Portfolio management: The use of alternative investments for the purpose of diversification. Johan Jacob Hattingh. Dissertation

Portfolio management: The use of alternative investments for the purpose of diversification Johan Jacob Hattingh Dissertation submitted in fulfillment of the requirements for the degree Magister Commercii in Investment Management in the Faculty of Economic and Management Sciences at the Rand Afrikaans University Johannesburg Study leader: Miss JE Pretorius May 2004 Co-study leader: Prof. CH van Schalkwyk

INDEX CHAPTER 1: INTRODUCTION 1 1.1 INTRODUCTION AND PROBLEM STATEMENT 1 1.2 RESEARCH OBJECTIVES 2 1.3 BENEFITS OF THE RESEARCH 3 1.4 STUDY METHODOLOGY 3 1.5 LIMITATION OF THE STUDY 4 CHAPTER 2: HISTORY AND DEVELOPMENT OF MODERN PORTFOLIO THEORY 5 2.1 INTRODUCTION 5 2.2 MARKOWITZ PORTFOLIO THEORY 5 2.2.1 Overview 5 2.2.2 Risk 6 2.2.3 Return 8 2.2.4 The efficient frontier 8 2.3 CAPITAL MARKET THEORY 12 2.3.1 Introduction 12 2.3.2 Overview of the capital market theory (CMT) 13 2.3.2.1 Assumptions of the theory 13 2.3.2.2 Addressing the information need 14 2.3.2.3 Inclusion of a risk-free asset 14 2.4 THE SIGNIFICANCE OF PORTFOLIO THEORY FOR THIS STUDY 16 2.5 CONCLUSION 17 I

CHAPTER 3: RISK DIVERSIFICATION: BASIC DEFINITIONS AND APPLICATIONS 19 3.1. INTRODUCTION 19 3.2. RISK 19 3.2.1 Overview 19 3.2.2 Types of risk 20 3.2.3 Risk measurement 21 3.3. RETURN 24 3.3.1 Overview 24 3.3.2 Required rate of return 25 3.3.3 Measurement 27 3.4. THE RELATIONSHIP BETWEEN RISK AND RETURN 27 3.5. PORTFOLIO CONSTRUCTION 29 3.5.1 Introduction 29 3.5.2 Constructing the capital market line (CML) 30 3.5.3 Optimal risky portfolio s 35 3.5.3.1 Diversification 35 3.5.3.1.1 Definition 35 3.5.3.1.2 Systematic vs. unsystematic risk 36 3.5.3.1.3 Methods of establishing diversification 37 I. Different asset classes 37 II. Different market sectors 38 III. Different entities 38 IV. Gaining global exposure 38 V. Exposure to different investment styles 38 3.5.4 The effect of diversification on the portfolio 39 3.5.4.1 Benefits of diversification 40 3.5.5 Constructing the optimal risky portfolio 41 3.5.5.1 Two-asset portfolio 41 3.5.6 Constructing complete portfolios 44 3.6 SUMMARY 47 II

CHAPTER 4: IDENTIFICATION OF ALTERNATIVE INVESTMENTS 49 4.1 INTRODUCTION 49 4.2 DEFINING AN INVESTMENT 49 4.3 DEFINING ALTERNATIVE INVESTMENT INSTRUMENTS 50 4.4 ALTERNATIVE INVESTMENT CATEGORIES 51 4.5 ANTIQUES 52 4.5.1 Definition of an antique 52 4.5.2 Furniture 52 4.5.3 Military memorabilia 55 4.5.4 Clocks and watches 56 4.5.5 Ceramics 58 4.5.6 Glass 59 4.5.7 Books 59 4.5.8 Silver and metalware 60 4.5.9 Firearms 62 4.5.10 Musical instruments 63 4.6 COLLECTABLES 64 4.6.1 Defining collectables 64 4.6.2 Stamps 64 4.6.3 Toys 65 4.6.3.1 Wood and die-cast toys 66 4.6.3.2 Dolls 66 4.6.3.3 Bears 67 4.6.4 Sports memorabilia 68 4.6.5 Precious stones 69 4.6.6 Cars 69 4.6.7 Oriental rugs 71 4.6.8 Autographs 71 4.6.9 Wine 72 4.6.10 Rare, collectable and bullion coins 75 4.6.10.1 Numismatic coins 75 4.6.10.2 Bullion coins 76 III

4.6.10.3 Kruger Rands 77 4.7. ART 77 4.8 GENERAL 80 4.8.1 Containers 80 4.8.2 Racehorses 81 4.8.3 Investing in people (Justin Wilson) 82 4.9. GENERIC FACTORS TO CONSIDER 83 4.9.1 Authenticity 83 4.9.2 Condition 84 4.9.3 Rarity 84 4.9.4 Provenance 85 4.9.5 Familiarity 85 4.9.6 Importance of the asset 86 4.9.7 Technique/workmanship 86 4.9.8 Buying what you like 86 4.10. ADVANTAGES AND DISADVANTAGES OF INVESTING IN ALTERNATIVE INVESTMENTS 87 4.10.1 Advantages 87 4.10.2 Disadvantages 88 4.11 SUMMARY 90 CHAPTER 5: EFFICIENT MARKETS AND ALTERNATIVE INVESTMENTS 91 5.1. INTRODUCTION 91 5.2. DEFINING A MARKET 91 5.3. CHARACTERISTICS OF GOOD MARKETS 93 5.4. EFFICIENT MARKETS 94 5.5. THE EFFICIENT MARKET HYPOTHESIS (EMH) 94 5.5.1 Weak form EMH 95 5.5.2 Semi-strong form EMH 96 5.5.3 Strong-form EMH 96 5.5.4 Implications of the EMH 97 5.6. EXISTING MARKETS 98 IV

5.6.1 Bond and equity markets 98 5.6.2 Real estate markets 99 5.7. ALTERNATIVE INVESTMENT MARKETS 100 5.7.1 Defining alternative investment markets 100 5.7.2 Are alternative markets good markets? 100 5.7.2.1 Availability of information 101 5.7.2.2 Liquidity 101 5.7.2.3 Transaction costs 102 5.7.2.4 Informational efficiency 103 5.7.3 Are alternative markets efficient markets? 103 5.8. CONCLUSION 104 5.9. SUMMARY 104 CHAPTER 6: APPLICATION OF PORTFOLIO THEORY 106 6.1 INTRODUCTION 106 6.2 INCLUSION OF KRUGER RANDS 107 6.2.1 Introduction 107 6.2.2 Assets used in the construction of the portfolios 108 6.2.2.1 Debt instruments 108 6.2.2.1.1 Definition 108 6.2.2.1.2 Assumptions 108 6.2.2.2 Equity investments 109 6.2.2.2.1 Definition 109 6.2.2.2.2 Assumptions 109 6.2.2.3 Cash 110 6.2.2.4 Kruger Rands 110 6.2.2.4.1 Definition 110 6.2.2.4.2 Assumptions 111 6.2.3 Analysis of the assets used to construct the diversified portfolios 111 6.2.3.1 Performance analysis 111 6.2.3.2 Risk analysis 118 V

6.2.4 Portfolio construction 121 6.2.4.1 Co-movements between the assets 121 6.2.4.2 Initial diversified portfolio 124 6.2.4.3 The alternative portfolio 126 6.3 INCLUSION OF WINE ASSETS 128 6.3.1 Introduction 128 6.3.2 Assets used in the construction of the portfolios 130 6.3.3 Analysis of the assets used in the construction of portfolios 131 6.3.3.1 Performance analysis 131 6.3.3.2 Risk analysis 136 6.3.4 Portfolio construction 138 6.3.4.1 Construction of the diversified portfolio 139 6.3.4.2 Construction of the Mouton-Rothschild (MR) portfolio 142 6.3.4.3 Construction of the Montrose portfolio 146 6.4 INCLUSION OF ART INVESTMENTS 150 6.4.1 Introduction and assumptions 150 6.4.2 Assets used in the portfolio construction 151 6.4.3 Analysis of the assets used in the construction of the portfolios 152 6.4.3.1 Performance analysis 152 6.4.3.2 Risk analysis 159 6.4.4 Portfolio construction 160 6.4.4.1 Construction of the diversified portfolio 161 6.4.4.2 Construction of the art portfolio 164 6.5 SUMMARY 166 CHAPTER 7: CONCLUSION 167 7.1 FURTHER RESEARCH POSSIBILITIES 170 BIBLIOGRAPHY 171 VI

LIST OF FIGURES Figure 2.1: Probability distribution of assets with different levels of risk 7 Figure 2.2: Risk/Return combinations of two different assets 9 Figure 2.3: Fictitious Efficient Frontier 10 Figure 2.4: Efficient Frontier and Utility Curves 11 Figure 2.5: Combining the Risk-Free Asset and a Risky Portfolio 15 Figure 3.1: Probability Distribution 21 Figure 3.2: The Relationship between Risk and Return 33 Figure 3.3: Systematic risk and Unsystematic risk 40 Figure 3.4: Expected return of the portfolio 43 Figure 3.5: Standard deviation of the portfolio 43 Figure 3.6: Efficient frontiers given different correlations 44 Figure 6.1: Weekly All Bond Index (ALBI) movements 1990 2003 112 Figure 6.2: Weekly All Share Index (ALSI) movements 1990 2003 112 Figure 6.3: Weekly 90-Day BA-rate (BA) movements 1990 2003 113 Figure 6.4: Weekly price movement of the Kruger Rand 114 Figure 6.5: Weekly HPY for the ALBI (1990 2003) 115 Figure 6.6: Weekly HPY for the ALSI (1990 2003) 116 Figure 6.7: Weekly HPY for the BA (1990 2003) 116 Figure 6.8: Weekly HPY for the Kruger Rand (1990 2003) 117 Figure 6.9: Probability distributions of assets 119 Figure 6.10: Scatter plot matrix of asset returns 122 Figure 6.11: Efficient frontier 125 Figure 6.12: Asset mix of a diversified portfolio 125 Figure 6.13: Efficient frontier of the alternative portfolio 126 Figure 6.14: Asset mix of the alternative portfolio 127 Figure 6.15: Monthly price movement of the ALBI 131 Figure 6.16: Monthly price movements of the ALSI 132 Figure 6.17: Monthly rate movements of the 90-day BA-rate 133 VII

Figure 6.18: Monthly price movements of the Mouton-Rothschild 1982 133 Figure 6.19: Monthly price movements of the Montrose 134 Figure 6.20: Holding Period Yields on assets 135 Figure 6.21: Probability distributions of assets 137 Figure 6.22: Scatter plot matrix of asset returns 139 Figure 6.23: Efficient frontier for the diversified portfolio 141 Figure 6.24: Asset mix of the diversified portfolio 142 Figure 6.25: Scatter plot matrix of the returns on the assets used in the MR portfolio 143 Figure 6.26: Efficient frontier for the MR portfolio 144 Figure 6.27: Asset mix of the MR portfolio 145 Figure 6.28: Scatter plot matrix of the returns on the assets used in the MR portfolio 147 Figure 6.29: Efficient frontier for the MR portfolio 148 Figure 6.30: Asset mix of the Montrose portfolio 149 Figure 6.31: All Bond Index Movements 152 Figure 6.32: All Share Index Movements 153 Figure 6.33: Annual BA-rates 154 Figure 6.34: Annual movements of the FAI 155 Figure 6.35: HPY movements of the ALBI 156 Figure 6.36: HPY movements of the ALSI 156 Figure 6.37: HPY of the BA 157 Figure 6.38: HPY of the FAI 158 Figure 6.39: Probability distributions of assets 159 Figure 6.40: Scatter plot matrix of asset returns 161 Figure 6.41: Efficient frontier for the diversified portfolio 163 Figure 6.42: Asset mix of the diversified portfolio 163 Figure 6.43: Efficient frontier for the art portfolio 164 Figure 6.44: Asset mix of the art portfolio 165 VIII

LIST OF TABLES Table 3.1: Test Data 32 Table 3.2: Test Data 34 Table 3.3: Standard deviation for a given correlation 42 Table 4.1: Performance of the AFPI versus various other assets 54 Table 6.1: Table of comparative performance measures 118 Table 6.2: Variance and standard deviations of assets 120 Table 6.3: Covariance matrix 123 Table 6.4: Correlation coefficient matrix 124 Table 6.5: Portfolio risk and return figures 128 Table 6.6: Table of comparative performance measures 136 Table 6.7: Variance and standard deviation of assets 138 Table 6.8: Correlation coefficient matrix of asset returns 140 Table 6.9: Correlation matrix 144 Table 6.10: Portfolio statistics of the diversified portfolio and the MR portfolio 146 Table 6.11: Correlation matrix 148 Table 6.12: Portfolio statistics of the diversified portfolio, MR portfolio and Montrose portfolio 150 Table 6.13: Table of comparative performance measures 158 Table 6.14: Variance and standard deviation of assets 160 Table 6.15: Correlation coefficient matrix of asset returns 162 Table 6.16: Portfolio statistics of the diversified portfolio and the art portfolio 165 IX

CHAPTER 1: INTRODUCTION 1.1 INTRODUCTION AND PROBLEM STATEMENT Modern financial markets are highly developed and efficient, with market participants having access to seemingly endless amounts of information related to an ever-growing asset selection. Amidst this the basic building blocks for sound investment portfolio construction remain the same. The high levels of efficiency make the process of attaining above-average risk-adjusted returns more difficult than ever before. This should prompt investors to shift their focus to effective portfolio diversification. The following quotes emphasise the importance and some of the characteristics of diversification: Modern portfolio theory suggests that diversification is rational (Dobbins, Witt and Fielding 1996:12). Diversification means that many assets are held in the portfolio so that the exposure to any particular asset is limited. (Bodie, Kane and Marcus 2003:10). This portfolio that includes all risky assets is referred to as the market portfolio because the market portfolio contains all risky assets, it is a completely diversified portfolio. (Reilly and Brown 2000:244) given that investors should only take on that part of risk for which they expect to be rewarded. (Dobbins et al 1996:12) These statements form the basis of any successful investment strategy, be it for the individual investor or the institutional investor. Most advisors, journalists and the like will recommend a healthy balance between cash, equities and debt (bonds), many even recommend holding property in one s portfolio. 1

The central question being addressed in this study is whether or not cash, equities, bonds and property should be the only assets that investors consider, or are there other alternatives that may hold significant benefits for investors in the form of increased diversification? 1.2 RESEARCH OBJECTIVES The aim of this study is to investigate the potential benefits of including alternative assets (used as a synonym for hard assets throughout the study) in portfolios that are constructed in such a way that they are diversified. The study will start by providing an overview of modern portfolio theory and its most important elements, namely risk, return and diversification. Secondly, the study will identify and discuss various alternative assets that are believed to hold the potential for better diversification and move investors closer to attaining a true market portfolio. Lastly, the study will attempt to prove the hypothesis that these investments do hold the potential of improving the diversification of existing portfolios. Specifically the paper will consider three assets, namely gold coins (in the form of Kruger Rands), wine and art. Focusing on these three aspects will allow the study to verify some of the findings of previous studies, which have shown that the correlations between different alternative investments, such as art and furniture, vary substantially from positive to negative (with more traditional assets), allowing for diversification via the use of these assets. 2

1.3 BENEFITS OF THE RESEARCH The study will aim to emphasise the following: Any benefits to be gained from using these alternative investments in the construction of a diversified portfolio. The viability of the use of alternative investments. Secondary benefits to be obtained from investing in these assets, for example emotional dividends to be gained. The result of this study is believed to be important because: It will show the diversification effects that these assets might have on existing portfolios. It will investigate the risk and return pay-off structures of these assets, thus allowing the reader to consider these alternative investments as viable additions to his/her portfolio. If the initial hypotheses are proven to be true, the reader s investment universe may broaden substantially, allowing for better and more efficient portfolios. 1.4 STUDY METHODOLOGY Initially the study will concentrate on a literature study and overview of basic portfolio theory, after which it will progress into a more detailed study of the key concepts of portfolio theory namely risk, return and diversification. The study will then aim to identify and discuss various alternative investments. This section will give consideration to, among others, the basic constituency of each of the identified asset classes, key characteristics of each of the asset classes and basic determinants of value. This section will also focus on the 3

identification of universal factors which may affect the investability of alternative assets in general. The study will conclude with an empirical analysis, aimed at proving the inherent benefits that these assets hold for prospective investors. This section will give consideration to three prominent alternative assets, namely gold coins, wine and art. The study will aim to prove that portfolios, which are believed to be diversified, may be diversified even further via the inclusion of these assets in the portfolio. Consideration, be it direct or indirect, will be given to the effects of changing investment time frames and asset quality. 1.5 LIMITATION OF THE STUDY The limitations of this study, which will become clear as the study progresses include the following: Due to the fact that none of these alternative assets are traded on formal exchanges (even though progress has been made in at least the wine market), the availability of historical information is limited in various ways. In some instances performance figures had to be generated (based on reasonable assumptions) whilst in other instances (such as was the case for art investments) the extent of the study was limited by the frequency of data, as well as the range of data. Literature and research aimed specifically at addressing the diversification benefits of alternative assets on existing portfolios, is very limited. 4

CHAPTER 2: HISTORY AND DEVELOPMENT OF MODERN PORTFOLIO THEORY 2.1 INTRODUCTION The purpose of this chapter is to serve as an introductory chapter to the history and development of what may be loosely termed modern portfolio theory. This chapter will consider the origination and development of some of the concepts which form the basis of this study. The chapter starts with an overview of the Markowitz portfolio theory and progresses into a discussion regarding later developments such as the Capital Market Theory. 2.2 MARKOWITZ PORTFOLIO THEORY 2.2.1 Overview According to Correia, Flynn, Uliana and Wormald (2000:90) modern portfolio theory is based on the culmination of the work of various researchers, but was pioneered by Markowitz (1959) and later Sharpe (1964). Harry Markowitz developed the basic portfolio model at a time when investors were seeking a measure to quantify the risk variable. Bodie, Kane and Marcus (2002:223) state that, according to Markowitz, the variance of the rate of return [on an asset] is a meaningful measure of portfolio risk under a reasonable set of assumptions. 5

These assumptions, as stated by Reilly and Brown (2003:260), are listed below: Investors will evaluate an investment opportunity as being presented by a probability distribution of expected returns over some period of time; Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth; Risk of a specific investment will be measured by the deviation from expected return; Investors utility curves are functions of risk and return only; and For a given level of risk, investors will aim to maximize the level of return. Therefore under the above-mentioned set of assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return. 2.2.2 Risk The risk associated with an asset may be graphically illustrated via a probability distribution that illustrates the dispersion of the realized returns around the mean level of expected return. Greater levels of dispersion imply greater levels of risk associated with the specific asset. (D Ambrosio 1976:301) Figure 2.1 illustrates the probability distributions of the returns on two different assets. 6

Figure 2.1 Probability distribution of assets with different levels of risk Source: Mason, Lind and Marchal From figure 2.1 it is clear that curve A represents the probability distribution of a low variance (low risk) asset, whilst curve B represents the probability distribution of a high variance (high risk) asset. This conclusion follows on the fact that curve B is flatter (platykurtic) than curve A and therefore the returns on the high risk asset will be more dispersed around a central value, normally the mean rate of return or expected rate of return. A standardized measure of variance around a mean is known as standard deviation. According to Reilly et al (2003:212) standard deviation is an appropriate measure of risk because it: Is somewhat intuitive; Is correct and widely recognized; and Has been used in most theoretical asset pricing models. According to Wilcox (1999:12) Harry Markowitz derived the formula for the calculation of the standard deviation of a portfolio. The formula shown below indicates that the standard deviation of a portfolio is a function of the individual asset variances as well as their weight in the portfolio. 7

n n n w 2 i σ 2 i + i= 1 i= 1 i= 1 σ = w w Cov [2.1] port i j ij Where: σ port w σ Cov i 2 i ij = the standard deviation of the portfolio = the weights of the individual assets as determined by their of value in the portfolio = the variance in the rates of return in i = the covariance between the rates of return for assets i and j proportion What is important is the fact that the variance of the portfolio is also a function of the covariance/co-movement of the assets included in the portfolio. Therefore when constructing a portfolio the focus of the investor should not be on the variance of the individual asset but more on the average covariance with the other assets in an existing portfolio. 2.2.3 Return The expected return on a portfolio may be calculated as the sum of the weighted average expected return on the individual assets that make up the total portfolio. (Farrell 1997:21) A detailed discussion on the concepts of risk and return follows in chapter 3. 2.2.4 The efficient frontier D Ambrosio (1976:326) states that the portfolios on the efficient frontier dominate all portfolios below the frontier, thus the efficient frontier represents 8

that set of portfolios that has the maximum rate of return for every given level of risk, or the minimum risk for every level of return. An efficient frontier may be derived by considering different risk and return scenarios resulting from different combinations of two or more assets. Figure 2.2.4.1 illustrates a fictitious example of a graph that may be derived in this way. Figure 2.2 Risk/Return combinations of two different assets Source: Fischer and Jordan (1975:511) The envelope curve that represents the best of all the possible combinations represents the efficient frontier. Figure 2.2.4.2 represents a fictitious efficient frontier. 9

Figure 2.3 Fictitious Efficient Frontier Source: D Ambrosio (1976:326) The benefits associated with diversification lead investors/theorists to believe that the efficient frontier will be made up of investment portfolios rather than individual assets, except at the extremes where a portfolio consists of only one asset or security. Investors will invest in combinations of assets (portfolios) that fall along the efficient frontier based on their utility curve. The investor s utility curves are functions of expected return and expected variance (this follows on the basic assumptions of the Markowitz portfolio theory). These utility curves indicate the trade-off that investors are willing to make between risk and return. According to Stevenson and Jennings (1976:233) the optimal portfolio on the efficient frontier for a given investor, lies at the point of tangency between the efficient frontier and the curve with the highest possible utility, as illustrated in figure 2.4. 10

Figure 2.4 Efficient Frontier and Utility Curves Source: Correia et al (2000:100) According to Correia et al the following can be seen from figure 2.4: Three discrete indifference curves of investor X (X 1, X 2, X 3 ). The curves proceed in an upward and leftward direction, depicting the increasing return being sought for increased risk. The slope of the curve reflects the risk preference of the investor. Curve X 1 doesn t encounter any investment opportunity. Curve X 2 offers less utility but is the first to contact the set of feasible portfolios on the efficient frontier. Curve X 3 encounters the efficient frontier on two occasions, but because curve X 2 offers a higher level of utility the investor will prefer to invest in portfolio X. The three indifference curves of investor Y (Y 1,Y 2,Y 3 ) represent the utility curves of a more risk-averse investor (as opposed to investor X). This is evident in the increased steepness of the utility curves of investor Y. The steepness implies that a greater level of return is required per unit of risk incurred (Francis 1986:798). In other words this means that a risk-averse investor will require return to increase by a substantial margin for an increase in the level of risk, as opposed to a 11

less risk-averse investor who might only require a small increase in the level of return given an increase in the level of risk. According to Bodie, Kane and Marcus (2002:157) investors assign a welfare or utility score to competing investment portfolios based on the expected return and risk of those portfolios. Portfolios with attractive risk-return characteristics are assigned higher utility values. The utility value of a portfolio will increase if expected return increased and decrease if expected variability (risk) increased. 2.3 CAPITAL MARKET THEORY 2.3.1 Introduction Following the development of the Markowitz portfolio theory, various studies have been performed to address some of the shortfalls identified in the theory. One of the major shortfalls associated with this theory is the need for large amounts of information. Specifically, the investor needs information on the risk and return characteristics of each asset considered as well as information relating to the covariance of each asset pair. The inclusion of a risk-free asset into a portfolio also gave rise to some of the major developments which followed the Markowitz portfolio theory. A breakthrough with regard to the problems posed by the Markowitz portfolio theory came in 1963 when Sharpe developed the Market (or single-index, or diagonal) model. 12

2.3.2 Overview of the capital market theory 2.3.2.1 Assumptions of the theory As was the case for the Markowitz portfolio theory, the Capital Market Theory (CMT) was developed on the basis of a set of assumptions. These assumptions as listed by Dobbins et al (1993:45) are reproduced below: All investors are Markowitz efficient investors who want to target points on the efficient frontier based on their utility curves. This assumption underlines the fact that the Capital Market Theory (CMT) builds on the Markowitz theory. Investors are able to borrow or lend any amount of money at the riskfree rate of return (RFR). All investors have homogeneous expectations. This implies that all investors have similar expectations of the probability distributions of future rates of return. All investors share the same investment time horizon i.e. 1 month, 6 months, 12 months etc. All investments in the market are infinitely divisible i.e. investors are able to buy and sell fractions of assets. There are no taxes or transaction costs involved in the purchase and sale of assets. There is no inflation or any change in interest rates, or inflation is fully anticipated. Capital markets are in equilibrium. The aim of the preceding section was not to question the reasonableness of the listed assumptions, but to provide an overview of the theory. 13

2.3.2.2 Addressing the information need As was mentioned in the introduction to this section, one of the shortfalls of the Markowitz portfolio theory identified by theorists is the need for large amounts of information. According to Sharpe, who developed the CMT, each security s price movement can be related to the price of the market portfolio that is, a portfolio comprising a weighted average of all the securities traded on the market. (Dobbins et al (1993:46). Furthermore returns on different securities within an asset universe are assumed to be related only through common dependence upon the market [portfolio], and as such the necessity to specify the covariance (and correlation coefficients) of returns between security pairs is eliminated. 2.3.2.3 Inclusion of a risk-free asset The concept of a risk-free asset has various implications on the basic Markowitz portfolio model and the inclusion of these types of asset have allowed for various developments in asset pricing. In order to fully appreciate the implication of the inclusion of a risk-free asset on the basic portfolio theory it is necessary to define the term. Francis (1986:760) defines a riskless asset (or risk-free asset) as an asset that produces a positive level of return, but it has zero variability and therefore future returns are known. This implies a standard deviation (Markowitz s measure of risk) that is equal to zero. It may also be proved that the covariance of these assets with any other asset or portfolio of assets will be equal to zero. 14

The effects of the inclusion of a risk-free asset into an existing portfolio are listed below: The rate of return generated by the portfolio, which now includes a riskfree asset, remains the weighted average of the returns generated by the assets included in the portfolio. According to Francis (1986:760) the standard deviation of a portfolio that combines a risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio. Because of the above-mentioned effects the graph that may be drawn to illustrate possible risk and return characteristics of the portfolio is a straight line between two assets, as illustrated in figure 2.5. Figure 2.5 Combining the Risk-Free Asset and a Risky Portfolio Source: Stevenson and Jennings (1976:257) The investors will be able to attain any point along line RFR-A and RFR-B by investing a portion of their wealth in the risk-free asset and a portion in the risky-portfolio on the efficient frontier (i.e. portfolio A or B). Any combination on these lines dominates all risky portfolios on the efficient frontier that fall below this point, the reason being that these portfolios have equal variance 15

and higher rates of return than any portfolio on the original efficient frontier that falls below the selected portfolio. From this it appears that all portfolios on line RFR-B will dominate all portfolios on line RFR-A. At the point of tangency between the line drawn from RFR to the efficient frontier, that portfolio (portfolio M) dominates all portfolios below that point. Because portfolio M lies at the point of tangency with the efficient frontier, it has the highest portfolio possibility line and therefore all investors will aim to invest in portfolios that lie on line RFR-CM. This portfolio represents a portfolio which includes all risky assets. This portfolio is referred to as the market portfolio and represents a completely diversified portfolio. According to Reilly et al (2003:244) this portfolio includes not only common stocks but also all risky assets such as bonds, options, real estate, coins, stamps, art or antiques. In effect line RFR-CM which may also be termed the Capital Market Line (CML), becomes the new efficient frontier, as all investors will want to target points on the CML. From the above discussions it follows that the relevant risk measure for all risky assets should be their covariance with the market portfolio, this is the risk often referred to as systematic risk. The concepts of risk (systematic and unsystematic) as well as diversification, will be discussed in a later chapter. 2.4 THE SIGNIFICANCE OF PORTFOLIO THEORY FOR THIS STUDY The general principles of portfolio theory as reviewed in this chapter will have a significant bearing on the remainder of the study. As will be seen in later chapters, portfolios constructed during the course of this study will be constructed based on the Markowitz portfolio theory. It should also become evident that the focus of this text will be to study those alternative assets which, at least in theory, form part of the fully diversified market portfolio. Often these assets are forgotten or ignored, but the question 16

arises whether or not these assets may add to the diversification of existing portfolios. The benefits that may be associated with these assets may include, in addition to diversification benefits, improved levels of portfolio return. 2.5 CONCLUSION This chapter provided a literature review of basic modern portfolio theory. From this review it followed that the Markowitz Portfolio Theory, as developed by Harry Markowitz, established the basis for modern portfolio theory, whilst subsequent developments in this field of study included amongst others the Capital Market Theory (CMT). Markowitz showed that the two most important factors to be considered when constructing a portfolio is risk and return. Prior to his findings most investors constructed portfolios based purely on expected return, or at best constructed portfolios based on separate considerations of risk and return. Markowitz provided a measure that enables investors to consider risk and return simultaneously. In addition to this Markowitz identified variance as the appropriate risk measure in a portfolio management and construction setting. Application of the Markowitz Portfolio Theory enable investors to derive Markowitz efficient frontiers which, in addition to their utility curves, would allow them to determine optimal portfolios given their unique risk preferences. Following the discussion on the Markowitz Portfolio Theory this chapter briefly considered the Capital Market Theory (CMT) which introduces the concept of the risk-free asset. In terms of this theory the investor has to decide to invest in a combination of the risky portfolio, i.e. that portfolio that contains all the risky assets traded in the market, and the risk-free asset. According to the CMT the only other decision that the investor has to make, other than deciding to invest, is the whether or not he or she would be willing to borrow 17

or lend at the risk-free rate. This set of decisions, namely the investing decision and the financing decision, gives rise to the separation theorem. For the purpose of this study the Markowitz Portfolio Theory was used in the construction and identification of optimal portfolios. This was done because only the market portfolio (risky portfolio) as identified by the CMT can be diversified. 18

CHAPTER 3: RISK DIVERSIFICATION: BASIC DEFINITIONS AND APPLICATIONS 3.1. INTRODUCTION In the modern investment setting investors are faced with the complicated task of selecting good investments, in addition to this they have to consider trade-offs between risk and return and finally they are required to combine various types of investments in optimal portfolios. This chapter builds on the basic concepts noted in the previous chapter and aims to introduce the reader to a more complicated and detailed discussion of concepts such as risk and return when selecting investments, the consideration of risk and return measurement for individual investments, as well as the impact that the inclusion of assets will have on the risk and return characteristics of existing asset portfolios. Furthermore, the chapter will serve as an overview of the concept of risk diversification and the bearing that diversification has on the investment decision. Diversification as a concept will be discussed as well as the methods used to measure and achieve diversification. 3.2. RISK 3.2.1 Overview Risk manifests itself in various forms, these include amongst others, business risk, country risk, exchange rate risk and financial risk. For the purpose of this chapter, risk (in its broadest sense) may be defined as the uncertainty of future returns, or alternatively as the uncertainty that an investment will earn its expected return. (Reilly et al 2000:1210). According to Mason et al (1997:99) the larger the range of expected returns of an asset, the larger the 19

dispersion of the returns and the riskier the asset. Therefore the higher the range of the expected returns (or historical returns) the riskier the asset will be. Bodie, Kane and Marcus 2002:155 reason that the presence of risk means that more than one outcome is possible. The risk associated with an asset consists of two elements, namely: Systematic risk. According to Francis and Archer (1979:155) systematic risk is the minimum level of risk that may be achieved by means of diversification. All assets carry this risk; as all assets are influenced to some greater or lesser extent by changes in factors such as money supply, interest rates, exchange rates and taxation (Dobbins et al 1996:8). Unsystematic risk. Cohen, Zinbarg and Zeikel define unsystematic risk as risk that is unique to a specific asset, derived from its particular characteristics. It can be eliminated in a diversified portfolio. The existence of the unique risks and the idea that investors should not expect to be rewarded for taking on risk which can be avoided (Dobbins et al 1996:8) warrants an investigation into the existence of alternative means of reducing unique risks. 3.2.2 Types of risk It is important to note that the term risk is a collective term encompassing various types of risks, including, amongst others, the following: Business risk; Financial risk; Market risk; Interest rate risk; Reinvestment rate risk; 20

Purchasing power risk; Exchange rate risk; and Country risk. 3.2.3 Risk measurement The most established measure of risk is standard deviation. Standard deviation may be defined as a measure of variability equal to the square root of variance (Mason, Lind and Marchal 1996:11). It is a measure of dispersion around a mean value. A larger dispersion around a mean value would indicate more variability and presents the investor with greater risk. Figure 3.1 Probability Distribution Source: Dobbins et al (1996:6). As mentioned earlier standard deviation measures dispersion around a mean. According to Dobbins et al. (1996:6) approximately 66.67% of all occurrences should, on average, lie within one standard deviation of the expected outcome, considering the figure 3.1 this implies that 66.67% of the 21

occurrences should fall within the 10% to 22% range and the implied standard deviation should be 6%. The aforementioned statement is made assuming that the return distribution is a normal distribution. Dobbins et al (1994:7) go on to say that 83.33% of occurrences (relating to the above mentioned figure) should fall above 22% (upside potential) and below 10% (downside risk). Variance may be calculated as follows: [ r( s R] σ = [3.1] n 2 ) s 2 Where : σ 2 = variance R = the mean rate of return on the asset r(s) = the realized return given scenario s From this formula (3.1) it follows that variance of an asset equals the sum of the probability of a given scenario multiplied by the squared difference between realised return and expected return (the mean value). It should be noted that this measurement of risk is based on the past performance of the asset, hence there is no guarantee that the risk characteristics of the asset will remain constant. To eliminate this shortfall as much as possible, investors should use as much historical data as possible. The mathematical equation for the calculation of standard deviation may be represented as follows: 2 σ = σ [3.2] 2 Where: σ = Variance as calculated using formulas 3.1. 22

According to Bodie, Kane and Marcus (2002:165) the risk of single assets in a portfolio should be measured in the context of the effect that their returns will have on the variability of the overall portfolio. The fact that the amount of risk associated with a portfolio is dependent on the extent to which the assets in the portfolio move together, results in the assumption that the risk of a portfolio isn t simply the weighted average risk of the assets that make up the portfolio. Bodie et al (2002:165) further state covariance measures how much the returns of two risky assets move in tandem. The following equation is used to calculate the covariance of two assets: Cov = [ r E( r )][ r E( r )] n [3.3] AB A A B B / Where :P(s) = the probability of scenario s r X E(r = the realized return on asset X X ) = the expected return on asset X From this it seems that the covariance of a portfolio is reliant on two factors, namely: Variability (standard deviation) of the individual assets, and The relationship (correlation) amongst different assets included in the portfolio. The correlation coefficient of returns is calculated by using the following formula: Cov ρ = AB AB σ Aσ [3.4] B Where : ρ = AB the correlation coefficient of assets A and B 23

The portfolio standard deviation, for a three-asset portfolio, may be calculated by using the following formula: σ 2 port = W σ + W σ + W σ + 2W W Cov + 2W W Cov + 2W W 2 A 2 A 2 B 2 B 2 C 2 C A B AB A C AC B C [3.5] Cov BC σ 2 σ = [3.6] port Where : W σ X 2 X Cov = the weight that each asset carries in the portfolio = the variance of asset X XY = the covariance between the returns on assets X and Y The formulae indicate that factors such as the weight that an asset carries in the portfolio, its standard deviation (risk), as well as the correlation/comovement of the asset with other assets in the portfolio, is essential in the calculation of portfolio variance. This study uses the risk measurement as identified by Harry Markowitz in his portfolio theory. This does not, however, mean that other measures of risk such as beta-coefficients aren t considered to be applicable or sufficient measures of risk. 3.3. RETURN 3.3.1 Overview Fischer and Jordan (1983:4) define an investment as the current commitment of Dollars for a period of time in order to derive future payments that will compensate the investor for (1) the time the funds are committed, (2) the expected rate of inflation, and (3) the uncertainty of future payments. 24

The return generated by an asset may be defined as the sum of all sources of income or capital gains realized on that asset during the holding period. From the definition mentioned above it follows that these income sources should compensate the investor in terms of time, inflation and risk. The rate of return demanded by an individual investor is known as the investor s required rate of return. 3.3.2 Required rate of return The investor s required rate of return is made up of three factors or alternatively it has three dimensions. According to Reilly and Brown (2000:16) these are: The time value of money during the period of investment, The expected rate of inflation during the period, and The risk involved. In order to analyse the investor s required rate of return, the investor has to take into account the real risk-free rate (RRFR). According to Reilly et al (2000:16) The real risk-free rate is the basic interest rate, assuming no inflation and no uncertainty about future [cash] flows. The RRFR is essentially the rate of return that investors would demand if they knew with certainty what cash flows they would receive and when. From this it appears that the RRFR addresses the time value of money dimension of the investor s required rate of return. Reilly and Brown (2000:17) identify two factors that will influence this rate of return namely, the time frame of the investor and the investment opportunities in the economy. Longer time frames or investment horizons results in the greater probability of opportunity cost and therefore a higher level of return will be commanded by investors. 25

Farrell (1997:126) states that [nominal] rates of interest that prevail in the market are determined by real rates of interest, plus factors that will affect the nominal rate of interest, such as the expected rate of inflation. From this it appears that the nominal risk-free rate of return is equal to the RRFR adjusted for conditions in the capital markets and the inflation rate. A required rate of return that exceeds the NRFR is said to have a risk premium. This risk premium represents the composite of all uncertainty, but certain fundamental elements are identifiable namely business risk, financial risk, liquidity risk etc. (as was discussed earlier). According to Dobbins et al. (1996:7) investors do not like risk, and the greater the riskiness of returns of an investment, the greater will be the return expected (or required) by investors. The required or expected rate of return of an investor may be calculated using the following equation: k i = r + β r r ) [3.7] f ( m f Where : k i = required rate of return on asset i r f = the nominal risk-free rate of return r m = the market return β = the beta of the asset Beta is a standardized measure of the extent to which an asset moves in relation to the market. Beta measures the covariance between the asset returns and the market returns in relation to the variance of the market. Therefore the market portfolio (the portfolio that consists of all risky assets) has a beta coefficient of 1. An asset with a beta of more than one has high levels of systematic risk, whilst assets with low or negative betas have low levels of systematic risk (Francis 1986:260). 26

3.3.3 Measurement According to Bodie, Kane and Marcus (2002:162) the mean or expected return of an asset is a probability-weighted average of its return in all scenarios. The equation may be written as follows: R( s) R = n [3.8] Where : R(s) = return in scenario s. Bodie et al (2002:163) state that [t]he rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio. Thus, when calculating the return generated by a portfolio of investments over a period of time, the investor uses the weighted average return of all assets in the portfolio. For example a portfolio made up of asset A and asset B (assume equal weighting), where asset A yielded a return of 40% and asset B a return of 10%, will yield a 25% return over the investment period. The equation for the calculation of the portfolio return will be as follows: R port = W r + W r +... + W r [3.9] A A B B Where : A = portfolio A B = portfolio B 3.4. THE RELATIONSHIP BETWEEN RISK AND RETURN According to Corgel, Ling and Smith (2001:149) investors are risk averse and because of this the relationship between risk (as measured by standard deviation) and return is positive. Dobbins et al. (1996:9) state that [t]he capital asset pricing model shows that the expected return of an investment is 27

a positive linear function of market risk (measured by beta). Basic investment theory states that as risk increases so too should return, in order to compensate the investor for the increased degree of uncertainty. The risk/return relationship is of critical importance to investors. Investors need to determine their individual risk preferences and should aim to minimize their risk exposure at their required rate of return. According to Bodie, Kane and Marcus (2002:157) investors who are risk averse reject investment portfolios that are fair games or worse. For the purposes of this text fair games may be defined as prospect[s] that have a zero risk premium. From this it seems that a risk averse investor will only accept those prospects which offer a positive risk premium. The Markowitz portfolio theory assumes that investors have utility curves, which are functions of risk and return. This needs to be explained. According to Bodie et al (2002:157) [m]any particular scoring systems are legitimate. One reasonable function that is commonly employed by financial theorists and the Chartered Financial Analyst Institute assigns a portfolio with expected return E(r) and variance 2 σ the following utility score: U 2 = E r ) 0.005A [3.10] ( σ Where: U = utility score A = an index of the investor s risk aversion The factor 0.005 is a scaling convention, allowing the use of absolute values. From the equation it is apparent that the utility score (U) is a function of expected return and variance. Furthermore it may be said that the effect that variance has on the utility score is dependent on the value of A. Larger values of A indicate a greater degree of risk aversion. The utility score derived in this fashion should be compared to the rate of return offered by risk-free investments. 28

The following should apply: If U minus rate of return offered by the risk-free investment > 0 the investor will opt for the risky portfolio. If U minus rate of return offered by the risk-free investment = 0 the investor should be indifferent towards the two options but by virtue of the definition of a risk averse investor, the investor should opt for the risk-free alternative. If U minus rate of return offered by the risk-free investment < 0 the investor will opt for the risk-free alternative. 3.5. PORTFOLIO CONSTRUCTION 3.5.1 Introduction It is widely accepted that investors should aim to maximize the level of return for a given level of risk. Alternatively they aim at minimizing the risk for a given level of return. This is done by constructing a portfolio of assets which as a whole is subject to the investor s risk appetite. According to Reilly and Brown (2000:259) an investor s portfolio includes all of his or her assets and liabilities, not only stocks or marketable securities, but also items such as houses, cars, antiques etc. As was noted earlier Corgel et al (2001:149) believe that all investors are risk averse and therefore the risk-return relationship is positive. It holds true that if investors were faced with a choice between two assets that promise the same level of return they would opt for the asset with the lower level of risk. In order to accept a higher level of risk, investors will demand a higher level of potential return to compensate them for the higher degree of uncertainty. When constructing diversified investment portfolios investors should consider different investments based on their risk and return pay-off structures. 29

Investors should identify their specific investment goals and constraints when determining their risk appetite. 3.5.2 Constructing the capital market line (CML) According to Bodie et al (2002:183) the make-up of any portfolio is subject to decisions being made on one of three levels namely: The capital allocation decision which refers to the choice investors need to make between investing in a risk-free asset and a risky asset portfolio. The asset allocation decision which describes the distribution of risky investments across different asset classes. (Construction of the optimal risky asset portfolio). Security selection decision which describes the choice of particular securities held within each asset class. This section is concerned with the first of these decisions. For illustration purposes certain assumptions need to be made. It should be kept in mind that the aim of this section is to find the balance between investing in a risk-free asset and a risky portfolio. The assumptions are as follows: The risky portfolio s composition remains constant throughout the exercise. The risky portfolio is made up of two assets, namely two mutual funds, one of which is invested in equities and one which is invested in bonds. The allocation between the two assets in the risky portfolio will be 60% invested in equities and 40% invested in bonds. 30