Copyright UCT A chronicle of the CAPM beta and the evaluation of three methods for modelling risk-return relationships on the JSE in the new South Africa A research report presented to The Graduate School of Business University of Cape Town In partial fulfilment of the requirements for the Master of Business Administration Degree by Carl Zietsman December 2011 Supervised by: Dr Chipo Mlambo
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Copyright UCT
A chronicle of the CAPM beta and the evaluation of three methods for modelling risk-return relationships on
the JSE in the new South Africa
A research report
presented to
The Graduate School of Business
University of Cape Town
In partial fulfilment
of the requirements for the
Master of Business Administration Degree
by
Carl Zietsman
December 2011
Supervised by: Dr Chipo Mlambo
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Declaration
I know that plagiarism is wrong. Plagiarism is to use another’s work and pretend that it is
one’s own.
I have used the American Psychological Association (APA) convention for citation and
referencing. Each significant contribution and quotation from the work(s) of other people has
been attributed, cited and referenced.
I certify that this submission is all my own work.
I have not allowed and will not allow anyone to copy my work with the intention of passing it
off as his or her own work.
This report is not confidential and may be used freely by the Graduate School of Business.
Carl Zietsman
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Abstract
Since their inception, both the capital asset pricing model (CAPM) and its associated beta
coefficient (beta) have received wide-spread empirical and theoretical criticism. Numerous
studies have shown that: (1) the CAPM beta exhibits significant time instability, (2) the
empirical performance of the CAPM appears to be poor and (3) valid testing of the empirical
performance of the CAPM is difficult, if not impossible.
Despite these challenges, the CAPM is still in widespread use, is arguably here to stay and
the practitioner is left with a potentially confusing array of information on the topic.
This study presents a chronicle of the CAPM beta and explores the time-variance of the
CAPM beta as well as a South African Rand beta factor for JSE Top 40 resource and non-
resource stocks in the new South Africa. In addition to this, the empirical performance of
three risk-return models used in practice for portfolio construction is compared.
The findings are as follows: (1) both classic beta and the South African Rand beta appear
to vary over time, (2) resource and non-resource shares behave differently, especially with
regard to their Rand betas, and (3) one of the two-factor alternatives to the single risk factor
market model has a poorer fit to the data than expected.
KeyWords
All Share Index; ALSI; beta; capital asset pricing model; CAPM; expected return;
1.1 Background ........................................................................................................................ 1 1.2 Research Area ..................................................................................................................... 2 1.3 Importance .......................................................................................................................... 2 1.4 Research Objectives and Scope .......................................................................................... 4 1.5 Research Assumptions ....................................................................................................... 5 1.6 Research Ethics .................................................................................................................. 5 1.7 Conclusion .......................................................................................................................... 5
2 Literature Review ............................................................................................................. 6
2.1 The Creation of a Risk Measure ......................................................................................... 6 2.2 What is Risk? ...................................................................................................................... 6 2.3 The Importance of Beta ...................................................................................................... 7 2.4 CAPM Assumptions ........................................................................................................... 7 2.5 The History of the CAPM and Beta ................................................................................... 9 2.6 Conclusion ........................................................................................................................ 22
3.1 Research Approach ........................................................................................................... 23 3.2 Data Collection, Research Design and Sampling ............................................................. 23 3.3 Data Analysis Method ...................................................................................................... 26
4 Findings, Analysis and Discussion ................................................................................. 35
4.1 Research Findings ............................................................................................................ 35 4.2 Comparison of the Models ............................................................................................... 35 4.3 Time Behaviour of Beta ................................................................................................... 43 4.4 Research Limitations ........................................................................................................ 45
GSB University of Cape Town Graduate School of Business
HML High Minus Low (Fama and French Three Factor Model)
IAPM International Asset Pricing Model
I-Net Bridge South African financial database
JSE Johannesburg Stock Exchange
M&M Miller and Modigliani/Modigliani and Miller
MAD Mean Absolute Deviation
MBA Master of Business Administration
ME Market Equity
NYSE New York Stock Exchange
OLS Ordinary Least Squares
PE or P/E Price Earnings Ratio
SA South Africa
SLB Sharpe, Lintner and Black
SMB Small Minus Big (Fama and French Three Factor Model)
TFM Fama and French’s Three Factor Model
[TR] Total Return (I-Net Bridge data type)
UK United Kingdom
USD United States Dollar
VMMRM Variable Mean Response Regression Model
ZAR South African Rand
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Acknowledgement
I would like to thank my supervisor, Dr Chipo Mlambo, for her support and guidance in the
preparation of this report. I would also like to thank Chris Holdsworth and Heidi
Raubenheimer for their statistical inputs and Dr Nicholas Marais for always volunteering to
be the test subject/sounding board for my various MBA-related projects. Finally, a big thank
you to my friends, family and everyone at Paterson & Cooke for their tremendous support
throughout the duration of my MBA.
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1 Introduction
This chapter introduces the CAPM and its associated beta coefficient. The problem of beta
instability and the apparent poor empirical performance of the CAPM is discussed. The
importance of the risk-return relationship is emphasised and two alternatives to the CAPM-
related market model, which utilise a Rand beta as an additional risk factor, are presented
for testing. The research objectives and scope are defined and research ethics are touched on
briefly just prior to the end of the chapter.
1.1 BackgroundIn his article, “The History of Finance”, Merton Miller (2000) traces the roots of
modern finance theory back to Harry Markowitz’s (1952) pivotal article, “Portfolio
Selection”. Miller refers to it as the “big bang” (2000, p. 9) of modern finance—i.e.,
where it all began.
Prior to this point, the academic study of the equity markets was not considered to
be a topic to be taken seriously (Ball, 1995). The little work that had been done on the
topic was done by statisticians who had more-or-less concluded that share prices were
completely unpredictable “random walks” (Ball, 1995, p. 6).
Markowitz’s work along with the subsequent work of Sharpe1 (1963, 1964) and
Lintner (1965a), led to the development of the now-famous capital asset pricing
model (CAPM). Further refinements to the model were made by Mossin (1966) and
Black (1972).
Put simply, the CAPM states that the expected return on a particular asset is a
function of the risk-free interest rate, the reward for bearing risk (also known as the
risk premium), and the amount of systematic risk present in a particular asset relative
to an asset of average riskiness (Firer, Ross, Westerfield, & Jordan, 2008).
The CAPM can be expressed mathematically as shown in Equation 1:
(Eq.1)
Where:
1 In some sources, Treynor (1961) is credited with the almost-simultaneous development of the CAPM along with Sharpe; e.g. Copeland, Weston, & Shastri (2005) and Black, Jensen and Scholes (1972). However, these sources list Treynor’s work as unpublished. It is therefore not included in the list of references at the end of this report.
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is the expected return on the asset;
is the risk free rate;
is the market risk premium;
is the amount of systematic risk present in an asset, relative to an asset of
average riskiness.
According to Damodaran (1997, p. 107), “the beta of any investment in CAPM is a
standardised measure of the risk it adds to the market portfolio” and, “the expected
return on an asset is linearly related to the beta of that asset”. In other words, the
CAPM states that return that investors expect on an asset is a function of only the
systematic risk (the risk which cannot be diversified away) of that asset.
1.2 ResearchAreaThe CAPM model and its simple, intuitive formulation of the relationship between
risk and return have become an integral part of the practice and theory of finance
(Siegel, 1995). However, since their inception, both the CAPM and beta have been
under regular empirical and theoretical attack in the academic literature (Dowen,
1988).
To be more specific, the subject of beta instability over time and the apparently
poor empirical correlation between an asset’s beta (systematic risk) and its actual
returns has become one of the great academic debates in the finance literature over
the last fifty-odd years. Subrahmanyam notes that, “why one stock’s expected return
might vary from that of another has preoccupied scholars for decades” (2010, p. 27).
Some argue that the empirical record of the CAPM is poor enough to invalidate the
way it is often applied (Fama & French, 2004). A number of alternative models have
been proposed, including Ross’ (1976) arbitrage pricing theory (APT) and Fama and
French’s (1996) three factor model (TFM).
Yet despite the TFM’s success at explaining stock returns (Fama & French, 1996)
efforts to date to explain its success in terms of the underlying economics of the
empirical relationships have been less than successful (Fama & French, 1995). In
other words, the situation is still somewhat unresolved, leading one to question the
importance of further investigation into the topic.
1.3 ImportanceThe relationship between risk and return is of vital importance as it is one of the
“central lines of research in finance” (Subrahmanyam, 2010, p. 27). Scott & Brown
(1980) ascribe the importance of the risk–return relationship to the fact that
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systematic risk is used for portfolio selection, while Fama and French (2004) mention
its application in estimating the cost of equity capital.
Fisher and Kamin also provide the following compelling and somewhat humorous
argument:
In the application of modem portfolio theory, the systematic risk of asecurity is of central importance. Beta (β), the future regressioncoefficientofthereturnofthesecurityonthereturnofthemarket,isanindexofthatrisk.Sincethefutureisyettoberevealed,nonclairvoyantpractitionersandresearchersmustrelyonestimatedratherthanactualvalues of beta and the estimates must be based on data that arecurrentlyavailable.(Fisher&Kamin,1985,p.127)
However, despite the importance of the risk-return relationship and the apparent
shortcomings of the CAPM, it is often the only asset pricing model taught in many
MBA investment courses (Fama & French, 2004, p. 25).
It is this very peculiarity which introduces the topic of this research report. This
report investigates alternatives to the single factor market model (Equation 2), which
uses CAPM beta as its sole measure of risk, and the performance of these alternatives
in the South African context. The two models which have been chosen for
comparison both originate from the work of Barr, Kantor and Holdsworth (BKH).
The “performance” measure by which the models are judged, is their adjusted
coefficient of determination—i.e. how well they fit historical data.
BKH have researched the link between the performance of South African stocks
and the Rand to US Dollar exchange rate, extensively (2003, 2007, 2011). Based on
their research, they propose two similar two-factor models which are similar in form
to the multi-factor models found in APT. The particular appeal of their models stems
from the fact that the economics are relatively easy to explain—unlike the TFM and
other ad hoc multiple regression models which “happen” to explain stock returns.
According to BKH, a factor analysis of the SA market shows that it is a function of
two factors. SA stocks tend to group into a number of clusters, particularly in terms of
performance versus the strength of the Rand (Holdsworth, 2011). The two most
distinct clusters are arguably: (1) the Rand leverage and (2) Rand play clusters. The
two clusters can be explained simply as follows (Barr, Kantor, & Holdsworth, 2007):
(1) The Rand leverages cluster is dominated by SA resource companies.
These companies tend to earn revenues in US Dollars while a large
portion of their costs (labour) are paid in local Rand. Therefore a weak
Rand can result in improved profits for a resource company.
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(2) Rand plays on the other hand are companies which earn local revenues
and have local costs. These companies tend to suffer when the Rand
weakens.
The two models proposed by BKH are shown in Equations 3 and 4 and are derived
from this sensitivity of SA companies to changes in the Rand-Dollar exchange rate.
The two models are very similar, the latter simply being a refinement of the former.
The first equation states that the return for a particular asset is related to asset
specific factors (unsystematic risk), fluctuations in the Rand-Dollar exchange rate and
fluctuations in the market proxy.
The second equation states that the return for a particular asset is related to asset
specific factors (unsystematic risk), fluctuations in the Rand-Dollar exchange rate and
fluctuations in the market proxy not explained by fluctuations in the Rand-Dollar
exchange rate (hence the ALSI prime2). That is, both BKH models are two factor
models as opposed to the single factor (risk measure) market model.
∙ (Eq.2)
∙ ∙ (Eq.3)
∙ ′ ∙ (Eq.4)
1.4 ResearchObjectivesandScopeResearch objectives are generally considered to, “lead to greater specificity than
research or investigative questions” (Saunders, Lewis, & Thornhill, 2009, p. 34) and
“are generally more acceptable to the research community as evidence of the
researcher’s clear sense of purpose and direction” (ibid.). For this reason, the goals of
this research have been presented in the form of objectives and not questions.
This research has three broad objectives:
(1) To present a chronicle of the “life” of the CAPM and CAPM beta;
(2) To test the “goodness of fit” (to historical data) of the market model
versus the two BKH models;
2 In other words, the second model attempts to correct for multicollinearity—the fact that movements in the ALSI are correlated with the ZAR.
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(3) To explore how classic beta and the Rand beta vary as a function of time.
This research does not have a distinct hypothesis, but it is supposed that the BKH
models will provide a better fit to historical data due to the fact that they have an
additional factor (risk measure) compared to the market model. It is also believed that
the BKH model which adjusts for multicollinearity (Model 3) will outperform the
BKH model which does not adjust for multicollinearity (Model 2).
The scope of the proposed research will be limited to the South African context and
date range of 1994 to 2011.
1.5 ResearchAssumptionsRichard Roll, in his famous “Critique of asset pricing tests” makes the following
statement:
No correct and unambiguous test of the theory has appeared in theliterature,and…thereispracticallynopossibilitythatsuchatestcanbeaccomplishedinthefuture.(Roll,1977,p.129&130)
Any test of any asset pricing model is bound to be a magnet for controversy for
reasons which will be discussed in the literature review in Chapter 2.
However, since the proposed research is not attempting to validate or disprove any
theoretical model, but rather simply to compare the practical performance of three
models in terms of how well they fit a set of data, Roll’s critique should not be of
great concern.
1.6 ResearchEthicsThe research undertaken in this research report involved a review of academic
literature and the analysis of quantitative data obtained from electronic databases—
human subjects were not involved. There were thus no potential ethical concerns. The
research was approved by the Graduate School of Business (GSB) Ethics in Research
Committee.
1.7 ConclusionBefore testing the various models, it is necessary to conduct a literature review. This
review of the literature is presented in the next chapter, Chapter 2.
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2 LiteratureReview
In this chapter, a literature review is conducted. Beta is defined formally, the CAPM
assumptions are explained and a comprehensive history of the CAPM and beta is presented.
The chapter concludes by noting the CAPM has an important place in the financial literature
due to the elegant, simple and intuitive way in which it defines the relationship between risk
and return.
2.1 TheCreationofaRiskMeasureAs described in the introduction, the debate around the validity of the CAPM model
and beta has become one of the great debates in the financial literature of the last
fifty-odd years, with perhaps only the Miller and Modigliani (M&M) propositions
regarding the irrelevance of capital structure (1958) and dividend policy (1961)
having gained more attention.
Damodaran (1997) acknowledges the widespread use of the model and notes that
despite the fact that it has become a “magnet for criticism” (1997, p. 93), the CAPM
“is the standard against which other risk and return models are measured” (1997, p.
93). The popularity of the CAPM is attributable to the fact that it is both simple and
intuitive (Lakonishok & Shapiro, 1984).
One of the key results of the CAPM was to create a clear measure for risk. Blume
(1971, p. 1) notes that:
The concept of riskhas sopermeated the financial community thatnoone needs to be convinced of the necessity of including risk ininvestment analysis….One [such]measure of riskwhichhas hadwideacceptance in the academic community [is] the coefficient of non‐diversifiable risk or more simply the beta coefficient in the marketmodel.
2.2 WhatisRisk?Damodaran defines risk informally as, “the deviation of actual returns from expected
returns” (2006, p. 56)—more formally, “the beta of any investment in the CAPM is a
standardized [sic] measure of the risk that it adds to the market portfolio” (2006, p.
69).
Since beta cannot be measured directly (Bundoo, 2006), it is usually3 estimated by
performing an ordinary least squares (OLS) regression of time series of monthly
3 The estimation of beta is a subject in itself. This literature review documents a high-level overview of a number of common statistical methods used, but is by no means an exhaustive review of the topic.
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returns for a particular security against the total returns of a suitable market proxy.
(Gooding & O'Malley, 1977). This is expressed mathematically in Equation 5.
, (Eq.5)
2.3 TheImportanceofBetaBeta is of great importance in the world of investment. The reason for its importance
is the relationship between risk and reward—i.e., that according to the CAPM,
investors are only rewarded for the systematic risk which they incur (Theobald,
1980).
Therefore, in order to form an opinion about the expected returns on an investment,
it is essential that beta be accurately estimated. Any inaccuracies in the estimate of
beta represent unsystematic risk, which, according to the theory, is not rewarded
(Theobald, 1980).
This is particularly important for portfolio construction. In order to construct and
optimum portfolio (one which maximises return for a given level of risk, or one
which minimises risk for a given level of return), one has to take a view on the
relationship between risk and reward.
If one has a more accurate way to model this relationship between risk and return,
then one should be able to construct better portfolios. In this case, a “more accurate
model” is considered to be one which provides a better fit to empirical data.
2.4 CAPMAssumptionsIn order to follow many of the arguments presented in the literature regarding the
validity of the CAPM and its single measure of risk (beta), it is essential to
understand the assumptions on which the CAPM is based. The assumptions are as
follows and are adapted from a number of sources including Sharpe (1964), Jensen
(1969), Friend and Blume (1970), Black, Jensen and Scholes (1972) and Copeland et
al. (2005):
Markets are mean-variance efficient in the Markowitz sense. A Markowitz
portfolio is a portfolio that yields the highest expected return for a specified
level of risk, or the minimum standard deviation for a specified expected
return. A sketch showing the Markowitz “efficient frontier” line is shown in
Figure 1.
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Figure1:MarkowitzPortfolios4
All investors have a common, single-period investment horizon;
All investors are risk-averse individuals who attempt to maximise their wealth
utility function;
All investors are price-takers—i.e. regardless of the amount of investment, a
particular investor is assumed to have no effect on the price of an asset;
A risk-free asset exists such that investors are able to borrow or lend unlimited
amounts of the asset at the risk-free rate of interest;
Investors have homogeneous expectations about asset returns that are
normally distributed.
Friend and Blume note the following about the assumptions underlying the CAPM:
Inreality,theseassumptionsarenotlikelytoholdcompletely,but…[theCAPM] may, nonetheless, be an adequate approximation of reality formostsecurities.(Friend&Blume,1970,p.562)
4 Adapted from Copeland et al. (2005) and Damodaran (1997).
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2.5 TheHistoryoftheCAPMandBeta
2.5.1 TheBeginningAlmost immediately after Fama (1968) showed that the work of Sharpe and Lintner
represented, “equivalent approaches to the problem of capital asset pricing under
uncertainty” (Fama E. , 1968, p. 40), the CAPM was thrust into the academic
limelight.
For example, Michael Jensen published a study in 1969 in which he presented a
model for, “evaluating the performance of portfolios of risky assets” (Jensen, 1969, p.
245). In this study, he developed his well-known alpha coefficient and extended the
ex ante5 Sharpe-Lintner model to its ex post6 form—i.e. expected returns could be
expressed as a function of the level of systematic risk, the risk free rate of return and,
“actual realised returns… on the market portfolio over any holding period” (Jensen,
1969, p. 241).
The ex post form of the CAPM equation is given by Equation 6 (Copeland,
Weston, & Shastri, 2005, p. 165):
(Eq.6)
Jensen concluded his study by noting that the CAPM seemed to have empirical and
theoretical justification and that a “major effort” (Jensen, 1969, p. 245) to test the
model was required. A major effort was certainly what subsequently transpired—in
fact, it could be argued that the ex post form of the CAPM which Jensen derived
opened the flood gates to relentless empirical testing of the CAPM.
2.5.2 TheEarly1970sOne of the first (Copeland et al., 2005) major empirical studies on the CAPM was
performed by Friend and Blume in 1970. In this study, they “questioned the
usefulness” (p. 574) of market-line theory to explain market behaviour, given the fact
that it seemed to give “seriously biased estimates of performance” (ibid.). They also
noted that bias was related to the level of portfolio risk—greater bias was observed
for portfolios which had levels of risk significantly different to the market level of
risk.
Beta then came under more direct scrutiny in 1971 in another study by Blume. In
this study, Blume examined the stationarity of beta and documented a tendency of,
5 Ex ante means expectations-based (Copeland et al., 2005). 6 Ex post means based on observations of actual or realised returns (Copeland et al., 2005).
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“risk measures to regress towards the mean over time” (Blume, 1971, p. 10). Blume
also noted that beta was more stable for portfolios consisting of greater numbers of
securities than smaller portfolios or individual securities. This link between portfolio
size and beta stability was also confirmed by Modigliani and Pogue (1974).
Black, Jensen and Scholes (1972) presented similar findings to those of Blume,
noting that the expected excess returns on assets were not always proportional to beta.
They found that stocks with high betas tended to have negative alphas and stocks with
low betas tended to have positive alphas—put more simply, stocks with high betas
tended to yield lower-than-expected returns and stocks with low betas tended to yield
higher-than expected returns.
Black, Jensen and Scholes also found evidence of measurement error bias and
noted that this could be partly overcome by specific methods of grouping stocks into
portfolios for testing purposes. Black, Jensen and Scholes went as far as suggesting
that their evidence was sufficient to, “warrant rejection of the traditional form of the
[CAP] model” (Black, Jensen, & Scholes, 1972, p. 5).
Initially, scholars attempted to explain the less-than-ideal empirical performance of
the model by the restrictive assumptions around infinite borrowing and lending of a
riskless asset e.g. Black (1972). However, this apparent inability of beta and the
CAPM to describe returns in empirical testing led to scholars beginning to investigate
reasons for the differences in beta observed between firms, and the reasons for the
observed instability of beta over time.
For example, Hamada (1972) found that the added risk of increased debt (i.e.
corporate leverage) could explain up to 24% of observed systematic risk in common
stocks. Blume and Husic found evidence of a share price effect, noting that, “the price
per share of a stock appears to be related to future returns even if risk as often
measured is held constant” (1973, p. 283).
Levy (1974) investigated the use of beta for portfolio construction (i.e. the ability
of beta coefficients to predict returns) and was one of the first to suggest that better
predictions might be attained via the calculation of separate betas for bear and bull
markets.
Investigations along the lines of trying to explain observed differences in beta
between firms, the variations in beta over time and especially the poor empirical
performance of the CAPM continued throughought the early 1970s. Many different
solutions were proposed.
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For example, Vasicek (1973) suggested that equity betas be estimated using a
Bayesian7 statistical approach instead of the usual ordinary least squares (OLS)
regression method popular with most authors, while Robicheck and Cohn (1974)
believed that beta instability could be explained by changing macroeconomic factors.
Solnik (1974) suggested that the poor empirical performance of the CAPM was
related to the use of domestic (i.e. national) proxies for the market. He believed that
the solution was an international asset pricing model (IAPM), based on his belief
that, “the true measure of risk should be the international risk of an investment”
(1974, p. 552).
Modigliani (of Miller and Modigliani fame) and Pogue maintained that difficulties
in testing the CAPM were related to the fact that the CAPM was stated in ex ante
terms:
Themajor difficulty in testing the CAPM is that themodel is stated interms of investors’ expectations and not in terms of [sic] realizedreturns.Thefactthatexpectationsarenotalwaysrealizedintroducesanerrorterm,whichfromastatisticalpointofviewshouldbezeroontheaverage,butnotnecessarilyzeroforanysinglestockorsingleperiodoftime.(Modigliani&Pogue,1974,p.77)
2.5.3 TheLate1970sThe period from 1975 to 1980 was characterised by the publication of two landmark
articles in the field of finance in the midst of a continued ballooning of the literature
on the empirical study of the CAPM.
The two pivotal articles which appeared in this time were as follows:
(1) Stephen Ross developed his famous arbitrage pricing theory (APT) as an
“alternative to the mean variance capital asset pricing model” (Ross, 1976,
p. 341);
(2) Richard Roll published his critique of tests on the CAPM in which he
noted that testing of the CAPM was essentially, “infeasible” (Roll, 1977,
p. 129).
While both articles are great milestones in the subject of modern finance, the latter
is of greater importance to the research proposed in this document. The implications
of Roll’s article must have rattled the CAPM academic community to its core when it
7 The Bayesian approach is a recursive (i.e. iterative) statistical procedure which aims to provide more accurate estimates of a “true” parameter.
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was published. The opening lines of the article are repeated below (Roll, 1977, p. 129
& 130):
The two‐parameter asset pricing theory is testable in principle; butargumentsaregivenherethat:(a)Nocorrectandunambiguoustestofthetheoryhasappearedintheliterature,and(b)thereispracticallynopossibilitythatsuchatestcanbeaccomplishedinthefuture.Thisbroadindictmentofoneofthethreefundamentalparadigmsofmodernfinancewill undoubtedly be greeted by my colleagues, as it was by me, withscepticismandconsternation.Thepurposeofthispaper istoeliminatethescepticism.(Noreliefisofferedfortheconsternation.)
In parallel to the release of Roll’s bombshell, there was continued investigation of
various factors related to beta and the CAPM. The continued investigation can be
summarised as follows:
Macroeconomic and microeconomic determinants of risk (beta) continued to
investigated. Beaver & Manegold (1975) proposed an accounting beta which
was related to firm-specific accounting measures. Turnbull (1977)
demonstrated that systematic risk was related to firm growth and maturity.
Livingston (1977) suggested that risk measures should be estimated from
multiple indices (as opposed to the single index used in the CAPM) to account
for the significant covariance of industries. Basu (1977) found evidence that
portfolios with low price earnings (P/E) ratios tended to, “earn superior
returns on a risk-adjusted basis” (Investment performance of common stocks
in relation to their price-earnings ratios: A test of the efficient market
hypothesis, 1977, p. 681), thus casting some doubt on whether or information
was being reflected in share prices as rapidly as “postulated by the semi-strong
form of the efficient market hypothesis” (Basu, 1977, p. 681)8. Bar-Yosef and
Brown (1979) investigated the relationship between share price level and
systematic risk for two sample groups, one in which share splits had occurred
and one in which share splits had not occurred and concluded that, “economic
as opposed to simple accounting events are necessary to alter the risk-return
characteristics of a firm’s common stock” (1979, p. 63). Litzenberger and
Ramaswamy (1979, p. 163) found “before-tax expected rates of return… [to
be] linearly related to systematic risk and to dividend yield”.
8 According to the EMH, it should be impossible to benefit from information that is in the public domain (e.g. P/E ratios which are published in annual financial statements and are relatively easy to calculate based on historical information). Therefore since P/E ratios are considered to be common knowledge to all investors, investing in low P/E stocks should not result in one earning returns in excess of those required to compensate the investor for the level of systematic risk (beta). However, Basu found that “low P/E portfolios seem to have, on average, earned higher absolute and risk-adjusted rates of return than the high P/E securities” (1977, p. 680).
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The variation of beta as a function of bull and bear markets was investigated
by a number of authors including Fabozzi & Francis (1977, p. 1098) who
concluded that, “neither the alpha nor the beta statistics… appear to be
significantly affected by the alternating forces of bull and bear markets”. This
finding was then refuted again by Kim and Zumwalt—“the results indicate…
that unlike the Fabozzi and Francis study, more securities exhibited
statistically significant differences between up market and down-market betas
than would occur by chance” (An analysis of risk in bull and bear markets,
1979, p. 1016).
The regression tendencies of beta over time continued to be investigated by a
number of authors including Blume (1975), Gooding and O'Malley (1977),
Elgers, Haltiner, and Hawthorne (1979). Francis (Statistical analysis of risk
surrogates for NYSE stocks, 1979, p. 995) concluded that, “even better beta
forecasts could probably be obtained from multiple regression”.
Statistical methods were subjected to increasing scrutiny, e.g. Fabozzi and
Francis (1978, p. 101) found that beta coefficients tended to vary randomly
over time and that “ordinary least-squares (OLS) regressions used in nearly
every instance…” tended to be “inappropriate” (ibid.). Cornell and Dietrich
(1978) investigated using mean absolute deviation (MAD) regression
procedures to estimate beta instead of the usual OLS methods and found that
MAD did not outperform OLS methods. Eubank and Zumwalt (1979) found
that forecast errors could be reduced by utilising beta adjustment procedures
and an optimal estimation period for a particular investment horizon.
2.5.4 The1980sBy the early eighties, it appeared to be relatively widely accepted that beta was
unstable. Roll and Ross summarised the situation rather succinctly when they noted
that there was, “more than a modest level of disenchantment with the CAPM” (1980,
p. 1073) in the financial community.
The studies in the 1980s seemed to continue along much the same lines as those in
the seventies, with most studies falling into one of the following categories:
Investigations into the statistical aspects related to the estimation of beta;
Investigations into the determinants of beta;
Investigations into alternatives for the market proxy (these studies were
arguably prompted by Roll’s critique).
Summaries of these various categories of studies are provided in the section below
in the same order in which they appear in the bullet-point list above.
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Theobald (1980) conducted a study on beta factors in the United Kingdom (UK) and
found evidence of autocorrelation and heteroscedasticity9. Scott & Brown then,
“demonstrate[d] that concurrent autocorrelated10 residuals and intertemporal
correlations between market returns and residuals can lead to biased, unstable, OLS
estimates of betas” (Biased estimators and unstable betas, 1980, p. 55).
Engle (1982) then arguably popularised the use of use of autoregressive conditional
heteroscedasticity (ARCH) methods for modelling time series data in financial fields
after he used such a method for modelling inflation in the United Kingdom.
Chen and Lee (1982) developed a variable mean response regression model
(VMMRM) which was later shown to be flawed by McDonald (1983).
Bos and Newbold (1984, p. 40) were an interesting exception to the
“autocorrelation and homoscedasticity movement”—they found little evidence of
autocorrelation and argued that beta appeared to display “purely random” behaviour.
Fisher and Kamin (1985, p. 128) presented a complex paper in which they
“develop[ed] a form of Kalman filter11” which could be used for the estimation of
betas when residual returns were observed to be heteroscedastic.
Bollerslev (1986, p. 307) proposed a “Generalized [sic] autoregressive conditional
heteroscedasticity [GARCH]” model which was an extension to Engle’s original
ARCH model.
Rahman, Kryzanowski and Sim (1987) found that a generalised least squares
(GLS) model out-performed the usual OLS regression model in cases where beta
displayed random behaviour.
Authors who did not focus on the statistical aspects of estimating generally12
tended to concentrate on the determinants of beta. Reinganum (1981, p. 19) presented
evidence which refuted Basu’s P/E ratio and suggested that the P/E ratio was simply a
9 Heteroscedasticity means that the variation around the regression equation is not the same for all variables—i.e. in simple terms it can be observed as increasing spread of data points around the regression equation and is therefore not desirable when regression equations are to be used as these equations are based on the assumption of homoscedasticity (Lind, Marchal, & Wathen, 2010). 10 Autocorrelation means that successive residuals are not truly independent—i.e. that the values of successive residuals are related to the values of previous residuals. Regression equations generally assume the absence of autocorrelation. Thus when autocorrelation is present, it can lead to poor/invalid regression results (Lind, Marchal, & Wathen, 2010). 11 A Kalman filter is an advanced statistical method used to eliminate noise from time series data in order to provide better estimates of the true parameters. The detail is beyond the scope of this document. 12 The bear-and-bull market beta debate also reared its head a number of times, e.g. Lindahl-Stevens (1980) who looked at defining bear and bull markets and Chen (1982) who again argued that there was some justification for the separate calculation of bear and bull market beta factors.
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proxy for the effect of firm size, which led him to conclude that “empirical anomalies
which suggest[ed] that either the simple one-period capital asset pricing model
(CAPM) is [was] misspecified or that capital markets are [were] ineffcient” (1981, p.
19).
Soon thereafter, Banz found evidence which supported the size effect:
The evidence presented in this study suggests that the CAPM ismisspecified.Onaverage,smallNYSE[NewYorkStockExchange] firmshavehadsignificantlylargerriskadjustedreturnsthanlargeNYSEfirmsoverafortyyearperiod.(Banz,1981,p.16)
Bildersee and Roberts (1981) found evidence that beta instability was linked to
changing interest rates, Mandelker and Rhee (1984, p. 56) found that, “degrees of
operating and financial leverage explain a large portion of the variation in beta”,
while Lakonishok and Shapiro (1984, p. 40) provided the following conclusion to
their important empirical study on the variance of beta:
In their investigation into explanations for the instability of equity beta, DeJong and
Collins (1985) found that a statisitically significant portion of betas variation could be
explained by a firms leverage and changes in the risk-free rate. Bhandari (1986)
provided some support for the findings of DeJong and Collins in that he found the
debt equity ratio (DER) to be an important determinant of beta. He also found
evidence of greater variation in risk premia in January—something commonly known
as the “January effect”.
Chung (1989) provides a good conclusion to the “determinant investigation” period
of the 1980s in which he acknowledges the importance of examining the determinants
of the systematic risk of common stocks but cautions that, “most of the previous
empirical studies lack theoretical justification for selecting the possible determinants
of beta… [and] thus, these studies suffer from problems of serious potential model
misspecification” (1989, p. 343).
2.5.5 The1990sBy the early 1990s interest in the beta/CAPM topic appeared to be subsiding, judging
by the relatively few articles found for the period from 1990 to 1994. Martikainen
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(1991, 1993) conducted some empirical studies on risk and return on the Finnish
stock market and presented results much in-line with the status quo.
Bhardwaj and Brooks (1993) resurrected two old debates, namely that of the “size”
effect and that of the bear-and-bull market debate. They argued that when allowing
beta to vary as a function of bear and bull markets, the “size” effect was reversed—
i.e. that, “prior evidence on the overall superior performance by small firm stocks
possibly results from a failure to adjust for the risk differential in bull and bear
markets” (1993, p. 270 & 271).
Gregory-Allen, Impson, & Karafiath (1994) presented an interesting paper which
challenged the “conventional wisdom” that large portfolios of securities had more
stable betas than smaller portfolios or individual securities. Their argument was that
larger portfolios of shares simply eliminated the estimation-hampering “background
noise” more effectively and that their instability is as large as those of individual
securities when viewed in the light of the smaller variance common to larger
portfolios. In the end they concluded that, “neither category exhibit[ed] stability over
time, even for intervals as short as 100 days” (1994, p. 915).
However, it was exactly during this very period of apparent subsiding interest that
Eugene Fama and Kenneth French’s landmark series of articles, which were destined
to fuel the CAPM debate with new intensity, were published.
2.5.6 ASummaryPriortoFamaandFrenchPrior to discussing the Fama and French articles, it is worth noting that aside from a
few exceptions, empirical studies prior to those of Fama and French agreed on the
following broad points (Copeland, Weston, & Shastri, 2005, p. 167):
Securities or portfolios with high (or low) betas will earn lower (or higher)
returns than predicted by the CAPM;
Beta is the dominating measure of risk when compared to models which
include squared-beta terms or measures of unsystematic risk;
When the CAPM is tested it is found to be approximated best by functions
linear in beta;
Over long periods of time the return on the market portfolio tends to out-
perform the risk free rate of return;
Factors other than beta are able to explain portions of realised returns not
captured by beta.
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2.5.7 FamaandFrenchandthe“Death”ofBetaIn a series of landmark articles which were published in the 1990s (Fama & French,
1992, 1993, 1995 & 1996), Eugene Fama and Kenneth Frrench dealt what seemed
like a tremendous blow to the proponents of the CAPM and beta, causing the
financial press to announce the “death of beta” (Adcock & Clark, 1999, p. 217).
In 1992, their extensive empirical study entitled, “The Cross Section of Expected
Stock Returns” was published. Their main result was that: “two easily measured
variables, size and book-to-market equity, seem to describe the cross-section of
average stock returns” (1992, p. 451). They concluded that,“[their] tests do not
support the central prediction of the SLB [Sharpe Lintner Black] model, [i.e.] that
average stock returns are positively related to market βs” (1992, p. 428).
In their 1993 article, “Common risk factors in the returns on stocks and bonds”,
Fama and French built a three-factor model based on their 1992 results. They found
that stock market returns are explained by “[1] an overall market factor and factors
related to [2] firm size and [3] book-to-market equity”. These findings caused them to
become quite outspoken against the use of the traditional CAPM:
Many continue to use the one‐factor Sharpe‐Lintner [CAPM]model toevaluate portfolio performance and to estimate the cost of capital.despitethelackofevidencethatitisrelevant.Ataminimum.theresultshereandinFamaandFrench(1992a)shouldhelptobreakthiscommonhabit.(1993,p.54)
However, in concluding their 1993 article, they ended with some open questions in
which they implied that more work needed to be done to discover the fundamental
economic processes captured by measures like size and book-to-market effects.
In 1995, another Fama and French article appeared—in which they attempted to
investigate the economic reasons behind the success of factors like size and book-to-
market effects at explaining returns. They found that, “return tests cannot tell a
complete economic story” and that “size and BE/ME [book equity to market equity]
remain arbitrary indicator variables that, for unexplained economic reasons, are
related to risk factors in returns” (1995, p. 131). Unfortunately they failed to draw
conclusive evidence from this study:
We suspect that our failure to findmore systematic evidence that thecommon factors in earnings drive retums is due to noisymeasures ofshocks to expected earnings. But we have no evidence on thematter.And our colleagues in behavioral finance will surely suggest anotherexplanation.(1995,p.154)
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In 1996 they summarised and re-stated their model in the form of two equations,
shown in Equations 7 and 8 below (1996, p. 55 & 56):
Themodelsaysthat theexpectedreturnonaportfolio inexcessof therisk‐freerate isexplainedbythesensitivityofitsreturntothree factors: (i) the excess return on a broad market portfolio
(ii)thedifferencebetweenthereturnonaportfolioofsmallstocksand the returnonaportfolioof large stocks ( , smallminusbig); and (iii) thedifferencebetween the returnonaportfolioofhigh‐book‐to‐market stocks and the return on a portfolio of low‐book‐to‐marketstocks( ,highminus low).Specifically, theexpectedexcessreturnonportfolioiis:
(Eq.7)
where E R R , E SMB , and E HML are expected premiums, andthe factor sensitivities or loadings, , , and , are the slopes in thetimeseriesregression:
(Eq.8)
2.5.8 RevivedInterestintheCAPMandBetaThe damning conclusions of the Fama and French studies arguably caused a massive
resurgence (again, judging by the increase in publications on related topics in the
latter 1990s compared to the early 1990s) in interest in beta and the CAPM and
caused many of the big names in finance to rush to beta’s aid. Roll and Ross (1994)
again emphasised the difficulty of testing the CAPM (which Roll previously
highlighted in his pivotal article published in 1977):
Aswehave seen, though, theempirical findingsarenotby themselvessufficient cause for rejection of the theory. The cross‐sectional OLSrelation is very sensitive to the choice of an index and indices can bequiteclosetoeachotherandtothemean‐variancefrontierandyetstillproducesignificantlydifferentcross‐sectionalslopes,positive,negative,orzero.(Roll&Ross,1994,p.115)
Ross and Roll go on to provide additional backing for the CAPM with the
following statement: “surely the idea of a trade-off between risk and expected return
is valid and meaningful” (Roll & Ross, 1994, p. 115).
Fischer Black, of Black and Scholes option-pricing (Black & Scholes, 1972) fame,
also comes out in strong defence of the CAPM, noting that it is a model of expected
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return which is totally different to a model designed to explain variance. He seems to
chastise Fama and French in the following extract:
Black appears to criticise Fama and French for relying on data in the absence of
sound theory:
Fama and French do not seem to believe much in theory when theyestimateexpectedreturn.They(andmanyothers)relyheavilyondata.Theylookataveragereturnsoncertainfactorsasevidenceofexpectedreturnsforthosefactors.(Black,1995,p.168)
Black advocates the use of theory, finding it to be “far more powerful than data”
(Black, 1995, p. 169). He dismisses large multivariate tests where t-statistics are used
blindly as “data mining” (ibid.), noting that the so-called “anomalies” (gold nuggets)
preyed upon by these data miners are not anomalous at all if one understands theory:
All these formsofdataminingaremadeworseby thehugenumberofminers, both academic and nonacademic. “There's gold in them tharhills,”sincepeoplewhofindgoodwaystoestimateexpectedreturnscanmakealotofmoney.
Black also refers to an “overpublication problem” (Black, 1995, p. 169)—suggesting
that academic journals have become cluttered with hundreds of articles containing
empirical studies on data comprising, “conventional tests of statistical significance”
which “are almost completely invalid”.
In his article, Black also makes the ironic statement, “if beta had been dead, the
Fama-French results would have revived it!” (Black, 1995, p. 170). This “revival” is
exactly what transpired—journals filled up again with articles refuting the death of
beta. In fact, that literature in the latter half of the 1990s was arguably largely
occupied with CAPM/beta revival studies.
Authors like Pettengill, Sundaram, and Mathur (1995) initiated a new wave of
attempts to validate the CAPM with conditional beta models in their article entitled,
“The Conditional Relation between Beta and Returns”. In this article, they employed
a conditional method that, “considers the positive relation between beta and returns
during up markets and the negative relation during down markets” (1995, p. 115)
after noting that, “a positive relation is always predicted between beta and expected
returns, but this relation is conditional on the market excess returns when realized
[sic] returns are used for tests” (ibid., italics added).
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Kothari, Shanken and Sloan (1995) adopted a different approach and tried to
invalidate Fama and French’s findings by showing that average returns did indeed
reflect substantial compensation for beta risk, provided that betas were measured at
the annual interval. They believed that Fama and French’s results were contaminated
by “survivorship bias” (1995, p. 186) but acknowledged that they did find evidence of
a size effect.
Jagannathan and Wang (1996) present a conditional CAPM model which includes a
“return on human capital when measuring the return on aggregate wealth” (1996, p.
3) and find that their specification “performs well in explaining the cross-section of
average returns” (ibid.).
Authors like Ashton & Tippett (1998) again highlight Roll’s original critique and
conclude that Fama and French’s results are easy to reconstruct if the market proxy is
not entirely efficient:
Amongst themost important of these is the currently fashionable ideathatempiricalresearchshowstheSharpe‐Lintner‐Blackbetacoefficienttohavelittle,ifanyassociationwithriskyassetexpectedreturns.Indeedifoneistobelievetheliterature,simpleaccountingbasedmeasuressuchas the ratio of book to themarket value of equity and size are muchbetterproxies for equity risk thanbeta itself (FamaandFrench, 1992,1993, 1995 and 1996). We show, however, that such conclusionspossiblystemfromempiricalproceduresbasedonbetasestimatedfrominefficient indexportfolios.Andinterestingly,whenthis isthecaseit isnothard toconstructexamplesunderwhichotherriskproxiesreplacebeta in cross sectional regressions involving expected returns as theindexbecomesprogressivelymore inefficient. (Ashton&Tippett,1998,p.1345)
Just prior to the end of the decade, Farber, Roll and Solnik (1997) investigate the
relationship between exchange rate regimes and risk and De Santis and Gerard
conclude that, “that currency risk is priced in addition to market risk” (1997, p. 1910),
which is of great importance to the proposed research outlined by this document.
Adcock and Clark (1999) then end the decade with an article appropriately titled,
“Beta Lives—Some Statistical Perspectives on the Capital Asset Pricing Model” in
which they conclude that the CAPM (beta) is immortal in the sense that it is a
“theorem that relates expected values” (1999, p. 217)—that is, “it is a statement only
about the parameters of a multivariate probability distribution” (ibid.).
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2.5.9 Beyond2000The CAPM/beta studies appearing post 2000 tend to diverge into an even greater
number of directions. The post-2000 studies can generally be grouped into the
following categories:
Studies that assess the effect of international factors (e.g. US business cycles)
on domestic industries. For example, Ragunathan, Faff, and Brooks (2000)
and Park and Kim (2009) find that US industry business cycles tend to affect
domestic risk in Australia and Korea, respectively.
Re-runs of the major empirical CAPM tests in various markets. For example,
Drew & Veerearaghavan (2003) compare the traditional CAPM with the Fama
French empirical models in Hong Kong, Malaysia and Philippines and come
to much the same conclusions as those of Fama and French. Bundoo (2008)
conducts and empirical test on the Stock Exchange of Mauritius using an
augmented Fama-French three factor model.
Summaries of the findings of the many years of research on the CAPM and
beta. For examples see Laubscher (2002) and Subrahmanyam (2010) who
presents an excellent reflection on the last 25 years of research on the cross-
section of expected stock returns.
Studies which still try to discover the statistical significance of other
determinants of beta/variables which explain actual returns. Subrahmanyam,
finds that, “more than fifty variables have been used to predict returns” over
the years (2010, p. 27). These include findings that factors related to
momentum, value and growth stock approaches appear statistically significant.
See Au & Shapiro (2010) and Karanthanasis et al. (2010).
Studies which resurrect the bear-bull market debate.
Continued investigation into the statistical methods underlying the estimates
of beta. For example the non-parametric estimation method of Eisenbeiss,
Kauermann & Semmler (2007).
Studies analysing the effects of various significant events on levels of
systematic risk. For example Paleari & Redondi (2005) studied the effects of
regulation changes on company risk (beta), while Choudhry (2005) studied the
impact of the September 11 terrorist attacks on volatility and business risk in
the United States.
Studies extremely relevant to the line of research proposed in this document.
These include the study on “102 Years of South African Financial Market
History” by Firer and Staunton (2002) and the studies of Barr, Kantor and
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Holdsworth (2003, 2007) on the effect of the Rand exchange rate on the JSE
top 40 stocks.
Studies directed at the behavioural aspect of finance—e.g. Karceski (2002)
who documents how fund managers tend to chase returns.
2.6 ConclusionOne of the “most fundamental issues in finance is the relation between risk and
return” (Milionis, 2011, p. 306). Furthermore, despite almost fifty years of research
since the birth of beta, the overall picture “remains murky” (Subrahmanyam, 2010, p.
27)—at least in the empirical sense.
What is clear (to me at least) is that while non-believers have “announce[d] that
beta is dead” (Karceski, 2002, p. 560); beta and the CAPM are in fact, “immortal”
(Adcock & Clark, 1999, p. 217)—at least in the theoretical sense.
Roll and Ross describe the CAPM has having an “intuitive grey eminence13” (1980,
p. 1074). Fama and French, despite probably being guilty of killing beta, support this
view in admitting that the CAPM “offers pleasing predictions about how to measure
risk and the relation between expected return” (2004, p. 25).
Another point which can be argued (and easily defended) after reviewing the
literature, is that the world does not need another “multi-factor” (Ebner & Neumann,
2008, p. 383) regression model with a “rather arbitrary” (ibid.)—i.e. unsubstantiated
selection of regression variables.
What would, however, be useful is a tool for the financial practitioner. Barr, Kantor
and Holdsworth (2007) provide such a tools (that is easy to explain in economic
terms) and this report attempts to test how well these tools fit the data relative to the
traditional CAPM beta-based market model.
The next chapter describes the test method, along with the data used for the test, in
more detail.
13 Eminence is defined as, “fame or acknowledged superiority within a particular sphere” by the Online Oxford Dictionary.
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3 Method
This chapter explains the research approach, motivates the choice of data and describes the
data collection method. A step-by-step explanation of the method for testing the fit of the
various models is given.
3.1 ResearchApproachThis report uses a deductive research approach, focusing on the testing of existing
theory rather than attempting to build new theory (as would be the case in an
inductive approach). A quantitative method is necessitated by the fact that numerical
data, in the form of historical stock market returns, are analysed.
As outlined in Section 1.4, this study has three main objectives:
(1) To present a chronicle of the “life” of the CAPM and CAPM beta;
(2) To test the “goodness of fit” (to historical data) of the market model
versus the two BKH models;
(3) To explore how classic beta and the Rand beta vary as a function of time.
Item (1) was dealt with in Chapter 2. Items (2) and (3) are the focus of this chapter
and the remainder of the report.
3.2 DataCollection,ResearchDesignandSamplingHistorical data from South Africa’s Johannesburg Stock Exchange (JSE) were
retrieved primarily from Thomson Reuters Datastream or from I-Net Bridge in cases
where Datastream did not have sufficient data in the desired format, or for the dates
required for the study.
Four sets of data were obtained, namely: (1) the selection of shares to be evaluated,
along with their return history; (2) a suitable proxy for the market, along with its
return history; (3) a history of the South African Rand to United States Dollar (ZAR:
USD) exchange rate; and (4) commercially available betas14 were used for
comparison with betas calculated using OLS regression methods.
The sample of shares consisted of the JSE Top 40 shares as at the end of October
2011. These shares are shown in Table 1 along with their share codes and market
sectors. The shares have been grouped into mining/resource and non-mining/non-
resource categories.
14 These betas were obtained from Cadiz FSG.
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Table1:JSETop40sharesgroupedbysectorJSE
CODE COMPANY NAME SECTOR
ACL ARCELORMITTAL SA. Industrial Metals & Mining AGL ANGLO AMERICAN (JSE) Mining AMS ANGLO AMERICAN PLATINUM Mining ANG ANGLOGOLD ASHANTI Mining ARI AFN.RAINBOW MRLS. Mining ASR ASSORE Mining BIL BHP BILLITON (JSE) Mining EXX EXXARO RESOURCES Mining GFI GOLD FIELDS Mining
HAR HARMONY GOLD MNG. Mining IMP IMPALA PLATINUM Mining KIO KUMBA IRON ORE Industrial Metals & Mining LON LONMIN (JSE) Mining SOL SASOL Oil & Gas Producers ABL AFRICAN BANK INVS. Financial Services APN ASPEN PHMCR.HDG. Pharmaceuticals & Biotech. ASA ABSA GROUP Banks BVT BIDVEST GROUP General Industrials CFR RICHEMONT SECS. (JSE) Personal Goods CSO CAPITAL SHOPCTS.GP.(JSE) Real Estate Investment Trusts FSR FIRSTRAND Banks GRT GROWTHPOINT PROPS. Real Estate Invstmnt./Services INL INVESTEC Financial Services INP INVESTEC (JSE) Financial Services
MND MONDI Forestry & Paper MNP MONDI (JSE) Forestry & Paper MSM MASSMART General Retailers MTN MTN GROUP Mobile Telecommunications NED NEDBANK GROUP Banks NPN NASPERS Media OML OLD MUTUAL (JSE) Life Insurance REM REMGRO General Industrials RMH RMB Banks SAB SABMILLER (JSE) Beverages SBK STANDARD BK.GP. Banks SHF STEINHOFF INTL. Household Goods SHP SHOPRITE Food & Drug Retailers SLM SANLAM Life Insurance TBS TIGER BRANDS Food Producers TRU TRUWORTHS INTL. General Retailers VOD VODACOM GROUP Mobile Telecommunications WHL WOOLWORTHS HDG. General Retailers
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The constituents of the JSE Top 40 were selected for three reasons, namely:
(1) The JSE Top 40 shares are relatively well traded—i.e. issues of thin
trading (Firer et al., 2008) and the statistical difficulties associated with
calculating betas for thinly-traded shares could be avoided.
(2) This study is particularly interested in differences between resource
(mining) and non-resource shares. Resource companies in the South
African context typically have large market capitalisations and therefore
are mainly confined to the Top 40 shares by market capitalisation.
(3) A sample of forty shares is both manageable in the time period allocated
for this study and large enough to yield reliable statistical results.
The time period chosen for the study is from the beginning of 1994 to the present
(end October 2011). The start date of 1994 was chosen since it provides both a
sufficient time period and a “convenient” date in the sense that this is when the new
South Africa came into being. The JSE also arguably became more efficient after this
date as it became part of the world economy—prior to 1994, South Africa had been
relatively isolated from the world economy for political reasons.
It was also decided to use monthly data and a sixty month period over which to
perform the necessary regressions to estimate beta. This is in-line with generally
accepted practice (Damodaran, 1997). Since sixty months of returns were required to
estimate beta at a particular point in time, share data were required for at least sixty
months prior to the beginning of 1994—i.e. back to 1989.
For use in the regression equations, it was necessary to convert closing share prices
and the ALSI index levels to percentage total return form (i.e. inclusive of dividends).
This is achieved either by downloading data directly in total return form, or
converting it using Equation 9 (adapted from Damodaran, 2006, p. 110).
In Equation 9, , refers to the total return of share at time . Also, refers to the
current time period, 1 refers to the previous time period and is the share
closing price. A similar equation is used for calculating total returns on an index,
except that the index closing level is substituted for the closing price of a share.
% ,, ,
,1 (Eq.9)
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The ZAR to USD exchange rate also had to be converted into a percentage change
from period-to-period to be compatible with the remainder of the data. This was done
with Equation 10 (note that “ZAR” will henceforth be taken to mean % ).
1 (Eq.10)
The input data sets are contained in the following appendices:
Appendix B contains the monthly percentage total returns of the JSE Top 40
shares from the end of February 1989 to the end of October 2011 as calculated
using Datastream total returns index operator [RI] for each share;
Appendix C contains the monthly percentage total returns of the JSE All Share
Index (ALSI) as downloaded from I-Net Bridge using the AJ203[TR] code
and data type operator;
Appendix D contains the monthly percentage fluctuation in the ZAR to USD
exchange rate as retrieved from I-Net Bridge using the USDZAR[CL] code
and data type operator;
Appendix E contains the Cadiz Financial Services Group (FSG) betas and
related parameters like number of months used to calculate the betas,
coefficients of determination, standard errors and the percentage of days on
which the share was traded.
3.3 DataAnalysisMethodLinear equations were fitted to the data using 60 month “rolling”15 OLS regressions
from the most recent date (end of October 2011) back to January 1994. The three
regression formulae are listed below and are henceforth referred to as Model 1
(Equation 11), Model 2 (Equation 12) and Model 3 (Equation 13).
∙ (Eq.11)
∙ ∙ (Eq.12)
∙ ′ ∙ (Eq.13)
15 By “rolling”, it is meant that a 60 month moving regression window is used—analogous to the way that a moving average calculation rolls from period-to-period.
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Model 1 consists of the month-on-month total return on share on the left side of the
equation and; an intercept term16, , beta coefficient, multiplied by the month-
on-month total return on the market proxy, , and an error term, , on the right
side of the equation.
The intercept term, , and the beta coefficient, are calculated for each rolling
sixty month period using ordinary linear regression17. The beta coefficient calculated
using this method is identical to the CAPM beta calculated using Equation 5 on
page 7. The equation is repeated below for comparative purposes.
, (Eq.14)
Model 2 and Model 3 are analogous to Model 1, except that they each have two
independent variables, namely and in the case of Model 2 and
′ (ALSI “prime”) and in the case of Model 3. Therefore, multiple
regressions18 are used for calculating the beta parameters for Model 2 and Model 3.
Model 3 (Holdsworth, 2011) is a refinement of Model 2 which is designed to deal
with the multicollearity19 between changes in the exchange rate of the ZAR and
changes in the level of the ALSI (see ² of 0.26 in Figure 2).
16 The symbol is used here instead of the Greek letter alpha ( ) to avoid possible confusion with Jensen’s Alpha (Jensen, 1969) which is defined as the difference between and 1 (Damodaran, 2006). 17 The general form of a linear OLS regression equation is: (Lind, Marchal, & Wathen, 2010, p. 469). 18 The general form of a multiple regression equation is: ⋯ (Lind, Marchal, & Wathen, 2010, p. 506). 19 Multicollinearity is when independent variables are correlated (Lind, Marchal, & Wathen, 2010).
y = ‐0.5179x + 0.0105R² = 0.2553
‐15%
‐10%
‐5%
0%
5%
10%
15%
‐15% ‐10% ‐5% 0% 5% 10% 15% 20% 25%
ALSI
ZAR
Monthly ALSI vs ZAR (2006/11/30 to 2011/10/31)
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In order to do this, an additional OLS regression is required where the ALSI
(dependent variable) is regressed against the ZAR (independent variable) and the
residuals of this regression are taken to form ALSI prime. In other words, ALSI prime
consists of movements in the ALSI not explained by movements in the ZAR.
Again, since 60 months of data are used for the regressions, data for the individual
shares are required back to 1989 and data for the ALSI and ZAR are therefore
required 60 months prior to this date (i.e. a total of ten years prior to 1994, or 1984)20.
The procedure is illustrated graphically in Table 2. The idea is to obtain the
intercept, coefficients and adjusted coefficient of determination for each regression
window, for each model and for each of the JSE Top 40 shares. This requires a total
of about 34 000 regressions—214 regression windows from 2011/10/31 back to
1994/01/31, multiplied by three models (with one model requiring two regressions),
multiplied by forty shares.
The block of data marked (1) applies to Model 1, the block of data marked (2)
applies to Model 2, and the blocks of data marked (3a) and (3b) apply to Model 3.
Block (3a) consists of the regression to obtain ′ while block (3b) makes use of
′ in the same way that block (2) makes use of . All of these data are for
Impala Platinum (IMP).
The detailed procedure (applied per block, but explained for block [1]) is as
follows:
(1) Select the sixty months of IMP returns (dependent variable) versus ALSI
returns (independent variable) from 2011/10/31 back to 2006/11/30, as
indicated by the green lines.
(2) Perform a linear regression to obtain the intercept , , the beta
coefficient , and the adjusted coefficient of determination21,
(see yellow highlights).
The reason for using the adjusted coefficient of determination and not
the normal coefficient of determination for comparative purposes is that
an increased number of variables in a multiple regression equation
automatically makes the coefficient of determination larger, regardless of
whether the actual “fit” to the data is any better or not. The adjusted
20 Data for the ALSI were only available back to 1986. This is not problematic—the regressions simply start once there is sufficient data, meaning that the study effectively starts from 1996. 21 The adjusted coefficient of determination is equal to the coefficient of determination for a linear equation with one independent variable and zero intercept.
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coefficient of determination (Equation 15) adjusts for this upward drift in
² with increasing numbers of independent variables (Lind et al., 2010).
Essentially this “levels the playing fields”, making for a fair comparison
between the fits of a regression equation with a single independent
variable versus a multiple regression equation with multiple independent
variables.
11
1
(Eq.15)
The detailed regression output for the regression equation just described
is shown in Table 3. A comparison of the figures highlighted yellow in
Table 3 and the figures in green text above the yellow highlights in block
NO. DATE X : ALSI Y : IMP β1 a1 SEβ1 SEa1 SEY R²1 R²adj1 NO. DATE X1 : ALSI X2 : ZAR Y : IMP β3 β2 a2 SEβ3 SEβ2 SEa2 SEy R²2 R²adj2
R²adj1, IMP_AVG (avg. of all R²adj1 for IMP from 1994/01/31 to 2011/10/31): 0.355 R²adj2, IMP_AVG (avg. of all R²adj2 for IMP from 1994/01/31 to 2011/10/31): 0.363
(3) Model 3, the “refined” BKH model, is similar to the second model, but
with an adjustment for multicollinearity between the ALSI and the ZAR.
24 The third model is a refinement of the second model which attempts to deal with the issue of a degree of multicollinearity between the ALSI and the ZAR. This refined model was presented by Holdsworth (Holdsworth, 2011) but it is not entirely clear whether this refinement of the model was done my Holdsworth in isolation or whether the original authors (Barr and Kantor) were involved. In this research report, credit for both models will be given to all three authors since it is fairly clear from the articles referenced that the three authors work(ed) together closely on the development of these models.
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In the figures that follow, Model 2 was first compared to Model 1 (Figure 5), then
Model 3 was compared to Model 1 (Figure 6), and lastly, Model 3 was compared to
Model 2 (Figure 7). In all of the figures, resource shares are grouped alphabetically
on the left side of chart according to their share codes, while non-resource shares are
grouped alphabetically on the right side of the chart. The share codes for resource
shares are also depicted in red font while share codes for non-resource shares are
depicted in black font.
Figure 5 shows that Model 2 appears to outperform Model 1. This is because the
median25 values of the differences between adjusted coefficients of determination of
Model 2 and Model 1, i.e. , , , are positive in seven of the thirteen26
resource stocks and in 24 of the 25 non-resource stocks. However, it is also clear from
visual inspection of the minimum to maximum range and the first to third inter-
quartile ranges that the data are largely positive, again suggesting that Model 2
outperforms Model 1.
What is interesting (and unexpected) is that Model 2 appears to perform better for
non-resource stocks than for resource stocks. This is unexpected because of the
positive ZAR betas of most resource stocks (Barr, Kantor, & Holdsworth, 2007)—i.e.
the fact that resource stocks tend to perform well when the ZAR is weak and the fact
that the ALSI tends to suffer when the ZAR is weak. Therefore, Model 1 should not
perform well in terms of providing a good fit to the performance of resource
companies since one would expect accuracy to suffer when attempting to capture two
inverse movements with a single factor.
It is possible (due to correlation between the ALSI and the ZAR) that the ALSI
tends to perform poorly when the ZAR is weak and that therefore changes in ZAR are
automatically captured in the ALSI factor without requiring an additional ZAR factor.
In Figure 6, the performance of Model 3 and Model 1 is compared. The median
values of the differences between adjusted coefficients of determination of Model 3
and Model 1, i.e. , , , are positive in two of the thirteen resource stocks
and in 22 of the 25 non-resource stocks. These are rather mixed results. Clearly
Model 3 does not perform well for resource stocks, but seems to perform relatively
well for the non-resource stocks. Since portfolios tend to be comprised of both
25 The median is regarded as a better measure for comparison than the mean since the mean may be skewed by one large positive or negative outlier (Lind, Marchal, & Wathen, 2010). 26 Shares for which there was insufficient data to get results were omitted—hence the fact that there are fourteen resource shares in the original sample, but only thirteen of which yield results.
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resource stocks and non-resource stocks, the overall performance of this model is
questionable.
In Figure 7 the performance of Model 3 is compared with the performance of
Model 2. In this case, it is relatively clear that Model 3 is a poorer fit to the data than
Model 2—the median values of the differences between adjusted coefficients of
determination of Model 3 and Model 2, i.e. , , , are negative for all of the
resource stocks and are negative for sixteen out of the 25 non-resource stocks.
Overall this seems to suggest that Model 2 provides the best relative fit to the data,
followed by Model 3 and Model 1 (not in any specific order as it is debateable
whether Model 3 is better than Model 1).
In order to get an idea of the absolute fits (i.e. up to now the comparisons have been
relative), the average adjusted coefficients of determination of each model for each
share (for the entire data range of the study) are presented numerically in Table 6 and
summarised by means of box plots in Figure 4.
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Table6:AveragecoefficientsofdeterminationforModels1,2&3No. JSE CODE COMPANY NAME R²adj1AVG R²adj2AVG R²adj3AVG 1 ACL ARCELORMITTAL SA. 0.292 0.300 0.3062 AGL ANGLO AMERICAN (JSE) 0.691 0.711 0.6633 AMS ANGLO AMERICAN PLATINUM 0.401 0.433 0.3834 ANG ANGLOGOLD ASHANTI 0.220 0.219 0.1885 ARI AFN.RAINBOW MRLS. 0.303 0.315 0.2676 ASR ASSORE 0.057 0.049 0.0397 BIL BHP BILLITON (JSE) 0.568 0.582 0.5578 EXX EXXARO RESOURCES 0.269 0.260 0.2429 GFI GOLD FIELDS 0.189 0.196 0.16310 HAR HARMONY GOLD MNG. 0.137 0.142 0.13611 IMP IMPALA PLATINUM 0.355 0.363 0.34112 LON LONMIN (JSE) 0.252 0.278 0.27513 SOL SASOL 0.374 0.397 0.3991 ABL AFRICAN BANK INVS. 0.122 0.172 0.2062 APN ASPEN PHMCR.HDG. 0.024 0.118 0.1203 ASA ABSA GROUP 0.268 0.316 0.3084 BVT BIDVEST GROUP 0.303 0.329 0.3365 CFR RICHEMONT SECS. (JSE) 0.362 0.371 0.3436 CSO CAPITAL SHOPCTS.GP.(JSE) 0.193 0.329 0.3047 FSR FIRSTRAND 0.257 0.327 0.3238 GRT GROWTHPOINT PROPS. 0.019 0.044 0.0469 INL INVESTEC 0.264 0.290 0.27610 INP INVESTEC (JSE) 0.286 0.289 0.29111 MSM MASSMART 0.060 0.159 0.17012 MTN MTN GROUP 0.256 0.339 0.32213 NED NEDBANK GROUP 0.247 0.305 0.28514 NPN NASPERS 0.277 0.338 0.32615 OML OLD MUTUAL (JSE) 0.374 0.400 0.33616 REM REMGRO 0.223 0.239 0.21717 RMH RMB 0.280 0.345 0.32918 SAB SABMILLER (JSE) 0.432 0.428 0.42319 SBK STANDARD BK.GP. 0.295 0.350 0.33820 SHF STEINHOFF INTL. 0.323 0.324 0.30521 SHP SHOPRITE 0.087 0.198 0.21522 SLM SANLAM 0.209 0.307 0.29323 TBS TIGER BRANDS 0.244 0.300 0.30524 TRU TRUWORTHS INTL. 0.077 0.202 0.21225 WHL WOOLWORTHS HDG. 0.159 0.238 0.244Resource Average 0.316 0.327 0.304Non-Resource Average 0.226 0.282 0.275Overall Average 0.257 0.297 0.285
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Figure4:BoxplotsofaverageR²adjforModels1,2&3
These results also support the conclusion that Model 2 provides the best fit to the
data. One must be cautious of coming to the conclusion that the differences are
statistically significant, since these differences may simply be attributable to chance.
This is the reason it is desirable to state statistical measures with a confidence
interval—although, as already discussed, these data present statistical problems in
terms of independence and therefore statistical t-tests and ANOVA tables have not
been used.
The fact that this study has been unable to draw statistical conclusions from the
data is a weakness.
R²adj1AVG R²adj2AVG R²adj3AVG0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Box Plots of Adjusted Coefficients of Determination
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AppendixA:Results—BetaPlotsforJSEALSITop40
The graphs in this appendix plot a number of different betas as a function of time (from 1994
to the present) for the shares which made up the JSE ALSI Top 40 at the end of
October 2011.
The bold black line indicates “classic” beta, calculated by means of sixty month rolling OLS
regressions, regressing the share in question against the market proxy, the ALSI. The bold red
line indicates beta as calculated by Cadiz FSG. Cadiz betas are published quarterly. The
classic betas are very similar to the Cadiz betas.