomes examned. The structure of dscusson of the desgn outcome s best represented graphcally n Fgure 5.1 below. Analyss of Each Component n the Test Portfolo n Secton 5.3.1. Balanced Conservatve Core Alternatve Core Md- Term Small Caps Analyss of Overall Test Portfolo Based on the Weghtng found n Table 4.3 Secton 5.3.2. Excludng Errors Includng Errors Fgure 5.1: Structure of Dscusson for Desgn Outcomes Analyses on the outcomes of each of the components, namely the balanced, conservatve, core alternatve, core, md-term and small caps components of the test portfolo are to be dscussed separately. Ths dscusson s found n secton 5.3.1. The outcomes of the components are to be combned by usng the weghtngs found n Table 4.3, nto the overall test portfolo result. The overall test portfolo results wll be dscussed n both 68

contexts, one to exclude the error terms and the other to nclude the error terms. Ths dscusson s found n secton 5.3.2. 5.3.1 Results of Components The results of each component of the test portfolo are revewed below. The reason for examnng each component separately s due to the presence of repeated shares n the test portfolo across components. Repeated shares have been double counted when vewng the test portfolo holstcally. Some examples of the repeated shares are MTN and Barloworld. MTN was chosen for both balanced and md-term components. Barloworld s present n both core and md-term components. An nvestor needs to decde on an allocaton between the securtes wthn a portfolo. It s suggested to start wth equal allocaton among the securtes n a portfolo. Ths s supported by Elton et al. (1997: p.417) who state, equal nvestment s optmum f the nvestor has no nformaton about future returns, varances and covarances. Therefore, an equal splt n nvestment has been assumed for each securty n the component. From Table 4.3, the nvestment compostons of each component were stated as 18.75% for balanced component, 12.50% for conservatve component, etc. These are the compostons used for combnng the overall test portfolo. The above mentoned equal splt refers to the equal splt of the amount nvested n each of the securtes. For example: there are sx securtes n the balanced component. The monetary value of amount nvested n balanced component s R15000. Ths means that the monetary value nvested n each of the securtes n balanced component would be R15000 dvded by 6, whch equals to R2500. R2500 s the monetary value nvested n each of the securtes n balanced component. Further nvestments n the same shares are made f the share s present n another component. Indvdual shares weghtng, n each component, are based on the actual unts held. The actual unts held are calculated by dvdng equal monetary value n nvestments of the component nto the ntal ndvdual share prces (Refer to Appx I Dvds & Weghtngs Used for Beta Calculaton, p. 188). 69

The outcomes generated by passng raw data through the MATLAB codes are the beta values, alpha values and expected returns of components. The returns on a portfolo may be decomposed nto two parts: beta of the portfolo, whch s lnked to the return on the market, and alpha of the portfolo. Ths part can be attrbuted to characterstcs of the ndvdual shares comprsng the portfolo. Beta s the rato of correlaton between the component and the market to the varance of the market; ths s as defned n Chapter 2. Practcally speakng, beta represents the correlaton between the portfolo and the market. If beta s postve, t represents postve correlaton wth the market. Ths means that the portfolo moves n the same drecton as the market. Alpha can be nterpreted as the values that can be added by human nterventons, an example of whch s a fund manager. Thus, when beta s hgh, t s expected that alpha would be low, when the expected returns stay constant. Therefore, there s an nverse relatonshp between alpha and beta. Ths was dscussed n Chapter 2. The raw data has been passed through two sets of MATLAB codes respectvely. The results obtaned are smlar n both beta and alpha values but not the expected returns. Ths devaton has been prevously mentoned, and t s due to ncluson of error terms from sngle ndex model. The reasons contrbutng to these errors are dscussed n secton 5.3.2. 1. Balanced Portfolo In ths secton, the results, namely the betas, the alphas and the expected returns from ths component are dscussed. The results of ths component have been wrtten nto results_balanced.xls whch can be found on the dsk provded. 70

5 Ordnary Least Square Merrll Lynch Bayesan Adjustment 4 Weghted Average Beta 3 2 1 0 0 50 100 150 200 250 300 350 400-1 Tme [Days] Fgure 5.2: Weghted Average Beta for Balanced Component over Test Perod From Fgure 5.2, the weghted average beta for balanced component has been plotted aganst the number of days worth of data analysed. That s, the number of days nto the test perod. The purpose of representng results over the entre test perod s to dentfy trs. Ths s appled to the analyss of all the components to come n ths document. It s observed that the beta values stablse around the 50 th day,.e. t = 50. The ntal fluctuatons, between t = 0 and t = 45, are nherent wthn the data. It s not unusual for data to fluctuate durng the ntal test perod. The hgh fluctuatons are assocated wth the choce of daly data used. The beta coeffcents of stocks t to move near 1 over tme (ths s shown by ML seres), whle OLS and BA seres stablsed near 0 over tme. Ths means that ML seres ndcate almost total correlaton wth the market whle OLS and BA seres ndcate almost no correlaton. The almost no correlaton for both OLS and BA seres mples that dversfcaton has been managed adequately for ths balanced component. The ML seres ndcates the almost total correlaton, whch s due to the constant 1/3 added onto ts beta adjustment as seen n equaton (2.15), otherwse the ML seres would stablse at approxmate values as that of BA seres. Also, over tme, all three seres, OLS, BA and ML beta values have stablsed. 71

The general tr dsplayed, n Fgure 5.2, s that ML seres has the hghest beta value followed by BA then OLS. BA results are hgher than OLS because there are weghtng factors ncorporated. Ths tr s due to the adjustments made. The adjustments made on beta values are dscussed n secton 2.6. The OLS seres has the lowest beta values; ths s explaned mathematcally by usng the equaton (2.6). To obtan a low beta value, ether the covarances 19 between the shares and the market are low, or the varance present n the market s hgh. The securtes were chosen from dfferent sectors. So securtes may have lttle smlarty wth each other. If securtes have lttle smlarty wth each other then ther covarance wll be low. 1.4 1.2 Ordnary Least Square Merrll Lynch Bayesan Adjustment 1 Weghted Average Alpha 0.8 0.6 0.4 0.2 0 0 50 100 150 200 250 300 350 400 Tmes [Days] Fgure 5.3: Weghted Average Alpha for Balanced Component over Test Perod From Fgure 5.3, the postve alpha trs ndcate that ths component has been postvely msprced. Ths suggests that ths component has exceeded the general market expectaton. Alpha values can also be nterpreted as the values added by human nterventons. The ratonale of ths tr s the underlyng consttuents of ths balanced 19 Covarance s an unbounded measure of assocaton between two random varables. (Tucker et al., 1994: p.579) 72

component, manly commodty and cyclcal shares. Cyclcal shares returns are n close relaton wth the economcal cycle. South Afrca s currently n the boom phase of the busness cycle; hence selectng shares whch are closely related to buldng nfrastructure s preferable. Also, durng the test perod, the commodty prces dsplay an upward ncrease tr globally. Ths suggests there s upward pressure on the commodty prces, whch explans the better performance. It s also observed that the relatonshp between beta and alpha t to be nversely related, because the lowest beta value s assocated wth the hghest alpha value. The results for expected returns over the entre test perod are shown below. The excluson and ncluson of error terms have been shown n separate fgures. Fgure 5.4 shows that there s a steady ncreasng proportonal tr for the portfolo over the test perod. 180 160 140 Portfolo Returns [%] 120 100 80 60 40 20 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0 0 50 100 150 200 250 300 350 400 Tme [Days] Fgure 5.4: Returns Excludng Errors for Balanced Component over Test Perod When the error terms are ncluded, the graphcal results are shown n Fgure 5.5. The troughs and rdges present are related to the local economc envronment durng the test perod. The relatonshp between ths component and the local economc envronment s 73

dentfed by comparng the pattern establshed from ths component, shown n Fgure 5.5, to that of the All Share Index, shown n Fgure 5.26. It s also noted that the tr dsplayed by alpha values s smlar to that of the returns, excludng errors, of ths component. Ths can be potentally explaned by the fact that the alpha values have sgnfcant mportance to the expected returns, as shown n equaton (2.5), where expected returns are partally depent on alpha values. Therefore, the smlar trs are dsplayed by alpha values and expected returns excludng error fgures. 60 50 Ordnary Least Square Merrll Lynch Bayesan Adjustment 40 Portfolo Returns [%] 30 20 10 0 0 50 100 150 200 250 300 350 400-10 Tme [Days] Fgure 5.5: Returns Includng Errors for Balanced Component over Test Perod By comparng Fgure 5.4 and Fgure 5.5, t s evdent that the sgnfcance of the error terms cannot be gnored, as error terms play a sgnfcant part of expected returns. Ths s emphassed by the error results dsplayed n Table 5.1. 74

Leadng from the dscusson of results of ths component over the test perod, t s relevant to summarse results 20 of ths component. These are tabulated below, n Table 5.1. Table 5.1: Summarsed Results for Balanced Component Returns Include Error [%] Returns Exclude Errors [%] Errors [%] Beta Alpha OLS 0.115968 0.68821 18.4622208 70.54795797 52.0857372 ML 0.937312 0.5549095 23.17856253 82.6535753 59.4750128 BA 0.243772 0.6711464 19.08132616 72.00865596 52.9273298 From Table 5.1, ML beta value s 0.937312. As ths value s close to one, ths suggests the almost total correlaton wth the market. Thus the returns of ths component are explaned by the returns of the market,.e. they move n the same drecton. Also from equaton (2.5), t s observed that the only parameter whch can be controlled by an nvestor s the beta value. Selectng a portfolo that has a hgh beta value would ncrease the return. Ths statement s evdent from Table 5.1, where the hghest beta value, shown by ML, s assocated wth the hghest returns. It s also observed that there s an nverse relatonshp between the beta and alpha, as the lowest beta value s assocated wth the hghest alpha value, as shown by OLS. The low beta values suggest the possblty of addng value by external means,.e. a fund manager. 2. Conservatve Portfolo Ths s the component that ncludes the share wth sparse data, VenFn Ltd. (VNF). Thus, the analyses have been separated nto two parts. In the frst part, VNF has been ncluded n the subportfolo up to the pont when t was de-lsted,.e. 1 st March 2006 and n the second part, VNF has been excluded from the analyss snce 1 st March 2006. The detaled outcomes can be found n the fle results_conservatves.xls on the dsk provded. 20 Summarse results refer to the average calculated over the entre test perod. 75

2.5 Ordnary Least Square Merrll Lynch Bayesan Adjustment 2 Weghted Average Beta 1.5 1 0.5 0 0 50 100 150 200 250 300 350 400-0.5 Tme [Days] Fgure 5.6: Weghted Average Beta for Conservatve Component over Test Perod The beta tr dsplayed n Fgure 5.6 s lower than the betas for the balanced component, shown n Fgure 5.2. The reason s that the securtes of ths component are the blue chp 21 and growth securtes, where stable securty prces are present, and therefore lower systematc rsk. The beta values stablse over the test perod. The ML seres stablses around 0.6, whch mples ths portfolo s less volatle than ALSI. Ths also means that ths component should return 6% when ALSI rses 10%, smlarly ths component should lose only 6% when ALSI drops by 10%. The OLS and BA seres stablse near 0 over the test perod. The tr dsplayed n Fgure 5.6 s that the ML seres has the hghest beta value followed by the BA seres then the OLS seres. The reason for ths has been dscussed under the secton of balanced component. From Fgure 5.7, the alpha tr dsplays a negatve slope between the 1 st and 40 th days,.e. t = 1 and t = 40. Ths means that expected returns over the same perod are negatvely msprced as predcted by ther beta correspondent. Ths means that ths component has 21 These are the stocks that were bought wth equal fervour and enthusasm by both nvestors and speculators at the same exalted prces. (Graham et al., 1962: p.410) 76

not exceeded the general market expectatons between t = 1 and t = 40. Around the 130 th day,.e. when t = 130, there s a sharp downward vertcal dscontnuty n the alpha values because of the de-lstng of VenFn Ltd. from JSE due to acquston by Vodafone. (VenFn Group, 2006: p.10) 0.6 0.5 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0.4 Weghted Average Alpha 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400-0.1 Tme [Days] Fgure 5.7: Weghted Average Alpha for Conservatve Component over Test Perod The weghted average alpha over the test perod s low. Ths means the securtes n ths component are prced relatvely accurately. Ths s as expected snce the majorty of ths component s made up of blue chp and growth securtes. From Fgure 5.8, the ntal downward slope from t = 0 to t = 30 suggests a decrease n securty prces over ths perod. When ths component s vewed n solaton, ts returns move from 0% to just over 70% at the of the test perod. There s a sudden drop at the 130 th day,.e. t = 130, agan due to the de-lstng of VenFn Ltd. from JSE. Ths drop shows the sgnfcance of VenFn Ltd. n ths component. Ths s caused by the 35% nvestment allocaton placed wth VenFn Ltd. when ths subportfolo was formed. 77

80 70 60 Portfolo Returns [%] 50 40 30 20 10 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0 0 50 100 150 200 250 300 350 400 Tme [Days] Fgure 5.8: Returns Excludng Errors for Conservatve Portfolo over Test Perod Furthermore, t s mportant to vew the subportfolo n a domestc economc envronment, where the uncertanty of the economy needs to be ncorporated. Ths s shown graphcally n Fgure 5.9. By ncludng the errors nto portfolo returns, there are more fluctuatons along the ncreasng tr. The pattern shown n Fgure 5.9 concdes wth the general movement of the All Share Index, from Fgure 5.26. The returns of ths component accumulate from over 5% on the 50 th day to below 30% at of test perod. Ths rate of return s conservatve n relaton to the balanced component dscussed prevously. 78

35.00 30.00 Ordnary Least Square Merrll Lynch Bayesan Adjustment 25.00 20.00 Portfolo Returns [%] 15.00 10.00 5.00 0.00 0 50 100 150 200 250 300 350 400-5.00-10.00 Tme [Days] Fgure 5.9: Returns Includng Errors for Conservatve Component over Test Perod The summarsed results for conservatve component over the test perod s tabulated below, Table 5.2. Table 5.2: Summarsed Results for Conservatve Component Returns Include Error [%] Returns Exclude Errors [%] Errors [%] Beta Alpha OLS 0.055261 0.3042962 12.96138258 31.320745 18.3593624 ML 0.622325 0.2129558 16.19153807 39.6195382 23.4280001 BA 0.103182 0.2975688 13.13097797 31.87917021 18.7481922 OLS has the lowest beta value as shown n Table 5.2. OLS has a beta value of 0.055261; ths value represents a flat slope and low rate of change. Therefore the market-related rsk s low. The low beta value also suggests the dversfcaton of securtes n ths component, where the covarances between securtes are low, meanng there s lttle smlarty between ths component and the market. 79

3. Core Alternatve Portfolo The detaled outcomes of ths subportfolo can be found n the fle results_corealternatve.xls on the dsk provded. 0.4 0.35 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0.3 Weghted Average Beta 0.25 0.2 0.15 0.1 0.05 0-0.05 0 50 100 150 200 250 300 350 400 Tme [Days] Fgure 5.10: Weghted Average Beta for Core Alternatve Component over Test Perod From Fgure 5.10, the beta of ths component s generally very low. The ML seres stablses below 0.1, and the OLS and BA seres stablse near 0. These values are very much lower than both the balanced and conservatve portfolos. Hence, ths suggests that there are lmted correlatons wth the general market. The possble reason for ths s the hgh degree of dversfcaton present n ths component, snce 3 out of 5 securtes ncluded are dual-lsted 22. Ths has effectvely dversfed across dfferent economes as well as sectors and has effectvely transferred the rsk across countres. 22 Dual-lsted means the share s lsted on two stock exchanges. 80

Because 3 out of 5 shares ncluded n ths component are focused n the fnancal sector, ths has ntroduced the potental of concentraton rsks. They are, however, exposed to dfferent magntudes and classfcaton of rsks due to ther dfferent market captalsaton. For example: SBK s the largest bank n Afrca based on the market captalsaton and manly operates n emergng markets, whle FSR s more focused on local markets whose market captalsaton s not as bg as that of SBK. From Fgure 5.11, the alpha values move from below 0.05 at t = 0 to just below 0.3 at the of the test perod. These low alpha values suggest that ths component has exceeded the general market expectatons slghtly, and mples that there s very lttle msprcng of these securtes. 0.35 0.3 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0.25 Weghted Average Alpha 0.2 0.15 0.1 0.05 0 0 50 100 150 200 250 300 350 400 Tmes [Days] Fgure 5.11: Weghted Average Alpha for Core Alternatve Component over Test Perod 23 23 OLS and BA seres shown n Fgure 5.11 concdes. Ths means that ther alpha values are very smlar. 81

It s observed that the pattern shown n Fgure 5.11 for the alpha values s smlar to that dsplayed for returns excludng errors, n Fgure 5.12. Ths component s returns ncrease n a proportonal manner, where ts returns ncreased from 0% at t = 0 to over 30% at of test perod. Ths rate of returns s expected snce the securtes n ths component are manly blue-chp and value securtes where these categores of shares represent consstent growth over tme. The consstent growth of shares s shown through ther stable securty prces; therefore t s unusual to see rapd and sudden growth n returns over a short test perod. These vews are emphassed by the low alpha values over the test perod. 35 30 25 Portfolo Returns [%] 20 15 10 5 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0 0 50 100 150 200 250 300 350 400 Tme [Days] Fgure 5.12: Returns Excludng Errors for Core Alternatve Component over Test Perod 24 By examnng the returns of ths component n the overall domestc economc envronment where errors are ncluded, Fgure 5.13 s generated. From Fgure 5.13, the rate of returns ncreased from above 0% at t = 0 to over 25%, shown by ML, at the of 24 OLS and BA seres shown n Fgure 5.12 concdes. Ths means that ther returns wthout errors values are very smlar. 82

the test perod. The pattern dsplayed concdes wth the All Share Index shown n Fgure 5.26. 30.00 25.00 Ordnary Least Square Merrll Lynch Bayesan Adjustment 20.00 Portfolo Returns [%] 15.00 10.00 5.00 0.00 0 50 100 150 200 250 300 350 400-5.00-10.00 Tme [Days] Fgure 5.13: Returns Includng Errors for Core Alternatve Component over Test Perod From Table 5.3, t s evdent that both alpha and beta values are low n ths component. The low beta values across the three seres suggest a steady rate of change between the covarance of securtes and the market wth the varance of the market. Therefore, ths results n a flatter slope. A flatter slope s expected snce ths component complments the core component, and no drastc changes are expected. Table 5.3: Summarsed Results for Core Alternatve Component Returns Include Error [%] Returns Exclude Errors [%] Errors [%] Beta Alpha OLS 0.00553 0.1953271 7.50754386 19.58359531 12.0760514 ML 0.08702 0.1822235 12.28867446 20.77674891 8.48807445 BA 0.014024 0.1942573 7.909994432 19.67691018 11.7669157 83

Another reason for low beta values s that ths component s well-dversfed, hence most of the systematc rsk (β) has been elmnated. The rate of return generated from ths component s reasonable. The reason for ths s that the rate of return has exceeded the government s target nflaton of maxmum 6%. 4. Core Portfolo The outcomes of ths subportfolo can be found n the fle results_core.xls on the dsk provded. At the ntal start up of the data process, beta fluctuates to a maxmum value of just below one; whch s seen n Fgure 5.14. The beta values stablse at just over 0.2 for ML, 0.05 for BA and nearly zero for OLS. 1.2 1 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0.8 Weghted Average Beta 0.6 0.4 0.2 0 0 50 100 150 200 250 300 350 400-0.2 Tme [Days] Fgure 5.14: Weghted Average Beta for Core Component over Test Perod 84

The low beta values are due to the low covarances between the market and ndvdual shares n ths subportfolo, resultng n effcent dversfcaton. The dversfcaton s evdent from the dual-lstng structure of 3 out of 5 securtes n ths component. The beta values of ths component are hgher than that of the core alternatve. Ths means that the systematc rsk of the core s hgher than the core alternatve component. The core alternatve s a component whch wll complement ths one. The reason for hgher beta values n core than core alternatve s the nature of securtes. In ths component, the nature of chosen securtes s blue chp and commodty related. Commodtes dep on varous factors whch cannot be controlled by ndvdual nvestors. From recent events occurrng n both the local and global envronment, t s observed that commodty related securtes experence a reasonable amount of volatlty. From Fgure 5.15, the tr of ncreasng alpha values over the test perod ts to be assocated wth a decreasng tr of beta values. Ths nverse relatonshp s evdent when comparson s done between Fgure 5.14 and Fgure 5.15. The reason for ths has been dscussed prevously. 85

0.8 0.7 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0.6 Weghted Average Alpha 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 Tmes [Days] Fgure 5.15: Weghted Average Alpha for Core Component over Test Perod 25 80 70 60 Portfolo Returns [%] 50 40 30 20 10 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0 0 50 100 150 200 250 300 350 400 Tme [Days] Fgure 5.16: Returns Excludng Errors for Core Component over Test Perod 26 25 OLS and BA seres shown n Fgure 5.15 concdes. Ths means that ther alpha values are very smlar. 26 All three seres, BA, OLS and ML seres shown n Fgure 5.16 concdes. Ths means that ther returns wthout errors values are very smlar. 86

Fgure 5.16 shows the steady proporton ncrease of returns over tme. The returns have ncreased from 0% to over 70% from the begnnng to the of the test perod. The relatonshp between returns and alphas was dscussed n the prevous sectons. Leadng from returns excludng errors for the core component, t s relevant to dscuss the returns ncludng errors for the same component. From Fgure 5.17, t s seen that the returns move from 5% at t = 0 to 35%, shown by ML seres, at of the testng perod. The rate of returns shown s reasonable, due to the nature of ths component. For a core component, t s mportant for ts consttuents to show steady growth over tme. The general pattern shown n Fgure 5.17 concdes wth the pattern of the All Share Index, dsplayed n Fgure 5.26. 40.00 35.00 Ordnary Least Square Merrll Lynch Bayesan Adjustment 30.00 Portfolo Returns [%] 25.00 20.00 15.00 10.00 5.00 0.00 0 50 100 150 200 250 300 350 400 Tme [Days] Fgure 5.17: Returns Includng Errors for Core Component over Test Perod From Table 5.4, the beta values of ths component are hgher than the core alteratve component but lower than both balanced and conservatve components. The lower beta 87

values are due to the hgh degree of dversfcaton present n ths component. Ths thought s supported by the mult-lstng of varous securtes n ths component. The mult-lstng securtes are AGL, LBT and BAW. Through mult-lstng, the rsks have been dversfed through dfferent economes. Table 5.4: Summarsed Results for Core Component Returns Include Error [%] Returns Exclude Errors [%] Errors [%] Beta Alpha OLS 0.022915 0.3436701 10.41249866 34.67313681 24.2606381 ML 0.231943 0.3098974 15.13263746 37.74705924 22.6144218 BA 0.050592 0.3403098 10.82066909 34.96655329 24.1458842 5. Md- Term Portfolo The outcomes can be found n the fle results_mdterm.xls on the dsk provded. Ths component conssts of 11 shares n total. Ths component was selected for md-term nvestments. Ths refers to the md-term tme horzon; hence varous major sectors on JSE have been selected. Dversfcaton s, thus, acheved. Ths exposes the nvestor to dfferent rsks n each ndustry. Thus, by summng up each rsk assocated wth sectors, t s clear that a hgher beta value s created. The beta of ths component, shown n Fgure 5.18, s hgher than conservatve, core alternatve and core subportfolos, but on par wth the balanced component. 88

4.5 4 Ordnary Least Square Merrll Lynch Bayesan Adjustment 3.5 3 Weghted Average Beta 2.5 2 1.5 1 0.5 0 0 50 100 150 200 250 300 350 400-0.5 Tme [Days] Fgure 5.18: Weghted Average Beta for Md- Term Component over Test Perod It s observed, from Fgure 5.18, that the ML seres stablses near 1, whle the OLS and BA seres stablse near 0. Ths suggests the almost total correlaton of ML seres wth the market and almost no correlaton of OLS and BA seres. The ML seres has the hghest beta value followed by the BA seres then the OLS seres. These dscussons can be found n the dscusson on the balanced component. It s also noted that the alpha values dsplayed n Fgure 5.19, are generally hgher when compared to the other components of the test portfolo. The ratonale behnd ths s that the securtes categores have been ncluded n ths component, namely blue-chp, value and cyclcal securtes. These are usually the securtes wth sold fundamentals, meanng the possbltes of exceedng general market expectatons can be expected. 89

1.8 1.6 Ordnary Least Square Merrll Lynch Bayesan Adjustment 1.4 Weghted Average Alpha 1.2 1 0.8 0.6 0.4 0.2 0 0 50 100 150 200 250 300 350 400 Tmes [Days] Fgure 5.19: Weghted Average Alpha for Md- Term Component over Test Perod 27 Shown n Fgure 5.20, the rate of returns of ths component ncreased from 0% at t = 0 to over 180%, shown by ML seres, at of test perod. Ths s due to the cyclcal nature of the securtes ncluded. Some of the cyclcal securtes ncluded n ths component are M&R, HLD, PPC and BAW. Currently, the domestc South Afrcan economy s preparng for the 2010 Soccer World Cup and varous nfrastructure needs to be bult, therefore constructon and cement frms would show rapd growth. 27 OLS and BA seres shown n Fgure 5.19 concdes. Ths means that ther alpha values are very smlar. 90

200 180 160 140 Portfolo Returns [%] 120 100 80 60 40 20 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0 0 50 100 150 200 250 300 350 400 Tme [Days] Fgure 5.20: Returns Excludng Errors for Md-Term Component over Test Perod 28 From Fgure 5.21, t s observed that the returns ncludng errors for ths component ncreased from 0% at t = 0 to over 50%, shown by ML seres, at t = 350. The troughs and rdges shown are n close correlaton wth the local economy. 28 OLS and BA seres shown n Fgure 5.20 concdes. Ths means that ther returns wthout errors values are very smlar. 91

70.00 60.00 Ordnary Least Square Merrll Lynch Bayesan Adjustment 50.00 Portfolo Returns [%] 40.00 30.00 20.00 10.00 0.00 0 50 100 150 200 250 300 350 400 Tme [Days] Fgure 5.21: Returns Include Errors for Md-Term Component over Test Perod From Table 5.5, the hghest beta value s assocated wth ML seres. The value s 0.944171, whch s close to one. Ths mples almost total correlaton, and that a far amount of return on the portfolo s explaned by the return on the market. Ths vew s supported by the cyclcal nature of securtes. Table 5.5: Summarsed Results for Md-Term Component Returns Include Error [%] Returns Exclude Errors [%] Errors [%] Beta Alpha OLS 0.096257 0.9094312 18.90735527 92.17136522 73.2640099 ML 0.944171 0.7721751 23.6266315 104.6525744 81.0259429 BA 0.194819 0.8964564 19.34284848 93.30559812 73.9627496 6. Small Caps Portfolo The outcome can be found n the fle, results_smallcap.xls on the dsk provded. 92

From Fgure 5.22, beta values stablse around 0.2 for ML, 0.05 for BA and 0 for OLS. The beta values are low for ths component, meanng there s low systematc rsk. The low systematc rsk can be explaned by the low market captalzaton held by the securtes of ths component. Small market captalzaton also means the low correlaton between the market and the frm. 1.6 1.4 Ordnary Least Square Merrll Lynch Bayesan Adjustment 1.2 Weghted Average Beta 1 0.8 0.6 0.4 0.2 0 0 50 100 150 200 250 300 350 400-0.2 Tme [Days] Fgure 5.22: Weghted Average Beta for Small Caps Component over Test Perod The securtes ncluded n ths component are of the small captalzaton nature. Securtes of ths knd are the securtes wth good potental, that may one day develop nto bluechp frms. The frms ncluded came from four of the major sectors dvson for the All Share Index. These sectors are consumer goods, consumer servces, ndustrals and technology. These are also the sectors that are closely related to the 2010 Soccer World Cup. From Fgure 5.23, the alpha values ncreased to 0.45 at t = 350 from 0 at t = 0. Alphas of ths component are generally lower than alphas of the other components. The ratonale 93

behnd ths s that the securtes of ths component are small captalzaton n nature, meanng the mpact of general market expectatons on ths component s lmted. 0.5 0.45 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0.4 0.35 Weghted Average Alpha 0.3 0.25 0.2 0.15 0.1 0.05 0 0 50 100 150 200 250 300 350 400 Tmes [Days] Fgure 5.23: Weghted Average Alpha for Small Caps Component over Test Perod Fgure 5.24 shows the steady proporton ncrease of returns over tme. The returns have ncreased from 0% to over 50%, shown by ML seres, from the begnnng to the of the test perod. The troughs and rdges shown are n close correlaton wth the local economy. The relatonshp between returns and alphas was dscussed n the prevous sectons. 94

60 50 Portfolo Returns [%] 40 30 20 10 Ordnary Least Square Merrll Lynch Bayesan Adjustment 0 0 50 100 150 200 250 300 350 400 Tme [Days] Fgure 5.24: Returns Excludng Errors for Small Caps Component over Test Perod 29 40.00 Ordnary Least Square Merrll Lynch Bayesan Adjustment 30.00 Portfolo Returns [%] 20.00 10.00 0.00 0 50 100 150 200 250 300 350 400-10.00-20.00 Tme [Days] Fgure 5.25: Returns Includng Errors for Small Caps Component over Test Perod 29 OLS and BA seres shown n Fgure 5.24 concdes. Ths means that ther returns wthout errors values are very smlar. 95

From Fgure 5.25, t s seen that the returns move from -10% at t = 0 to 20% at of testng perod. The general pattern shown n Fgure 5.25 concdes wth the pattern of the All Share Index, dsplayed n Fgure 5.26. Table 5.6: Summarsed Results for Small Caps Component Returns Include Error [%] Returns Exclude Errors [%] Errors [%] Beta Alpha OLS 0.016919 0.3010414 5.738617277 30.27626406 24.5376468 ML 0.194612 0.2723105 10.44972671 32.89085395 22.4411272 BA 0.04451 0.2978093 6.343006695 30.55413004 24.2111233 From Table 5.6, the beta values are lower than the other components. Ths means that there are lmted correlatons between ths component and the market. The returns from ths component are low relatve to other components n the test portfolo. Ths s as expected snce the postons of the small captalsaton securtes are not sgnfcant enough to contrbute to or make a sgnfcant mpact on the market. 5.3.2 Results of Overall Test Portfolo In ths secton, the outcomes from each of the components have been combned to dsplay the overall results. Below, the overall outcomes have been represented, one to exclude the error from the sngle ndex model and the other to nclude t. Components are combned usng weghtngs. The weghtngs 30 are based on the fractonal nvestment n each component, as shown n Table 4.3. 30 Weghtngs refer to the percentage nvested n each subportfolo. These values can be found n Table 4.3. 96

Exclude Errors 100.000 90.000 80.000 R194 Bond All Share Index Ordnary Least Square Merrll Lynch Bayesan Adjustment 70.000 Expected Returns [%] 60.000 50.000 40.000 30.000 20.000 10.000 0.000 28-May-05 5-Sep-05 14-Dec-05 24-Mar-06 2-Jul-06 10-Oct-06 18-Jan-07 28-Apr-07 Date Fgure 5.26: Daly Comparson of Expected Returns Excludng Errors of Test Portfolo over Test Perod From Fgure 5.26, the R194 Bond acts as a benchmark to whch each of the seres models s compared. Expected returns of the R194 Bond start off from approxmately 7.3% and ncrease to 8.8% at of the test perod. The determnant of bond return s n close proxmty wth annual nflaton predcted by the government. In comparson wth others, the R194 Bond dsplays a relatvely steady tr throughout the test perod. The adjustment models, OLS, ML and BA and the All Share Index, all start off at 0% because the ntal share prces are beng used as the reference pont to whch the daly returns are compared. The results fluctuate untl November 2005, and then all adjustment models dsplay a reasonably postvely proportoned relatonshp. Ths mples that the expected returns have accumulated over tme, and hence ndrectly showed that the test portfolo performed better than the rsk-free nstrument. If the All Share Index outperforms the rsk-free nstrument, ths mmedately suggests that the test portfolo has 97

also performed better than the rsk-free nstrument, as there are postve correlatons between the test portfolo and the market shown by the beta values. Ths can be demonstrated by conductng a basc return calculaton on the All Share Index between the start and the of the test perod. The data used for ths calculaton s dsplayed below. All Share Index Value Start of Test Perod 1 st September 2005 15646.47 End of Test Perod 31 st January 2007 25481.25 The basc return calculaton s based on the followng formula: Re turn [%] End Po nt Start Start Po nt Pont 100 Therefore, return of the All Share Index s equal to 62.86% over the test perod. Ths result shows that the ALSI has outperformed the chosen rsk-free nstrument, the R194 bond, as expected. Also, the test portfolo generates better returns than that of the market,.e. the All Share Index, provded that the random error present n the market s not consdered. Ths suggests that an nvestor could outperform the market f the securtes were selected wth cauton. Wth every nvestment comes rsks, hence nvestments should be conducted cautously, ths also refers to process pror to makng the decsons. 98

Include Errors 60.000 50.000 R194 Bond All Share Index Ordnary Least Square Merrll Lynch Bayesan Adjustment 40.000 30.000 20.000 10.000 0.000 28-May-05 5-Sep-05 Expected Returns [%] 14-Dec-05 24-Mar-06 2-Jul-06 10-Oct-06 18-Jan-07 28-Apr-07 Date Fgure 5.27: Daly Comparson of Expected Returns Includng Errors of Test Portfolo over Test Perod When the nvestor ncludes the error terms nto the expected returns of the portfolo, as shown n Fgure 5.27, the test portfolo results are stll hgher than the government bond R194, but lower than the All Share Index (market benchmark). The dfferent outcome s due to the error term. The error term cannot be gnored n an economc envronment, snce by excludng t, the results would be dstorted. Ths dstorton arses from vewng the results n solaton, wthout the error terms, nstead of n a broad economc envronment. Ths s supported by Gleser (1998: p. 278), who says devatons 31 from measured mean due to mprecsely determned contextual condtons are now of a magntude that they cannot be gnored. 31 Devatons can be referred to as errors. 99

Also, Chen et al. (1983) suggest, sample estmators are usually treated as f they were true values of unknown parameters. Thus, by treatng the estmated error vector, generated by usng equaton (2.10), as a true value, ths wll greatly affect the outcome, as seen n Fgure 5.27. Ths dea s emphassed by Fsher et al. (1997: p.43), that optmsed mean-varance portfolos are extremely senstve to even subtle changes n the estmaton of the parameters. The error term cannot be estmated accurately as t s random n nature. Ths randomness s parametrc n nature and nherent n the market tself. Ths parametrc uncertanty plays a sgnfcant role n portfolo returns over tme, snce ths uncertanty should also be consdered as a measure of busness rsks (Israelsen et al. 2007: p. 419). Uncertanty assocated wth the error vector can be fundamentally explaned by supply and demand. A supply and demand relatonshp could be altered by varous factors, whether t be macro- or mcro- economcally related. Some of the most common economcal reasons are (Standard Bank Group, 2007): 1. The health of the US economy As the US s the most mportant economy globally, ts performance would drectly affect other natons. If the US economy s n a boom phase of the busness cycle, ths would mply the same goes for the rest of the world. In the context of ths desgn, when the US economy s blossomng, the South Afrcan economy would also blossom, thus creatng a healthy and actve stock exchange. As a drect consequence, the market performs better and there s an ncrease tr n securty prces. 2. Offcal nterest rate dctated by Reserve Bank Interest rate s part of the monetary polcy of a country. It drectly affects companes earnngs, because when nterest rates ncrease t would ncrease cost of debt payments and hence affect earnngs. 100

An ncrease n nterest rates would affect the level of economc actvty and consumer spng. It would reduce consumer spng, snce debt payments would be hgher and less dsposable ncome would be avalable for nvestment purposes. Ths would potentally result n less demand for the securtes. Thus securty prces would decrease n order to reach a new equlbrum pont between supply and demand. From Fgure 5.28, showng repo rate 32 changes, the ncreasng repo rate puts a downward pressure on share prces, snce there s less dsposable ncome to be spent on nvestments. 9 8 7 7.5 8 8.5 9 Repo Rate [%] 7 6 5 4 3 14-Apr-05 8-Jun-06 3-Aug-06 13-Oct-06 8-Dec-06 2 1 0 Fgure 5.28: Repo Rate Changes over Test Perod (Source: South Afrcan Reserve Bank, 2007a) 3. Exchange rate, or how the Rand fares aganst other currences If a frm exports or mports products or servces from other countres, or has payments or recepts n other currences, t s affected by the exchange rate 32 Repo rate s the nterest rate at whch the Reserve Bank ls money to the fnancal nsttutons. 101

between the Rand and other currences. A few currences of partcular nterest to the Rand are the US Dollar, the Brtsh Pound and the Euro. From Fgure 5.29, there s a clear deprecaton n South Afrcan currency between May and October 2006. Ths would affect frms whch are multlsted across countres by puttng upward pressure on expenses, leadng to reduced earnngs on ther fnancal statements, thus reducng EPS and potentally reducng share prces. 16 14 Exchange Rate [Rand Per Currency] 12 10 8 6 4 Rand Per US Dollar Rand Per Pound Rand Per Euro 2 0 9/1/2005 10/1/2005 11/1/2005 12/1/2005 1/1/2006 2/1/2006 3/1/2006 4/1/2006 5/1/2006 6/1/2006 7/1/2006 8/1/2006 9/1/2006 10/1/2006 11/1/2006 12/1/2006 1/1/2007 Date Fgure 5.29: Exchange Rate over Test Perod (Source: South Afrcan Reserve Bank, 2007b) 4. Inflaton rate The securty market dslkes nflaton as t pushes up the operatng, fnancal and nvestng costs for companes. The companes cannot pass the ncreased 102

costs to consumers quck enough due to some of the regulatons, thus nflaton drectly affect the company s earnngs. The nflaton rate s usually represented by the Consumer Prce Index (CPI). An ncrease n nflaton suggests a decrease n the purchasng power of consumers. So, f the consumers want to mantan ther current lvng standards, more money needs to be spent. Ths acton would lead to less dsposable ncome that can be used for nvestment purposes. Thus, the stock exchange may become less actve, snce supply s greater than demand.e. less people are buyng shares, leadng to the declne n share prces. 5. Rate of growth of South Afrca s Gross Domestc Product (GDP) The GDP s the value of all goods and servces produced n an economy. When GDP ncreases, the economy expands and a frm s earnngs wll rse and vce versa. When the frm s earnngs ncrease, ths leads to a hgh EPS. Therefore, share prces would ncrease. The dscrepances between the expected returns whch exclude and nclude error terms have, thus, been dscussed. The averages over the entre test perod wll now be compared. 103

Fgure 5.30: Average Returns Excludng Errors Comparsons Over Test Perod From Fgure 5.30, the R194 Bond performed at an average of 7.92% over the test perod, whle the OLS at 46.44%, the ML at 52.18%, the BA at 47.02% and the All Share Index at 27.69%. Ths suggests that the OLS and the BA can be approxmated, thus the BA adjustment model was unnecessary. The yeld of the R194 Bond s 7.92%. Ths fgure s only slghtly above the proposed nflaton target of 6% by the government. (Statstcs South Afrca, 2007) Ths suggests that f an nvestor doesn t wsh to encounter any rsk and s satsfed wth keepng the present monetary value of the nvestment, government bonds should be consdered. 104

Fgure 5.31: Average Returns Includng Errors Comparsons Over Test Perod From both Fgure 5.30 and Fgure 5.31, t s observed that the test portfolo selected has outperformed the R194 bond. Ths mples the purchasng power of money has been sustaned n ths desgn report. 5.4 Summary From the demonstraton, the followng was found: the computer programme developed, based on the proposed crtcal lterature revew as dscussed n Chapter 2, can be used to perform calculatons on the components (these nclude the balanced, the core, the core alteratve, the conservatve, the md-term and the small cap components) over the perod analysed: o beta values t to stablse around t = 50, the ML seres stablses above 0.5, the BA and the OLS seres stablse near zero 105

o the ML seres has the hghest beta values, followed by the BA seres then the OLS seres o alpha values t to rse and show a postve tr o alpha and beta values t to be nversely related, o alpha and expected returns dsplay a smlar tr o expected returns, for both excluson and ncluson of error terms, are hgher than the proposed annual nflaton rate. 106

Chapter 6 Conclusons & Further Work 6.1 Conclusons For any nvestor to generate returns on ther securtes portfolos, they need to gan the necessary nvestment-related knowledge. There are many models that can be used; the fundamentals of MPT have been wdely used by passve nvestors and they have been used n ths desgn to serve as the bass for the automated model. Wth the model developed, the objectve s accepted as acheved wthn the accuracy of ths desgn. However, ths desgn s based towards a partcular type of securty, namely shares and selected ndustres. The detals of these are dscussed below. The objectves of ths desgn have been met, namely: To develop a model for passve portfolo management usng MPT tools va a crtcal lterature revew. Ths s acheved by develop a complete methodology that asssts nvestors n the management of ther portfolos. The proposed methodology s represented graphcally n Fgure 1.1. The pertnent model was acheved through a crtcal lterature revew as outlned n Chapter 2, by usng both Markowtz s mean-varance framework and Sharpe s sngle ndex model. To develop a computer programme where the model s valdated through the use of a test portfolo. Ths s explaned by the automaton of the above-mentoned passve portfolo management model va a computer programme whch was developed as outlned n Chapter 3. The structure of the test portfolo was outlned n Chapter 4. The computer programme developed has acheved ts purpose whch s to demonstrate the automaton of the model. Ths s shown by the results generated by the computer programme, whch was dscussed n Chapter 5. 107

The MATLAB software selected for the development of the model has acheved the stated objectves. Therefore, the model developed n ths desgn has acheved the objectves as stated n Chapter 1. The desgn questons, as stated n Chapter 1, have also been answered. Frstly, the reasons for portfolo selecton have been nvestgated, namely the macroeconomc factors of an economy, an nvestors preferences and profles and the use of both fundamental and techncal analyss. Secondly, the fundamentals and models assocated wth MPT have been understood, namely Markowtz s Portfolo Theory and Sharpe s Sngle Index Model. The author has developed fundamental knowledge n the mean-varance framework and the sgnfcance of ths framework, thus a prvate nvestor can do the same based on ths desgn report. Thrdly, a rsk-return relatonshp has been establshed on the test portfolo. Ths s acheved by analysng the relatonshp between beta values wth expected returns, whch s dscussed n Chapter 5 desgn outcomes. The model developed s valdatng through the use of a selected test portfolo. It s relevant to examne the consttuents of the test portfolo, where the selected portfolo has been categorsed nto dfferent components due to the nature of ther consttuents. The reasons that were consdered for the test portfolo were dscussed. Sharpe s Sngle Index Model was used for determnng the portfolo returns. The test portfolo was dvded nto sx components, namely balanced, conservatve, core alternatve, core, md-term and small-cap, accordng to the nature of consttuents and nvestment tme horzon. In more detals, the components results were dscussed n Chapter 5. Betas are reasonable measures for rsk exposure and they gve approxmate drectons n whch the systematc rsks wll move. If the beta values are postve, they wll move n the same drecton to that of the market and vce versa. The low beta values generated from the components mpled low covarances, thus hgh levels of dversfcaton. The dversfcaton was manly acheved through the dual- or mult-lstng of the securtes on other stock exchanges. It was noted that both beta and alpha values ted to stablze around tme seres contanng 50 data values,.e. around t=50. Ths s due to the ntal startng up fluctuatons,.e. the use of daly data. 108

Alphas can be nterpreted as the human nterventons that can be added to components n an attempt to ncrease the returns. Alphas and betas have an nversely proportoned relatonshp. The patterns of alpha, for each component, are dentcal to that of the correspondng fgures for returns excludng errors. The troughs and rdges of graphs assocated wth returns ncludng error over the test perod, concde wth the All Share Index pattern. From the dscusson n Chapter 5, secton 5.3.2, t was observed that there were postve returns generated by the test portfolo. Two sets of outcomes were analyzed, one excludes and the other ncludes the error term from the sngle ndex model respectvely. The two sets of results do not concde. In the set of results that excludes the error term, the test portfolo outperforms both the government R194 bond and the market. Whle n the set of results that ncludes the error term, the test portfolo underperforms relatve to the market but outperforms the government R194 bond. The reasons for these dfferences could be due to the state of the US economy, the nflaton rate wthn the domestc economy, nterest rates, exchange rates relatve to other currences and GDP growth statstcs. Each of the pertnent reasons has been dscussed n more detal n secton 5.3.2. The average rate for the R194 bond s 7.57% over the test perod. Ths value s slghtly hgher than the government-proposed nflaton rate. Therefore, bonds may be used as an alternatve choce for rsk-averse nvestors. Ths was dscussed n secton 5.3.2. Generally, the returns generated by the OLS and BA adjustments were smlar, thus the Bayesan adjustments carred out on the ntal OLS results may be unnecessary. It s concluded that OLS s an adequate estmaton of BA for ths test. Fndngs from ths desgn ndcate that ths desgn has contrbuted to enable prvate nvestors to make sound nvestment decsons based on ths document. 109

In concluson, ths desgn has acheve ts objectves by provdng some useful nformaton that can be used by prvate nvestors to determne what aspects can be nvestgated pror to ther portfolo selectons and the relatonshps between the market and ther portfolos can be examned. 6.2 Drectons for Further Work The followng areas for further work are dentfed: 1) The models used n ths research gave statc estmaton of beta values. An approach can be taken to estmate beta values dynamcally; such an approach could be the use of Kalman flterng. 2) Hypothess formaton on the superorty of the Sngle Index Model over others. 3) Hypothess formaton on effcent market, testng for the type of market present. 4) Attempts can be made to deal wth mplcatons and lmtatons assocated wth MPT. 5) There are sgnfcant dscrepances between the results wth the error term from Sharpe s sngle ndex model and the results wthout t. An mplcaton for further research may be a detaled nvestgaton nto the error term from the sngle ndex model usng a neural network. A neural network s a recommed technque to dentfy the patterns and flter out nose from the errors. 6) In ths desgn, the short-sellng of securtes has not been mentoned. For further work, short-sellng cases can be nvestgated. 7) Personalsaton of the data set. User nterface can be mproved from what s proposed n ths desgn report. Currently, an nvestor needs to nsert a new column for a new securty n front of the All Share Index n the raw data workbook. He must then open the Excel workbook Weght Factors for Calculaton Beta on the CD provded, nsert an addtonal row for ncluson of new securty, enter the actual number of unts held and annual dvds; then a new percentage held by each of the portfolo consttuents needs to be calculated. Once these are establshed, the MATLAB codes must be run, the outcomes wll 110

be wrtten nto the prescrbed Excel workbooks. A drecton for further development would be that an Excel model can be developed wth user nterface. Ths model can replace the proposed MATLAB one n ths desgn. 8) Improvements on Sharpe s sngle ndex model. These are manly related to the assumptons assocated wth the model; hence ther valdty could be verfed. 111

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Appces 121

Appx A: MATLAB Code for Analysng Components of the Test Portfolo Wth Error Terms % Fnal Code: Use Smple Dscrete Return Wth Dvds % Acknowledgement must be pad to Mr. Randall Paton, who has asssted n wrtng of the followng code. % Some components from Ms. Hobbs' code had also been modfed for ths % research report functon Data = FnStats format long; = 1; % ntalse varables j = 2; k = 1; m = 1; weghttot = 0; %Select name of fle to process [fle, path] = ugetfle('*.xls', ' Orgnal Data Fle'); % Select fle from whch the raw data wll be read from [fle2, path2] = ugetfle('*.xls', 'Ouput Data Fle'); % Select fle from whch the results wll be wrtten to % Set up communcaton wth Excel DDE_Total = xlsread(strcat(path, '/',fle)); % Retrve data from a spreadsheet n an Excel workbook,.e. read from the frst spreadsheet n the workbook [a,b] = sze(dde_total); % a rows by b columns, b essentally represents the number of securtes ncludng the benchmark ndat = b - 1; % ndat s equal to b securtes less one, snce 1 refers to the date column presented n the worksheet ndatt = b; DataRows = ones(ndat, 1); % Create arrays of all ones, returns a ndat by 1 matrx of ones whle <= ndat % for s smaller or equal to ndat Name{1, } = ['Data Set' num2str() 'Abbrevaton']; % Convert numbers to strngs = + 1; % ncrementng = 1;% rentalse Abbcell = nputdlg(name, strcat('please specfy the portfolo data abbrevaton for data n', fle), DataRows); Allsname{1} = 'Composte Index Abbrevaton'; Allscell = nputdlg(allsname, 'Composte Index Detals', 1); % Create user-nterphase for user nvolvements % Defne company abbrevatons whle <= ndat Data().name = Abbcell{}; = + 1; = 1; 122

Data(ndatt).name = Allscell{1}; % The weght assgned to each share n the portfolo % Ensure the total weghts add up to 1 for the portfolo whle weghttot ~= 1 % Enter predetermned weghtng factors for each share - use weghts % determned from portfolo optmsaton whle <= ndat NameWeghts{, 1} = ['Data Set'' ' Abbcell{} ' ''Weght n percentage or decmal s' fle]; = + 1; = 1; Weghtcell = nputdlg(nameweghts, strcat('please specfy the weght n', fle), DataRows); % Defne the weght factors for beta calculatons - these are the % ndvdual percentages hold of each securtes n the portfolo whle <= ndat Data().weghtfactor = str2num(weghtcell{}); % Convert strngs to numbers weghttot = weghttot + Data().weghtfactor; = + 1; = 1; f weghttot ~= 1 warnh = warndlg('the specfed weghtngs do not add up to 1. Please re-enter the desred weghtngs', 'Improper Weghtngs'); weghttot = 0; watfor(warnh); % block executon and wat for event = 1; % Tme seres data for each of the shares n the portfolo whle <= ndatt Data().ddedata = DDE_Total(:, ); = + 1; = 1; % Defne the number of data ponts dpts = 1; % ntalse whle dpts <= a-2 % less 2, one s for the frst name row, and the other for unbased sample varance dpts = dpts + 1; A = cumsum(ones(dpts,1)); % create an array that counts the sample sze % Total number of observatons possble after calculatng returns N = a-1; % Total number of shares n the portfolo numshares = ndat; % Settng up the matrx for the ndepent varables 123

X = zeros(n, 2); % Create a zero matrx of N by 2,.e. N rows wth 2 columns X(1:N, 1) = ones(n,1); % Calculatng the returns for each shares n the portfolo % Enter dvds receved per share n cents durng perod examned,.e. dvds % declaraton date have been used as the reference whle <= ndat NameDv{, 1} = ['Data Set'' ' Abbcell{} ' '' Dvd Receved Per Share n Cents over test perod', fle]; = + 1; = 1; Dvcell = nputdlg(namedv, strcat('please enter dvds per share over the test perod', fle), DataRows); % Take nto accounts of the dvd pad per share n cents for each of % the securtes whle <= ndat Data().dvd = str2num(dvcell{}); = + 1; = 1; % Returns beng expressed n percentages whle <= ndatt data = Data().ddedata; b = length(data); f sempty(data().dvd)==1 dv() = 0; else dv() = Data().dvd./length(data); % get dvds nto daly form, thus t s assumed that t wll be consdered on a daly base Data().returns = ((data(2:b)-data(1)+dv())./data(1)).*100;% Equaton used here s the holdng perod yeld (HPY), how t dffers daly = + 1; = 1; % Returns on the ndex - the ndepent varable X(:,2) = Data(ndatt).returns; % Settng up the matrx for the depent varables Y = zeros(n, numshares); % create a zero matrx of N by numshares whle <= numshares Y = Data().returns; Data().Y = Y; = + 1; = 1; % Performng the regresson whle <= numshares 124

Data().betahat = nv(x'*x)*x'*data().y; Data().alphaestmate = Data().betahat(1); Data().betaestmate = Data().betahat(2); = + 1; = 1; % Calculatng the vector of resduals,.e. the error term whle <= numshares error = Data().Y - X*Data().betahat; Data().error = error; = + 1; = 1; % Calculaton of arthematc averages, ths s consstent wth the pertanng returns % calculaton, snce t was assumed to be dscrete smple compoundng returns, % nstead of contnuous compoundng whle <= ndatt returns = Data().returns; b = length(returns);% defne length for returns vector averages(1) = returns(1); averages(1) = averages(1); whle j <= b averages(j) = returns(j) + averages(j - 1); averages(j) = averages(j)./j; j = j + 1; j = 2; Data().averages = averages';% transpose nto column vector = + 1; = 1; % Calculaton of varances.e. sample varances, they are unbased, hence % the denomnator s the number of data ponts, j, less 1 whle <= ndatt returns = Data().returns; averages = Data().averages; vard(1) = ((returns(1) - averages(1)).^2); var(1) = vard(1); whle j <= b % use of column vector calculatons vard(j) = ((returns(j) - averages(j)).^2) + vard(j - 1);% gves cumulatve results var(j) = vard(j)./a(j, :); j = j + 1; j = 2; Data().var = var'; = + 1; = 1; 125

% Standard Devatons whle <= ndatt Data().stddev = sqrt(data().var); = + 1; = 1; % Covarances whle <= ndatt returns = Data().returns; averages = Data().averages; b = length(returns); whle k <= ndatt f k ~= % for k s not equal to ret = Data(k).returns; aves = Data(k).averages; ret = ret(2:);% ndcate the last ndex of array aves = aves(2:); returns = MakeCol(returns); % make returns vector nto ts column vector, f t s not already n the column form averages = MakeCol(averages); ret = MakeCol(ret); aves = MakeCol(aves); covar = (returns - averages).*(ret - aves); covar(1) = covar(1); whle j <= b covar(j) = covar(j)./a(j, :); j = j + 1; j = 2; Names{k} = Data(k).name; Index(k) = k; CoVars(:,k) = covar; k = k + 1; Indtake = VecClean(Index); Data().covarnames = CellClean(Names); Data().covars = MatClean(Indtake,CoVars); Data().CoVarInd = Indtake; k = 1; = + 1; clear Names Index CoVars % free up the system memory = 1; % Correlaton coeffcents calculatons whle <= ndatt ndces = Data().CoVarInd; CoVars = Data().covars; stddev = Data().stddev; b = length(stddev); whle k <= ndat covar = CoVars(:,k); stddev = Data(ndces(k)).stddev; rho(1) = 0; 126

= 1; whle j <= b rho(j) = covar(j)./(stddev(j).*stddev(j)); j = j + 1; Rhos(:,k) = rho; j = 2; k = k + 1; k = 1; Data().rhos = Rhos; = + 1; % Coeffcent of Varaton, ths s a measure of rsk/ volatlty whle <= ndat Data().cv = sqrt(data().var)./data().averages; = + 1; = 1; % Calculatons of betas - ordnary least squares method (ols) whle <= ndat covars = Data().covars; covar = covars(:,ndat); f Data(ndatt).var ~= 0 Data(ndatt).var = Data(ndatt).var; else f Data(ndatt).var ==0 Data().beta = 0; Data().betaols = covar./data(ndatt).var;% Equaton of beta calculaton = + 1; = 1; % Calculatons of alphas - ordnary least squares method (ols) whle <= ndat Data().averages = MakeCol(Data().averages); Data().betaols= MakeCol(Data().betaols); Data(ndatt).averages = MakeCol(Data(ndatt).averages); Data().alphaols = Data().averages - ((Data().betaols).*(Data(ndatt).averages)); = + 1; = 1; % Beta Adjustments % Merrll Lynch (ml) whle <= ndat Data().betaml = 2.*Data().betaols./3 + 1/3; = + 1; = 1; 127

% Vascek's technque: Bayesan's Adjustment (ba) % Calculatons on averages of betas b = length(data().betaols); Porto = zeros(b,1);% Returns an b, where b s the length of Data().beta, by 1 matrx of zeros,.e. a column vector whle <= ndat beta = Data().betaols; % Defne the length betasum(1) = 0; % Assgn ntal values betasum(1) = 0; whle j <= b betasum(j) = beta(j) + betasum(j - 1);% cumulatve averages of beta betasum(j) = betasum(j)./a(j, :); j = j + 1; j = 2; Data().avebeta = betasum'; Porto = Porto + betasum'; % Ensure the addton s between two column vectors,.e. of the same dmenson = + 1; = 1; avebetaporto = Porto./ndat;% presume equal-weghted betas for the securtes n the portfolo whle <= ndat Data().avebetaporto = avebetaporto; = + 1; = 1; % Varances of ndvdual betas.e. sample unbased varances whle <= ndat beta = Data().betaols; avebeta = Data().avebeta; varbeta(1) = 0; varbeta(1) = 0; whle j <= b varbeta(j) = (beta(j) - avebeta(j)).^2 + varbeta(j - 1); varbeta(j) = varbeta(j)./a(j, :); j = j + 1; Data().varbeta = varbeta'; j = 2; = + 1; = 1; % Cross - sectonal varance of all the estmates of beta n portfolo, %.e. the average used for calculaton s the average of ALL betas of % ndvdual shares n the portfolo at a partcular tme varbetaporto = zeros(b,1); whle <= ndat varbetaporto = varbetaporto + ((Data().betaols - Data().avebetaporto).^2); = + 1; 128

= 1; whle <= ndat Data().varbetaporto = varbetaporto./a(j, :); = + 1; = 1; %Calculate weght factors for Bayesan adjustments whle <= ndat Data().weght = Data().varbetaporto./(Data().varbetaporto + Data().varbeta); = + 1; = 1; % Calculaton of Bayesan adjustments whle <= ndat Data().betaba = (Data().weght).*(Data().betaols) + (1 - Data().weght).*(Data().avebetaporto); = + 1; = 1; % Alpha calculatons for adjustments % Merrll Lynch (ml) whle <= ndat Data().alphaml = Data().averages - ((Data().betaml).*(Data(ndatt).averages)); = + 1; = 1; % Vascek's technque: Bayesan's Adjustment (ba) whle <= ndat Data().alphaba = Data().averages - ((Data().betaba).*(Data(ndatt).averages)); = + 1; = 1; % Portfolo Betas betaportools = zeros(b,1); betaportoml = zeros(b,1); betaportoba = zeros(b,1); whle <= ndat betaportools = betaportools + Data().betaols; betaportoml = betaportoml + Data().betaml; betaportoba = betaportoba + Data().betaba; = + 1; = 1; whle <= ndat weghtfactor = Data().weghtfactor; betaportoolswthweghts = betaportools.*weghtfactor; betaportomlwthweghts = betaportoml.*weghtfactor; betaportobawthweghts = betaportoba.*weghtfactor; = + 1; 129

; = 1; betaportools = betaportoolswthweghts; betaportoml = betaportomlwthweghts; betaportoba = betaportobawthweghts; whle <= ndat Data().betaportools = betaportools; Data().betaportoml = betaportoml; Data().betaportoba = betaportoba; = + 1; = 1; % Portfolo Alphas averagesporto = zeros(b,1); whle <= ndat averagesporto = averagesporto + Data().averages; = + 1; = 1; whle <= ndat Data().averagesporto = averagesporto./a(j, :); = + 1; = 1; whle <= ndat Data().alphaportools = Data().averagesporto - (Data().betaportools).*(Data(ndatt).averages); Data().alphaportoml = Data().averagesporto - (Data().betaportoml).*(Data(ndatt).averages); Data().alphaportoba = Data().averagesporto - (Data().betaportoba).*(Data(ndatt).averages); = + 1; = 1; whle <= ndat Data().alphaportoolsmod = Data().alphaportools./100; Data().alphaportomlmod = Data().alphaportoml./100; Data().alphaportobamod = Data().alphaportoba./100; = + 1; = 1; % Expected returns of ndvdual shares whle <= ndat Data().returnsols = Data().alphaols + (Data().betaols).*(Data(ndatt).returns) + Data().error; Data().returnsml = Data().alphaml + (Data().betaml).*(Data(ndatt).returns) + Data().error; Data().returnsba = Data().alphaba + (Data().betaba).*(Data(ndatt).returns) + Data().error; = + 1; 130

= 1; Results_returnsols = zeros(n,ndat);% Defne the empty matrx,.e. to defne the matrx sze Results_returnsml = zeros(n,ndat); Results_returnsba = zeros(n,ndat); % Defne the outcomes Results_Beta = [Data(1).betaportools, Data(1).betaportoml, Data(1).betaportoba]; Results_Alpha = [Data(1).alphaportools, Data(1).alphaportoml, Data(1).alphaportoba]; Results_Alphamod = [Data(1).alphaportoolsmod, Data(1).alphaportomlmod, Data(1).alphaportobamod]; R_names{1} = 'ols'; R_names{2} = 'ml'; R_names{3} = 'ba'; whle <= ndat R_sharenames{} = Abbcell{}; Results_returnsols(:, ) = Data().returnsols; Results_returnsml(:, ) = Data().returnsml; Results_returnsba(:, ) = Data().returnsba; = + 1; = 1; % Export the results nto Excel spreadsheet wthout openng up the % worksheet xlswrte(strcat(path2, '/', fle2), R_names,'Beta', 'A1'); xlswrte(strcat(path2, '/', fle2), Results_Beta,'Beta', 'A2'); xlswrte(strcat(path2, '/', fle2), R_names,'Alpha', 'A1'); xlswrte(strcat(path2, '/', fle2), Results_Alphamod,'Alpha', 'A2'); xlswrte(strcat(path2, '/', fle2), R_sharenames,'Indvdual Returns OLS','A1'); xlswrte(strcat(path2, '/', fle2), Results_returnsols,'Indvdual Returns OLS','A2'); xlswrte(strcat(path2, '/', fle2), R_sharenames,'Indvdual Returns ML','A1'); xlswrte(strcat(path2, '/', fle2), Results_returnsml,'Indvdual Returns ML','A2'); xlswrte(strcat(path2, '/', fle2), R_sharenames,'Indvdual Returns BA','A1'); xlswrte(strcat(path2, '/', fle2), Results_returnsba,'Indvdual Returns BA','A2'); functon B = MakeCol(A)% Make the data set a column vector f t's not [a,b] = sze(a); f a == 1 f b > 1 B = A'; else 131

B = A; else B = A; functon B = CellClean(A);% Clean the cells = 1; j = 1; [a,b] = sze(a); pos = b + 1; whle <= b [a2,b2] = sze(a{}); f a2 == 0 pos = ; = + 1; = 1; whle j <= b - 1 f j == pos = + 1; B{j} = A{}; = + 1; j = j + 1; functon B = MatClean(Ind,A) = 1; [a,b] = sze(ind); whle <= b B(:,) = A(:,Ind()); = + 1; functon B = VecClean(A) = 1; j = 1; [a,b] = sze(a); pos = b + 1; whle <= b f A() == 0 pos = ; 132

= + 1; = 1; whle j <= b f j == pos = + 1; f <= b B(j) = A(); = + 1; j = j + 1; 133

Appx B: MATLAB Code for Analysng Components of the Test Portfolo Wthout Error Terms % Fnal Code - Use Smple Dscrete Return Wth Dvds wth Statstcal Analyss % Acknowledgement must be pad to Mr. Randall Paton, who has asssted n the wrtng of the followng codes % Some components from Ms. Hobbs' code had also been modfed for ths % research report functon Data = FnStats = 1; % assgn ntal values to varables j = 2; k = 1; weghttot = 0; % Select name of fle to process [fle, path] = ugetfle('*.xls','orgnal data fle'); % Select fle from whch the raw data wll be read from [fle2, path2] = ugetfle('*.xls','output data fle');% Select fle to whch the results wll be wrtten to % Setup communcaton wth Excel DDE_Total = xlsread(strcat(path,'/',fle)); % Retrve data and text from a spreadsheet n an Excel workbook,.e. read from the frst spreadsheet n the workbook [a,b] = sze(dde_total);% a rows by b columns, b essentally represents the number of securtes ndat = b - 1;% ndat s equal to b securtes less one, snce the 1 refers to the date column presented n the worksheet ndatt = b; DataRows = ones(ndat,1);% Create arrays of all ones, returns an ndat by 1 matrx of ones whle <= ndat % for s smaller or equal to ndat Name{1,} = ['Data Set ' num2str() ' Abbrevaton']; % convert numbers to strng = + 1; % ncrementng = 1;% rentalse Abbcell = nputdlg(name,strcat('please specfy the portfolo data abbrevatons for the data n ',fle),datarows); Allsname{1} = 'Composte ndex abbrevaton'; Allscell = nputdlg(allsname,'composte Index Detals',1); % Defne the number of data ponts dpts = 1; % ntalse whle dpts <= a-2 % less 2, snce one s for the frst name row, and the other s for the unbased sample varance dpts = dpts + 1; A = cumsum(ones(dpts, 1));% create an array that counts the sample sze 134

% Create user-nterphase for user nvolvements % Defne company abbrevatons whle <= ndat Data().name = Abbcell{}; = + 1; = 1; Data(ndatt).name = Allscell{1}; % Tme seres data for each of the shares n the portfolo whle <= ndatt Data().ddedata = DDE_Total(:,);% read drectly from the selected fle wthout openng the fle = + 1; = 1; % Ensure the total weghtng factors add up to 1 for the portfolo whle weghttot ~= 1 % Enter predetermned weghtng factors for each share - for beta calculaton for the portfolo % n percentages - should use the weghtng created from portfolo optmsaton whle <= ndat Name3{,1} = ['Data Set ''' Abbcell{} ''' Weghtng Factor In Percentage/ Decmal s ' fle]; = + 1; = 1; Weghtcell = nputdlg(name3, strcat('please specfy the weghtng factor n', fle), DataRows); %Defne the weghtng factors for beta calculatons - these are the %ndvdual percentages hold of each securtes n the portfolo whle <= ndat Data().weghtfactor = str2num(weghtcell{}); % Convert strngs nto numbers weghttot = weghttot + Data().weghtfactor; = + 1; = 1; f weghttot ~= 1 warnh = warndlg('the specfed weghtngs do not add up to 1. Please re-enter the desred weghtngs','improper Weghtngs'); weghttot = 0; watfor(warnh);% Watng for condton before executon = 1; % Enter the annual dvd receved per share n cents whle <= ndat Name4{,1} = ['Data Set''' Abbcell{} ''' Dvd Receved Per Share In Cents over test perod', fle]; = + 1; 135

= 1; DvCell = nputdlg(name4, strcat('please enter dvds per share over the test perod', fle), DataRows); % Take nto accounts of the dvd pad per share n cents for each of % the securtes whle <= ndat Data().dvd = str2num(dvcell{}); = + 1; = 1; % Calculaton of returns - captal gan returns wth dvds, returns beng expressed n decmals - the returns values % are rather small snce t s calculated per share whle <= ndatt data = Data().ddedata; b = length(data); f sempty(data().dvd)== 1 % testng array to see f t s empty dv() = 0; else dv() = Data().dvd./length(data);% get dvds nto daly form, thus t s assumed that t wll be consdered on a daly base Data().returns = ((data(2:b)-data(1)+dv())./data(1)).*100;% equaton used here s the holdng perod yeld (HPY), how t dffers daly = + 1; = 1; % Calculaton of arthematc averages, ths s consstent wth the pertanng returns % calculaton, snce t was assumed to be dscrete smple compoundng returns, % nstead of contnuous compoundng whle <= ndatt returns = Data().returns; b = length(returns);% defne length for returns vector averages(1) = returns(1); averages(1) = averages(1); whle j <= b averages(j) = returns(j) + averages(j - 1); averages(j) = averages(j)./j; j = j + 1; j = 2; Data().averages = averages';% transpose nto column vector = + 1; = 1; 136

% Calculaton of varances.e. sample varances, they are unbased, hence % the denomnator s the number of data ponts, j, less 1 whle <= ndatt returns = Data().returns; averages = Data().averages; vard(1) = ((returns(1) - averages(1)).^2); var(1) = vard(1); whle j <= b % use of column vector calculatons vard(j) = ((returns(j) - averages(j)).^2) + vard(j - 1);% gves cumulatve results var(j) = vard(j)./a(j, :); j = j + 1; j = 2; Data().var = var'; = + 1; = 1; % Standard Devatons whle <= ndatt Data().stddev = sqrt(data().var); = + 1; = 1; % Covarances whle <= ndatt returns = Data().returns; averages = Data().averages; b = length(returns); whle k <= ndatt f k ~= % for k s not equal to ret = Data(k).returns; aves = Data(k).averages; ret = ret(2:);% ndcate the last ndex of array aves = aves(2:); returns = MakeCol(returns); % make returns vector nto ts column vector, f t s not already n the column form averages = MakeCol(averages); ret = MakeCol(ret); aves = MakeCol(aves); covar = (returns - averages).*(ret - aves); covar(1) = covar(1); whle j <= b covar(j) = covar(j)./a(j, :); j = j + 1; j = 2; Names{k} = Data(k).name; Index(k) = k; CoVars(:,k) = covar; k = k + 1; 137

Indtake = VecClean(Index); Data().covarnames = CellClean(Names); Data().covars = MatClean(Indtake,CoVars); Data().CoVarInd = Indtake; k = 1; = + 1; clear Names Index CoVars % free up the system memory = 1; % Correlaton coeffcents calculatons whle <= ndatt ndces = Data().CoVarInd; CoVars = Data().covars; stddev = Data().stddev; b = length(stddev); whle k <= ndat covar = CoVars(:,k); stddev = Data(ndces(k)).stddev; rho(1) = 0; whle j <= b rho(j) = covar(j)./(stddev(j).*stddev(j)); j = j + 1; Rhos(:,k) = rho; j = 2; k = k + 1; k = 1; Data().rhos = Rhos; = + 1; = 1; % Coeffcent of Varaton, ths s a measure of rsk/ volatlty whle <= ndat Data().cv = sqrt(data().var)./data().averages; = + 1; = 1; % Calculatons of betas - ordnary least squares method (ols) whle <= ndat covars = Data().covars; covar = covars(:,ndat); f Data(ndatt).var ~= 0 Data(ndatt).var = Data(ndatt).var; else f Data(ndatt).var ==0 Data().beta = 0; Data().beta = covar./data(ndatt).var;% Equaton of beta calculaton = + 1; = 1; 138

% Calculatons of alphas - ordnary least squares method (ols) whle <= ndat Data().averages = MakeCol(Data().averages); Data().beta= MakeCol(Data().beta); Data(ndatt).averages = MakeCol(Data(ndatt).averages); Data().alpha = Data().averages - ((Data().beta).*(Data(ndatt).averages)); = + 1; = 1; % Beta Adjustments % Merrll Lynch (ml) whle <= ndat Data().betaml = 2.*Data().beta./3 + 1/3; = + 1; = 1; % Vascek's technque: Bayesan's Adjustment (ba) % Calculatons on averages of betas b = length(data().beta); Porto = zeros(b,1);% Returns an b, where b s the length of Data().beta, by 1 matrx of zeros,.e. a column vector whle <= ndat beta = Data().beta; % Defne the length betasum(1) = 0; % Assgn ntal values betasum(1) = 0; whle j <= b betasum(j) = beta(j) + betasum(j - 1);% cumulatve averages of beta betasum(j) = betasum(j)./a(j, :); j = j + 1; j = 2; Data().avebeta = betasum'; Porto = Porto + betasum'; % Ensure the addton s between two column vectors,.e. of the same dmenson = + 1; = 1; avebetaporto = Porto./ndat;% presume equal-weghted betas for the securtes n the portfolo whle <= ndat Data().avebetaporto = avebetaporto; = + 1; = 1; % Varances of ndvdual betas.e. sample unbased varances whle <= ndat beta = Data().beta; avebeta = Data().avebeta; varbeta(1) = 0; varbeta(1) = 0; 139

whle j <= b varbeta(j) = (beta(j) - avebeta(j)).^2 + varbeta(j - 1); varbeta(j) = varbeta(j)./a(j, :); j = j + 1; Data().varbeta = varbeta'; j = 2; = + 1; = 1; % Cross - sectonal varance of all the estmates of beta n portfolo, %.e. the average used for calculaton s the average of ALL betas of % ndvdual shares n the portfolo at a partcular tme varbetaporto = zeros(b,1); whle <= ndat varbetaporto = varbetaporto + ((Data().beta - Data().avebetaporto).^2); = + 1; = 1; whle <= ndat Data().varbetaporto = varbetaporto./a(j, :); = + 1; = 1; %Calculate weght factors for Bayesan adjustments whle <= ndat Data().weght = Data().varbetaporto./(Data().varbetaporto + Data().varbeta); = + 1; = 1; % Calculaton of Bayesan adjustments whle <= ndat Data().betaba = (Data().weght).*(Data().beta) + (1 - Data().weght).*(Data().avebetaporto); = + 1; = 1; % Alpha calculatons for adjustments % Merrll Lynch (ml) whle <= ndat Data().alphaml = Data().averages - ((Data().betaml).*(Data(ndatt).averages)); = + 1; = 1; % Vascek's technque: Bayesan's Adjustment (ba) whle <= ndat Data().alphaba = Data().averages - ((Data().betaba).*(Data(ndatt).averages)); = + 1; 140

= 1; % Portfolo Betas betaportools = zeros(b,1); betaportoml = zeros(b,1); betaportoba = zeros(b,1); whle <= ndat betaportools = betaportools + Data().beta; betaportoml = betaportoml + Data().betaml; betaportoba = betaportoba + Data().betaba; = + 1; = 1; whle <= ndat weghtfactor = Data().weghtfactor; betaportoolswthweghts = betaportools.*weghtfactor; betaportomlwthweghts = betaportoml.*weghtfactor; betaportobawthweghts = betaportoba.*weghtfactor; = + 1; ; = 1; betaportools = betaportoolswthweghts; betaportoml = betaportomlwthweghts; betaportoba = betaportobawthweghts; whle <= ndat Data().betaportools = betaportools; Data().betaportoml = betaportoml; Data().betaportoba = betaportoba; = + 1; = 1; % Portfolo Alphas averagesporto = zeros(b,1); whle <= ndat averagesporto = averagesporto + Data().averages; = + 1; = 1; whle <= ndat Data().averagesporto = averagesporto./a(j, :); = + 1; = 1; whle <= ndat Data().alphaportools = Data().averagesporto - (Data().betaportools).*(Data(ndatt).averages); Data().alphaportoml = Data().averagesporto - (Data().betaportoml).*(Data(ndatt).averages); Data().alphaportoba = Data().averagesporto - (Data().betaportoba).*(Data(ndatt).averages); = + 1; 141

= 1; whle <= ndat Data().alphaportoolsmod = Data().alphaportools./100; Data().alphaportomlmod = Data().alphaportoml./100; Data().alphaportobamod = Data().alphaportoba./100; = + 1; = 1; % Expected portfolo returns whle <= ndat Data().returnsportools = Data().alphaportools + (Data().betaportools).*(Data(ndatt).returns); Data().returnsportoml = Data().alphaportoml + (Data().betaportoml).*(Data(ndatt).returns); Data().returnsportoba = Data().alphaportoba + (Data().betaportoba).*(Data(ndatt).returns); = + 1; = 1; % Statstcal Analyss % Confdence nterval s a range of values around the expected outcome % wthn whch we xpect the acutal outcome to be some specfed percentage % of the tme. A 95 percent confdence nterval s a range that we expect % the random varable to be n 95% of the tme. For a normal dstrbuton, % ths nterval s based on the expected value (sometmes called a pont % estmate) of the random varable and on ts varablty, whch we measure % wth standard devaton - Determne the range n whch the outcome would % le usng dfferent level of confdence % Before confdence nterval for portfolo returns can be calculated, ts % averages and varances need to be establshed n order for the % calculaton on ts standard devaton % Calculaton of Portfolo Averages whle <= ndat returnsportools = Data().returnsportools; returnsportoml = Data().returnsportoml; returnsportoba = Data().returnsportoba; b = length(returnsportools); whle j <= b B = cumsum(returnsportools(2:j)./a(j)); C = cumsum(returnsportoml(2:j)./a(j)); D = cumsum(returnsportoba(2:j)./a(j)); averetportoolssum(j) = B(j - 1); averetportomlsum(j) = C(j - 1); 142

= 1; averetportobasum(j) = D(j - 1); j = j + 1; j = 2; Data().averetportools = averetportoolssum'; Data().averetportoml = averetportomlsum'; Data().averetportoba = averetportobasum'; = + 1; % Calculaton of Portfolo Varances whle <= ndat Data().varportools = ((Data().returnsportools - Data().averetportools).^2)./A(j); Data().varportoml = ((Data().returnsportoml - Data().averetportoml).^2)./A(j); Data().varportoba = ((Data().returnsportoba - Data().averetportoba).^2)./A(j); = + 1; = 1; % Calculaton of Portfolo Standard Devatons whle <= ndat Data().stddevportools = sqrt(data().varportools); Data().stddevportoml = sqrt(data().varportoml); Data().stddevportoba = sqrt(data().varportoba); = + 1; = 1; % 90% Percent Confdence Interval for pont estmates on portfolo returns whle <= ndat % Ordnary Least Squares Data().returnsols_upper90 = Data().averetportools + 1.65*Data().stddevportools; Data().returnsols_lower90 = Data().averetportools - 1.65*Data().stddevportools; % Merrll Lynch Data().returnsml_upper90 = Data().averetportoml + 1.65*Data().stddevportoml; Data().returnsml_lower90 = Data().averetportoml - 1.65*Data().stddevportoml; % Bayesan Adjustments Data().returnsba_upper90 = Data().averetportoba + 1.65*Data().stddevportoba; Data().returnsba_lower90 = Data().averetportoba - 1.65*Data().stddevportoba; = + 1; = 1; % 95% Percent Confdence Interval for pont estmates on portfolo returns 143

whle <= ndat % Ordnary Least Squares Data().returnsols_upper95 = Data().averetportools + 1.96*Data().stddevportools; Data().returnsols_lower95 = Data().averetportools - 1.96*Data().stddevportools; % Merrll Lynch Data().returnsml_upper95 = Data().averetportoml + 1.96*Data().stddevportoml; Data().returnsml_lower95 = Data().averetportoml - 1.96*Data().stddevportoml; % Bayesan Adjustments Data().returnsba_upper95 = Data().averetportoba + 1.96*Data().stddevportoba; Data().returnsba_lower95 = Data().averetportoba - 1.96*Data().stddevportoba; = + 1; = 1; % 99% Percent Confdence Interval for pont estmates on portfolo returns whle <= ndat % Ordnary Least Squares Data().returnsols_upper99 = Data().averetportools + 2.58*Data().stddevportools; Data().returnsols_lower99 = Data().averetportools - 2.58*Data().stddevportools; % Merrll Lynch Data().returnsml_upper99 = Data().averetportoml + 2.58*Data().stddevportoml; Data().returnsml_lower99 = Data().averetportoml - 2.58*Data().stddevportoml; % Bayesan Adjustments Data().returnsba_upper99 = Data().averetportoba + 2.58*Data().stddevportoba; Data().returnsba_lower99 = Data().averetportoba - 2.58*Data().stddevportoba; = + 1; = 1; % Plottng the statstcal results % Plottng 90% Confdence nterval results % Ordnary Least Sqauare fd1 = fgure(1); subplot(2,2,1); plot(data(1).returnsols_upper90', 'b'), grd hold on plot(data(1).returnsols_lower90', 'g'), grd hold on plot(data(1).returnsportools', 'r'), grd hold off ttle('expected Returns Over Tme - OLS [90% Confdence]') xlabel('t = 0 to 360') 144

ylabel('expected Returns n %') % Merrll Lynch subplot(2,2,2); plot(data(1).returnsml_upper90', 'b'), grd hold on plot(data(1).returnsml_lower90', 'g'), grd hold on plot(data(1).returnsportoml', 'r'), grd hold off ttle('expected Returns Over Tme - ML [90% Confdence]') xlabel('t = 0 to 360') ylabel('expected Returns n %') % Bayesan Adjustments subplot(2,2,3); plot(data(1).returnsba_upper90', 'b'), grd hold on plot(data(1).returnsba_lower90', 'g'), grd hold on plot(data(1).returnsportoba', 'r'), grd hold off ttle('expected Returns Over Tme - BA [90% Confdence]') xlabel('t = 0 to 360') ylabel('expected Returns n %') leg('upper Bound', 'Lower Bound', 'Expected Return'); % Plottng 95% Confdence nterval results % Ordnary Least Sqauare fd2 = fgure(2); subplot(2,2,1); plot(data(1).returnsols_upper95', 'b'), grd hold on plot(data(1).returnsols_lower95', 'g'), grd hold on plot(data(1).returnsportools', 'r'), grd hold off ttle('expected Returns Over Tme - OLS [95% Confdence]') xlabel('t = 0 to 360') ylabel('expected Returns n %') % Merrll Lynch subplot(2,2,2); plot(data(1).returnsml_upper95', 'b'), grd hold on plot(data(1).returnsml_lower95', 'g'), grd hold on plot(data(1).returnsportoml', 'r'), grd hold off ttle('expected Returns Over Tme - ML [95% Confdence]') xlabel('t = 0 to 360') ylabel('expected Returns n %') % Bayesan Adjustments subplot(2,2,3); plot(data(1).returnsba_upper95', 'b'), grd hold on plot(data(1).returnsba_lower95', 'g'), grd hold on plot(data(1).returnsportoba', 'r'), grd hold off 145

ttle('expected Returns Over Tme - BA [95% Confdence]') xlabel('t = 0 to 360') ylabel('expected Returns n %') leg('upper Bound', 'Lower Bound', 'Expected Return'); % Plottng 99% Confdence nterval results % Ordnary Least Sqauare fd3 = fgure(3); subplot(2,2,1); plot(data(1).returnsols_upper99', 'b'), grd hold on plot(data(1).returnsols_lower99', 'g'), grd hold on plot(data(1).returnsportools', 'r'), grd hold off ttle('expected Returns Over Tme - OLS [99% Confdence]') xlabel('t = 0 to 360') ylabel('expected Returns n %') % Merrll Lynch subplot(2,2,2); plot(data(1).returnsml_upper99', 'b'), grd hold on plot(data(1).returnsml_lower99', 'g'), grd hold on plot(data(1).returnsportoml', 'r'), grd hold off ttle('expected Returns Over Tme - ML [99% Confdence]') xlabel('t = 0 to 360') ylabel('expected Returns n %') % Bayesan Adjustments subplot(2,2,3); plot(data(1).returnsba_upper99', 'b'), grd hold on plot(data(1).returnsba_lower99', 'g'), grd hold on plot(data(1).returnsportoba', 'r'), grd hold off ttle('expected Returns Over Tme - BA [99% Confdence]') xlabel('t = 0 to 360') ylabel('expected Returns n %') leg('upper Bound', 'Lower Bound', 'Expected Return'); % Defne the portfolo results Data_Outbeta(:,1) = Data(1).betaportools; Data_Outbeta(:,2) = Data(1).betaportoml; Data_Outbeta(:,3) = Data(1).betaportoba; Data_Outalpha(:,4) = Data(1).alphaportoolsmod; Data_Outalpha(:,5) = Data(1).alphaportomlmod; Data_Outalpha(:,6) = Data(1).alphaportobamod; Data_Outreturn(:,7) = Data(1).returnsportools; Data_Outreturn(:,8) = Data(1).returnsportoml; Data_Outreturn(:,9) = Data(1).returnsportoba; % Export the results nto Excel spreadsheet wthout openng up the % worksheet xlswrte(strcat(path2, '/', fle2),data_outbeta,'beta', 'A2'); 146

xlswrte(strcat(path2, '/', fle2),data_outalpha, 'Alpha', 'A2'); xlswrte(strcat(path2, '/', fle2),data_outreturn, 'Return', 'A2'); functon B = MakeCol(A)% Make the data set a column vector f t's not [a,b] = sze(a); f a == 1 f b > 1 B = A'; else B = A; else B = A; functon B = CellClean(A);% Clean the cells = 1; j = 1; [a,b] = sze(a); pos = b + 1; whle <= b [a2,b2] = sze(a{}); f a2 == 0 pos = ; = + 1; = 1; whle j <= b - 1 f j == pos = + 1; B{j} = A{}; = + 1; j = j + 1; functon B = MatClean(Ind,A) = 1; [a,b] = sze(ind); whle <= b B(:,) = A(:,Ind()); = + 1; 147

functon B = VecClean(A) = 1; j = 1; [a,b] = sze(a); pos = b + 1; whle <= b f A() == 0 pos = ; = + 1; = 1; whle j <= b f j == pos = + 1; f <= b B(j) = A(); = + 1; j = j + 1; 148

Appx C: Instructons for Runnng MATLAB Codes It s mportant to note that MATALB s needed to be nstalled on the computer, pror to the runnng of the codes. Also, It s extremely mportant to enter the asked nformaton, as t appears n the excel workbook Weghtng Factors for Calculatons Beta, n the correct order. Otherwse the results wll be altered. 1) Put the CD, that accompaned ths report, nto the CD- RAM. 2) Run the CD and vew the fles that are on the CD. Ths s done by frstly, double clck on My Computer con on the desktop. Secondly double clck on CD- RAM. The fles on the CD are now vsble. 3) Select MATLAB Codes and Fnal Results folders. Copy and Paste these onto the desktop. In MATLAB Codes folder, there are two sets of codes present, one set to nclude error terms and the other exclude the errors. In Fnal Results folder, there are two folders present namely, FINAL PORTFOLIO Exclude Error Terms and FINAL PORTFOLIO Include Error Terms. Also present s an excel workbook named, Weghtng Factors for Calculatons Beta. 4) Double clck on the workbook, Weghng Factors for Calculatons Beta. The followng screen should appear: In the workbook, there are eght worksheets present. The frst sx worksheets are assocated wth the correspondng component n the overall test portfolo. These 149

are namely Balanced, Conservatves, Core Alternatves, Core, Mdterm and Smallcap. In each of these worksheets, the followng nformaton are found:. Stock names that are the consttuents of each subportfolo.. Percentage. Ths refers to the weghtng factors that are used for beta calculaton n the MATLAB Code.. Dvds over Test Perod n Cents. These refer to the dvds pad to the nvestor over the test perod. Keep ths workbook open, snce the pertnent excel nformaton s needed for runnng the codes. 5) Now, open MATLAB programme. Ths may be done by ether double clckng on the MATLAB shortcut on the desktop, or by clckng just once on start, at the bottom left hand corner of the screen, select all programs, then clck on MATLAB. When MATLAB s opened, the followng screen s observed: 6) Copy and paste the two sets of codes found n MATLAB Codes folder nto the Current Drectory on the left hand sde of the above screen. 7) Decded on whch sets of codes that you want to run frst. Then double clck on the fle. For demonstraton purpose, the author has decded to run the codes that nclude error terms. (The smlar method s used for runnng the other sets.) If the user now double clcks on MATLABCodeWthErrorTerm.m. The followng screen should appear: 150

8) Once the above screen has appeared, the user s now ready to run the codes. The codes may be run by ether pressng F5 or pressng the run con, as t appears so: on the top toolbar. 9) By pressng F5 or pressng run con. The followng screen appears: 151

The wndow that appears on the left hand sde of the above screen reads Orgnal Data Fle. Ths refers to the raw data assocated wth each of the components n the test portfolo. For demonstraton purpose, the author has decded to run Balanced component. It s mportant to fnd the Balanced component on the desktop. Go to Look In on top of the wndow, go to desktop, and double clck on Fnal Results folder, then double clck on FINAL PORTFOLIO Include Error Terms The followng screen appears: 152

Double clck on Balanced Portfolo folder. There are two excel workbook present, one refers to as the raw data and the other results. Ths s shown below: Select the excel workbook named, balanced_raw data, snce ths s assocated wth Orgnal Data Fle. 153

10) Once Step (9) s done. The followng screen appears: Ths tme, the wndow that appears on the left hand sde of the above screen reads Output Data Fle. Ths refers to the fle, to whch the results from MATLAB, are to be wrtten to. It s mportant to select the results workbook whch corresponds to the above component, n ths case, Balanced. 11) Go to Look In on top of the wndow, go to desktop, and double clck on Fnal Results folder, then double clck on FINAL PORTFOLIO Include Error Terms. A smlar screen to the one under step (9) appears. Double clck on Balanced Portfolo folder. There are two excel workbooks present, select the excel workbook named, results_balanced, snce ths s assocated wth Output Data Fle. 12) Wat, whle MATLAB processes the code, then the followng screen appears: 154

There are 6 shares present n Balanced, therefore there are 6 abbrevatons that need to be entered. These abbrevatons are found under Stock names as descrbed n step (4). Data set 1 refers to the frst stock, as t appears n (4), n the subportfolo. Once the requred nformaton are entered, t looks as below: 155

Clck on OK. 13) The followng screen appears: The composte ndex abbrevaton refers to the benchmark chosen n ths research. It s the ALL SHARE ndex. Type ALSI n. Clck on OK. 14) Then the followng screen appears: 156

The computer s now askng for the weght factors that are assocated wth each of the components. These are found n Percentage, as descrbed n step (4). Enter the weght. The screen wll now appear as below: Clck on OK. 157

15) The followng screen appears: The computer s now requestng for the dvd nformaton assocated wth the correspondng shares. These nformaton are found under Dvds over Test Perod as dscussed n step (4). Enter the nformaton, the followng then appears: 158

Clck on OK. 16) Wat, whle MATLAB processes the entered nformaton. Ignore the warnng messages n the MATLAB wndow, shown below: 17) When the processng s complete, the followng screen appears: 159

18) Repeat the above mentoned steps for all 6 subportfolos n the overall test portfolo. Remember separate codes are used for the fnal portfolo folders whether t s to exclude or nclude the error terms. 19) After step (18), one can open the FINAL RESULTS folder. Double clck on the workbook present. The graphs present are dentcal to that of the man body of report. 160

Appx D: MATLAB Code for Valdatng The Computer Programmes % The followng codes were used to valdate the computer programme % wrtten. The computer programme were valdated n parts. The % followng codes were then modfed to gve rse to the general % computer programme as seen n Appx A and B % Select the fle to whch the results wll be exported to. [fle, path] = ugetfle('*.xls', 'Output Fle'); % Let A be refer to as the P1 (Data value/ prce of a securty) A = [12, 13, 10, 9, 20, 7, 4, 22, 15,23]'; % Let B be refer to as the PM (Data value/ prce of the market) B = [50, 54, 48, 47, 70, 20, 15, 40, 35, 37]'; % Defne the number of observatons dpts = 1; b = length(a); whle dpts <= b -2 dpts = dpts + 1; C = cumsum(ones(dpts, 1)); % Create an array that counts the sample sze % Calculate the returns of each of the pertnent tme- seres (A and B). % The returns are beng expressed n percentages returnsofa = ((A(2:)-A(1))./A(1)).*100; returnsofb = ((B(2:)-B(1))./B(1)).*100; % Calculate the arthematc averages of A and B averagesofa = mean(returnsofa); averagesofb = mean(returnsofb); % Calculate the varances of A and B vardofa = ((returnsofa - averagesofa).^2); vardofb = ((returnsofb - averagesofb).^2); varanceofa = vardofa./c; varanceofb = vardofb./c; % Calculate the covarances of A and B covar = (returnsofa - averagesofa).*(returnsofb - averagesofb); cov = covar./c; % Calculaton of OLS beta for A betaofa = cov./varanceofb; % Calculaton of OLS alpha for A alphaofa = averagesofa - (betaofa*averagesofb); % Adjustments done to Beta % Merrll Lynch's Adjustment betaofaml = 2.*betaofA./3 + 1/3; 161

% Bayesan's adjustments: there are a few parameters need to be calculated % pror to the adjustment. The followng parameters need to be establshed, % the average of OLS beta, varance of beta estmate and cross- % sectonal standard devaton of all beta estmate n the portfolo. In % ths demonstraton, there are only two securtes. % Calculate the average of OLS beta averagebetaofa = mean(betaofa); % Calculaton of varance of OLS beta estmate vardofabetaestmate = ((betaofa - averagebetaofa).^2); varanceofabetaestmate = vardofabetaestmate./c; % Calculaton of cross- sectonal standard devaton of all beta estmate averagebetaportoofa = averagebetaofa; % In ths demonstraton, there s only one securty n the portfolo, the other securty s the benchmark used,.e. the market ndex varbetaofa = ((betaofa - averagebetaportoofa).^2); varancebetaportoofa = varbetaofa./c; % Weght factor calculaton weght = varancebetaportoofa./(varancebetaportoofa + varanceofabetaestmate); % Beta calculaton based on Bayesan's adjustment betaofaba = (weght.*betaofa) + (1-weght).*averagebetaofA; % Modfed alpha values based on Merrll Lycnh's adjustments done to beta alphaofaml = averagesofa - (betaofaml*averagesofb); % Modfed alpha values based on Bayesan's adjustments done to beta alphaofaba = averagesofa - (betaofaba*averagesofb); % Export results to Excel % Defne the headngs for each column Results_names{1} = 'Number of Observatons'; Results_names{2} = 'A'; % data value for ndvdual securty Results_names{3} = 'B'; % data value for the benchmark Results_names{4} = 'Returns of A'; Results_names{5} = 'Returns of B'; Results_names{6} = 'Average of A'; Results_names{7} = 'Average of B'; Results_names{8} = 'Varance of A'; Results_names{9} = 'Varance of B'; Results_names{10} = 'Covarance'; Results_names{11} = 'OLS beta'; Results_names{12} = 'BA beta'; Results_names{13} = 'ML beta'; Results_names{14} = 'OLS alpha'; Results_names{15} = 'BA alpha'; 162

Results_names{16} = 'ML alpha'; % Wrte the outcomes to the chosen excel workbook xlswrte(strcat(path, '/', fle), Results_names,'MATLAB Outputs', 'B2'); xlswrte(strcat(path, '/', fle), C, 'MATLAB Outputs', 'B3'); xlswrte(strcat(path, '/', fle), A, 'MATLAB Outputs', 'C3'); xlswrte(strcat(path, '/', fle), B, 'MATLAB Outputs', 'D3'); xlswrte(strcat(path, '/', fle), returnsofa, 'MATLAB Outputs', 'E3'); xlswrte(strcat(path, '/', fle), returnsofb, 'MATLAB Outputs', 'F3'); xlswrte(strcat(path, '/', fle), averagesofa, 'MATLAB Outputs', 'G3'); xlswrte(strcat(path, '/', fle), averagesofb, 'MATLAB Outputs', 'H3'); xlswrte(strcat(path, '/', fle), varanceofa, 'MATLAB Outputs', 'I3'); xlswrte(strcat(path, '/', fle), varanceofb, 'MATLAB Outputs', 'J3'); xlswrte(strcat(path, '/', fle), cov, 'MATLAB Outputs', 'K3'); xlswrte(strcat(path, '/', fle), betaofa, 'MATLAB Outputs', 'L3'); xlswrte(strcat(path, '/', fle), betaofaba, 'MATLAB Outputs', 'M3'); xlswrte(strcat(path, '/', fle), betaofaml, 'MATLAB Outputs', 'N3'); xlswrte(strcat(path, '/', fle), alphaofa, 'MATLAB Outputs', 'O3'); xlswrte(strcat(path, '/', fle), alphaofaba, 'MATLAB Outputs', 'P3'); xlswrte(strcat(path, '/', fle), alphaofaml, 'MATLAB Outputs', 'Q3'); 163

Appx E: Valdaton Results The followng results are found n ths secton: Table E1 represents the results that were obtaned by runnng the valdatng computer programme. Ths computer programme can be found n Appx D. Table E2 represents the results that were obtaned by manually calculatng the results usng the equatons found n Chapter 2. Table E3 represents the error by comparng Table E1 and Table E2. 164

Table E1: Outcomes from Valdatng Computer Programme Data # A B Returns of A Returns of B Varance of A Varance of B Covarance OLS beta BA beta ML beta OLS alpha BA alpha ML alpha 1 12 50 8.33 8.00 30.86 711.11-148.15-0.21-3.73 0.19 10.00-55.69 17.52 2 13 54-16.67-4.00 466.82 107.56-224.07-2.08-4.66-1.06-25.00-73.19-5.81 3 10 48-25.00-6.00 504.12 53.48-164.20-3.07-5.16-1.71-43.42-82.40-18.10 4 9 47 66.67 40.00 696.37 860.44 774.07 0.90-3.17 0.93 30.68-45.35 31.31 5 20 70-41.67-60.00 617.28 341.69 459.26 1.34-2.95 1.23 38.98-41.20 36.84 6 7 20-66.67-70.00 1081.53 439.19 689.20 1.57-2.84 1.38 43.18-39.10 39.64 7 4 15 83.33-20.00 688.93 0.25-13.23-52.08-29.66-34.39-958.33-539.86-628.04 8 22 40 25.00-30.00 15.43 16.06-15.74-0.98-4.11-0.32-4.41-62.90 7.91 9 15 35 91.67-26.00 672.15 5.98-63.37-10.61-8.93-6.74-184.09-152.74-111.88 23 37 Ave. 14-19 165

Table E2: Outcomes from Manual Calculatons Data # A B Returns of A Returns of B Varance of A Varance of B Covarance OLS beta BA beta ML beta OLS alpha BA alpha ML alpha 1 12 50 8.33 8.00 30.86 711.11-148.15-0.21-3.73 0.19 10.00-55.69 17.52 2 13 54-16.67-4.00 466.82 107.56-224.07-2.08-4.66-1.06-25.00-73.19-5.81 3 10 48-25.00-6.00 504.12 53.48-164.20-3.07-5.16-1.71-43.42-82.40-18.10 4 9 47 66.67 40.00 696.37 860.44 774.07 0.90-3.17 0.93 30.68-45.35 31.31 5 20 70-41.67-60.00 617.28 341.69 459.26 1.34-2.95 1.23 38.98-41.20 36.84 6 7 20-66.67-70.00 1081.53 439.19 689.20 1.57-2.84 1.38 43.18-39.10 39.64 7 4 15 83.33-20.00 688.93 0.25-13.23-52.08-29.66-34.39-958.33-539.86-628.04 8 22 40 25.00-30.00 15.43 16.06-15.74-0.98-4.11-0.32-4.41-62.90 7.91 9 15 35 91.67-26.00 672.15 5.98-63.37-10.61-8.93-6.74-184.09-152.74-111.88 23 37 Ave. 14-19 166

Table E3: Errors Comparson Between Table E1 and Table E2 Returns of A 8.53E-16 0 0-2.1E-16 Returns of B Average of A Average of B Varance of A Varance of B Covarance -8.9E- -2.65E- -1.53477E- 16 0 0 15-4.8E-16 15-8.9E- 2.53681E- 16 0 5.29E-16 16-8.9E- 5.19284E- 16 0 9.3E-16 16 1.78E- 16-4.9E-16 5.29E-16 0 OLS beta -1.07E- 15-2.13E- 16-4.34E- 16-4.94E- 16 1.71E-16 0 0 0 0 0-3.29911E- -2.83E- -2.1E-16 0-4.2E-16 0 16 16 1.78E- 2.68585E- -2.59E- 1.71E-16 16 3.3E-16 5.25E-15 15 15-1.2E- -4.43E- -3.38553E- 1.132E- 0 16 0 16 16 16-3.38E- -2.24236E- -1.6E-16 0 16 0 16 BA beta ML beta OLS alpha -2.3E- 8.6E- 5.3E- 15 16 16-1.9E- -4.2E- -1E- 15 16 16-1.7E- -5.2E- -3E- 15 16 16-2.4E- -2.4E- -2E- 15 16 16-2.7E- 15 0 0-2.8E- -1.6E- -2E- 15 16 16-2.5E- -2.7E- -3E- 15 15 15-1.9E- 3.5E- 15 16 0-3E- 16-3.35E- 16-1E-15-4E-16 BA alpha ML alpha -3E- 2E- 15 16-2E- -1E- 15 15-2E- -8E- 15 16-3E- -1E- 15 16-3E- 15 0-4E- 15 0-3E- -3E- 15 15-2E- -3E- 15 16-1E- -4E- 15 16 167

Appx F: Sample Sze of Test Portfolo It s mportant to establsh whether the sample sze chosen s good representaton of the populaton. Total Sample Sze n 250 securtes 166 data ponts per 41500 securty n 203.7155 Standard Devaton of s 5676.55 Sample Standard Error of Sample Means s 33 x s 27.86509 The followng equaton s then used to determne the sample sze: n 2 z s n (F1) 34 E Where E s the allowable error Z s the z score assocated wth the degree of confdence selected s s the sample devaton of the plot survey, n ths case mean value of the standard devaton had been used From equaton (F1), t s seen that sample sze s depent of E. There are two unknowns n the equaton, so the standard error of sample means s used as the allowable error n the sample, thus remove one unknown. From Table F1: Calculaton of Sample Sze n Terms of Confdence Intervals, for the E = 28, the sample sze ranges from 9 to 21, depng on the degree of confdence selected. Thus the number of securtes ncluded n portfolo beng 27, wthout repeatng any securtes, t s a decent representaton of the equty market. Also, the securtes chosen are the consttuents of headlne ndces; ths mples the mertocracy of these frms. The frms chosen also account for more than 1/3 of the stock exchange market captalsaton. These renforces the sample chosen s a good representaton of the market as a whole. 33 Mason, R.D. and Lnd D.A, 1996, Statstcal Technques n Busness & Economcs, Nnth Edton, Irwn, p.329, Equaton (8-11) 34 Mason, R.D. and Lnd D.A, 1996, Statstcal Technques n Busness & Economcs, Nnth Edton, Irwn, p.330, Equaton (8-12) 168

Table F1: Calculaton of Sample Sze n Terms of Confdence Intervals 90% Confdence Interval 95% Confdence Interval 99% Confdence Interval z 1.65 z 1.96 z 2.58 s 49.60 s 49.60 s 49.60 E n E n E n 1 6697.786 1 9450.951 1 16375.81 2 1674.446 2 2362.738 2 4093.952 3 744.1984 3 1050.106 3 1819.534 4 418.6116 4 590.6844 4 1023.488 5 267.9114 5 378.038 5 655.0324 6 186.0496 6 262.5264 6 454.8836 7 136.6895 7 192.8765 7 334.2002 8 104.6529 8 147.6711 8 255.872 9 82.68871 9 116.6784 9 202.1705 10 66.97786 10 94.50951 10 163.7581 11 55.3536 11 78.10703 11 135.3373 12 46.5124 12 65.6316 12 113.7209 13 39.63187 13 55.92278 13 96.89828 14 34.17238 14 48.21914 14 83.55005 15 29.76794 15 42.00423 15 72.78137 16 26.16323 16 36.91778 16 63.968 17 23.17573 17 32.70225 17 56.6637 18 20.67218 18 29.1696 18 50.54262 19 18.55342 19 26.17992 19 45.36235 20 16.74446 20 23.62738 20 40.93952 21 15.18772 21 21.43073 21 37.13335 22 13.8384 22 19.52676 22 33.83432 23 12.66122 23 17.86569 23 30.95616 24 11.6281 24 16.4079 24 28.43022 25 10.71646 25 15.12152 25 26.20129 26 9.907967 26 13.9807 26 24.22457 27 9.187635 27 12.96427 27 22.46339 28 8.543094 28 12.05478 28 20.88751 29 7.964073 29 11.23775 29 19.47183 30 7.441984 30 10.50106 30 18.19534 31 6.9696 31 9.834496 31 17.04038 32 6.540806 32 9.229444 32 15.992 33 6.1504 33 8.678559 33 15.03747 34 5.793932 34 8.175563 34 14.16592 35 5.46758 35 7.715062 35 13.36801 169

Appx G: Ratonale for Shares Inclusons n the Test Portfolo The most commonly used ratos such as Prce Earnng Rato, Earnngs Per Share, Dvd Per Share have been consdered for shares nclusons. The shares chosen have dsplayed ether consstent or an ncreasng tr n ther PE, EPS and DPS per share. (Profle Group (Pty) Ltd., 2006b) Table G1: Ratonale for Shares Inclusons Code Name Sector Subsector Ratonale AFB Alexander Forbes Lmted Fnancal Insurance Internatonal fnancal & rsk servces provder Major shareholder n VenFn Ltd. wth 24.7% shares AGL Anglo Amercan plc Basc Materals Mnng - General Mnng Global leader n mnng and natural resource sector Prmarly lsted on London Stock Exchange; varous lstng on other stock exchanges AMS Anglo Platnum Ltd. Basc Materals Mnng - Platnum World's largest platnum produce, thus can effectvely affect commodty prce Gold, Copper, Nckel and Cobalt are recovered as by-products Dual lsted on London Stock Exchange ASA Absa Group Ltd. Fnancal Banks Foregn nvestor, Barclays plc, s the major shareholder, holds 56.4% of the frm BAW Barloworld Lmted Industrals Industral Goods and Servces - General Dversfed ndustral brand management BCX Busness Connexon Group Lmted Technology Software and Computer Servces Also lsted on both London and Namban Stock Exchange Afrca's leadng ntegrator of compettve, nnovatve and practcal busness solutons based on nformaton and communcaton technology 170

BDE BIDBEE Other Securtes - Industral Industral Goods and Servces - Busness Support Servces BVT The Bdvest Group Ltd. Industrals Industral Goods and Servces - Busness Support Servces Good corporate governance Internatonal servces, tradng and dstrbutons CLH Cty Lodge Consumer Servces Lesure and Hotels Hgh qualty affordable hotels targeted at busness communty & lesure travelers; however doesn't offer 5 star servces 2010 Soccer World Cup, spectators & toursts need accommodaton DST Dstell Group Lmted Consumer Goods Food & Beverages Leadng SA producer n wne & sprts ERP ERP.com Holdngs Ltd. Technology Software and Computer Servces Prncpal busness actvty s to act as an nvestment holdng company, wth subsdares FBR Famous Brand Lmted Consumer Servces Lesure and Hotels Operate n all major segments of quck servce restaurant FSR FrstRand Lmted Fnancal Banks Blurrng of boundares n fnancal servces ndustry and convergence of products and servces Dfferentated by ts de-centralzed structure and owner-manager culture Dual lsted on Namban Stock Exchange IPL Imperal Holdngs Ltd. Industrals Industral Goods and Servces - Transportaton Subsdares and assocates n bankng, lfe assurance, short-term nsurance, leasng and fleet management, avaton leasng, logstcs and transport, etc 171

LBT MTN Lberty Internatonal plc Fnancal Real Estate Major UK property group MTN Group Ltd. Telecommuncatons Tele. Servces Property market started to regress snce 1997 economc depresson Dual lsted on London Stock Exchange Afrcan- focused holdng, provdng telecommuncaton nfrastructure Ad SA transton from developng to developed country MUR Murray and Roberts Holdngs Lmted Industrals Constructon & Buldng Materals Industral holdng company and mult-faceted global character PIK Pck n Pay Stores Lmted Consumer Servces Food & Drug Retalers PPC Pretora Portland Cement Company Ltd. Industrals Constructon & Buldng Materals PPC Cement s the leadng suppler of cement n southern Afrca Cement s an mportant raw materal for all constructons/ nfrastructure REM Remgro Lmted Industrals Industral Goods and Servces - General Interests n luxurous goods among other economc sectors n SA RLO Reunert Lmted Industrals Industral Goods and Servces - Electrcal Played a major role n SA economy development Holds shares n Afrcan Cables and Semens Telecommuncaton SAB SABMller plc Consumer Goods Food & Beverages One of the world's largest brewers SA have been experencng healthy economy, thus steady ncreasng demands for luxurous goods/ drnks Dual lsted on London Stock Exchange 172

SBK Standard Bank Group Ltd. Fnancal Banks Wde representaton n Afrca and emergng markets nternatonally In 2005, undergoes nternal restructurng to ncrease the frm's compettveness Dual lsted on Namban Stock Exchange SHP Shoprte Holdngs Ltd. Consumer Servces Food & Drug Retalers Investment holdng company wth nvestments n supermarket chan, property, fresh produce and furnture, therefore dversfcaton Dual lsted on Namban Stock Exchange TBS Tger Brands Lmted Consumer Goods VNF VenFn Ltd. Fnancal Food & Beverages Investment Companes Balanced spread of Afrcan & selected nternatonal operatons n manufacturng, processng & dstrbuton of branded food and healthcare products Hold USD 100 mllon worth of Dmenson Data Convertble Bond Operatng actvtes have spread over telecommuncatons, technology and meda nterests WHL Woolworths Holdngs Ltd. Consumer Servces General Retalers Focus on qualty, value and customer servce. e.g. Frst retal store wthout stocks, well managed queues, etc (Source: Profle Group (Pty) Ltd., 2006a) 173

Appx H: Ordnary Shares Lsted Based on Market Captalzaton The fundamental reason for selectng shares based on ts market captalsaton s that ths would nclude all the ordnary shares lsted on JSE, thus ths gves a better representaton of market. The overall market value of ordnary shares on JSE s R 2,566,352,039,068. Table H1: Ordnary Shares Lsted Based on Market Captalzaton ALPHA CODE EQUITY_NAME EQUITY STATUS DATE MARKET_CAP % AGL ANGLO AMERICAN PLC C 20041231 199,373,508,019 7.7688 BIL BHP BILLITON PLC C 20041231 162,897,702,132 6.3474 RICHEMONT SECURITIES RCH DR C 20041231 98,136,000,000 3.8239 SAB SABMILLER PLC C 20041231 95,875,522,495 3.7359 STANDARD BANK GROUP SBK LTD C 20041231 88,968,730,548 3.4667 SOL SASOL LTD C 20041231 81,546,428,425 3.1775 FSR FIRSTRAND LTD C 20041231 73,108,938,523 2.8487 MTN MTN GROUP LTD C 20041231 72,291,401,202 2.8169 OML OLD MUTUAL PLC C 20041231 55,084,404,818 2.1464 TKG TELKOM SA LTD C 20041231 54,589,118,262 2.1271 ANGLOGOLD ASHANTI ANG LTD C 20041231 52,630,760,534 2.0508 ASA ABSA GROUP LIMITED C 20041231 49,777,635,073 1.9396 REM REMGRO LTD C 20041231 45,905,540,814 1.7887 AMS ANGLO PLATINUM LTD C 20041231 45,002,937,672 1.7536 SLM SANLAM LTD C 20041231 35,978,418,671 1.4019 GFI GOLD FIELDS LTD C 20041231 34,193,508,843 1.3324 LBT LIBERTY INTERNATIONL PLC C 20041231 33,937,587,939 1.3224 IMP IMPALA PLATINUM HLGS LD C 20041231 31,957,627,894 1.2453 NED NEDBANK GROUP LTD C 20041231 30,667,368,936 1.1950 174

MLA MITTAL STEEL SA LTD C 20041231 29,196,764,646 1.1377 RMH RMB HOLDINGS LTD C 20041231 25,846,714,483 1.0071 BVT BIDVEST LTD ORD C 20041231 25,518,679,284 0.9944 BAW BARLOWORLD LTD C 20041231 23,696,729,250 0.9234 NPN NASPERS LTD -N- C 20041231 23,591,152,500 0.9192 IPL IMPERIAL HOLDINGS LTD C 20041231 22,829,130,584 0.8896 HAR HARMONY G M CO LTD C 20041231 20,224,049,561 0.7880 SAP SAPPI LTD C 20041231 19,842,967,036 0.7732 LGL LIBERTY GROUP LTD C 20041231 18,421,087,606 0.7178 EDGARS CONS STORES ECO LTD C 20041231 16,476,202,431 0.6420 TBS TIGER BRANDS LTD ORD C 20041231 16,353,199,623 0.6372 PPC PRETORIA PORT CEMNT C 20041231 15,321,953,115 0.5970 STEINHOFF INTERNTL SHF HLDGS C 20041231 14,297,163,741 0.5571 LON LONMIN P L C C 20041231 14,020,343,469 0.5463 INP INVESTEC PLC C 20041231 13,538,561,524 0.5275 KMB KUMBA RESOURCES LTD C 20041231 13,281,585,284 0.5175 JDG JD GROUP LTD C 20041231 11,729,400,000 0.4570 PIK PIK N PAY STORES LTD C 20041231 11,278,306,062 0.4395 WHL WOOLWORTHS HOLDINGS LTD C 20041231 10,936,382,144 0.4261 DSY DISCOVERY HOLDINGS LTD C 20041231 10,185,632,145 0.3969 NPK NAMPAK LTD ORD C 20041231 10,046,317,039 0.3915 FOS FOSCHINI LTD ORD C 20041231 9,619,929,640 0.3748 MSM MASSMART HOLDINGS LTD C 20041231 9,021,346,667 0.3515 ABL AFRICAN BANK INVESTMENTS C 20041231 8,731,946,839 0.3402 LIBERTY HOLDINGS LTD LBH ORD C 20041231 8,693,928,778 0.3388 AFRICAN OXYGEN LTD AFX ORD C 20041231 8,588,469,754 0.3347 NTC NETWORK HEALTHCARE HLDGS C 20041231 8,518,187,676 0.3319 175

TRU TRUWORTHS INTERNATIONAL C 20041231 8,302,632,768 0.3235 SNT SANTAM LTD C 20041231 8,179,528,517 0.3187 INL INVESTEC LTD C 20041231 7,963,914,387 0.3103 AVI AVI LTD C 20041231 7,836,719,942 0.3054 RLO REUNERT ORD C 20041231 7,202,231,850 0.2806 SHP SHOPRITE HLDGS LTD ORD C 20041231 7,010,885,034 0.2732 MET METROPOLITAN HLDGS LTD C 20041231 6,993,381,259 0.2725 APN ASPEN PHARMACARE HLDGS. C 20041231 6,870,953,611 0.2677 SUI SUN INTERNATIONAL LTD C 20041231 6,634,395,471 0.2585 MUTUAL AND FEDERAL MAF INS C 20041231 6,061,653,388 0.2362 PWK PIK N PAY HOLDINGS LTD C 20041231 5,799,739,902 0.2260 DDT DIMENSION DATA HLDGS PLC C 20041231 5,638,727,346 0.2197 TNT TONGAAT-HULETT GROUP ORD C 20041231 5,525,478,731 0.2153 ARI AFRICAN RAINBOW MINERALS C 20041231 5,416,368,231 0.2111 SPG SUPER GROUP LTD C 20041231 5,181,991,795 0.2019 GRT GROWTHPOINT PROP LTD C 20041231 5,067,604,510 0.1975 AFB ALEXANDER FORBES LTD C 20041231 4,997,609,233 0.1947 MEDI-CLINIC CORP LTD MDC ORD C 20041231 4,988,440,386 0.1944 ALT ALLIED TECHNOLOGIES C 20041231 4,909,116,915 0.1913 DST DISTELL GROUP LTD C 20041231 4,908,915,900 0.1913 AEG AVENG LTD C 20041231 4,753,750,896 0.1852 HIVELD STEEL AND HVL VANADUM C 20041231 4,730,284,704 0.1843 AFE A E C I LTD ORD C 20041231 4,584,283,002 0.1786 MUR MURRAY AND ROBERTS H ORD C 20041231 4,563,523,511 0.1778 CAT CAXTON CTP PUBLISH PRINT C 20041231 4,467,529,659 0.1741 ELH ELLERINE HOLDINGS LTD C 20041231 4,399,035,200 0.1714 ALLAN GRAY PROPERTY GRY TRST C 20041231 4,103,697,493 0.1599 176

LEW LEWIS GROUP LTD C 20041231 3,900,000,000 0.1520 SPP THE SPAR GROUP LTD C 20041231 3,629,438,349 0.1414 JNC JOHNNIC HOLDINGS LTD C 20041231 3,620,731,156 0.1411 GND GRINDROD LTD C 20041231 3,591,401,304 0.1399 JOHNNIC JCM COMMUNICATIONS C 20041231 3,542,436,676 0.1380 NCL NEW CLICKS HLDGS LTD C 20041231 3,476,071,204 0.1354 WAR WESTERN AREAS LTD C 20041231 2,963,709,475 0.1155 UTR UNITRANS LTD C 20041231 2,939,375,604 0.1145 MVELAPHANDA GROUP MVG LTD C 20041231 2,863,721,245 0.1116 MPC MR PRICE GROUP LTD C 20041231 2,803,249,491 0.1092 GOLD REEF CASINO GDF RESORTS C 20041231 2,783,033,636 0.1084 ARL ASTRAL FOODS LTD C 20041231 2,676,038,280 0.1043 HOSKEN CONS INVEST HCI LTD C 20041231 2,628,378,170 0.1024 ILV ILLOVO SUGAR LTD C 20041231 2,600,617,900 0.1013 ITE ITALTILE LTD C 20041231 2,521,433,205 0.0982 AFR AFGRI LTD C 20041231 2,516,605,000 0.0981 SYC SYCOM PROPERTY FUND C 20041231 2,484,477,038 0.0968 MVELAPHANDA MVL RESOURCES LD C 20041231 2,412,845,530 0.0940 PTG PEERMONT GLOBAL LTD C 20041231 2,326,500,000 0.0907 HYPROP INVESTMENTS HYP LTD C 20041231 2,303,460,729 0.0898 TRE TRENCOR LTD C 20041231 2,235,005,654 0.0871 OMN OMNIA HOLDINGS LTD C 20041231 2,167,614,578 0.0845 AQP AQUARIUS PLATINUM LTD C 20041231 2,151,601,192 0.0838 DRD DRDGOLD LTD C 20041231 2,093,598,539 0.0816 ASR ASSORE LTD C 20041231 2,058,000,000 0.0802 RBW RAINBOW CHICKEN LTD C 20041231 2,056,499,775 0.0801 NHM NORTHAM PLATINUM LTD C 20041231 2,049,425,475 0.0799 177

PMN PRIMEDIA LTD -N- C 20041231 2,030,228,043 0.0791 MARTPROP PROPERTY MTP FUND C 20041231 1,957,996,184 0.0763 APA APEXHI PROPERTIES -A- C 20041231 1,888,215,030 0.0736 APB APEXHI PROPERTIES -B- C 20041231 1,869,142,151 0.0728 EMI EMIRA PROPERTY FUND C 20041231 1,868,673,630 0.0728 SAE SA EAGLE INSURANCE CO C 20041231 1,802,566,000 0.0702 TSX TRANS HEX GROUP LTD C 20041231 1,725,960,207 0.0673 DEL DELTA ELECRICAL IN C 20041231 1,686,378,467 0.0657 VUKILE PROPERTY FUND VKE LTD C 20041231 1,679,333,328 0.0654 OCE OCEANA GROUP LTD C 20041231 1,667,040,310 0.0650 CRM CERAMIC INDUSTRIES LTD C 20041231 1,661,982,413 0.0648 WES WESCO INVESTMENTS LTD C 20041231 1,646,151,000 0.0641 ALLIED ELECTRONICS ATN CORP C 20041231 1,613,090,309 0.0629 PAP PANGBOURNE PROP LTD C 20041231 1,591,714,958 0.0620 REDEFINE INCOME FUND RDF LTD C 20041231 1,585,886,270 0.0618 SA RETAIL PROPERTIES SRL LTD C 20041231 1,559,605,650 0.0608 KGM KAGISO MEDIA LTD C 20041231 1,542,101,454 0.0601 ILA ILIAD AFRICA LTD C 20041231 1,537,522,693 0.0599 TIW TIGER WHEELS LTD C 20041231 1,534,592,175 0.0598 CORONATION FUND CML MNGRS LD C 20041231 1,529,099,720 0.0596 CLH CITY LODGE HTLS LTD ORD C 20041231 1,487,333,318 0.0580 WBO WILSON BAYLY HLM-OVC ORD C 20041231 1,470,498,250 0.0573 RAH REAL AFRICA HLDGS LTD C 20041231 1,412,678,975 0.0550 DTC DATATEC LTD C 20041231 1,398,281,430 0.0545 BYTES TECHNOLOGY GRP BTG LTD C 20041231 1,313,168,953 0.0512 CPL CAPITAL PROPERTY FUND C 20041231 1,288,389,264 0.0502 PALABORA MINING CO PAM ORD C 20041231 1,274,197,500 0.0497 178

APK ASTRAPAK LTD C 20041231 1,259,423,250 0.0491 KAP KAP INTERNATIONAL HLDGS C 20041231 1,256,160,000 0.0489 AMA AMALGAMATED APPL HLD LTD C 20041231 1,241,309,680 0.0484 RES RESILIENT PROP INC FD LD C 20041231 1,211,731,082 0.0472 IFR IFOUR PROPERTIES LTD C 20041231 1,206,119,392 0.0470 KWV KWV BELEGGINGS BEPERK C 20041231 1,197,000,000 0.0466 BPL BARPLATS INVESTMENTS ORD C 20041231 1,168,336,806 0.0455 TRT TOURISM INV CORP LTD C 20041231 1,162,490,424 0.0453 BUSINESS CONNEXION BCX GROUP C 20041231 1,158,228,781 0.0451 GRF GROUP FIVE LTD ORD C 20041231 1,114,631,298 0.0434 METBOARD PROPERTIES MPL LTD C 20041231 1,081,484,727 0.0421 METAIR INVESTMENTS MTA ORD C 20041231 1,058,905,980 0.0413 HDC HUDACO INDUSTRIES LTD C 20041231 1,044,627,725 0.0407 TDH TRADEHOLD LTD C 20041231 1,041,991,323 0.0406 BAT BRAIT S.A. C 20041231 986,767,813 0.0385 MST MUSTEK LTD C 20041231 984,817,395 0.0384 GMB GLENRAND M.I.B. LTD C 20041231 982,106,446 0.0383 MRF MERAFE RESOURCES LTD C 20041231 940,817,313 0.0367 DISTRIBUTION AND DAW WAREHSG C 20041231 929,570,862 0.0362 CAPITEC BANK HLDGS CPI LTD C 20041231 917,087,253 0.0357 IVT INVICTA HOLDINGS LTD C 20041231 913,884,356 0.0356 CLIENTELE LIFE CLE ASSURANCE C 20041231 905,800,000 0.0353 ACP ACUCAP PROPERTIES LTD C 20041231 869,441,051 0.0339 PSG PSG GROUP LIMITED C 20041231 850,465,000 0.0331 RNG RANDGOLD AND EXP CO S 20041231 822,944,408 0.0321 CDZ CADIZ HOLDINGS LTD C 20041231 812,954,286 0.0317 DLV DORBYL LTD ORD C 20041231 795,834,144 0.0310 179

CMH COMBINED MOTOR HLDGS LTD C 20041231 790,964,375 0.0308 CSB CASHBUILD LTD C 20041231 783,837,405 0.0305 NWL NU-WORLD HOLDINGS LTD C 20041231 748,276,587 0.0292 PGR PEREGRINE HOLDINGS LTD C 20041231 744,985,504 0.0290 ATS ATLAS PROPERTIES LTD C 20041231 739,905,095 0.0288 MBN MOBILE INDUSTRIES -N- C 20041231 721,471,600 0.0281 MERCANTILE BANK MTL HLDGS LD C 20041231 709,005,334 0.0276 ADR ADCORP HLDGS LTD ORD C 20041231 697,264,458 0.0272 ART ARGENT INDUSTRIAL LTD C 20041231 670,229,609 0.0261 BRANDCORP HOLDINGS BRC LTD C 20041231 656,326,983 0.0256 COM COMAIR LTD C 20041231 630,000,000 0.0245 PMA PRIMEDIA LTD C 20041231 625,035,312 0.0244 FBR FAMOUS BRANDS LTD C 20041231 618,369,132 0.0241 FREESTONE PROPERTY FSP HLDGS C 20041231 611,034,370 0.0238 SUR SPUR CORPORATION LTD C 20041231 590,678,639 0.0230 BEL BELL EQUIPMENT LTD C 20041231 584,327,680 0.0228 SFN SASFIN HOLDINGS LTD C 20041231 574,265,714 0.0224 JCD JCI LTD S 20041231 567,621,986 0.0221 PREMIUM PROPERTIES PMM LTD C 20041231 545,313,787 0.0212 AGI AG INDUSTRIES LTD C 20041231 543,428,767 0.0212 PHM PHUMELELA GAME LEISURE C 20041231 532,629,637 0.0208 DCT DATACENTRIX HOLDINGS LTD C 20041231 530,738,231 0.0207 ZAMBIA COPPER INV LD ZCI ORD C 20041231 530,028,920 0.0207 OCT OCTODEC INVEST LTD C 20041231 509,239,698 0.0198 PARAMOUNT PROP FUND PRA LTD C 20041231 508,968,333 0.0198 MTX METOREX LTD C 20041231 498,459,374 0.0194 BCF BOWLER METCALF LTD C 20041231 486,797,169 0.0190 180

MCP MICC PROPERTY INCOME FND S 20041231 481,007,571 0.0187 ADH ADVTECH LTD C 20041231 472,397,863 0.0184 ENV ENVIROSERV HOLDINGS LTD C 20041231 459,673,975 0.0179 SPE SPEARHEAD PROP HLDGS LTD C 20041231 452,407,728 0.0176 BDEO BIDVEST CALL OPTIONS C 20041231 432,000,000 0.0168 CUL CULLINAN HOLDINGS ORD C 20041231 430,913,122 0.0168 TGN TIGON LTD S 20041231 404,837,161 0.0158 PCN PARACON HOLDINGS LTD C 20041231 397,096,378 0.0155 VLE VALUE GROUP LTD C 20041231 396,514,166 0.0155 MOB MOBILE INDUSTRIES ORD C 20041231 367,827,080 0.0143 MCU M CUBED HLDGS LTD C 20041231 367,500,000 0.0143 ABT AMBIT PROPERTIES LTD C 20041231 357,820,127 0.0139 SBO SAAMBOU HOLDINGS LTD S 20041231 338,958,403 0.0132 BARNARD JACOBS BJM MELLET C 20041231 333,484,003 0.0130 ACH ARCH EQUITY LTD C 20041231 330,345,552 0.0129 DGC DIGICORE HOLDINGS LTD C 20041231 313,650,989 0.0122 UCS UCS GROUP LTD C 20041231 311,394,474 0.0121 SRN SEARDEL INVST CORP -N- C 20041231 310,852,296 0.0121 GOODHOPE DIAM (KIM) GDH LTD S 20041231 305,000,000 0.0119 ERP ERP.COM HOLDINGS LTD C 20041231 292,801,869 0.0114 CONTROL INSTRUMENTS CNL GRP C 20041231 261,232,120 0.0102 SCN SCHARRIG MINING LTD C 20041231 256,187,318 0.0100 YOMHLABA RESOURCES YBA LTD S 20041231 240,000,102 0.0094 LAF LONRHO AFRICA PLC C 20041231 236,358,132 0.0092 PIM PRISM HOLDINGS LTD C 20041231 225,032,509 0.0088 BSB THE HOUSE OF BUSBY LTD C 20041231 219,922,039 0.0086 EOH EOH HOLDINGS LTD C 20041231 215,322,895 0.0084 181

CKS CROOKES BROS LTD C 20041231 214,385,600 0.0084 CNC CONCOR LTD RCON C 20041231 206,887,707 0.0081 LAN LA GROUP LTD -N- C 20041231 200,451,750 0.0078 IDION TECHNOLOGY IDI HLDGS C 20041231 194,997,127 0.0076 DTP DATAPRO GROUP LTD C 20041231 192,342,886 0.0075 WNH WINHOLD LTD ORD C 20041231 178,974,877 0.0070 MMG MICROMEGA HOLDINGS LTD C 20041231 177,489,530 0.0069 SOV SOVEREIGN FOOD INVEST LD C 20041231 172,114,394 0.0067 TPC TRANSPACO LTD C 20041231 168,808,354 0.0066 BRN BRIMSTONE INVESTMENT -N- C 20041231 159,179,372 0.0062 SKJ SEKUNJALO INVESTMENTS LD C 20041231 155,536,073 0.0061 LAR LA GROUP LTD ORD C 20041231 154,473,965 0.0060 ELR ELB GROUP LTD ORD C 20041231 145,598,000 0.0057 HOWDEN AFRICA HLDGS HWN LTD C 20041231 144,604,039 0.0056 PPR PUTPROP LTD C 20041231 143,964,805 0.0056 EXL EXCELLERATE HLDGS LTD C 20041231 129,333,485 0.0050 SETPOINT TECHNOLOGY STO HLDG C 20041231 127,596,004 0.0050 SPS SPESCOM LTD C 20041231 126,028,886 0.0049 JASCO ELECTRONICS JSC HLDGS C 20041231 125,515,945 0.0049 GIJ GIJIMA AST GROUP LTD C 20041231 107,332,928 0.0042 SBL SABLE HLDGS LTD ORD C 20041231 101,040,000 0.0039 CRG CARGO CARRIERS LTD C 20041231 100,000,000 0.0039 MTZ MATODZI RESOURCES LTD C 20041231 96,458,678 0.0038 SER SEARDEL INVEST CORP LTD C 20041231 96,196,987 0.0037 MAS MASONITE AFRICA LTD ORD C 20041231 94,243,242 0.0037 AFG AFGEM LTD C 20041231 93,468,516 0.0036 PET PETMIN LTD C 20041231 93,155,555 0.0036 182

AME SWL MTE RAG KIR AFRICAN MEDIA ENTERTAIN C 20041231 90,597,234 0.0035 SHAWCELL TELECOMM LTD S 20041231 90,000,000 0.0035 MONTEAGLE SOCIETE ANONYM C 20041231 88,200,000 0.0034 RETAIL APPAREL GROUP LTD S 20041231 84,750,000 0.0033 KAIROS INDUSTRIAL HLDGS C 20041231 83,188,181 0.0032 RTN REX TRUEFORM CL CO -N- C 20041231 79,813,570 0.0031 SUM SPECTRUM SHIPPING LTD C 20041231 76,500,000 0.0030 PSC PASDEC RESOURCES SA LTD C 20041231 75,551,340 0.0029 PNC PINNACLE TECH HLDGS LTD C 20041231 74,563,293 0.0029 SAL SALLIES LTD C 20041231 71,962,492 0.0028 LNF LONDON FIN INV GRP PLC C 20041231 71,829,321 0.0028 WLN WOOLTRU LTD-N- C 20041231 71,784,095 0.0028 DEC DECILLION LTD C 20041231 71,294,647 0.0028 COMPU CLEARING OUTS CCL LTD C 20041231 69,666,894 0.0027 OLG ONELOGIX GROUP LTD C 20041231 69,160,830 0.0027 SVN SABVEST LTD -N- C 20041231 65,204,664 0.0025 BRIMSTONE INVESTMNT BRT CORP C 20041231 61,689,917 0.0024 SBG SIMEKA BSG LTD C 20041231 60,750,000 0.0024 SCH STOCKS HOTELS AND RESORT S 20041231 59,000,000 0.0023 FVT FAIRVEST PROPERTY HLDGS C 20041231 58,705,802 0.0023 JDH JOHN DANIEL HOLDINGS LTD C 20041231 58,019,759 0.0023 CNX CONAFEX HLDGS SOCIE ANON C 20041231 56,911,596 0.0022 BSR BASIL READ HLDGS LTD C 20041231 56,202,000 0.0022 WLO WOOLTRU LTD ORD C 20041231 54,534,832 0.0021 ENTERPRISE RISK ERM MNGMENT C 20041231 52,504,061 0.0020 PPE PURPLE CAPITAL LTD C 20041231 52,267,500 0.0020 AFRICAN AND OVERSEAS - AON N- C 20041231 50,687,205 0.0020 183

AFO AFLEASE GOLD LTD C 20041231 50,589,573 0.0020 ABO ABSOLUTE HOLDINGS LTD C 20041231 50,287,767 0.0020 LAB LABAT AFRICA LTD C 20041231 46,103,637 0.0018 VIKING INV AND ASSET VKG MAN S 20041231 45,458,051 0.0018 SIM SIMMER AND JACK MINES C 20041231 44,964,749 0.0018 HWA HWANGE COLLIERY LD ORD C 20041231 41,546,920 0.0016 TREMATON CAPITAL INV TMT LTD C 20041231 39,312,000 0.0015 DIAMOND CORE DMR RESOURCES C 20041231 38,876,357 0.0015 SBV SABVEST LTD C 20041231 38,118,407 0.0015 STA STRATCORP LTD C 20041231 37,407,499 0.0015 YRK YORK TIMBER ORG C 20041231 34,225,540 0.0013 KING CONSOLIDATED KNG HLDGS C 20041231 32,634,877 0.0013 NCS NICTUS BEPERK C 20041231 32,066,100 0.0012 IFW INFOWAVE HOLDINGS LTD C 20041231 31,051,188 0.0012 MKX MILKWORX LTD C 20041231 29,442,124 0.0011 FRT FARITEC HOLDINGS LTD C 20041231 28,977,378 0.0011 HAL HALOGEN HLDGS SOC ANON C 20041231 27,960,390 0.0011 ALX ALEX WHITE HOLDINGS LTD C 20041231 27,186,921 0.0011 ICC INDUS CREDIT CO AFRICA H C 20041231 26,833,352 0.0010 ALJ ALL JOY FOODS LTD C 20041231 26,565,000 0.0010 PMV PRIMESERV GROUP LTD C 20041231 26,412,548 0.0010 CAE CAPE EMPOWERMENT TRUST C 20041231 26,014,586 0.0010 NMS NAMIBIAN SEA PRODUCTS LD C 20041231 25,571,817 0.0010 VTL VENTER LEISURE AND COMM C 20041231 25,247,547 0.0010 EUR EUREKA IND LTD ORD C 20041231 25,224,000 0.0010 DON DON GROUP LTD C 20041231 23,558,824 0.0009 ITR INTERTRADING LTD C 20041231 23,000,000 0.0009 184

BDM BUILDMAX LTD C 20041231 22,993,098 0.0009 MONEY WEB HOLDINGS MNY LTD C 20041231 22,950,000 0.0009 MSS MARSHALLS LTD C 20041231 20,029,152 0.0008 SOUTHERN ELECTRICITY SLO CO C 20041231 19,231,860 0.0007 EXO EXXOTEQ LTD S 20041231 19,200,000 0.0007 NEW AFRICA INVESTMNT- NAN N- C 20041231 18,388,758 0.0007 BIC BICC CAFCA LTD C 20041231 18,360,000 0.0007 BEG BEIGE HOLDINGS LTD C 20041231 16,925,931 0.0007 ISA ISA HOLDINGS LTD C 20041231 16,721,079 0.0007 KLG KELGRAN LTD C 20041231 15,045,470 0.0006 IDQ INDEQUITY GROUP LTD C 20041231 14,604,000 0.0006 TOP INFO TECHNOLOGY TOT HLDG S 20041231 13,767,055 0.0005 RCO RARE EARTH EXTRACTION CO S 20041231 13,662,000 0.0005 IND INDEPENDENT FINANCIAL SE C 20041231 13,600,000 0.0005 SPA SPANJAARD LTD C 20041231 13,395,000 0.0005 REX TRUEFORM CLOTH RTO ORD C 20041231 13,221,412 0.0005 PAL PALS HOLDING LTD C 20041231 12,000,000 0.0005 AFRICAN DAWN CAPITAL ADW LTD C 20041231 10,604,953 0.0004 SJL S AND J LAND HOLDINGS C 20041231 10,520,000 0.0004 CORWIL INVESTMENTS CRW LTD S 20041231 9,749,580 0.0004 CND CONDUIT CAPITAL LTD C 20041231 9,522,751 0.0004 GLOBAL VILLAGE HLDGS GLL LTD C 20041231 9,409,783 0.0004 CMA COMMAND HOLDINGS LTD C 20041231 9,000,000 0.0004 QUY QUYN HOLDINGS LTD C 20041231 8,400,894 0.0003 ICT INCENTIVE HOLDINGS LTD S 20041231 8,243,973 0.0003 ALC AMLAC LTD S 20041231 8,190,000 0.0003 RNT RENTSURE HOLDINGS LTD S 20041231 8,087,586 0.0003 185

SQE ILT TBX MFL HCL SQUARE ONE SOLUTIONS GRP C 20041231 7,920,000 0.0003 INTERCONNECTIVE SOLUTION C 20041231 7,847,000 0.0003 THABEX EXPLORATION LTD C 20041231 7,653,099 0.0003 METROFILE HOLDINGS LTD C 20041231 7,407,741 0.0003 HERITAGE COLLECTION HLDG C 20041231 7,113,030 0.0003 SNG SYNERGY HOLDINGS LTD C 20041231 7,073,287 0.0003 NORTHERN ENG IND AFR NEI LTD S 20041231 6,721,449 0.0003 APE APS TECHNOLOGIES LTD S 20041231 6,575,000 0.0003 ITG INTEGREAR LTD S 20041231 6,282,095 0.0002 AOO AFR AND OSEAS ENTER ORD C 20041231 5,937,500 0.0002 VST VESTA TECHNOLOGY HOLDNGS C 20041231 5,922,000 0.0002 ZPT ZAPTRONIX LTD C 20041231 5,792,221 0.0002 SFA SHOPS FOR AFRICA LTD S 20041231 5,769,177 0.0002 VIL VILLAGE MAIN REEF G M CO C 20041231 4,854,756 0.0002 DYM DYNAMIC CABLES RSA LTD C 20041231 4,673,425 0.0002 STI STILFONTEIN G M CO LTD S 20041231 4,572,022 0.0002 SLL STELLA VISTA TECHNOL LTD C 20041231 4,200,000 0.0002 SMR SAMRAND DEVELOP HLDGS LD S 20041231 4,084,897 0.0002 SAM SA MINERAL RESOURCES COR C 20041231 3,742,749 0.0001 BEE BEGET HOLDINGS LTD C 20041231 3,578,832 0.0001 ADONIS KNITWEAR ADO HOLDINGS C 20041231 3,516,250 0.0001 CCG CCI HOLDINGS LTD S 20041231 3,475,576 0.0001 AWT AWETHU BREWERIES LTD ORD C 20041231 3,382,276 0.0001 MLL MILLIONAIR CHARTER LTD S 20041231 3,019,500 0.0001 ALD ALUDIE LTD S 20041231 2,661,275 0.0001 BRY BRYANT TECHNOLOGY LTD S 20041231 1,960,000 0.0001 BNT BONATLA PROPERTY HLDGS S 20041231 1,853,469 0.0001 186

ORE ORION REAL ESTATE LTD C 20041231 1,823,385 0.0001 CORVUS CAP (SA) HLDG CVS LTD C 20041231 1,640,882 0.0001 PAC PACIFIC HLDGS LTD S 20041231 1,448,578 0.0001 TRF TERRAFIN HOLDINGS LTD S 20041231 954,435 0.0000 PFN CONSOL PROP AND FIN LTD S 20041231 900,000 0.0000 AEC ANBEECO INVESTMENT HLDGS C 20041231 898,682 0.0000 CYB CYBERHOST LIMITED S 20041231 838,158 0.0000 CMG CENMAG HOLDINGS LTD C 20041231 768,000 0.0000 RHW RICHWAY RETAIL PROP LTD S 20041231 653,021 0.0000 NAI NEW AFRICA INVEST LD ORD C 20041231 625,262 0.0000 TRX TEREXKO LTD S 20041231 493,525 0.0000 CALULO PROPERTY FUND CLO LTD C 20041231 316,542 0.0000 (Source: Johannesburg Securtes Exchange) 187

Appx I: Dvds & Weghtngs Used for Beta Calculatons The actual unts hold was calculated by dvdng the ntal nvestment of each component equally nto ther respectve ntal share prce. The actual unts hold per share n each subportfolos were summed, the weghtngs (n ths case s the percentage of the unts hold n portfolo) were then determned. The dvds were determned based on data provded by Standard Bank Group (2006). Table I1: Dvds & Weghtngs for Balanced Portfolo Stock Name Actual Unts Hold Percentage Dvds over Test Perod [Cents] AMS 7.81 0.02 2100 CLH 63.13 0.13 238 MTN 52.25 0.10 65 PPC 9.47 0.02 3840 SHP 151.7 0.30 73 WHL 214.59 0.43 63 TOTAL 498.95 1.00 Table I2: Dvds & Weghtngs for Conservatves Portfolo Stock Name Actual Unts Hold Percentage Dvds over Test Perod [Cents] Percentage Wthout VNF ASA 21 0.13 503 0.20 BVT 22.73 0.14 369 0.21 IPL 16.29 0.10 474 0.15 RLO 46.51 0.28 433 0.44 VNF 60.5 0.35 0 TOTAL 167.03 1.00 1.00 TOTAL Wthout VNF 106.53 From Table I2, there were two sets of weghtngs used, one set wth VNF and the other wthout VNF. It s because ths share was de-lsted on 1 st March 2006, thus the analyses of the subportfolo have been separated nto two parts, one that ncludes 188

VNF up to the pont before t was de-lsted on 1 st March 2006, and the other wthout VNF. The weghtngs wthout VNF have been re-calculated by dvdng the actual unts hold nto the TOTAL Wthout VNF, ths would not affect the market value of ths portfolo yet t would consder the excluson of VNF due to de-lstng. Table I3: Dvds & Weghtngs for Core Alternatve Portfolo Stock Name Actual Unts Hold Percentage Dvds over Test Perod [Cents] AFB 143.37 0.44 59 FSR 123.84 0.38 61 SAB 17.13 0.05 314 SBK 27.78 0.08 289 TBS 15.05 0.05 839 TOTAL 327.17 1.00 Table I4: Dvds & Weghtngs for Core Portfolo Stock Name Actual Unts Hold Percentage Dvds over Test Perod [Cents] AGL 18.24 0.08 1260 BAW 29.13 0.14 1314 LBT 26.53 0.13 340 PIK 109.09 0.52 114 REM 27.78 0.13 875 TOTAL 210.77 1.00 189

Table I5: Dvds & Weghtngs for Md-Term Portfolo Stock Name Actual Unts Hold Percentage Dvds over Test Perod [Cents] BAW 17.65 0.03 1314 FSR 112.58 0.17 60.5 MUR 106.95 0.17 90 MTN 38 0.06 65 PPC 6.89 0.01 3840 RLO 42.28 0.07 433 SAB 15.57 0.02 314 SHP 110.33 0.17 73 SBK 25.25 0.04 289 TBS 13.69 0.02 839 WHL 156.07 0.24 63 TOTAL 645.26 1.00 Table I6: Dvds & Weghtngs for Small Caps Portfolo Stock Name Actual Unts Hold Percentage Dvds over Test Perod [Cents] BCX 328.95 0.18 52 BDE 37.29 0.02 0 DST 67.8 0.04 153 ERP 1212.12 0.65 8 FBR 232.56 0.11 30 TOTAL 1878.72 1.00 190