A two-factor style-based model and risk-adjusted returns on the JSE. A Research Report presented to

Transcription

1 A two-factor style-based model and risk-adjusted returns on the JSE A Research Report presented to The Graduate School of Business University of Cape Town In partial fulfilment of the requirements for the Masters of Business Administration Degree by Xavier Ducasse December 28 Supervisor: Dr Francois Toerien

2 Abstract: A two-factor style-based model and risk-adjusted returns on the JSE In 1992 Fama and French documented an empirical challenge to finance theory in general and the CAPM in particular. Their three-factor model used size and book-to-market value in addition to systematic risk as factors to estimate excess returns. On the JSE, Van Rensburg and Robertson (23), followed by Auret and Sinclaire (26) have documented the existence of a two-factor style-based model where size and price-to-earnings ratio can be used to predict share returns. This study tests this model and analyses cross-sections riskadjusted returns. Returns including dividends, total risk and systematic risk are analysed for JSE-listed shares ranked on a monthly basis for the period Systematic risk is estimated using a simulation methodology aimed at testing the diversification properties of the different cross-sections. This report highlights the dangers of considering average returns in isolation of a well identified risk and concludes that if anything, larger market capitalisation JSE-listed shares earn higher risk-adjusted returns over the period under investigation. A secondary observation documents the importance of the choice of the return horizon in carrying out empirical research on cross-sections. ii

3 GLOSSARY OF TERMS: APT Arbitrage Pricing Theory BTM Book-to-Market value CAPM Capital Asset Pricing Model HML - High Minus Low (book-to-market value) I-NET Intelligent Network Portfolio of all shares in the Population JSE Johannesburg Securities Exchange P/E Price-to-earnings ratio SMB - Small Minus Big (market capitalisation) iii

4 Table of Contents Introduction Problem definition Area of study and definitions Literature review Returns Portfolio diversification and systematic risk Methodology Data coverage Data cleaning Ranking Systematic risk and portfolio construction Share returns Analysis method Result analysis Returns Risk-adjusted returns Total risk or average risk of holding one random share Sharpe s index Total risk Diversified risk Sharpe s index Diversified risk Influence of horizon Returns Risk Limitations of the study... 1 iv

5 Conclusion... 2 Bibliography... 3 Appendix 1 Share codes... 7 Appendix 2 Return distributions... 8 Appendix 3 Risk distributions... 9 Appendix 4 Distribution Sharpe s index total risk... 6 Appendix - Ratio of portfolio risk to the average risk of holding one random share Appendix 6 Risk free rate (source Inet-Bridge) Appendix 7 Distribution of Sharpe s index, diversified risk Appendix 8 Influence of the horizons on risk v

6 Introduction Various studies carried out internationally but also on the Johannesburg Securities Exchange (JSE) have highlighted the existence of style-based effects where firm specific factors can be used to predict returns. Most of these recent studies observed that the Capital Asset Pricing Model (CAPM) framework failed to explain returns, and that additional factors such as size, book-to-market ratio or price-to-earnings ratio (P/E) can be used to explain the shares behaviour. In South Africa, some recent studies (Graham and Uliana, 21; Van Rensburg, 21; Van Rensburg and Robertson, 23; Auret and Sinclaire, 26) have stressed that size and P/E could be used to model share average returns. They indicated that low P/E, small market capitalisation shares earned higher average returns. The primary aim of this study is to verify whether a two-factor characteristic-based model applies when returns are adjusted for risk and to understand the cross-sectional risk/reward characteristics. After defining the problem in more detail and reviewing the literature to date, cross-sectional returns will be tested over a three year period from 1/2 to 12/27. Firstly, average returns will be compared to confirm the two-factor model established by Van Rensburg and Robertson in 23. Secondly, the study will adjust these returns for total risk and the carry out the same comparisons. Thirdly, an estimate of systematic risk will be drawn from the diversification properties of each group to understand cross-sectional returns adjusted for systematic risk. Finally, because P/E and to a lesser extent market capitalisation are dynamic variables, the study will investigate the impact of the choice of the return horizon on the results. 1. Problem definition While anomalies to the one-factor CAPM framework had been explored for a long time, the article of Fama and French in 1992 was considered as an important milestone in the understanding of market behaviour. Their study of the US stock markets concluded that two additional variables could be used as factors to explain excess returns. Their research, similar to that previously carried by Banz (1981) and Basu (1983), clearly highlighted the deficiency

7 of using β as a single factor to explain returns. They identified size and book-to-market value as additional and independent relevant variables. In South Africa, similar studies were carried out on the JSE. The most significant one is probably the study by Van Rensburg and Robertson published in 23. Their analysis on the JSE reached a slightly different conclusion to what had empirically been observed in the US. They identified that size and the price-to-earnings ratio (P/E) had the most power in explaining excess returns on the South African stock market. One month later, they published a second paper where, using a methodology similar to the one used by Fama and French in 1992, they confirmed that small P/E and small size outperformed the market. Furthermore, they documented an inverse relationship between β and returns on the JSE. β is a measure of systematic risk used as the single factor within the CAPM framework to explain excess returns. Thus, the conclusion of Robertson and Van Rensburg highlighted risk vs. return anomalies on the JSE which suggested that either higher returns could be achieved while not bearing more risk, or that β was a poor measure of systematic risk. The objectives of this study stem from these last observations. This study will examine and compare risk-adjusted returns across four cross-sections of the JSE. These cross-sections will be formed of shares according to their relative P/E and market capitalisation values. Thus the four groups will entail low P/E and small market capitalisation shares, low P/E and large market capitalisation shares, high P/E and small market capitalisation shares, and finally high P/E and large market capitalisation shares. The study will first look at the average returns for each of the groups and test whether small size and small P/E shares earn higher returns as per the findings of Van Rensburg and Robertson. Hypothesis n 1: Null hypothesis: P/E and market capitalisation do not explain average returns. Alternative hypothesis: A two-factor model exists and low P/E, small market capitalisation shares earn higher returns on average. In their study, Van Rensburg and Robertson looked at CAPM adjusted returns. However, if β gives a poor estimate of risk, such adjustment can be misleading. Thus, this study will adjust returns to the observed risk rather than its theoretical value. Firstly, each group s return per 2

8 unit of total risk will be compared; total risk being represented as the average risk of holding one random share for each group. Hypothesis n 2: Null hypothesis: Returns adjusted for total risk are similar across the four cross-sections of the JSE. Alternative hypothesis: Some cross-sections earn higher returns per unit of total risk than others. In finance theory, investors are only rewarded for bearing systematic risk. This study will therefore try to look at returns adjusted for systematic risk. Systematic risk will be estimated using the diversification properties for each group. A large number of random portfolios of increasing sizes will be simulated to understand the nature of and ultimately remove the idiosyncratic risk for each group. Systematic risk-adjusted excess returns will be compared for the four sections in order to understand whether the style-based returns properties of the JSE observed by Van Rensburg and Robertson hold. Hypothesis n 3: Null hypothesis: Returns, adjusted for systematic risk (or diversified risk in this case), are the same across the groups. Alternative hypothesis: Some groups earn higher returns per unit of systematic risk than others. Finally, because P/E as a variable is subject to change over time, the influence of the return horizon will be analysed. Intuitively, as a low P/E share outperforms the market, its P/E is likely to increase. Earnings are an accounting based measure, adjusted on a yearly basis while price constantly fluctuates. Thus, in the case of a two-factor model, where P/E is one of them, the return horizon used to carry the research is likely to have some significance and can potentially constitute a bias. For this reason, the study will systematically check the previous three hypotheses for three, six and twelve month returns. Moreover, the last section of the document will statistically compare the differences observed across the 3 horizons. 3

9 Hypothesis n 4: Null hypothesis: The choice of the return horizon does not influence the risk/reward properties of the 4 groups. Alternative hypothesis: Time horizon is a significant parameter and could introduce a bias. 2. Area of study and definitions Return Return is defined as follow: Where: = return on the share s over the period (t, t+1) expressed as percentage of the price of the share s at the time t. is the dividend paid to the share s over the period (t,t+1) is the market price of the share s at the time t is the market price of the share s at the time t+1 Risk In finance, the risk of a given asset is generally characterised by the standard deviation of its returns around a mean or expected value. Such definition of risk implies that any variation around a mean or expected value is treated equally. In other words, excess returns, despite being good news for the investor, are considered in the same way as losses. The mathematical measure of volatility is the standard deviation 1 or σ, defined as the square root of the sum of the square distances to an expected value divided by the number of intervals: 1 A large enough sample of returns is necessary to assume that their distribution is normal. 4

10 Where: is the number of returns over the period. is the return for = i. is the average of the returns over the period. From a more conceptual point of view, risk is defined by the likelihood of receiving an unanticipated return (Damodaran, 22, p.6). If returns could be fully predicted, then the notion of risk would be irrelevant. This concept is not shared by all. Duxburry and Summers, in 24, concluded that risk is better defined by a loss aversion than a variance aversion. Markowitz, in his early studies, looked at only calculating the downside risk by the use of semi-variance. However, despite being conceptually simple, such approach proved extremely complex from a computational view point. Representing risk by the variance or standard deviation around an expected value also assumed that returns are normally distributed. Some distributions can for example include an element of skewness with a higher probability to get large gains than large losses (Lofthouse, 27, p.31). Harvey and Sidique (2) have, for example, shown that skew is relevant in portfolio selection and that risk should adjusted for such phenomenon. Traditionally, in finance theory 2, the risk associated with an asset or a company can be of two kinds. The first one is the risk affecting the market as a whole. For example, a rise in interest rates would affect the whole economy and would impact all firms using debt as a means of finance. It is to be noted that if a large number of firms would be impacted, the impact would differ from one firm to another depending on their level of leverage. The second part of the risk is specific to the company or to a small group of companies. A company could for 2 Specifically CAPM and APT. Even though the first one links systematic risk to the market portfolio, while the latter attributes it to macro-economic factors, the discussion will be carried out in general terms.

11 instance decide to make an investment based on a flawed sales forecast, leading to value destruction. Such a company specific risk is also called unsystematic or diversifiable. Diversification principle The principle of diversification tells us that the unsystematic risk of an asset can be eliminated by holding a large portfolio. The reason for that phenomenon is two-fold. Firstly, the percentage variation associated with one asset within a diversified portfolio is much smaller than it would be if only that particular asset was held. Secondly, positive and negative asset specific variations will average out to zero in a large portfolio as these effects will tend to cancel each other out (Firer et al, 24, p.399). The latter point can be illustrated statistically. The risk of a portfolio is partially defined by the correlation between the different assets constituting the portfolio (Markowitz, 192, pp.77-91) and takes the expression: Where: is the weight of asset i is the weight of asset j is the standard deviation of asset i is the standard deviation of asset j is the correlation coefficient of assets i and j The correlation coefficient of assets i and j varies between one and minus one. It will be one when the assets i and j are perfectly correlated, which means that they move simultaneously in the same direction. Alternatively it will be minus one when the assets i and j move simultaneously in opposite directions. It may be seen from the formula that the lower the correlation is between assets i and j, the lower the portfolio risk will be. 6

12 Intuitively, the diversification properties of a portfolio are associated with its size, on the basis that the larger the portfolio is, the less the variations of a particular asset will affect the portfolio as a whole. Building on Markowitz s work on portfolio theory, Elton and Gruber (1977) mathematically characterised the relationship between portfolio size and risk. They confirmed that risk goes down as size increases and also found that risk goes down at a slower rate as more securities are added to the portfolio. The relationship between size and risk is of crucial importance to an investor operating in a market with transaction costs: as size goes up, risk goes down at a slower rate and transaction costs increase. Risk and Return Markowitz (192) was the first to identify and make explicit the trade-off between risk and return. Following on his work, Sharpe (1964), Lintner (196), Mossin (1966) and Black (1972) developed the CAPM theory. Their model defines assets returns as a linear function of the systematic risk an investor is willing to bear. Where: is the expected return of capital asset i. is the risk free rate, defined as the interest rate that can be earn from a security without default risk. is the expected return of the market. (beta) defines the systematic risk of an asset compared to the market portfolio, or also:, also noted β. This model relies on important assumptions: - Investors have the same expectations and are risk-averse - Investors have access to the risk-free rate - There are no transaction costs - Markets are in equilibrium 7

13 Empirical studies have also confirmed the positive relationship between risk and return. The 21 international study of Dimson, Marsh and Staunton confirmed that investors were rewarded over time for bearing risk. In 22 Firer and Staunton showed in their study of the South African financial market history, that such findings applied to South Africa. Despite these results, a large number of studies have highlighted the imprecise nature of the CAPM to predict future returns. First, from a fundamental point of view, the single-factor CAPM indicates that systematic risk is only influenced by the market portfolio. However, a share s systematic risk is more likely to come from different macro-economic or industry specific factors. This intuitive notion is at the origin of the Arbitrage Pricing Theory (APT) model proposed by Ross in The APT model, which also relies on market equilibrium, defines returns as a linear function of different systematic factors. Such factors, varying in number and type, are dependent on the asset under consideration and can only be determined empirically. This stresses the main problem in applying the APT to predict future returns. With risk factors being unspecified, Lofthouse indicates that APT has not set the fundmanagement industry alight (28, p.7). To date, one of the most influential researches has been the one conducted by Fama and French in 1992 which resulted in the famous three factors model. The three factor model adds to an analogous beta, the book-to-market value and market capitalisation as explanatory variables of future returns. Roll argued in 1977 that the CAPM could not be empirically tested for practical reasons as the market portfolio is unobservable. Roll s view is that the stock index is only a proxy of the market portfolio and that this latter includes works of art and real estate for example. Studies on factor-based returns only bring empirical evidence and are not yet backed up by any theory. For this reason, the CAPM is still, despite its unrealistic assumptions and repeated failure to explain reality, one of the pillars of finance theory. 8

14 3. Literature review 3.1. Returns The one-factor CAPM is still considered to be an important part of finance theory and is still widely used to calculate companies cost of capital, for example. The principle that an asset s future returns is a linear function of its covariance with the market portfolio has been empirically challenged. First Nicholson, in 196, observed that firms with low price-toearnings ratios generated abnormally higher returns. This was confirmed by Basu in 1977, who observed that returns for shares with high price-earnings ratio were higher than predicted by the CAPM. Size was also empirically identified by researchers as a potential factor to explain higher returns. Banz (1981) noticed on the NYSE that small capitalisation firms also showed greater returns than predicted by the CAPM and introduced the idea of a potential size effect. In 198, Stattman found a correlation between average returns on the US stock market and book-to-market value. These results were later confirmed by Rosenberg, Reid and Lanstein (198). Internationally, Chan, Hamao and Lakonishok, in 1991, reached similar conclusions on the Japanese market. If the results of these studies indicated overall that returns could potentially be predicted using style-based factors, the empirical nature of the methods limited their ability to be fully conclusive. Researchers could not find a consensus. Some argued that the P/E effect subsumed the size effect (Basu, 1983) while others thought they were independent (Jaffe et al., 1989). To date the most influential empirical challenge to the CAPM has come from the work of Fama and French (1992, 1993). Their three-factor model included firm book-tomarket equity and size as explanatory variables of future returns. Expanding on the quintile methodology developed by Jaffe et al., their empirical research allowed then to formulate the following return model: 9

15 Where: is the portfolio s return; is the risk-free asset return; is analogous to the traditional ; is the return of the market; SMB stands for the Small (market capitalisation) Minus Big ; HML stands for the High (book-to-market ratio) Minus Low ; are coefficients obtained from regressions; for big market capitalisation and 1 for small ones; for assets with a large book-to market value. is a random idiosyncratic risk Fama and French s research showed that the CAPM underestimated the performance of small market capitalisation and small book-to-market ratio stocks. To correct this imperfection of the CAPM, they added two additional factors that accounted for size and book-to-market characteristics. They argued that investors cared little about beta and that they used other means to measure risk. Commenting on their study, Haugen (1996) noted that small stocks carry bigger expected returns and tend to be riskier, but their superior returns are driven by size and relative trading costs rather than market risk. Despite being widely acclaimed, Fama and French s model has not found a consensus amongst researchers, and critics have been in agreement arguing that their findings were backward looking and the results of data mining (Black, 1993, and Kothari, et al., 199). On the JSE, evidences of a small size effect have not been clearly observed. De Villiers et al. (1986) concluded their study of the period by observing that if anything, large market capitalisation shares outperformed smaller ones. Later in 1988, Bradfield et al. did not observe any style-based factor on the JSE and argued that overall the CAPM was successful in estimating returns. A further study from Bradfield and Barr, in 1989, found no evidence of a size effect, even a reverse one as argued by De Villiers et al. (1986). Page and Palmer (1991), in a study of ten year returns from 1978 to 1988, documented the existence of an earnings effect but did not find any correlation between size and excess returns. Their conclusions were in line with those of Basu in the US who had documented an argument around size being a proxy for an earnings-to-price effect. In an unpublished MBA report, 1

16 Matiwaza in 1998 concluded that the relationship between size and average returns was not significant enough to support a small size effect theory on the JSE. In 21, Van Rensburg observed the impact of style-based factors to explain returns of industrial shares on the JSE over the period His results documented CAPM anomalies and the influence of three style-based factors: earnings to price, market capitalisation, and twelve months past positive returns. He further pointed out the existence of a size effect where smaller market capitalisation earned higher risk-adjusted returns. However, the fact that returns were adjusted for risk using the CAPM to later prove the CAPM wrong raises some questions about the validity of the methodology. During the same period, Graham and Uliana (21) analysed the performance of value shares (low market-to-book value of ordinary shares) versus growth shares (high market-to-book value of ordinary shares) for the period Their findings established a strong relationship between higher returns and value shares for the post-1992 period and again provided some empirical evidence against systematic risk being the only explanatory factor for returns. In 23, Van Rensburg and Robertson established that returns could be predicted using price-to-earnings and size as independent factors. Their study over the period 199-2, using a methodology similar to the one developed by Fama and French in 1992, showed that low P/E and small size earned higher returns, a finding which contradicted the CAPM. Auret and Sinclaire (26) also tried to understand the effect of firm specific factors to explain returns. Their study examined the influence of five different factors: size, price-to-earnings, cash-flow-to-price, dividend-yield, price-to-net asset value and book-to-market value. In line with the conclusions of Fama and French (1992), they observed that book-to-market value played a strong role in explaining returns; however, their results also showed that BTM almost completely subsumes the effect of size and PE. They then concluded that book-to-market value failed to improve on the Van Rensburg and Robertson 23 model using size and PE Portfolio diversification and systematic risk Diversification as a way to decrease portfolio risk was first introduced by Harry Markowitz (192, 199). He was the first to make the trade-off between risk and return mathematically explicit and to characterise the effect of diversification. His work on portfolio selection laid the foundation of Modern Portfolio Theory (MPT). One of the most practical aspects of Markowitz s 192 paper was his definition of the efficient frontier. This line, also known as 11

17 the Markowitz Efficient Frontier, is defined in the space of all possible portfolios by first plotting for any level of risk, the highest level of expected return and then, for any level of expected return, the lowest level of risk. Markowitz documented that the two previously defined lines are indeed the same and constitutes the set of optimal portfolios. Building on the efficient frontier, Tobin (198) introduced the notion of the super-efficient portfolio. His work demonstrated that if investors can either borrow or invest in the risk-free asset, they have the possibility to beat the risk-reward ratio of the efficient frontier. The different weighting of risk-free asset and efficient portfolios define the capital market line. Investing in the risk-free asset can be considered as lending money to the government, assuming that the risk-free asset is a long-term government bond. This line goes from the risk-free rate on the vertical axis and tangent the efficient frontier as represented below. Borrowing Lending Figure 1 - Efficient frontier and Capital market line Sharpe (1964) and Lintner (196) extended on Tobin and Markowitz s works to formalise the Capital Asset Pricing Model. Sharpe proved that under significant assumptions, a portfolio made up of all the assets available on the market was part of the Capital Market Line. Sharpe s work on two thousand portfolios in 1963 highlighted the difficulty of practically using the theoretical framework developed by Markowitz. Sharpe tried to by-pass the computational problems raised by the calculation of the co-variance matrices by developing a diagonal model. The complexity of such model for a large number of securities proved nonconclusive. One of the first influential empirical studies on the relationship between the size of a portfolio and its risk was led by Evans and Archer in Their work, based on a simulation methodology, established a clear relationship between the number of securities and the 12

18 variations of portfolio returns. They also concluded at the time that the benefit of increasing portfolio size beyond ten shares was questionable. Their findings were confirmed by Fisher and Lorie in 197 who, also using a large scale simulation method, observed that eighty percent of the variability could be removed when holding a portfolio of eight stocks. Solnik, in 1974, took the diversification properties out of the American boundaries and looked at the effect of diversifying internationally. His study showed that an internationally well diversified investor would bear half the risk of a well diversified investor in the US. Up until then all the studies on diversification were empirical using generic simulation methods. In 1977, Elton and Gruber moved away from the simulation method to analyse further the relationship between portfolio size and risk reduction. Their study was conceptually different from that of Evans and Archer. First, their work aimed at establishing an analytical representation of the link between risk and portfolio size and identifying the leverage of the different risk factors. By comparison, the simulation method of Evans and Archer was highly dependent of the time period under study and did not allow the identification of any risk explanatory factor. Secondly, Elton and Gruber s conceptual views of risk were different and they added another dimension to risk. They argued that taking the volatility of returns as a measure of risk gave an incomplete picture and that the risk of holding a portfolio with a different return from the market should also be considered. Mathematically, their concept translated as indicated below, where risk is split into a systematic, unsystematic and third component: Systematic risk Unsystematic risk Portfolio risk Where: N = the number of securities in the portfolio and; M = number of securities in the population of securities under consideration Elton and Gruber s findings also differed from those of previous researches as they concluded that increasing the number of securities from one to ten only removes fifty-one percent of a portfolio standard deviation. This result was highly different from the 13

19 recommendation of Evans and Archer who argued that eight securities removed eighty percent of the risk. Statman in 1987 took a different approach and factored in transaction costs to understand the limits of the benefits coming from diversification. His work on five hundred stocks consisted of identifying the point where transactions costs outweighed the benefits of risk reduction. Statman concluded that a well-diversified portfolio should include at least thirty shares for a borrowing investor and forty shares for a lending investor. These findings challenged for the first time the reference articles from Evans and Archer and the well established idea that ten securities were enough to remove most of the unsystematic risk. The idea that eight to ten shares were enough was further challenged by Newbould and Poon in Their views tended to agree with those of Statman. They concluded that a risk-averse investor needed more than twenty shares to diversify its risk. In a further study (1996), they determined that one hundred securities were necessary for an investor who wanted to be within five percent of the average return and twenty percent of the average risk. It should be noted that their definition of risk was similar to the one expressed earlier by Evans and Archer. In 21, Campbell et al. observed that the idiosyncratic risk of US stocks had increased over the last thirty years and that a larger number of securities were necessary to remove it. In recent years, the large increase of computational power has greatly helped academics in simulating the behaviour of portfolios. Despite not bringing any significant element to the theory field, they all tend to agree that eight to ten securities, as initially widely accepted, are not enough to significantly benefit from diversification. As an example, Statman who recommended forty shares in 1987 re-assessed his threshold to three hundred securities in 24. On the Johannesburg Stock Exchange, different studies have empirically tested this risk/size relationship. In 1997, Firer and Neu-Ner in their study over the period concluded that 9% of the benefits of diversification may be obtained by holding a random portfolio of thirty shares. They also observed that diversification seemed to be more efficient on the JSE than on other stock markets and noticed that the South African market had the particularity to be highly concentrated. Interestingly, they also looked at cross-sectional risks and noticed that portfolios made up of low beta shares carried a lower systematic risk, lower risk variations while earning higher returns. Their findings were recently confirmed in an MBA 14

20 research report by Webbstock and Wessels in 23. They considered the period 2-23 and compared the diversification properties of the JSE with previous studies. They first observed that the risk of holding a single share had increased when compared to previous studies. This conclusion seemed to confirm the trend observed earlier on the US market by Campbell et al. Despite observing an increase in the risk of holding one share, they concluded that a random portfolio of twenty-five shares would allow for an effective diversification on the South African market. 4. Methodology 4.1. Data coverage This study examined the average monthly risk and return of portfolios made up of the totality of the JSE shares ranked according to their P/E and market capitalisation over three years. The years under consideration were 2, 26, and 27. These years were chosen because they offered the most recent data with the number of companies varying from two hundred and ninety-two at the beginning of the period to four hundred and ten at the end of 27 (after data cleaning). It was also believed that with IFRS reporting requirements entering into force in SA on 1/1/2 for listed companies, data, such as price-to-earnings, will have gained in consistency. Risk and return monthly profiles were analysed under three investment horizons: three, six and twelve months. The sample size and data coverage for each horizon is detailed in Table 1. Horizon 3 months 6 months 12 months Period of investment 1/1/ - 1/12/7 1/1/ - 1/12/7 1/1/ - 1/12/7 First return period 1/1/ - 31/3/ 1/1/ - 3/6/ 1/1/ - 31/12/ Last return period 1/12/7 28/2/8 1/12/7 2//8 1/1/7 3/9/8 Sample size Table 1 - Data coverage Monthly closing prices, price-to-earnings (P/E) ratios, traded volumes, and market capitalisations were collected from Inet-Bridge. As returns were calculated inclusive of dividends, ex-dividend dates and values were collected from DataStream. 1

21 4.2. Data cleaning A thin trading filter was used to eliminate the shares poorly traded during each month. The filter shared some similarities with the one used by Van Rensburg in 21. For each month, shares were ranked according to their traded value. For each particular month, all the shares that cumulatively accounted for less than five percent of the median of the total traded value were removed. The method differed with Van Rensburg as the filter was discretely applied to individual months and not throughout the entire period. This change in the method was justified by the dynamic character of the study and the fact that shares are not the same from one month to another. The five percent value seems to have empirical origins and led to rather conservative results removing an average of forty eight shares per month. The monthly detail of the thin trading impact is summarised in Table 2. 1/4/ 1/3/ 1/2/ 1/1/ 1/8/ 1/7/ 1/6/ 1// 1/12/ 1/11/ 1/1/ 1/9/ 1/4/6 1/3/6 1/2/6 1/1/6 1/8/6 1/7/6 1/6/6 1//6 1/12/6 1/11/6 1/1/6 1/9/6 1/4/7 1/3/7 1/2/7 1/1/7 1/8/7 1/7/7 1/6/7 1//7 1/12/7 1/11/7 1/1/7 1/9/7 Total number of sh ares Thinly traded shares Total Table 2 - Thin trading results Two additional shares with abnormal return values were also removed from the dataset: codes LAF and ITE. The share codes of the shares analysed in this study are listed in Appendix Ranking For each month, P/E ratios and market capitalisations of all the JSE shares were collected from Inet-Bridge. Shares were ranked according to their P/E and market capitalisation at the beginning of each month, and then classified in four different groups, as shown below. The ranking was performed on a monthly basis according to the relative position of the factors P/E and market capitalisation to the monthly median value. Group 1: shares with small P/E and small market capitalisation Group 2: shares with small P/E and large market capitalisation Group 3: shares with high P/E and small market capitalisation Group 4: shares with high P/E and large market capitalisation 16

22 Groups are summarised in Figure 2: Market capitalisation Group 2 Group 4 P/E Group 1 Group 3 Figure 2 - Group ranking The ranking was performed on a monthly basis. Thus, the rank of a given share could change from one month to another. The ranking being dynamic and related to the median value, each addition of a new share or change in the P/E or market capitalisation value of one share could affect the ranking. It then must be noticed that the large increase in the number of listed companies over the period under study (plus forty percent) must have resulted in a highly dynamic dataset. Table 3 indicates the size of each group across the period under study. Groups 1 and 4 are systematically the largest groups and have a similar size. Groups 2 and 3 are also of a similar size while less populated than the formers. On average, Groups 1 and 4 respectively regroup 3.7% of the shares and Groups 2 and 3, 19.2%. 17

23 Table 3 Ranking 4.4. Systematic risk and portfolio construction The simulation method The objective of this study is to understand cross-sectional risk-adjusted properties. To fully understand these characteristics, risk needs to be broken into its components: systematic and idiosyncratic. The systematic part of the risk can be estimated by analysing the portfolio diversification characteristics. A simulation method was used to randomly build and analyse the behaviour of portfolios made-up of ranked shares. Simulation is a rather generic process that finds application in a lot of fields. It generally consists of drawing and computing random inputs from a domain in order to observe aggregate behaviours. In portfolio theory, the simulation method has mainly found some applications in analysing the diversification properties of a large number of random portfolios of different sizes. From a process view point, portfolios are made up of shares that are randomly selected. Each selected share is then not replaced in the initial domain to ensure that a share does not appear twice in a given portfolio. Evans and Archer in 1968 were some of the pioneers of the simulation method. Their analysis of diversification properties on the Standard and Poor s index was built on a total of two thousand, four hundred portfolios ranging from one to forty shares in size. This method came as an alternative to the method elaborated by Markowitz (199). An ever increasing number of 18

24 asset rendered Markowitz method rather intricate as the integration of assets co-variances involves complex algorithms and computational power. Markowitz framework involves the calculation of co-variances. Thus, the calculation for a thirty share portfolio entails the calculation of four hundred and thirty-five co-variances. Moreover, co-variances are not stable over time! The simulation method, despite being widely used, is not the only alternative to Markowitz s method. Elton and Gruber introduced an analytical relationship in Their model aimed at establishing an estimate of the mathematical relationship between portfolio size and risk. This method presents the advantage, when compared to the simulation, to better identify the influence of the different portfolio risk factors. It must also be noted that their model relies on a different risk definition as stressed in the literature review. The risk definition used in this study is similar to the one used by Evans and Archer and does not include the risk of holding a portfolio with a different return from the market. Portfolio building For each group and each month, the totality of the shares was randomly drawn and ranked one hundred times. Portfolios of N shares 3 were built by selecting the first N shares of each of the one hundred random rankings. Excel was used as the software to perform these tasks and a VBA application was developed. The core function of the application was to draw random numbers without replacement, as in the model used by Webbstock and Wessels (23). For each group and each month, one hundred random portfolios of size one to thirty shares 4 were built. For a given return horizon, the total number of portfolios analysed equalled: 4 groups x 36 months x 1 random portfolios x 3 sizes = 432 portfolios Portfolio size was limited to thirty shares as per previous studies. This value reflects the findings of Webbstock and Wessels in 23 who concluded that the benefits of diversification were achieved on the JSE for portfolios of twenty-five shares. Firer and Neu- Ner had previously recommended holding thirty shares in order to reduce the risk by seventy 3 N varying from 1 to ,6 and 12 month return horizons are systematically considered. It must be noted that for the 12 month return horizon, only 34 months are observed which reduces the number of portfolios to

25 percent (7%). Based on these studies, it is assumed that the average risk of a large number of thirty share portfolios should give a relatively close estimate of the systematic risk component for each group. It must also be noted that the aim of the study is to establish a comparison basis between group and not to precisely estimate the systematic risk. Another limitation of portfolio size came from the size of Groups 2 and 3 which varied from thirty-nine to fifty shares in Share returns Share returns were calculated using the formula: Where: is the return on the share s over the period (t, t+1) expressed as percentage of the price of the share s at the time t. is the dividend paid to the share s over the period (t,t+1) is the market price of the share s at the time t is the market price of the share s at the time t+1 It must be noted that unlike most studies on the JSE, dividends were systematically included in the calculation of returns at the ex-dividend date. Dividend amounts and ex-dates were sourced from Datastream and manually added to the calculation of returns. It has to be noted that previous studies on the JSE ignored dividends or used dividend-adjusted returns. Such methodologies are likely to introduce a bias when risk-adjusted returns are considered and were therefore rejected. Portfolio returns The formula of share returns, where total return is divided by the initial share price, ensures that shares are equally weighted when placed in a portfolio. 2

26 The return of the portfolio is the weighted average of the return of the individual assets (shares in this case). With asset returns being calculated as indicated above, the return of a portfolio made up of N assets was calculated as follow: Where is the return of the portfolio; is the return from the asset i; is the number of securities. Portfolio risk As previously indicated, for each group and each month, the returns of 1 random portfolios of size 1 to 3 were computed. The monthly risk for a portfolio of size N was estimated by calculating the standard deviation around the average return of the one hundred random portfolios of size N. The number of portfolios used to estimate the risk associated with a given size, for each group, over the time period under consideration is indicated in Table 4. 3 months 6 months 12 months Group Group Group Group Table 4 - Sample size for a portfolio of size N 4.6. Analysis method The analysis is divided into four parts. First, the returns of each group are analysed and compared. Secondly, the analysis examines the total risk for each group and compares returns on a total risk-adjusted basis. Thirdly, the diversification properties and the systematic risk are detailed in order to document the reward to systematic risk profile for each group. Finally, the influence of the investment horizon is analysed. 21

27 For each section, comparisons are made between the groups using graphic representations and two sample test-statistics. The two sample test-statistic s aim is to establish a ranking between group. Thus the following pairs are systematically compared: Group 1&2 Group 1&3 Group 1&4 Group 2&3 Group 2&4 Group 3&4 The choice of the test-statistic being critical to the validity of the results, an analysis of the different data distribution is systematically performed. Total Risk The total risk for each group is represented by the average risk of holding one random share. This value was obtained for each group by calculating the standard deviation around the average return of all the shares on a monthly basis. This value is called for Group i and month k. Over the total period, the average risk of holding one random share of Group i is represented by averaging the monthly risks for k=1 to 36 for three and six month horizons, and k=1 to 34 for twelve month horizon. This average total risk is called for Group i. Diversification and systematic risk The diversification properties of the different groups are compared by calculating the average of the monthly standard deviation around the average monthly returns for each portfolio size. This value is called for Group i and portfolio size j. In other words, is the arithmetic average of the monthly values, where is the standard deviation around the average return of the one hundred random portfolios of size j, for the Group i and during the month k. The risk profile associated with each group is further analysed by looking at the relationship between holding a portfolio rather than a single share. For each group and each return horizon, this relationship is established by calculating the following ratio: 22

28 Where: is the risk ratio of holding a portfolio of size j to the average risk of holding one random share of Group i. With: 1 i 4; and 2 j 3. Sharpe s index The study analyses risk-adjusted return rates across the four groups. This analysis is done for total risk ( ) and systematic risk ( ). Several indexes are available to perform risk-adjusted return comparisons. The most commonly used ones are Jensen s index, Treynor s index and Sharpe s index. As this study intends to understand the risk components across the four crosssections, Sharpe s index will be used. Both Jensen and Treynor s index involve the computation of beta while Sharpe s ratio measures express returns per unit of risk (σ). Bodie et al., in 1996 noted that the important aspect of the ratio is that it takes the total risk of the portfolio into account. It is noted that Modigliani and Miller introduced a variation of the Sharpe ratio in 1997 named the M-squared. This measure multiplies the Sharpe ratio by the standard deviation of the market portfolio. This extra feature is not used in the analysis as the M-squared did not add any value to the analysis. The following formula was used to calculate Sharpe s index: Where is the expected return of the portfolio; is the risk free rate, and σ is the standard deviation of the portfolio. 23

29 The detail of the risk-free rates used for the calculations is provided in Appendix 6. The riskfree rates were sourced on a monthly basis from Inet-bridge and consist of the lowest longterm government bond rates. The risk-free rate over a period of N months was estimated by averaging the N monthly riskfree rates. The Sharpe s index was used to analyse the total risk-adjusted returns ( ) and diversified risk-adjusted returns ( ) across the four groups. Time horizon The influence of time horizon is studied by simulating three types of investors investing in thirty share portfolios. The first one re-invests every three months in actualised groups, the second will do the same thing on a six month basis, while the third only invests and actualises the groups every twelve months. The three investors invest for a period of twelve months. The simulation method was used to model the influence of these different behaviours. Twelve month risk and returns were calculated as previously and then compared. Long term bonds in the case of this study are to 1 year government bonds. 24

30 . Result analysis The empirical analysis will be split into four parts. First, the returns of each group will be compared. This first analysis will seek to verify the conclusions reached by Van Rensburg and Robertson in 23 and later confirmed by Auret and Sinclaire in 26. These two studies stressed the existence of a two-factor model where low P/E and market capitalisation were identified as predictors for higher returns. Secondly, the average risk associated with holding one random share, also called total risk, of each group will be analysed. This risk will be compared to the respective returns in order to evaluate returns adjusted for total risk. Sharpe s index will be used to represent this relationship. Thirdly, the diversification properties of each group will be analysed in order to understand the systematic risk for each of the groups. Again the systemic risk was compared to returns using Sharpe s index. Finally, the potential influence of time horizons will be detailed. This part does not aim to reach any conclusions but to highlight the potential incidence of the return horizons on the results. The first three parts of the analysis will be identically structured. Time series of results will first be presented for the three return horizons. The results will then be statistically analysed and discussed..1. Returns Average returns For each group, average returns were calculated on a monthly basis for the three investment horizons by averaging the returns of all the shares. The values for Group i and month k will be presented in the time series analysis. The overall average returns for group i was obtained by averaging the monthly returns. This procedure was repeated for the three investment horizons. 2

31 Results are presented on a monthly basis for ease of comparison. The formula used to obtain these results is detailed hereunder: Where is the monthly average return for Group i; is the average of the for Group i; n is the investment horizon under consideration. With: i = 1 to 4; k = 1 to 36 for 3 and 6 month horizons, and k = 1 to 34 for 12 month horizon; n = 3, 6 or 12 months. The monthly average returns are summarised in Table : Horizons Groups Group 1 Group 2 Group 3 Group 4 3 months 3.3% 2.33% 2.63% 2.63% 6 months 2.97% 2.16% 2.6% 2.62% 12 months 2.78% 1.89% 2.49% 2.48% Table - Monthly average returns It may be observed that the average monthly returns of Group 1 are higher than those of Group 3, and Group 4, which in turn are higher than those of Group 2. It has to be noted that returns across the three time horizons cannot be compared. The consistent decreasing trend observed between three and twelve month returns is explained by the fact that almost one third of the twelve month returns are calculated using closing stock prices positioned in 28 where very negative returns were observed. 26