MA831 EZEKIEL PEETA-IMOUDU DISSO

MA831 FINAL YEAR PROJECT
Testing the validity of the CAPM on the Volatility.
Ezekiel Peeta-Imoudu
1201252
BSc Economics and Mathematics
May 2015
Supervisor: Dr Haslifah Hasim
Department of Mathematical Sciences

Ezekiel Peeta-Imoudu 1201252
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Ever since the derivation of The Capital Asset Pricing Model (CAPM), a large number of
studies have been dedicated to investigating and assessing its validity and performance. These
studies have significantly impacted the field of financial economics, with some empirical
studies supporting the model, while others disputing and opposing the model. This paper
provides a comprehensive review of the CAPM, with the first part discussing the theory, as
well as the main literature on the continuing academic debate of its validity. The second part
is empirical, paying particular attention to testing validity by estimating Beta for 12 selected
companies on the London Stock Exchange (LSE) over the period 2001-2010, and comparing
it to actual Beta results by a paired sample t-test, regression analysis, Pearson correlation and
a Kruskal-Wallis test.
Research results show that the CAPM is almost completely valid in the LSE, with 11 out of
12 of the companies showing no significant differences between the estimated Betas and the
actual beta, hence providing evidence in support of the model. The overall result demonstrates
the CAPM is valid in the LSE.
Keywords: Asset, Pricing Model, CAPM, Security, Beta, Risk, Expected return, and
Market portfolio.

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Completing this dissertation without support and encouragement from friends and family
would have been impossible. For that reason I would like to thank them. I also wish to thank
Dr. Aris Peperoglou, for always willing to help and give his best suggestions.
Massive thanks to Dr. Haslifah Hasim, my supervisor, for her excellent guidance, caring,
patience, and provision throughout the course of this project. My deepest gratitude goes
towards her. She encouraged and challenged my growth and development as a writer,
researcher, experimentalist, a problem solver and someone who thinks independently. Not a
lot of supervisors give their students the opportunity to develop and embrace a sense of self-
sufficiency and individuality, by letting them carry out independent work, whilst still being
constantly available in case of help or clarity; however, you have done for me this remarkably.
Therefore for this and everything you did for me, I appreciate you. Thank you once again for
agreeing to work with me on this project; it definitely would not have been possible without
your help.
Furthermore, I would like to thank the Department of Mathematical Sciences of Essex
University for this wonderful opportunity to enable students like me to contribute to the world
of research, both theoretically and empirically whilst developing and learning new skills in
the process. Thank you for this challenge.
Finally, I would like to thank the Almighty God for his grace and strength, which saw me
through this project from start to finish, despite the difficulties and problems encountered. For
this I say, Glory Be to God.

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1.1 Overviewof Asset Pricing Models............................................................................................................7
1.2 CAPM Introduction...............................................................................................................................11
1.3 Research Objectives...............................................................................................................................13
2.1 Introduction..........................................................................................................................................15
2.2 CAPM Theory.......................................................................................................................................16
2.3 Evidence and Critique of the CAPM .....................................................................................................23
2.4 Advancements to the CAPM .................................................................................................................28
2.5 CAPM and Arbitrage Pricing Model (APT) Debate..............................................................................31
2.6 Conclusion.............................................................................................................................................35
3.1 Sample Selection...................................................................................................................................37
3.2 Data Selection........................................................................................................................................38
3.3 Beta Estimating and Testing Method....................................................................................................39
4.1 Empirical Results..................................................................................................................................43
4.2 Descriptive Statistics .............................................................................................................................45
4.3 Kruskal-Wallis Test..............................................................................................................................46
4.4 Regression and Time-Series Analysis ....................................................................................................46
5.1 Future Research....................................................................................................................................51

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1. Introduction
Asset Pricing Theory
Accurately measuring the trade-off that exists between the expected return and risk is one of
the main problems in a financial market, and the Theory of Asset Pricing helps to address this.
Professional investors and other people who in their daily life invest their money in one form
or the other will need to make key decisions from time to time; the behaviour of asset prices is
useful for this. To expand on this, an individual saving or investing in one financial form or
the other will make that choice depending on what they think of the risks and returns that are
related with the various forms of investment. Risk is an important factor for investors when it
comes to making investments as for example, the greater the risk of the investment, the less
likely it is for a person who is risk-averse to want to take the risk and make that investment,
unless the amount they get in return is large enough to compensate them for taking on board
the high risk investment. The Asset Pricing theory is aimed at recognising and measuring
these risks, and also assigning rewards for subsequently bearing these risks.
The theory explains and lets us understand important things for example, why expected
returns can change overtime, why two totally different stocks can have completely different
expected returns, and also why we can calculate for example, an expected return much higher
on like a stock than that of a short-term government bond. It also helps us to evaluate various
reasonable rates of return for various assets, as through financial awareness we are made to
recognize that investors like to hold well-diversified portfolios, and do not like lower
expected returns in comparison to higher expected returns. Well-diversified portfolios are
portfolios that contain a variation of securities with risks, which are closely approximated to
the market systematic risk, with the unsystematic ones diversified out.

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The information on the rate of return for the particular asset is crucial, as investors need to
understand the risk they encounter with a particular portfolio. This will help in their critical
investment decisions, which could possibly range from evaluating projects to forming
investment portfolios. These investment portfolios can even be assessed and the overvalued
and undervalued assets can be identified. If we view it from a corporate scenery, we realise
that companies can also look at the characterised risks of their potential acquisitions and
projects, whilst allocating a discount rate to reflect the risk, and then choose the project that
has a higher promised rate of return than what would be presumed by the risk theory, to help
them create value.
Asset pricing as a finance theory, helps to answer the fundamental question of how an
investment’s expected return is affected by its risk. Contrasting macroeconomic events and
frictions in the financial market are also linked with the risk related with the returns of asset
price, as a lot of significant decisions in economics that have to do with consumption and
physical investments rely on this. The fundamental role asset pricing plays is high, as the
mispricing of assets could contribute to financial crises, which could cause damage to the
economy, for example the recent economic recession in the UK.
In order to provide some insights into this and aid in in the prevention of mispricing assets, a
variety of Asset Pricing models have been produced to achieve this, although not every model
can be said to be faultless or impractical. The most dominating and significant of the Asset
Pricing models is the Capital Asset Pricing Model (CAPM); I will be discussing the
framework of the model, both theoretically and empirically in a lot more detail later on in the
paper, especially with regards to its validity.

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The remainder of this paper is organised as follows. The first part of the paper will give an
overview of Asset Pricing Models. Accordingly, the next part will introduce the CAPM
briefly and then research objectives will follow next, where I’ll briefly describe the main
purpose of the research. The Literature Review will then follow suit and comprise of further
expansion on CAPM by reviewing literature, discussing the background theory in a lot more
detail, and highlighting the various critiques of the model. I will correspondingly be outlining
observations made by previous researchers who came up with predictions of the Model after
theoretical and empirical evidence to support or contradict to the model.
The next section, Methodology, will address and discuss how and why we intend to achieve
the research objectives. Furthermore, we’ll move on to Analysis and Results, where we
examine the calculations and empirical analysis and explain the implications. Finally, the final
part of the paper will then be the conclusion section, which concludes the paper and gives an
answer to the relevant parts of the research objectives.
1.1 Overview of Asset Pricing Models
Asset Pricing Models formalize the accurate relationship between a financial asset’s expected
return and the way its risk is measured. They have majorly contributed to the world of finance
by their attempt to understand how these two variables are formalized, through calculating
and determining the appropriate return of a financial asset. The prices or the returns we expect
from financial assets in financial markets are described by the models, by using one or more
variables to determine an asset’s fundamental worth. By financial assets we mean assets such
as bonds, futures contracts, common stocks, etc.

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Investors who invest in financial assets do so in the hope of attaining some return for their
investments or financial wealth without losing the worth of their investment. With this there
exists a risky asset, and a risk-free asset; the asset labelled as risk-free having a return which
is certain, and the risky asset the asset with variability in expected returns which brings about
uncertainty, and the asset pricing models help explain and measure these risks so the investors
can make their decisions based on that information.
Moreover, these models are quite distinctive, for example, their different assumptions, and
they fit at least one of the specific circumstances. However amongst this, they still possess
few similarities, which are established on one or more of the subsequent concepts; the law of
one price, no arbitrage principle and the financial market principle, which are 3 economic
concepts.
Asset Pricing Models can be single or multifactor, depending on how many factors the model
looks at, with single factor models like the Capital Asset Pricing Model (CAPM) that uses a
single factor Beta to compare a portfolio to that of the whole market. This was proposed by
William Sharpe, as he realised that the return we expect on an asset relies only on its Beta. He
figured this by deducing that we cannot diversify a systematic risk, and the unsystematic risk
is specific to the earnings of a company, which can be moved through appropriate
diversification.
This single factor in the model is used to determine the expected return of an investment by
calculating the amount of risk in the investment. Single factor models use systematic risk,
according to William Sharpe, which we define as the risk of being in the market.

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Multi-factor models, on the other hand are quite different from Single factor models; they are
suggested to be an alternative to single factor ones, as they incorporate more than one risk, as
they allow the asset to have more than one measure of systematic risk, portfolios built could
contain either a risk factor itself or one that that contained a lot of stocks that had a relation
with a risk factor which was not observable. A model such as the CAPM with extra factors
added its formula can afterwards be considered as a multifactor model. An example of a
multi-factor model would be the Intertemporal CAPM (ICAPM) developed by Robert Merton.
These Asset-Pricing Models have been an important contribution to the financial world; proof
of this is in its wide usage and its continuous research by academics and constant effortless
domination in financial textbooks, but due to the simplicity, rational presumptions and
imaginative observations, the CAPM has still been the main utilised model and has not really
still found much competition, although it has come under rigorous testing on its validity and
validity of its assumptions. This has generally resulted in some people favouring it, and others
not.
Consequently through this, new inventive models have also emerged, which try to side-step
the problems of CAPM through contrasting approaches in how they compute their
calculations of asset prices alongside their presumptions, which challenges prevalent models.
To give a well-rounded overview of some of the major Asset Pricing Models, especially the
mainly few recent asset pricing models which are innovatively developed from the basis of
CAPM, I was able to construct the following table;

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Model Developed
By
Year Single or
Multifactor
Model
Brief Outline
Capital Asset
Pricing
Model
(CAPM)
Sharpe and
Litner
1964 and
1965
respectively
Single
Explores risk and return
relationship by expected return
using the Beta (, and measures
risks with the return of the stock
market covariance to that of the
securities
Consumption
based CAPM
(CCAPM)
Rubinstein,
Lucas and
Breeden
1976, 1978
and 1979
respectively
Single Similar to CAPM but here Beta
sensitivity is measured in relation
to the changes in aggregate
consumption.
Intertemporal
CAPM
Merton 1973 Multifactor The model assumes that there is a
continuous flow of time.
Arbitrage
Pricing
Theory
(APT)
Stephen
Ross
1976 Multifactor Has the assumption that the return
of each asset back to the investor
has several factors that control it,
but these factors are independent.
International
CAPM
Stulz 1981, 1995 Single Here the expected return is
calculated by measuring the
sensitivity is measure to the world
market index.
Conditional
CAPM
(Cond-
CAPM)
Jaggannatha
n and Wang
1996 Multifactor
In this model the return we expect
on an asset is related to the degree
of responsiveness of changes in
the economic state.
Fama and
French 3
Factor Model
Fama and
French
1992, 1993 Multifactor For this model, Fama and French
figured out that beta did not
explain the cross section of the
returns on stocks. They did this by
outspreading the traditional
CAPM to include explanatory
variables such as size and book-
to-market in explaining this stock
returns cross-section.
Carhart 4CH
Model
Carhart 1997 Multifactor The only difference between this
model and the Fama and French
three-factor model is the addition
of a price momentum factor.
Liquidity-
Adjusted
Asset Pricing
Model
Archaya
and
Pederson
2005 Multifactor They both discovered through
studies that liquidity also affects
the portfolio investment
performance, so to incorporate
this, they devised this model to
help explain the effect of liquidity
risk on asset prices.

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Fig 1. Table of some Asset Pricing Models including brief general background information.
From Fig 1 we can really see how Asset-Pricing Models as a whole, have acquired a long
history of theoretical and empirical investigation, which indicates how much authors have
attempted and successfully contributed to the development of the models, in order to progress
Asset Pricing in the world of finance and financial markets. It also shows the contribution
overtime of Researchers to Asset pricing in general through their studies and development of
models, through mostly the CAPM.
1.2 CAPM Introduction
Brief History
The Capital Asset Pricing Model (CAPM) was one of the major contributions to the financial
economics that transpired in the 1960s. The CAPM, as the first asset-pricing model, is
deemed as the most conspicuous model in the history of asset pricing. During this period
(1960s), a couple of researchers studied and used the Markowitz’s portfolio theory, to
formulate for financial assets, a theory of price information, which subsequently derived the
Asset Pricing Model famously known as CAPM. For this contribution to economic sciences,
in 1990, the researchers involved in this-Harry Markowitz were awarded the Alfred Nobel
Memorial prize.
Markowitz portfolio selection theory (1952), forms and originates the basis of the CAPM.
Briefly, it is a theory that examined the ways in which risks could be reduced, and even
though assets might differ in terms of their risks returns that we expect, we can still optimally
invest our wealth in them.

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Before Sharpe, the theory was developed by Tobin (1958), who presented the Separation
theorem and the efficient linear set theorem. Years later after this, Sharpe and Litner finally
developed the relationship between determining an asset’s return whilst taking into account its
risk, hereby introducing the CAPM. Merton Miller (who introduced the Intertemporal CAPM
later on in 1973) and William Sharpe were also presented the famous Nobel Prize in 1990
alongside Markowitz.
Furthermore, the introduction of CAPM has overtime brought about a wave of empirical
studies done by scholars and people in the field of research. The studies have all included
heavy debates concerning the results, which brings about disputes on the model, which I will
consequently highlight later on in the detailed literature review besides the paper’s own
approach to checking its validity.
Brief Overview
The CAPM is an equilibrium theory on expected return and risk measurement. It integrates a
Beta () factor or beta value of a share, which is very significant to the CAPM, particularly in
the formula. This Beta value contributes to the volatility and risk of the whole portfolio of the
market, which contains risky securities. Therefore any share which has an assigned beta co-
efficient value of below 1, will not have as much impact on the total portfolio of the market,
whereas we would expect shares that have an assigned beta coefficient above 1 to have an
higher than average effect on the total market portfolio.
Through the manner in which the equilibrium price is formed on the capital market that is
efficient, we can generate the relations between an asset’s Beta value, its expected return and
its risk premium, and state that the latter two will change in direct proportion to the former
one. An accurate composed portfolio, which contains risky securities, can allow an investor to
choose to bare himself to a sizeable amount of risk. For the attitudes of the investors towards

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risk, this can be seen in their selection of a risky portfolio combination and an investment
which is risk-free, but in CAPM in regards to the model stressing on what the optimal risky
portfolio should be composed of, it should depend on the investors’ future predicted
calculations of various securities. If the investors do not have as much information as their
counterparts with regards to investing, they are better off holding the same portfolio of shares
as the other investors. We choose to call this the market portfolio of shares.
One of the main significant contributions of CAPM is the measure of risk, which it provides
for an individual security, which is said to be very constant with the portfolio theory. (Weston
and Copeland, 1986). This can also then allow us to evaluate in a well-diversified portfolio,
the risks which are not diversifiable, known as un-diversifiable risk. (Weston and Copeland,
1986).
1.3 Research Objectives
The main purpose of this project is to fully review the CAPM model by providing a detailed
well rounded research on the CAPM, and also test the accuracy, by checking its validity to see
if it holds. For this research/project, the main objectives are to:
1. Provide a full comprehensive and detailed understanding of the Capital Asset Pricing
Model (CAPM) – Overall revision of its theoretical and empirical framework through
existing literature (Theory and Evidence, including disputations and advancements) in the
Literature Review.

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2. Estimate and Analyse Beta, where Beta is a way of measuring risk based on Volatility of
the stocks. So here we will be predicting estimates of Beta of these 12 companies listed on
the London Stock Exchange for a 10 year period and compare it to actual results. Also, we
will be able to analyse the Beta values, observing its various levels of volatility, in order
to explain and interpret the relative risk and return level of the stocks in question in
comparison to the market (FTSE100) and the actual Beta values for the time period
selected.
3. Further analyse the relationship between the estimated Betas and actual Betas graphically
and statistically by performing tests, to further explain the results/findings observed, and
further test the accuracy of the CAPM.

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2.
2.1 Introduction
Asset pricing models produced from Finance theory all have quite a long history of theoretical
and empirical investigation, and the existing literature for these models is vastly increasing, as
studies are continuously carried out on them. If we consider the Capital Asset Pricing Model
(CAPM) in particular, which was the first ever known asset-pricing model and an uncommon
revolution and valuable addition to economics, we notice that ever since its conception,
enormous efforts have been devoted to evaluating its validity. This research on the CAPM
also led to a lot academic researchers in the field of economics developing and advancing it,
hence emanating the innovation of other asset-pricing models, with CAPM as the basis from
which they were developed. Extraordinarily, studies conducted on CAPM over time have
also appeared to have given rise to numerous debates which has led to some people becoming
advocates to CAPM principles, with others not in full support of the model.
These distinctions in the already led studies have right now served as a real fortifying element
to this paper’s interest in the CAPM, but in order to undertake a test of its validity, we would
need to revise the underlying economic theory and existing literature for the CAPM- ranging
from the main literature of its derivation all the way to innovative advancements to the model.
Literature also containing evidence and critiques of the model by some academic researchers
who carried out theoretical and empirical studies to contribute to the evaluation of the validity
of the model, albeit in support or in opposition of it, will all be explored within the review.
The review will consequently address 4 main themes (which are adequately interlinked),
which we think have proved to be the most prevalent aspects within the scope of CAPM’s
literature. Ultimately, we will discuss how this research can contribute to the topic.
These themes are as follows:

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 CAPM Theory - Theory, Economic Intuition and Assumptions, Graph and Formulae
 Critique of the CAPM - Predictions, Evidence, Theory and Empirical tests
 Advancements to the CAPM
 CAPM and The Arbitrage Pricing Model (APT) Debate
2.2 CAPM Theory
The relationship between the risk and the return on an asset was advanced by Sharpe and
Litner in 1965 through the Capital Asset Pricing Model (CAPM). Originally, the model was
derived from the Markowitz theory of portfolio selection (1952), and although the Mean
Variance Analysis was developed by Tobin (1958) when he presented the concepts of the
Linear Efficient set and Separation theorem, the theory which involves mean variance
analysis, is an essential basis of the CAPM.
The MVA follows the assumption that during asset/portfolio selection by investors, the ones
which delivers the least possible variance for an expected return that is given or offers the
biggest expected return for a level of variance that is given are chosen. We call any selection
of this sort Mean variance efficient, else not efficient. Showing all the available combinations
of assets that either provide minimum amount of variance for given expected return, or
provide maximum amount of return for a given variance level, we get what we call the
Efficient Frontier (EF) which allows for all asset combinations in portfolio;
Figure 2.1- Diagram of The Efficient Frontier - Source: R. E. Bailey (2005)

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The point MR, the minimum risk portfolio, signifies for all potential expected return values
the minimum variance, and the FF curve on the upper part of the MRP represents the actual
efficient frontier portfolios there are considered to be efficient, and those on the inside
inefficient. The portfolios outside the FF curve are not considered to be feasible. This moves
us forward to the diagram of the mean variance with a slope called Sharpe ratio, which
measures the amount one-unit risk that can be compensated by excess return. So with this risk
level we are given we can see that, the lower the ratio, the lower a portfolios’ excess return,
and vice versa. (Bailey R, 2005)
Therefore if we were to consider which portfolio was efficient, we would look at the one that
has the highest slope. Here the efficient frontier is illustrated by the Capital Market Line
(CML), which can be described as the steepest tangent line to the FF frontier for risky assets,
as seen below.
Figure 2.2- Diagram of the Capital Market Line (CML) - Source: R. E. Bailey (2005)
From Figure 2.2, we see that the optimal portfolios of different investors, are located along
the CML, with the investors holding distinct amounts of the risk-free asset depending on their
behaviour towards risk, and this is what the CAPM envisages. The point M represents the
expected rate and standard deviation of return, depicted by MM respectively. This is also

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the point that identifies the market portfolio where the share of each asset that is risky equates
its share in the whole market. The CML goes through the r0 point on the vertical axis and lies
in tangent at the point M to FF, which is the portfolio frontier for risky assets only. The MV
Efficient portfolios are those laying along the CML, the only difference between them is their
proportion out of the total portfolio which got invested in the risk free asset. The tangent point
is the portfolio that only has the risky asset, with the portfolio consisting of just the asset that
is risk-free asset being the point on it. From the diagram we see how efficient portfolios are in
market equilibrium if every investor had the same conviction on means and variances. (Bailey
R, 2005)
This derivation of the Efficient Frontier clues in the Separation Theorem, a proposition
according to Sharpe (1964, pp.426). This states that by merging any other two portfolios in
the frontier, we can acquire every portfolios expected return and variance. So we can hence
breakdown the process of selecting portfolios by firstly selecting a combination of unique and
optimum risky assets, and secondly making a choice which is separate concerning how the
funds are allocated in the combination as well as the riskless single asset. Hence, the portfolio
which is the most preferred is distinct from the individuals risk attitude. Investors ready to
combine the market portfolio with the risk-free asset will find the relationship tracing out the
efficient combinations and risk (CML), available. So through the CML, these investors are
wise enough to identify the advantages of building a well-diversified portfolio which will
trace out all their optimal risk-return combinations (Richard Pike and Bill Neale, 2009).
Richard Pike and Bill Neale (2009) speaks about how in order for a theory to simplify an
analysis and expose the vital relationships between key variables, it relies on assumptions.
Generally, validating a theory will not depend on the practicality of its assumptions, but on
the empirical correctness of its predictions, which if they do not match with reality due to no
empirical errors or random influences, then we can re-assess the assumptions.

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Hence why the assumptions of a theory are important. Therefore, outlining the most important
assumptions of the CAPM for all investors according to Richard Pike and Bill Neale (2009),
they are;
 Maximizing expected utility enjoyed from wealth-holding is an aim for all investors.
 Common single-period planning horizon is what all investors operate on.
 All investors choose from alternative investment opportunities by considering the
risks and expected return.
 All investors are rational and are risk-averse.
 All investors arrive at comparable assessments of the probability distributions of
expected returns from securities, which are traded.
 Expected returns are normal for all such distributions.
 Unlimited amounts can be borrowed or lent by all investors at a similar common
interest rate.
 In trading securities, there are no transaction costs involved.
 Both dividends and capital gains are taxed at the same rates.
 All investors are price takers which means that not one investor can influence the
market price through the scale of their own transactions.
 All securities are highly divisible meaning they can be traded in small portions.
The CAPM calculates the Expected return of a security/asset with the emphasis of investors
needing to know the Risk Premium for the total portfolio, which is the extra amount that is
needed to compensate an investor for taking a risky investment, and Beta of the security
against the market, which is the degree that the security is an alternative for market investing.
The premium of this security is calculated by the part of its return that perfectly correlates
with the market, with the parts that do not perfectly correlate diversified away without
demanding a risk premium. According to the CAPM model, the expected return of an
investment to an investor will equal to the Risk-free rate, which is just the rate of return on an
investment with no risk (return of that investment acknowledged with certainty); for example
Government treasury bills, plus a whole security premium which is greater than that of the
risk-free rate, times the risk factor for bearing the investment.

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Mathematically, this can be expressed as;
j = Ro + βj (m – Ro)………………………… Equation (2.1)
Where
1. j is the Expected Return on the asset j,
2. Ro represents the Risk-Free rate,
3. βj as the value of the Beta of asset j which signifies its risk,
4. m signifying the market’s expected return, and
5. m – Ro) representing the market Premium.
This relationship is what the CAPM predicts, revealing a function of the expected return of an
asset j and the market expected return with a slope of j, and this can be further illustrated as
the Characteristic line, as shown next;
Figure 2.3- Diagram of the Characteristic line - Source: R. E. Bailey (2005)
This characteristic line shown above uses the prediction of the CAPM we know to be j – r0 =
(M – r0) j, as a linear association amongst (j- r0) and (M – r0) with slope of j, and with
every of the assets having its own characteristic line, but distinct, depending to the value of j.
(Bailey. R, 2005)
Equation (2.1) above makes us understand that a linear combination of portfolio M return and
the return that is risk-free, will give us the expected return on a security (j), with the co-

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efficient Beta (measuring the security’s risk and relating it to the security’s covariance
with the portfolio M tangency. So therefore, this expected return on the security will be equal
to the addition of the risk premium and the risk-free asset which would depend on how risky
the security is. This equation of the CAPM, depicted by Equation (2.1), is commonly denoted
as the Security Market line (SML).
Here (SML), we consider the returns we expect to be linear, and we can mathematically
express the co-efficient Beta value for an asset j as:
j = jm / 
2
m……………………………………Equation (2.2)
This SML line shows the associations which must be fulfilled amidst the beta and return of
the security, and also the return from the M portfolio. It is illustrated in 2 diagrams below;
Figure 2.4 and 2.5- Diagrams depicting the Security Market Line - Source: R. E. Bailey (2005)
In Figure 2.4, the SML predicts that the all the beta-coefficients for all assets, its portfolios
and the average rates of return will be located along the SML, and it interprets as a linear
association between the Beta (j) and the expected return (j) of an asset (j), the CAPM
prediction, which is j = r0 + (M – r0)j. Figure 2.5 on the other hand, highlights the
disequilibrium in the CAPM, as from the diagram we can see 2 points A and B. The point A
lies above the SML, meaning the return rate on the asset A is higher than what the CAPM
predicted given the co-efficient of the Beta, whilst for asset B, it is lower than the prediction
of the CAPM. Hence, from this we say asset A and asset B are under-priced and overpriced

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respectively, both of them conditional on the CAPM’s validity. (Bailey. R, 2005). When that
happens the CAPM is said to be in disequilibrium, as the points are not on the line.
All three main diagrams I have highlighted; The Capital Market Line (CML), Characteristic
line (CL) and the Security market Line (SML), all depict 3 key relationships for the CAPM.
The CML indicates the risk premium that is required for any portfolio that comprises of the
the market portfolio of risky assets and the risk-free asset, while the CL helps to express the
relationships between the expected return on a particular security for expected return values
on a portfolio that are given, and lastly the SML denotes the suitable appropriate return on
separate assets (and portfolios which are inefficient) (Richard Pike and Bill Neale, 2009).
Since the CAPM assumes for investors that when they calculate their rate of return, only risks
that are systematic will be considered, which just means any risk which affects a great amount
of assets either each one a greater or lesser degree (non-systematic only affects a single one
without effect on all the assets), the CAPM gives the ensuing 3 implications stated by Fama
and Macbeth (1974), some of which are considered to be testable and are as follows;
 On a security, there exists a linear relationship between the risk (Beta) of a return and
its expected return.
 Beta is seen as the total measure of risk of a security, so other variables would not be
explanatory so the intercept of the equation should be the risk-free return.
 There is an association of Higher risk (high values of Beta) with Higher expected
return i.e. Rm) – Rg > 0.
Shortly after the birth of the CAPM, Black derived a modified version. Black (1972)
narrowed down the assumptions into three sets of conditions, which are:
 Assets markets are in equilibrium.
 The behaviour of investors are in line with the principle of mean-variance

23
 All investors possess homogenous beliefs about their decisions on the values of mean
variances and covariance means.
Black’s CAPM brought out the assumption of investors being able to lend or borrow any sum
with the notion that the risk-free asset does not exists. Black derived this same prediction by
mathematically expressing any asset j’s equilibrium expected return in equilibrium as;
j = R0 + j (m – R0) ………………………………...Equation 2.3
From Equation 2.3, R0 represents the return on assets with Zero value of Beta. The 0 beta
value imply that the there is no correlation with the market portfolio and the asset return.
Therefore the insinuations of Black CAPM, which are testable, are alike to the previous one
apart from the point that the intercept does not have to be the risk-free rate.
Overall, the CAPM proposes a way for investors and others to evaluate their investments, by
just assessing and comparing expected return and required return. If they find out that the
former is not favourable, then they would not embark on potential investment in that actual
security.
2.3 Evidence and Critique of the CAPM
As some considerable research has been conducted over the years in order to test validity of
the CAPM, some of the results have provided evidence backing the Capital Asset Pricing
Model (CAPM), while other findings have conferred substantial evidence, which has led to
questions on the validity of the model. The studies which provide support of the model are the
two classic studies; Black, Jensen and Scholes (1972) and Fama and Macbeth (1973), whilst 2
of the main prominent studies providing evidence not in favour of the CAPM are the Fama
and French study (1992) and the Powerful critique of all CAPM empirical tests made by
Richard Roll (1977).

24
Majority of these studies focused on the how effective Beta is in explaining the historical
returns of the portfolio, and tested on the New York Stock Exchange (NYSE), and there will
be a brief discussion and summary of these studies, alongside a few others that gave responses
on these studies, as these were the most prominent as discovered.
Black, Jensen and Scholes (1972) utilised the portfolio of the entirety of stocks which were
traded on the stock exchange as their market portfolio proxy, which was equally weighted.
They tested both the CAPM and the Black version (zero Beta) using time-series and cross
section methods, with the cross section methods carried out for the sub and whole period due
to how big the data was. They were able to compute the relationship between the portfolios
betas and the monthly average return between the years of 1926 to 1966 (40 years). Using this
massive amount of data, they came up with findings from this study, which showed an
astonishing close-fitting relationship between both variables.
Although the slope and intercept from their study appeared to be significantly distinct and
larger than the mean risk-free rate of return across the period studied, they observed and
deduced a positive linear relationship between the beta and average return, which is why they
chose not to reject this prediction of linearity by the CAPM. Also, the beta value explained
most of the variations in the returns as the R-square value of 0.98. Even in the time-series test
where there was a regression of every portfolio excess return versus that of the market, results
consistent with the CAPM prediction were found, as the results of 10 regressions done has
intercepts done for every portfolio which were mostly insignificantly distinct from zero.
There was however the intercept change which was not random; high risk portfolios were
negative whilst other portfolios were positive which led to strict CAPM rejection, but Black,
Jensen and Scholes demonstrated the model with zero-beta having a positive relation, which
holds therefore the empirical results suit the Black Model better, as with the intercept of the
Black CAPM, there is no worry.

25
The Fama and MacBeth study (1973, 1974) also provided evidence similar which favours the
CAPM. Fama and Macbeth (1973, 1974), carried out the test using about 20 portfolios from
NYSE securities (similar time period as Black, Jensen and Scholes) to address the problem of
measurement error on beta. The tests concentrated on two of CAPM implications, which were
i) how linear was the expected return of portfolio beta, and ii) whether the expected return
was determined only by the portfolio beta and not the portfolio residual variance.
The portfolios are built such that the portfolio beta has a minimised measurement error. In
summary, the study divided the period of 15 years of the stocks into three 5-year periods and
grouped the stocks according to Beta ranks, then ensured unbiased beta estimation in every
portfolio after which regressions were done portfolios betas in the third sub-period
The regression, which was cross-sectional was carried out separately each month to obtain co-
efficients each month, with the all monthly average values calculated for each co-efficient and
then a t-test to was further carried out for testing significance. The main logic of the test that
one of the values would be equal to the average risk-free rate, the second equal to excess
return on market and the last two from the equation developed equal to 0. However the first
value did not equal the risk-free rate, which is a fail but did not contradict the Black CAPM,
while the second and last 2 values where greater than 0 (positive) and not considerably
distinct from 0 respectively. Therefore the overall study favoured the CAPM theory, as it
revealed the positive relationship between the betas and returns whilst even variables like the
residual variance did not explain the variation in the return.
Despite these evidences to support the model, the CAPM was still subject to critique, which
arose as a result of the conducted empirical studies. One of the most prominent contradictions
to the CAPM was done by Fama and French (1992, 2004). This study was also done with
traded stocks on the NYSE and other stock markets, but between 1963 and 1990.

26
Here the same method as the Fama and Macbeth (1973) was used by regressing the returns on
the various combinations of the explanatory variables which range from beta, Earnings to
price ratio (E/P), debt to equity ratio, the Book-market ratio (B/M), etc. which resulted in
different results to the Fama and Macbeth study.
By analysing the beta size and market size, they get an insignificant and negative co-efficient
value of beta, with the market size co-efficient also negative but significant. The other study
found the same consistent result. However, leaving the Beta alone as the sole variable brings
about worse results as it leads to a t-statistic, which is smaller, implying that in explaining
returns, Beta does not play a significant role, and this challenges the early study of CAPM.
Fama and French insinuated an explanation for this, saying that in representing the population
one of the sample periods may not be appropriate so; the difference in results is brought about
by the difference in time period. Also, regressing the (B/M) and the (E/P) alone show that in
explaining returns, they all play an important role, which is in dissimilarity to when the Beta
was left alone. To conclude, Fama and French deduced that size of market and (B/M) are
decisive variables in explaining returns, with Beta not as significant in explaining returns.
Therefore size of the firm and variables other than beta predict returns observed better, which
is why they oppose the CAPM, as it lays emphasis on Beta as the main explanatory variable
for returns.
One famous critique, presented by Richard Roll, famously called Roll’s Critique provided
additional dispute to all attempts to test the CAPM, and gave a conclusion that the CAPM
cannot be tested. He uses the basis for this critique as the efficient of the market portfolio
insinuation in CAPM, since market portfolios include all type of assets held as an investment
by anyone, ranging from bonds to stocks etc.

27
When applying, it means that the market portfolio is considered to be one that is not
observable, and individuals regularly use a stock index as a proxy replacement for the market
portfolio that is true. This substitution was debated by Roll as not safe and could result to
false interpretations of CAPM’s validity. Due to this inability, he believes that the CAPM
might not be testable empirically, and empirical tests done for the CAPM must comprise of all
assets available to investors.
Further evidence of Rolls worry of the CAPM not only being an empirical problem is
presented by Campbell et al. (1997). Here some of the studies indicated rejecting the CAPM
with a proxy would also imply the rejecting the market portfolio CAPM that is true, provided
correlation between true market and proxy market return surpasses around 0.70.
This shows that Rolls worry of the CAPM is not an empirical problem as even Stambaugh
(1982) conclusions, which were similar even when a proxy containing just stocks or also
stocks, bonds, etc. were used. This implied that a proxy that is good does not essentially have
to comprise of a wide range of assets.
Closer to that same year of Stambaugh, Banz (1981) and Reinganum (1981) both published
studies which provides evidence against the CAPM, with Reinganum’s study analysing
various anomalies that challenge CAPM such as the grouping of the portfolios by the market
value or the price earnings ratio (P/E) and the Banz (1981) study regressing returns on 25
portfolios on their Betas from a period of 1926- 1975.
By usually splitting his data into 6 sub-periods he realised a negative and significant market
size coefficient from 4 of 6 of the sub and the whole periods. His studies was one of the first
studies to find the “small-firm” effect, which was that of the CAPM predicting lower returns
than the returns on firm stocks that possessed market value that was relatively small.
Therefore the value of the Beta as a whole does not apprehend the whole risk.

28
Reinganum also came up with a conclusion which was consistent with Banz, which was that
the firms size which related the price earning ratio and market value is a likely missing factor,
as he found out in his portfolio groupings that the portfolios with a either a low P/E or a low
market value inclines to higher returns than prediction. He also found out that the effect of the
P/E is negligible we run together with Beta, market value and P/E. This is also consistent with
Banz.
2.4 Advancements to the CAPM
To further highlight the significance of the CAPM, numerous different expansions of this
model were presented after it. I will now provide a synopsis on some of the various literatures
I have researched on a few of the innovative asset pricing models derived from the basis of
the CAPM. The first two expansions of the CAPM we observed from the research on CAPM
literature are the Intertemporal CAPM (ICAPM) and the Arbitrage Pricing Theory (APT).
They are multifactor models introduced through Merton (1973) and Ross (1976) respectively.
The Intertemporal CAPM was Introduced by Merton as the CAPM was viewed as a static
model, as in CAPM the amount put into assets was set for a given time period, and this was
deemed to be improbable for the reason that the portfolio of investors can be rebalanced at
any time. Therefore according to Merton, he developed an equilibrium asset pricing model
(ICAPM) which had the same straightforwardness and empirical accordance of the CAPM,
consistent with the limited liability of assets and expansion of expected utility, and also makes
sure there is a consistency between the empirical evidence and the specification of
relationship among yields. The APT of Ross (1976) approached the idea that the long average
return is only affected by a minimal amount of systematic influences. The APT has been
presumed as a better alternative to CAPM by some, which will be discussed in the next
section.

29
The Consumption Capital Asset Pricing Model (CCAPM) of Rubinstein (1976), Lucas (1978)
and Breeden (1979) is another advancement of the CAPM, but based on the consumption in
the economy. Breeden (1979), did not base this model on financial wealth, as in comparison
to consumption, consumption was an adequate statistic for a dollar’s marginal utility.
Meaning that when we are not comfortable financially, we would more than welcome a few
more dollars to spend on consumption, than if we were comfortable. The model is similar to
the conventional CAPM as it allows assets to be priced with a single beta, but it measures this
beta by using the covariance of the assets return with aggregate future consumption, rather
than wealth.
Also, Fama and French in 1996 disputed that the CAPM Beta cannot explain the expected
return of Stocks, alone. Hence to address this, Fama and French (1993, 1996 and 1998)
derived a multifactor model commonly known as the 3-Factor model to help use numerous
factors like book to market and market capitalisation to explain the average return on an asset.
In 1996, another version of the CAPM was derived by Jagannathan and Wang. They derived
this by considering a dynamic economic and an unobservable return on portfolio of total
wealth, which according to them they subsequently denoted that the assumption of the CAPM
as being dynamic allows its betas and expected return to differ over time, then the statistical
rejections and effect of model requirements become weaker. They justified this by taking the
assumption of Beta (change with the business cycle. This version is called the Conditional
CAPM (CCAPM).
Literature on stock market efficiency by Jegadeesh and Titman (1993) posited the one year
momentum and also stated how stocks which normally perform best (worst) over a three- to
12-month period will usually continue to perform well (poorly) over the subsequent three to

30
12 months, as they possess the tendency to produce positive abnormal returns of about
1percent in the following year, per month. This one year momentum ideology was captured
by Carhart (1997), and he added a fourth factor (price momentum factor) to the Fama and
French 3 factor model, extending it. He extended this model by adding a price momentum
factor, which would explain the abnormal returns in momentum-sorted portfolios, wherein
which the price momentum represents the tendency of firms with positive past returns to earn
positive future returns, and vice versa. We know this model as the 4-factor model.
Since investors not only invest domestically but abroad, some researchers tried to address this,
as they felt that existing asset pricing models all considered domestic assets with the
assumption of investors all based in the same country. One researcher called Stulz (1981),
mentioned his ideology of the proportionality of the real expected return of an risky asset with
the covariance of the assets’ home country with changes in the rate of consumption in the
world. His argument also highlighted the Traditional CAPM, as being only appropriate for an
asset in traded in a closed financial market. This subsequently brought about an extension of
the CAPM known as The International Capital Asset Pricing Model (Int-CAPM), which has
given a theoretical structure for incorporating investments made abroad in asset pricing.
Finally, liquidity risk has an effect on stock pricing, and there is pretty much strong evidence
for this (Amihud and Mendelson (1986) and Archarya and Pedersen (2005)). Acharya and
Pederson (2005), derived one of the most recent asset pricing models,which gives an intuition
of the effect of liquidity risk on asset prices. Archarya and Pederson (2005) made a reference
in their paper to Chordia et al., (2000); Hasbrouck and Seppi, (2001); Huberman and Halka,
(1999), which are previous studies that comment that the performance of an investment
portfolio, which is important in calculating expected returns, are affected by liquidity. The
finance term “liquidity” in this case, said to be generally denoted as the ability to exchange
vast amounts rapidly, and effortlessly without moving the price (Pastor and Stambaugh, 2003).

31
The study suggested that if this liquidity is important for an investor, since it affects the
portfolio investment performance, it should be priced, which is why it got developed from the
traditional CAPM to become the Liquidity Adjusted Pricing Model (LAPM).
2.5 CAPM and Arbitrage Pricing Model (APT) Debate
There have been various remarks towards both the CAPM and the APT; a few studies
comparing and contrasting both asset pricing models, and other studies proposing the APT as
a better or demonstrable alternative to the traditional Sharpe-Litner CAPM, as they see the
APT as new and different approach to asset pricing determination, due to the differences in
certain parts of its assumptions, insinuation and method. Both models show are in the market
equilibrium, risky assets are priced and also provide investors with estimates on required rate
of return on their respective investments, or securities. Both models are built on the standard
of Capital Market Efficiency. However, the main difference between both however, would be
one of CAPM being a single factor model with systematic risk (Beta) being the sole
determinant of expected return, whilst the expected return of the APT on the other hand has
more than one single factor as its determinant, so the APT is considered to be a multi-factor
model. The APT does not try to explicate the causes of the returns of securities, while the
CAPM does, which is another difference.
The theory was developed by Stephen Ross (1976), the economist, and is represented by the
equation;
rj = ojC FC jD FDjK FK j……….. Equation (2.4)
Identifying the variables in the equation, we have rj as the rate of return expected on random
security variable j, FC and FD representing non-diversifiable factors C and D respectively
which could continuously go on, hence the “…”, osignifies the estimated return levels for j,

32
with all indices possessing a zero value, jK indicates the sensitivity to factor k of the j
security return; and finally j is the residual term or characteristic risk, independent across
securities.
In John Wei (1988), the assumptions that are for the most part utilized in the derivation of the
APT are i) All Investors display homogenous beliefs that the stochastic properties of the
capital assets return are consistent with a K factors structure which is linear, ii) Its either a
competitive equilibrium is what the capital markets are in, or no arbitrage opportunities, iii)
The large numbers theory are applied to the amount of securities in that economy, as they are
either so huge or infinite, and iv) the amount of factors, k, can either be already known (in
advance) or the investigator estimates it accurately.
This APT model and the CAPM model both differ on what they both emphasize in regards to
returns on asset and the role of the covariance in this; the former stresses the role between
asset returns and exogenous, whilst the other accentuates the co-variance between asset
returns and the endogenous market portfolio instead (John Wei, 1988).
Creating a well-diversified portfolio with no senilities to each factor is possible with adequate
securities, hence enabling the portfolio to offer a zero risk premium as it is effectively risk
free. This is what the APT illustrates, and literature from Brealy and Myers (1981) also
deduced a difference between the CAPM and the APT in what the risk premium depends on.
With the APT, the risk premium of securities bank on the sensitivity of these securities to
each of the factors and how related the risk premiums are with one another.

33
This is different from the instance of the CAPM, where the risk premium in this model is
decided by the securities risk level which we know to be systematic, times the risks’ market
price. The systematic risk specified is total risk times the extent of the correlation of its
returns to that of the market portfolio.
Brealy and Myers (1981) further made an interesting statement, by stating that the CAPM and
the APT can be equal if there is proportionality between the risk premium expected from each
portfolios and the market risk of the portfolio. In regards to similarity, Weston and Copeland
(1986) highlighted that the CAPM and the APT are very similar in the aspect of its
application; in the sense of using the models to determine how much capital cost for
estimation and its budgeting, both of them can be utilised as part of literally the same way.
When testing the APT empirically, we get a process called factor analysis which has been
used to identify applicable factors, and a range of these factors have risen to be likely
determinants of the actual value of common security returns (Emmery and Finnerty, 1991).
The factors could not be identified easily, and the APT does not really say which of the
factors have relevance in regards to the economy and their behaviour, but it suggests the
relationship between a limited amount of factors and returns on securities (Van Horne, 1989).
Moreover, the APT has lot of Beta factors and is derived in a totally different way than the
CAPM, and according to Weston and Copeland 1986), the return of an asset cannot be easily
analysed against randomly factors found, but in order to extract the factors underlying all
security returns, the same factor analysis mentioned earlier must be employed. This I believe
is quite different from the CAPM as with CAPM we regress the return of an asset that of the
market portfolio when estimating it.

34
Literature studies on both the CAPM and the APT have debated which model is preferred to
the other. Although some have viewed the CAPM to have simplicity as its plus side, on
testing it, they come across 2 problems, one being the CAPM only focusing on expected
returns as its concern and also the issue that all risky investment should be involved in the
market portfolio, whereas majority of the market indexes only hold a sample of common
stocks (Brealy and Myers, 1981).
With respect to the debate of how much advancement can be achieved when utilising the APT
instead of using the CAPM, the study of Roll and Ross (1980) claimed that APT is more
susceptible to testing than CAPM. The study claimed this because all assets returns it’s not
essential to test them, and also, there is no distinct role for the market portfolio.
Weston and Copeland (1986) deemed the CAPM as not a good tool for making decisions and
with the CAPM; accurate beta estimation is tough since the Betas tend to change overtime.
Moreover, despite these comments on CAPM, Cooley and Roden (1986) still believe that in
any occasion the CAPM offers us a comprehension of how investors act, and their market
dynamics. The model might not be considered as perfect, however it is useful as it provides a
few noteworthy perspectives in to the main considerations of security price determination.
This has also aided in predictive ability for companies for instance, the idea that low beta
securities are less volatile and produce a much lower return than high beta securities.
However, studies of Chen (1983) and Roll and Ross (1983) looked to suggest that the APT is
an upgrade of the CAPM, notably when some CAPM anomaly has been found in the security
returns.

35
Regardless of this conjectural dispute, and the struggle of finding correct Betas, the CAPM is
still used by investors as they like how it shows the interaction of key variables, and how it
relates risk and returns systematically (Cooley and Roden, 1986). Even additional studies
such as the Brown and Weinstein (1983) study did not really find any noteworthy differences
between both the CAPM and APT.
The question of replacing CAPM with APT has been subject to much debate by many, and I
believe that to determine this, sufficient research is compulsory. Both models are two hugely
important asset pricing theories; nevertheless, according to Van Horne (1989), the APT could
well become the main asset pricing theory, with the CAPM as a theme of it, according to Van
Horne (1989).
2.6 Conclusion
In a lot of empirical studies on the CAPM, the CAPM was advised as invalid and not able to
explain expected return by its market Beta alone. However, Beta as a measure of risk is still
useful and despite all the empirical evidence against using the CAPM, it still remains a
valuable tool for approximating cost of capital as well as studying market efficient events, and
evaluation of performance (Laubscher, 2002). In relation to this, another extensively held
argument was the market proxies utilised adequately representing the efficient market
required by CAPM, and this further gave rise to the argument of the unachievable CAPM
equilibrium.
Furthermore, the ATP now outdoes the CAPM as it has been proven to succeed empirically,
and the mere fact that it incorporates multiple factors affecting asset returns rather than just
one like the CAPM, makes it more realistic in financial markets.

36
Also, the ATP enables investors to select variables, making it more aligned to the
requirements of a pricing model which can be considered as universal. However in cases
where appropriate the same factor could result to misuse of the APT.
A thorough review of the existing literature for CAPM has revealed that there is still a
considerable amount of research that could still be done. Some of the literature chosen and
discussed casted light on both theoretical and empirical studies on CAPM, addressing and its
validity. This is one of the major reasons which encouraged and inspired me to carry out
research/study in an area (London Stock exchange) where I can also contribute to this testing,
empirically. Therefore, the contribution of this study will be to examine CAPM, by
empirically estimating the Beta of 12 company Stocks listed on the London Stock Exchange
(LSE) over a 10 year period, and analysing it to see if the CAPM holds in its ability to
accurately predict Beta, in that industry by comparing it with the actual Beta for the same
time period.

37
3. Methodologyand Data
This section introduces the methods of testing the validity of the CAPM. It presents the data
outsourced and describes the method applied to conduct studies. The methodology is used to
obtain outcomes for further analysis.
3.1 Sample Selection
A period of ten (10) years is covered by the data utilised in this study, from 2001-01-01 to
2010-12-01. This period was selected as a result of historical Beta which was not available for
a few Bank Stocks required from the financial databases. Initially the focus of the study was
up to 10 UK banks with duration of 5 years, but it was realised that having a longer
duration/span of years rather than number of companies will make the result more accurate
and efficient. Therefore in order to provide an extended time frame and adequate number of
observations, 12 firms and 10 years was chosen. It was also soon realised that there were no
12 UK banks with each of them having at least 10 years’ worth of historical prices/Beta on
the London Stock Exchange (LSE). Therefore it was decided to instead have a combination of
the few banks which had the data required, alongside other companies from other working
sectors. The companies selected include some of the most prestigious and internationally
renowned companies in the UK. The list of companies comprise of 4 Banks, 2 oil companies,
2 pharmaceutical companies, 2 supermarkets, one tobacco firm and one huge retailer, all
listed on the LSE. They are; Barclays Bank, RBS, Standard Chartered, HSBC, Tesco,
Sainsbury, Glaxo-Smith Kline, AstraZeneca, BP, Premier Oil, Imperial Tobacco and Marks
and Spencer Group.

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3.2 Data Selection
In the course of this study, data for the selected range of companies listed on the LSE for a
period of ten (10) years is required, and Yahoo Finance was able to provide this. One of the
key data required for the stocks from Yahoo Finance was the “Adjusted Close” in particular,
as it is an essential part in the calculation for the Beta estimates because it helps calculate the
return on the stocks.
Subsequently, actual Beta values for the selected companies which aid in testing and
comparing the accuracy of the estimated Beta prediction needed to also be obtained but
Yahoo Finance did not have this unfortunately. This would have caused complications for the
research, however luckily they were able to be obtained from software called DataStream.
After learning to use DataStream, we were able to source out actual Beta values required for
the selected companies for the time period. During this process, the well-known UK Bank
“Santander” and oil company “Royal Dutch Shell” were even omitted, as they both did not
meet the periodic requirements; they possessed incomplete data which did not stretch back to
the time period needed, which is why they were never included as part of the 12 companies.
With regards to the Risk-Free rate needed to be selected for the research, the United Kingdom
Government Gilt 10 Year Bond Yield was decided as the proxy. It was a value of 1.65%
which was obtained from http://www.bloomberg.com/markets/rates-bonds/government-
bonds/uk/. This seemed the appropriate risk-free rate to use for the research as it will better
reflect long-term changes in the financial market, and also since the duration of the data is a
10 year period, it seemed fit to select a risk-free rate that matches it.
For the market portfolio proxy, the FTSE 100 index was used, from which the returns would
be utilised by carrying out the same calculations as similarly done with the Stocks to
collectively aid in the beta estimation. This is explained more in the next section.

39
3.3 Beta Estimating and Testing Method
The Sharpe-Litner CAPM formula for Expected Return is the main equation used to compute
the Beta estimates and return. It is also called the SML or CAPM Equation. To help compute
estimates for Beta, historical data from Yahoo Finance (https://uk.finance.yahoo.com/) had to
be extracted. After downloading the adjusted monthly stock prices for the selected companies,
the monthly index of the FTSE100 (market portfolio) for the same period was also
downloaded.
Since the adjusted prices table was sorted by date in descending order, to help conveniently
calculate the returns, it was better to have the table in increasing order by date. So the entire
table was selected and sorted in ascending order by date. Since for the data, we do not have
the price for the month preceding the first month in the table, we will not be able to calculate
the price for the first month, but for the subsequent months the formula for return of an asset
was used, which is mathematically depicted as:
Return = [ (Adjusted close t+1 / Adjusted close t) – 1 ] ………………….Equation 3.1
This equation (3.1) is well interpreted as the current month’s stock price divided by the last
month’s stock price.
The above equation was successful used to compute returns for each asset and the market to
provide the expected rate of return of each of the companies for that time period, as well as
the market portfolio (FTSE100) .After getting values from applying Equation 3.1 to the data
for each stock, excel syntax was used to compute an average for the values, which was
“=AVERAGE (value1: valuez)”, where z means the zth value of the data. The same average
was also computed for the monthly FTSE100 data as well as that helps generate the required
market portfolio for estimation of CAPM Beta.

40
Following this, we then use CAPM formula to calculate the Beta (i) for each of the 12
companies. Since from CAPM we know that the expected rate of return, denoted by i for
example, of one of the company assets will satisfy:
i – 0 = i (m – 0)…………………………………Equation 3.2
Equation 3.2 was re-arranged to make Beta the subject, that way we would be able to compute
our Beta values, using the risk-free rate, the market value and the expected return on the asset,
with the latter 2 needed to have previously been computed to aid in the new computation. Re-
arranging this, we have for the Beta of a Company asset i:
i= (m – 0)/i – 0 ……………………………………….Equation 3.3
Computing the above using excel syntax, we were able to get results for estimates for Beta for
each of the companies. For the actual Beta values obtained from Data Stream, they range
from the same monthly 10year period as well, so to be able to compare it to the estimates we
have to take averages for the Beta value, so we could get a general average for each of the
companies. That way we can compare the averaged actual Beta to the estimated Beta.
Looking at the estimated Beta and the actual Beta, we will then analyse both Beta values in
comparison to one another and in relation to the market as well.
When analysing Beta in general, we generally start by assuming that the Beta for the risk-free
rate is equal to 0, and the Beta of the market has a value of 1, therefore if the Beta of any of
the selected companies stock is greater (lesser) than this, we can say that overtime, it
fluctuates more (less) than the market. Moreover, that high (low) value of Beta will mean that
the stock is considered to be more (less) riskier, hence meaning that returns on that stock
would be potentially higher (lower).

41
Different values of Beta have different implications. For example, values of Beta that equal 0
do not necessarily mean that it’s the risk-free rate as it could also mean that the return on the
stock is not associated with the market’s movement. When the value is equal to 1, it means
that the stock for example the bank Barclays, moves similarly to the market (FTSE100). If
greater than 1, then we say the stock overtime is more volatile than the market’s, which means
that a rising of the market will mean that we anticipate the stock to also rise, but at a higher
level, and vice versa.
When the stock is between 0 and 1, it means that we presume the stock to also move with the
market, but at a slower rate. Therefore, a rise in the market will mean the stock rises too, but
not as significantly as the market. So here we have volatility which is less than that of the
market. Finally, when the value of the beta of the stock is less than 0, we can say that the
movement of the stock with the market has an inverse relationship.
This inverse relation depicts that when the market rises the stock would decrease instead of
increase. This does not really generally happen unless the stock is for example e.g. Gold. So
this is one major basic way in which we will be analysing the Beta values obtained from
DataStream and also the Beta values we have estimated using the method suggested earlier.
We will do this by utilising and observing the values of estimated Beta obtained to deduce
and explain how much they diverge from the market, since Beta is way to measure the
volatility of a stock in comparison to the market.
Furthermore, to gain more insight on the differences or similarities between the results of the
Beta estimates in comparison to the actual Beta values to aid in examining the accuracy of the
CAPM (final research objective), statistical analysis is required. Therefore we will be doing
the following.

42
1. Descriptive Statistics for both the estimated and actual betas of the 12 selected
companies is examined.
2. Pearson correlation co-efficient.
3. Kruskal-Wallis test, to help find out statistically if a significant difference between
the values of beta estimated by CAPM and actual beta values from DataStream exists.
4. Conduct a T-test, to check both actual and predicted beta estimates and see if there’s
a significant difference between them by using the t-statistic and the p-value. The hypothesis
for this is stated in chapter 4, and to undergo this t-test we would need to run a regression on
the data to also find the p-value and t value and comment on them afterwards. Using
Microsoft Excel Data Analysis tool, regressions to obtain t-statistic and p-values per company
will be run, by using the yearly expected return of the companies (average of the monthly
returns) in order to calculate yearly estimated betas for each stock using the CAPM formula.
So one company would have 10 estimated betas, one for each year and to match the actual
beta values to this, we’ll also average these values from Data stream. Therefore we will have
10 yearly predictions and actual values of Beta from CAPM and data stream respectively. We
had to do this as it will enable us to regress the data, as it needed more observations per
company in order to perform a regression. It was initially planned to average the whole 10
years “adjusted close” value, as well as the whole 10 year average of estimated Beta, but this
would not help us in running a t-test. From these we would be able to obtain the results of the
t-test, and subsequently analyse and comment on the results of rejecting or accepting the H0,
and its implications.

43
4. Empirical Results and Analysis
For this part, results attained from the using the empirical methods as basis for the test of the
CAPM conferred in the previous chapter are displayed. Correspondingly, obtained results
will be analysed within this section. Firstly initial results are presented alongside descriptive
statistics and Pearson Correlation. Subsequently we will present results from the Kruskal-
Wallis test, paired sample t-test and the results of regression analysis of Estimated and Actual
Beta results of 12 firms on the London Stock Exchange from 2001-2010.
4.1 Empirical Results
To help calculate estimates for Beta using CAPM, we had to use the risk-free rate and also the
computed expected market return of FTSE100 over the 10 year period using the methods
outlined in the methodology. Their values are seen in the table below;
Table 4.1 – Risk-free rate and Computed Expected Market Return (FTSE100) (Source: Authors Calculation)
Computing the expected return on each stock for the 10 year time period as a whole, using the
methodology outlined in previous chapter gave me results of estimated beta in table 4.2,
which also includes the Actual beta taken from data stream that was averaged over the 10year
period as well to match the same approach of calculating estimated. The results are;
Table 4.2 – Table of Return, CAPM Estimated Beta and Actual Betas (Source: Authors Calculation)
10 year Risk-free rate 0.016800
10 year Expected Mkt. return 4.3581E-04
Stock, i
Expected return on
asset i
10 year Beta (i)
Estimate CAPM
Actual Beta (DataStream)
10 year Avg.
Barclays -0.003376 1.232906 1.442583333
Royal Bank ofScotland(RBS) -0.012171 1.770420 1.32175
StandardChartered 0.012766 0.246534 1.686833333
HSBC 0.004025 0.780669 1.29075
Tesco 0.008576 0.502557 0.60525
Sainsbury 0.005300 0.702784 0.73025
Glaxo-SmithKline (GSK) 0.001126 0.957816 0.48675
AstraZeneca 0.004864 0.729390 0.616583333
BP 0.003102 0.837050 0.878
BG Group 0.015241 0.095289 0.655083333
ImperialTobacco 0.016571 0.014013 0.375166667
Marks andSpencer Group(M
and S) 0.011498 0.323975 0.514166667

44
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
10 year CAPM Estimated
Beta
Actual Beta (Datastream)
10 year Avg.
Analysis of these beta results from the table shows us that majority of the selected stocks have
the lowest values of beta co-efficient are the values calculated with CAPM. This is
contrasting to the Actual Beta values from DataStream, which has more of the majority of its
stocks having highest results of coefficients of beta. The lowest value of beta overall is the
stock Imperial Tobacco (0.014013) and this is from the CAPM estimate, and the CAPM
estimate also has the highest beta stock which is RBS (1.770420). None of the stocks in this
sample have negative Beta. Approximately ¾ of the stocks from the whole sample have
values of Beta between 0 and 1, which imply that these stocks move in the same direction like
the market does but not as volatile as the value is between 0 and 1. The remaining ¼ of beta
values in the table have Beta which is higher than 1 and this implies that their volatility in
comparison to the market is higher in general.
Graphically showing the relationship between the estimated and actual values, we have;
Figure 4.1- Graph showing the relationship between Estimate Beta and Actual Beta

45
4.2 Descriptive Statistics
Table 4.3 shows us descriptive statistics of the CAPM estimated Beta and the DataStream
actual betas for the selected stocks. We can see from this table that the highest average of
beta is by the DataStream, whilst the lowest average of beta is from CAPM estimates, which
is similar to the earlier conclusions.
The estimated betas calculated by CAPM range from 0.014 (Imperial Tobacco) to 1.770
(RBS) for the selected stocks whilst Betas from DataStream for the same period range from
Imperial Tobacco company again (0.375) to 1.686 (Standard Chartered). Standard deviation is
higher in CAPM estimated Beta (0.499) than the Actual beta (0.436), but both values are
not too far off.
CAPM Estimated Beta Actual Beta
Mean 0.682784052 0.883597222
Median 0.716087078 0.692666667
Minimum 0.014013976 0.375166667
Maximum 1.770420854 1.686833333
Sample Variance 0.249193213 0.190219285
Standard Deviation 0.499192561 0.436141359
Skewness 0.73812501 0.71535939
Kurtosis 0.712839148 -0.972448648
Sum 8.193408625 10.60316667
Sum Sq. Dev. 2.741126 2.092412729
Observations 12 12
Table 4.3 – Descriptive Statistics of CAPM Estimated Beta and Actual Averaged Beta from
DataStream (Source:Authors Calculation)
Moreover, Results of Pearson Correlation between the CAPM Estimated beta and Actual
Beta have indicated a fairly positive correlation which exists between the variables, with a
value of 0.420 (rounded). Table 4.4 shows this.

46
CAPM Estimated Beta Actual Beta
Capm Estimated Beta 1
Actual Beta 0.419631957 1
Table 4.4- Pearson Correlation between Estimated Beta and Actual Beta. (Source: Authors Calculation)
4.3 Kruskal-Wallis Test
Results of this Test confirm that there is no statistically significant difference between the
estimated betas and the actual betas. Since KW= 0.853 and p= 0.3556 > 0.05, we do not reject
the H0 as there is not enough evidence to claim that some of the population medians are
unequal at the a = 0.05 significance level, which means that the samples come from
populations with equal medians. Also the X2= 0.853 which is less than or equal to X2
u which
is 3.841 so we do not reject the null, as the rejection region for this Chi-Square test is R = {X2:
X2 > 3.841} at the degree of freedom which is one. This is summarised in Table 4.5.
Method degreesoffreedom Value of H-statistic p-value
Kruskal-Wallis 1 0.853 0.3556
Table 4.5- Kruskal-Wallis test (Source: Authors Calculation)
4.4 Regressionand Time-Series Analysis
For this section, we had to estimate 10 Yearly Beta values per company (since the Data is 10
years monthly), by averaging the expected return values into yearly and plugging it into the
CAPM equation to find estimates for Beta. Actual Beta values from DataStream were also
averaged to match these estimates, by having 10 values (observations) for that. To statistically
analyse the CAPM model further to test the accuracy of the prediction of Beta, t-tests was
used for rejecting and accepting the hypothesis. The t-test applied by this study was a paired
sample t-test to find the p-value which would mainly indicate the significance of the
difference between the CAPM return and the actual return.

47
To analyse the results of the regression analysis, we choose a significance level of 95% as we
believe the statistics to be good enough to support the result with a confidence of 95%,
leaving the remaining 5% as the chance of being rejected. From the t-distribution, the critical
value for this 95% is 2.262. The table showing the results of the tests is shown below;
Stock, i
Expected
return on
asset i
10 year
CAPM
Beta (i)
Estimate
10
year
Actual
Beta
Avg. Difference
t-statistic
from yearly
averages
regression
p-value
from
yearly
averages
regression
T-Stat
for "9"
degrees
of
freedom
Hypothesis Not
rejected/Rejected
Barclays -0.003376 1.232906 1.4426 0.209676 -1.039833 0.328820 2.262
Accepted Null
Hypothesis
Royal Bank
of Scotland
(RBS) -0.012171 1.770420 1.3218 -0.448670 -0.802048 0.445693 2.262
Accepted Null
Hypothesis
Standard
Chartered 0.012766 0.246534 1.6868 1.440299 -0.221402 0.830325 2.262
Accepted Null
Hypothesis
HSBC 0.004025 0.780669 1.2908 0.510080 0.912665 0.388094 2.262
Accepted Null
Hypothesis
Tesco 0.008576 0.502557 0.6053 0.102692 0.106529 0.917785 2.262
Accepted Null
Hypothesis
Sainsbury 0.005300 0.702784 0.7303 0.027465 -0.563222 0.588716 2.262
Accepted Null
Hypothesis
Glaxo-Smith
Kline 0.001126 0.957816 0.4868 -0.471066 1.701840 0.127198 2.262
Accepted Null
Hypothesis
AstraZeneca 0.004864 0.729390 0.6166 -0.112806 1.292644 0.232215 2.262
Accepted Null
Hypothesis
BP 0.003102 0.837050 0.8780 0.040949 0.251152 0.808026 2.262
Accepted Null
Hypothesis
BG Group 0.015241 0.095289 0.6551 0.559793 0.099317 0.923330 2.262
Accepted Null
Hypothesis
Imperial
Tobacco 0.016571 0.014014 0.3752 0.361152 2.394902 0.043521 2.262
Rejected Null
Hypothesis
Marks and
Spencer
Group 0.011498 0.323975 0.5142 0.190191 1.580541 0.152638 2.262
Accepted Null
Hypothesis
Table 4.6- Time-Series Regression Analysis computed on Microsoft Excel. (Source: Authors
Calculation)
For the time series analysis, the selected 12 individual stocks are given the appropriate
hypothesis. The Hypothesis Test for which would make us accept the null hypothesis is;
Null Hypothesis, depicted by Ho: , which implies that there’s not that much of a
significant difference between CAPM Beta and actual Beta, and we have the Alternative
hypothesis, depicted by H1: ≠, which means that there’s a significant difference.

48
Here the test statistic is given as: tstat = b0/Standard error (b0) ~ tn-g-1. We decide to reject the
null hypothesis if the tstat is > t n-g-1, α/2 or < - t n-g-1, α/2. We know t n-g-1, α/2 to be 2.262 from
looking at the t-table as t n-g-1, α/2 = t9, 0.025 = 2.3011 where α = 5%, therefore if we take for
example the tstat of Glaxo Smith-Kline which we found out to be 1.701840, evaluating this, we
see if tstat (1.701840) > tn-g-1, α/2 (2.262) or tstat (1.701840) < - tn-k-1, α/2 (-2.262). This is the
decision rule that helps us accept the null hypothesis against the alternative hypothesis which
is 2-sided, at a 5% level of significance, as the inequality does not satisfy. And from this we
can say that the CAPM does hold for the stock.
To further confirm this we can look at the P-value, where we know that if we have a
subsequent p-value which is <0.05 or <0.10, then we can conclude that there is a significant
difference between both Betas at 5% or 10% level of significance, which would lead us to
reject H0 in favour of H1, and vice-versa. However, the p-value for this stock is 0.127198;
therefore we do the opposite and accept the null-hypothesis instead as it implies that there is
no significant difference between both Betas. We can also say that 12.7198% is the lowest
significance level that the null hypothesis can be rejected by given the sample data we have.
Furthermore, The Hypothesis Test for which would make us reject the null hypothesis is
still; Null Hypothesis, depicted by Ho: , which implies like said earlier that there’s not
that much of a significant difference between CAPM Beta and actual Beta, and we have the
Alternative hypothesis, depicted by H1: ≠, which means that there’s a significant
difference.
The test-statistic decision rule is also the same. We decide to reject H0 if the tstat is > t n-g-1, α/2
or < - t n-g-1, α/2. As we know t n-g-1, α/2 to be 2.262 from looking at the t-table since t n-g-1, α/2 = t9,
0.025 = 2.3011 where α = 5%, if we take as another example from Table 4.1, the tstat of Imperial

49
Tobacco which we found out to be 2.394902, evaluating this, we also check if tstat (2.394902)
> tn-g-1, α/2 (2.262) or tstat (2.394902) < - tn-k-1, α/2 (-2.262). Since the inequality is satisfied this
time around, we reject the null hypothesis in favour of the alternative hypothesis which is 2-
sided, at a 5% level of significance. And from this we can say that the CAPM does not hold
for that stock. To further confirm this result we also look at the P-value, which we find to be
0.043521; therefore we reject the null-hypothesis. The value 0.043521 is < 0.05, therefore
showing that there is some significance difference between both betas, hence why we reject
H0. We can also say that 4.3521% is the higher significance level that the null hypothesis can
be accepted by, given the sample data. Therefore, from table 4.6 which includes the
regression analysis, we figure out that that the CAPM Beta holds for about 91.6% of the data
and 8.3% does not hold in the CAPM. Therefore in this case, we can say that overall the
CAPM holds in regards to its ability to accurately predict beta.
Also from the report calculation, the values of the computed R2 of the individual stocks range
from 41.7570% (maximum) to a very low 0.60901% imply that for the stock with 41.7570%
R2 , 41.7570% of the total variation is explained by the regression line for that stock. The
closer the value of R2 for the stocks to be closer to 1, the better it is as it would mean that the
regression line would perfectly fit with the data. From Figure 4.1 we can see this, and this
means that the regression model is quite good at explaining the statistically significant beta,
so therefore from this we can deduce that the CAPM does hold despite the differences, as the
differences are not significant enough to disprove the model. The adjusted R2 does not really
matter as there’s only one variable being compared, if it was more than one then I would have
also given attention to the adjusted R squared coefficient of determination.
To round up, in this case based on these results, we can say that the CAPM for an investor
would have been a useful investment procedure due to its fairly accurate prediction ability.

50
5. Conclusion
In this section of the paper, a summary of the obtained findings from the analysis is presented.
These findings form the basis of conclusion on the CAPM’s validity, from examined data in
the investigation. Comments will also be made on the reliability of the results. Finally, we will
present areas of interest for further future research purposes.
This study has been established to contribute to CAPM by providing a detailed
comprehensive review of the model, and also investigating its validity on London Stock
Exchange with regards to testing its accuracy of Beta estimations. For this study a sample of
12 firms listed on the London Stock Exchange, was selected with data dating from 2001-2010.
For each of the stocks monthly Beta are calculated and compared to the actual beta values
from DataStream to test the CAPM prediction. Research suggests through the Pearson
correlation that fairly positive correlation exists between both values. This suggests that there
is probably not much significant difference between estimated Betas and actual Betas.
These results are confirmed with the results from the Kruskal-Wallis test, which concludes
that there is not enough evidence of significant difference to conclude that the CAPM does
not hold.
Furthermore, the t-test aimed at sampling revealed results of analysis of significant difference
in only 1 out of the 12 companies; Imperial Tobacco. As results of 11 out of the 12 show that
there is no significant difference between actual return and CAPM return.
Therefore these findings confirm that the CAPM is valid on the London Stock Exchange for
11 of the 12 companies during the 10 year period selected. However it has to be pointed out
that this study is based on 12 companies, so more extensive study probably comprising of lot
more stocks and longer time period will need to be undertaken to draw final conclusions.

51
Moreover, even though the model from the results has predicted the Beta fairly accurately, it
means that the CAPM model holds, however that might not necessarily be the case, reason
being that external factors which actually happened could have contributed to the fluctuations
or differences in actual Beta from the estimates, which the model would not be aware about.
These are called market anomalies, so even though the CAPM estimate was not too far off, it
could have been worse, or not worse.
5.1 Future Research
Interest and inquisitiveness into CAPM stirred the selection of this research topic. At this
point, I am even more interested and would probably like to conduct further research on the
same topic (the validity of the CAPM), and as I mentioned earlier, if we used more stocks and
a longer time period, the estimates could have been much more accurate. In addition the t and
p values from statistical testing would have been more accurate as well if we computed more
than 10 estimates and actual values of Betas for each Bank, 10 each was only used, as a
monthly average was taken.
As an evaluation point, an aspect that could be considered for the selected data for the Beta
calculation must be the compromise between the accuracy and the significance. Explaining
this, the more years we use might increase the accuracy of Beta, so using a 15 year period
instead of 10 could have made the results more accurate, however, how relevant those data
points are, are in query. This is because the old trends from that period might not have much
bearing on new trends for the same stock. And although we could have been more accurate by
using either daily data or using weekly data to get more observations and more accuracy,
monthly was used instead as we thought that might provide some increased relevance because
of the nature of most investors needs. So that could have been one of the reasons.

52
Moreover, in Brown and Warner (1985) the daily prices are said to better in event
methodology for auto correlation, so data collected weekly and monthly might not provide a
very meaningful relationship between Beta and returns in general. So we could have probably
tested the CAPM model on daily data instead e.g. 90 days in a particular year, to see if we
would have gotten far better result.
For investment purposes to further expand the research, we could have created a portfolio
split into the 12 companies, and then calculate weighted average beta of the entire portfolio.
This would help any investor who has planning to invest, as it gives them a larger picture of
the volatility of Beta anticipated compared to the market, so they can make a wiser investment,
as according to Pellet, 47 (2004), on the individual stock level, Beta does not work very well.
Finally for future research purposes, we could also try and focus on another industry other
than the Bank industry so for example, supermarkets or oil companies, and see how the
CAPM performs there. Or by testing another sample to go along with the sample of company
stocks that would have enhanced the research further. With regards to enhancing and
strengthening the research objective, we could have aimed to also try and predict expected
returns of assets using the Betas and empirically checking it with the actual to further test the
accuracy of the CAPM by testing this relationship between Beta and expected return. Also, it
will be of interest to consider the alternative models which have been previously mentioned in
this paper e.g. the APT and study the results in relation to CAPM.

53
6. Appendix
Table 6.1 List of 12 companies and their ticker/Abbreviation on the London Stock
Exchange
Company/Stock, i Ticker
Barclays BARC.L
Royal Bank of Scotland(RBS) RBS.L
StandardChartered STAN.L
HSBC HSBA.L
Tesco TSCO.L
Sainsbury SBRY.L
Glaxo-SmithKline (GSK) GSK.L
AstraZeneca AZN.L
BP BP.L
Royal Dutch Shell BG.L
Imperial Tobacco IMT.L
Marks and SpencerGroup(M and S) MKS.L

54
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MA831 EZEKIEL PEETA-IMOUDU DISSO

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