msc final dissertation final pdf

Queen Mary, University of London School of Economics and Finance

Supervisor: Dr Sarah Mouabbi Student: Shane, Abeywardana Wickramasinghe

Student Number: 150257712 Dissertation submitted in partial fulfillment of the requirements

for the degree of Master in Investment and Finance

ACADEMIC YEAR 2015- 2016

Word Count (Excl. References): 5,238

Dissecting Anomalies in Asset Pricing: The Equity Premium Puzzle

DissectingAnomaliesinAssetPricing:TheEquityPremiumPuzzle

Student ID 150257712 Shane, Abeywardana Wickramasinghe

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Abstract This paper studies the reasons behind large equity risk premia observed in US markets

over the past century. Prior empirical research has pointed to preference assumptions

such as risk aversion and market inefficiency, among others, being modeled to

explain this “phenomenon”. We test a variety of contemporary asset pricing models

such as the CAPM to observe whether they are able to fully explain the variability of

expected stock returns. The findings of the study indicate to evidence against the

Capital Asset Pricing Model (CAPM) hypothesis, in that the intercept (alpha) is not

equal to zero meaning that arbitrage opportunities do seem to exist. The inclusion of

additional factors using the Fama-French 3 and 5-factor models also showed that the

CAPM did not “hold” and that the beta alone was not sufficient to determine the

expected returns on stocks/portfolios.



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Dedication

This dissertation is lovingly dedicated to my grandparents Ronald and Hema Colonne who sadly passed away in 2010 and 2013 respectively. Their

support, encouragement, and constant love have given me the impetus to take this journey and have been my

guiding light.



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Acknowledgements

I would like to thank my supervisor, Dr. Sarah Mouabbi for her valuable input and

guidance throughout the dissertation writing process. I would also like to thank my

teaching assistant, Giovanni Nappi who provided me with advice on running

regressions as I found this to be the most intricate part of my dissertation.



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Table of Contents ABSTRACT.......................................................................................................................................2DEDICATION ..................................................................................................................................3

ACKNOWLEDGEMENTS............................................................................................................4TABLE OF CONTENTS ...............................................................................................................5

1.INTRODUCTION........................................................................................................................62. LITERATURE REVIEW...........................................................................................................73. EMPIRICAL METHODOLOGY...........................................................................................10

3.1 UNDERSTANDING THE RISK AND RETURN RELATIONSHIP............103.2 EVOLUTION OF ASSET PRICING MODELS.................................................10

3.2.1 THE CAPITAL ASSET PRICING MODEL.........................................103.2.2 TESTING THE CAPITAL ASSET PRICING MODEL......................123.2.3 FAMA FRENCH THREE AND FIVE-FACTOR MODEL................133.2.4 TESTING THE FAMA-FRENCH 3 AND 5-FACTOR MODEL........14

4. EMPIRICAL ANALYSIS........................................................................................................164.1 OBJECTIVES OF THE STUDY..........................................................................164.2 DATA SELECTION................................................................................................164.3 DATA ANALYSIS..................................................................................................16

4.3.1 FURTHER INTERPRETATION OF THE RESULTS..........................224.3.2 LIMITATIONS OF THE DATA ANALYSIS........................................23

5. CONCLUSION...........................................................................................................................245.1 RECOMMENDATIONS FOR FUTURE RESEARCH....................................25

REFERENCES................................................................................................................................26APPENDIX A..................................................................................................................................29



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1. Introduction The “equity premium puzzle”, which was a term coined by Mehra and Prescott

(1985), referred to the phenomenon that observed returns on stocks over the past

century were much higher than government bonds. They concluded that the difference

in returns was strikingly large to be explained by economic models.

More than 30 years after it was first introduced, the equity premium puzzle is still not

close to being fully explained in the financial literature. Not only has the model not

been able to fully describe the anomaly of such large excess returns but the puzzle

also questions modern economic theories such as the random walk hypothesis

(Malkiel, 1973), since the empirical results of Mehra and Prescott (1985) show that

stock prices follow a trend wise movement rather than a random one.

The question addressed in this paper examines whether modern asset pricing models

such as the CAPM, can fully explain the variability of expected stock/portfolio returns

and whether abnormal returns (alpha) can be generated by random stock selection and

exposure to various factors; thus creating the possibility of arbitrage opportunities.

The remainder of the dissertation is structured as follows: Section 2 provides a

theoretical framework and summarizes research on the “puzzle” between 1975-2015.

Section 3 specifies the empirical methods that will be used to test the data. The

importance of the risk/return relationship in financial decision-making is also briefly

touched on. The results of the empirical tests are reported in Section 4. Section 5

concludes the paper.



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2. Literature Review Historically, the average return on equity has far exceeded the average return on

short-term effectively risk-free debt. “Over a ninety-year period (1889-1978), the

average real annual yield on the Standard and Poor 500 Index was seven percent,

while the average yield on short-term debt was less than one percent” (Mehra and

Prescott, 1985). There has been an abundance of literature seeking to explain the

equity risk premium (ERP) (which is calculated by deducting the aggregated return on

a market index from the theoretical risk-free rate) obtained from asset pricing models

by examining the determinants behind asset price fluctuations.

Many academics have since attempted to explain the large disparities between the two

returns arguing that while there was an innate belief that stocks were more volatile

than bonds, it was not an adequate explanation to explain the extent of this disparity.

A direct inference of this was that an implausibly high level of risk aversion was

required that was fundamentally incompatible with economic models.

In light of the work carried out by Mehra and Prescott (1985), two possible

resolutions to the “equity premium puzzle” emerged where “either the standard

models were wrong or the historical premium was misleading”, and that future premia

were expected to be lower (Dimson et al, 2008). Over the past two decades,

academics have attempted to resolve the “puzzle” by simplifying and modifying the

Mehra–Prescott (1985) model. Their efforts have focused on alternative assumptions

about preferences, including “market inefficiency, risk aversion, aggregate

consumption, state separability, habit formation and precautionary savings, leisure,

market imperfections such as transaction costs and borrowing constraints” (Dimson et

al, 2008). Kocherlakota (1996), Mehra and Prescott (2003) and Mehra (2008) provide

excellent pragmatic rationalizations to some of the preference assumptions discussed

in this paper.

Previously, Lucas (1978) had observed the “stochastic behavior of equilibrium asset

prices in a one-good, pure exchange economy with identical consumers” attempting

to understand the relationship between exogenously determined productivity changes



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and market determined movements in asset prices. His paper provided an illustration

of the use of some methods that could “bring financial and economic theories closer

together” (Lucas, 1978) and formed the basis for future research that utilized his

model, which was later adapted in Mehra and Prescott’s work. Their work would later

act as the catalyst for an extensive research effort in attempting to solve this “equity

premium puzzle”.

Constantinides and Duffie (1996), however, strongly argued that the Lucas (1978) –

type exchange economy model faired poorly in explaining security prices. Mehra and

Prescott (1985) used an adaptation of the model Lucas derived; pointing out that the

model predicted “a mean equity premium that was too low and a mean interest rate

that was too high, given the observed low variability of aggregate consumption

growth” (Constantinides and Duffie, 1996).

In formal tests of the conditional Euler equations (which helps explain how interest

rates and growth rates in the economy are closely related), Hansen and Singleton

(1982) and Ferson and Constantinides (1991) rejected the model even though “no

priori upper bound was imposed on the relative risk aversion coefficient. Thus, the

poor performance of the model was unmitigated even by unconventionally high

values of the relative risk aversion coefficient” (Constantinides and Duffie, 1996).

The Euler equations were also rejected by the diagnostic tests carried out by Hansen

and Jagannathan (1991).

Friend and Blume (1975) had previously attempted to systematically exploit cross-

sectional data on household asset holdings to “assess the nature of households’ utility

functions”, suggesting that a relationship between utility functions and wealth of all

investors was an “obvious” foundation in constructing an aggregate demand function

for risky assets. Their results indicated that irrespective of wealth level, “the

coefficients of proportional risk aversion for households were on average well in

excess of one and probably two”, suggesting that investors required “a substantially

larger premium to hold equities or other risky assets” (Friend and Blume, 1975, as

cited in, Gowing, 2009). This agreed with Shiller’s (1982) calculations showing that

either a large risk aversion coefficient or “large consumption variability was required

to explain the means and variances of asset returns” (Shiller, 1982).



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In more recent studies, Fama and French (2002) use a rendition of the Gordon growth

model using dividend and earnings growth rates from 1951 to 2000 to measure the

expected return. Their findings indicated that the expected returns calculated were

much lower than the equity premium produced by the average stock return. They

argued that the reasons for the higher average return were due to “declining discount

rates that produced large unexpected capital gain” (Fama and French, 2002). This was

similar to the findings of Campbell (1991), Cochrane (1994) and Campbell and

Shiller (1998).

Dimson et al (2003) conducted research on equity premia in the United Kingdom

where they found a strong value premium in UK markets between 1955 and 2001.

Further research by Dimson et al (2011) on the equity premium was subsequently

observed for markets around the world in a “19-country world index” which included

markets in the USA, Canada, France, Germany and the UK, among others. They

concluded that the equity premium was positive and substantial in all markets.



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3. Empirical Methodology

3.1 Understanding the Risk and Return Relationship Many financial academics will agree that the treatment of risk is the main element in

financial decision-making. “Prior to 1952 the risk element was usually either assumed

away or treated qualitatively in the financial literature” (Modigliani and Pogue, 1973).

Risk is hard to define, let alone measure. However, a measure of risk can be seen as

the extent to which the future security values are likely to deviate from the expected

or predicted value. The higher/lower the risk that is borne, the higher/lower the

expected return required by an investor to compensate them for taking on additional

risk. For example, a portfolio comprised of common stock is generally seen as risky

as it is difficult to predict exactly the value of a portfolio at a future date. In contrast,

an investor who holds a portfolio of treasury bonds faces very little uncertainty about

the monetary outcome and as such has borne no risk (Modigliani and Pogue, 1973).

3.2 Evolution of Asset Pricing Models

3.2.1 The Capital Asset Pricing Model The CAPM is a financial model that defines the relationship between risk and

expected return, which is used in pricing risky assets. Jack Treynor (1961), William

Sharpe (1964), John Lintner (1965a and 1965b), Jan Mossin (1966) and Eugene Fama

(1968) were among the first to introduce the model, adding to the earlier research

efforts of Harry Markowitz (1952) and James Tobin (1958a and 1958b) on

diversification and modern portfolio theory.

The general notion behind the CAPM is that investors need to be compensated in two

ways: time value of money and risk. The time value of money concept captures the

idea that money today will be worth more in the future (as interest can be earned) and

is represented by the theoretical risk-free rate (𝑅!) where an investor simply holds a

portfolio of government bonds that pay fixed coupon payments. The risk element is

the return that an investor requires by taking on additional risk (by investing in



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stocks) and is measured by beta that acts as a sensitivity measure to the returns to the

market premium 𝑅!" − 𝑅!" over a period of time.

A time series test of the CAPM was presented by Black, Jensen, and Scholes (1972).

The test is based on the time series regressions of excess portfolio returns and excess

market returns, which can be expressed by the equation below:

𝑅!" − 𝑅!" = 𝛼! + 𝛽! 𝑅!" − 𝑅!" + 𝜀!" (1) Where:

"𝑅!" 𝑖𝑠 the rate of return on stock i at time t, 𝑅!" 𝑖𝑠 the risk free rate at time t, 𝑅!" is the

rate of return on the market portfolio at time t, 𝛽! 𝑖𝑠 the beta of stock i", a sensitivity

measure of the expected excess asset return to the expected excess market return,

where 𝛽! =!"# !!", !!"!"# !!"

and 𝜀!" is the random disturbance term in the regression

equation (Black, Jensen, and Scholes, 1972, cited in, Yang and Xu, 2006) .

While the equity risk premium 𝑅!" − 𝑅!" gauges the excess return of a stock

(absolute return), 𝛼! represents the ‘abnormal’ return on a stock (relative return). In

other words, it is the excess return relative to the return of a benchmark index. For

example, if a portfolio (collection of securities) has an alpha (𝛼!) of 6%, the portfolio

is said to have positive alpha and is outperforming the benchmark index. This would

invalidate the efficient market hypothesis (EMH), developed by Eugene Fama, who

argued that asset prices fully reflected all available information and that it was

impossible to “beat the market” consistently on a risk-adjusted basis (Fama, 1970).

Since beta is readily available in financial markets, investment managers are

constantly looking to “seek alpha”.

The major difficulty in testing is that the CAPM is stated in terms of investors’

expectations, not in terms of realized returns. From a statistical perspective, this

introduces an error term 𝜀!", which theoretically should be zero on average, but not

necessarily zero for any single stock of a single period of time.



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3.2.2 Testing the Capital Asset Pricing Model The null and alternative hypothesis during the CAPM test is: 𝐻!:𝛼! = 0 (2)

𝐻!:𝛼! ≠ 0 The first step is to estimate the excess returns for each security from a time series of

returns of S&P 500 stocks (𝑅!" − 𝑅!" ) and the excess return of S&P 500 index,

which is defined as the proxy for the market portfolio (𝑅!" − 𝑅!").

The second step is to estimate the beta coefficient for each stock using their adjusted

monthly returns. The beta is estimated by regressing each stock’s adjusted monthly

return against the market index (S&P 500) (Choudhary and Choudhary, 2010):

𝑅!" − 𝑅!" = 𝛼! + 𝛽! 𝑅!" − 𝑅!" + 𝜀!" (3)

The third step is to divide the sample of twenty-five stocks into five equally weighted

portfolios based on their beta. The stocks are ranked by their beta with the first

portfolio comprising stocks with the largest value of betas and portfolio five

comprising stocks with the smallest value of betas (Fama and Macbeth, 1973, as cited

in, Theriou, Aggelidis, and Spiridis, 2016). Fama and Macbeth (1973) found that this

technique “reduced the error-in-variables (EIV) problem that arises in asset pricing

forecasting”.

The fourth step is to calculate the mean monthly returns for each of the 5 portfolios:

𝑅!"

!

!!!

=1𝑁

𝑅!! + 𝑅!! + 𝑅!! + 𝑅!! + 𝑅!! (4)

Where:

Rpt is the return of a portfolio at time t, Ri1-5 is the return of each stock in a portfolio at

time t and N is the number of stocks in each portfolio.

The beta coefficients of the 5 portfolios are also calculated:



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𝑅!" = 𝛼! + 𝛽!𝑅!" + 𝜀!" (5)

Where:

Rptistheaveragereturnonportfolio, 𝛽! is the estimated portfolio beta and;

𝜀!" is a random disturbance term.

The fifth and final step is to calculate the excess return on portfolio and the excess

return on the market by deducting the theoretical risk-free rate (US 3 month T-bill):

𝐸𝑅!" = 𝛼! + 𝛽!𝐸𝑅!" + 𝜀!" (6)

If the CAPM is true, 𝛼! should be equal to zero and 𝛽! should be equal to the market

portfolio’s average risk premium.

3.2.3 Fama French three and five-factor model Fama and French (2004) argued that the “failure of the CAPM in empirical tests

implies that most applications of the model are invalid”. The Fama French three-

factor model was designed by Eugene Fama and Kenneth French to resolve some of

the problems found in the CAPM by introducing more variables to explain the returns

of a portfolio or stock with the returns of the market as a whole. Fama and French

(1993) found that two classes of stocks had generally performed better than the

market as a whole: (i) small caps and (ii) stocks with a low Price-to-Book ratio. Two

additional factors, SMB and HML, were subsequently introduced to the CAPM to

reflect a portfolio's exposure to these two classes (Fama and French, 1993):

𝑅!" − 𝑅!" = 𝛼 + 𝛽! 𝑅!" − 𝑅!" + 𝛽! ∗ 𝑆𝑀𝐵 + 𝛽! ∗ 𝐻𝑀𝐿 + 𝜀!" (7)

Where:

“SMB (small minus big) is the difference between the returns on diversified portfolios

of small and big stocks and HML (high minus low) is the difference between the

returns on diversified portfolios of high and low B/M stocks” (French, 2016b).



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In a more recent study, Fama and French (2015) added two more coefficients (to

capture profitability and investment patterns in average stock returns) to the original

3-factor model:

𝑅!" − 𝑅!" = 𝛼 + 𝛽! 𝑅!" − 𝑅!" + 𝛽! ∗ 𝑆𝑀𝐵 + 𝛽! ∗ 𝐻𝑀𝐿 + 𝛽! ∗ 𝑅𝑀𝑊 + 𝛽! ∗ 𝐶𝑀𝐴 + 𝜀!" (8) Where:

“RMW (robust minus weak) is the average-return on two robust operating profitability

portfolios minus the average-return on two weak operating profitability portfolios and

CMA (conservative minus aggressive) is the average return on two conservative

investment portfolios minus the average return on two aggressive investment

portfolios” (French, 2016b).

3.2.4 Testing the Fama-French 3 and 5-factor model Testing the Fama-French-3 and 5-factor model is similar to the CAPM in that much

of the CAPM equation is kept intact, differing only with the number of coefficients

used to explain the dependent variable. By doing so, a greater degree of understanding

of the variability in average stock returns (and portfolios) can be achieved.

Unlike the study by Fama and French (1993), which used 25 portfolios, this paper

will comprise 5 portfolios and are constructed to address the short time constraints,

making it a more reliable analysis.

The null and alternative hypothesis during the 3-factor test are as follows:

𝐻!:𝛼! = 0 And 𝛽! = 0,𝛽! = 0 (9)

𝐻!:𝛼! ≠ 0 And 𝛽! ≠ 0,𝛽! ≠ 0

The null and alternative hypothesis during the 5-factor test are as follows:

𝐻!:𝛼! = 0 And 𝛽! = 0,𝛽! = 0,𝛽! = 0,𝛽! = 0 (10)

𝐻!:𝛼! ≠ 0 And 𝛽! ≠ 0,𝛽! ≠ 0,𝛽! ≠ 0,𝛽! ≠ 0



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We follow steps 1-5 used to test the CAPM and then add the additional coefficients

into our equation:

𝐸𝑅!" = 𝛼! + 𝛽! ∗ 𝐸𝑅!" + 𝛽! ∗ 𝑆𝑀𝐵 + 𝛽! ∗ 𝐻𝑀𝐿 + 𝛽! ∗ 𝑅𝑀𝑊 + 𝛽! ∗ 𝐶𝑀𝐴 + 𝜀!" (11)

For the purposes of this paper, the additional factors, SMB, HML, RMW and CMA

have been taken from Kenneth French’s US research returns data (French, 2016a).

3.2.5 Significance Levels The significance level used in this paper to accept or reject the null hypothesis will be

0.05 denoting a confidence interval of 95%. The asterisk rating system will be used to

present P values:

P < 0.10*

P < 0.05**

P < 0.01***



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4. Empirical Analysis

4.1 Objectives of the Study The objective of this dissertation is to observe whether the CAPM and Fama-French

three and five-factor model hold true in the US stock market. We observe whether:

1. Stocks/portfolios are not able to produce abnormal returns (alpha) given exposure

to various factors.

2. Asset pricing models are able to fully explain the variability of expected stock

returns.

4.2 Data Selection The study uses monthly adjusted closing stock prices for a selection of 25 firms (see

Appendix A) on the S&P 500 index listed on the New York Stock Exchange for the

period of January 1990 to January 2016 (26 years). The S&P 500 index represents

approximately 80 per cent of the available market capitalisation. The data was

obtained directly from Bloomberg and Kenneth French’s data library (French, 2016a).

The monthly closing values of the S&P 500 Index are used as a proxy for the market

portfolio. In addition, the yields on 3-month US Treasury bills are used for the

theoretical risk- free return.

4.3 Data Analysis Based on 26 years of monthly returns on each of the five portfolios computed as

explained in section 3 of this paper, the next step was to calculate the least-squares

estimates of the parameters 𝛼! and 𝛽! for each of the five constructed portfolios (i =

1… 5) using all 26 years of monthly data (312 observations) (Black, Jensen and

Scholes, 1972). Three models were tested using the above data, the CAPM, the Fama-

French 3-factor model and the more recent Fama-French 5-factor model. The CAPM

results are summarised in Tables 1 and 2 while the results for the Fama-French 3

factor and 5-factor models are summarised in Tables 3 to 6.



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CAPM Results

Portfolio number 1 comprises the highest-risk stocks (high betas), and portfolio

number 5 comprises the lowest-risk stocks (low betas). The estimated risk coefficients

for the CAPM range from 1.4414 for portfolio 1 to 0.6152 for portfolio 5. We can

note that the coefficient associated with market exposure (β1) is positive across all

portfolios and is between 1.44 and 0.61 (see Table 1). The interpretation of the

coefficient is such that for a 1% increase in the market return, the excess return on the

portfolio (dependent variable) is expected to increase by its respective β1 value. For

example, portfolio 1 would increase by 1.44% for a 1% move in the market.

The critical intercepts, 𝛼! , are provided in Table 1 and the probability; along with the

respective “t” values. The t-statistic tells us whether the test is statistically significant

or not. For any value, which is not greater than the significance level (α), “we may

safely say that the t-statistic is significant and that the estimated coefficient is

significantly and statistically different from zero” (Leonida, 2016). In our regression,

considering the excess return on the market, we can notice that the t-ratio probability

value associated with the coefficient is 0.0000 across all portfolios. Therefore, we

may conclude that the exposure to the market does statistically affect the excess return

on the portfolio.

𝑅! is a measure of the “goodness of fit” of our model and tells us how well the

sample regression line fits the data set. It measures the proportion of the total

deviation in the dependent variable that can be explained by the regression model.

Since 𝑅! increases, as the number of explanatory terms are added, there is a risk that

𝑅! will be inflated. Adjusted 𝑅! is an adjustment that takes into account the degrees

of freedom and provides a better indicator as to how well the sample regression line

fits the data set. In our regression, the adjusted 𝑅! ranges between 0.4288 and 0.7290,

which means that our model can explain between 42.88% and 72.90% of the total

variability of the dependent variable.

The Durbin-Watson (DW) statistic is a test for the existence of autocorrelation (a

relationship between observations as a function of the time lag between them) in the



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residuals. In our regression, the DW statistic ranges between 1.8644 and 2.0957.

Since it is close to 2, we can conclude that we have evidence of no autocorrelation in

the residuals.

The F-statistic measures the overall significance of the estimated model. “Such a

statistic tests the hypothesis that all non-constant coefficients in the regression

equation are jointly equal to 0” (Leonida, 2016). In other words, the F-statistic

compares the estimated model with a model that includes only a constant (intercept-

only model). The reported F-statistic probability for each portfolio is shown on the

sixth line on Table 1 and it can be noted that all portfolios have a probability value

equal to 0.0000 meaning that the test is significant at a 0.01 confidence level.

Table 2 shows the results of the null hypothesis using the Wald test coefficient

restriction. As mentioned earlier, the null hypothesis for this test is that the coefficient

α is equal to zero. The F-statistic results from the Wald restriction test show that the

p-values are below our significance level of 0.05 that we are using to accept/reject our

1 2 3 4 5 R_mt

β1* 1.4414 1.1475 1.0243 0.6748 0.6152 1.0000

α 0.7494 0.7914 1.1040 0.5027 0.6851

p(β1) 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***

p(α) 0.0086*** 0.0000*** 0.0000*** 0.0075*** 0.0000***

f-stat 464.2699 837.5351 367.8062 234.4215 248.3549

p(f-stat) 0.0000 0.0000 0.0000 0.0000 0.0000

AdjustedR^2 0.5983 0.7290 0.5412 0.4288 0.4430

DW 1.9765 1.9433 1.8644 2.0957 1.9038

Table1

SummaryofStatisticsforCAPMTimeSeriesTests,EntirePeriod(January,1990-January,2016)

SampleSizeForEachRegression=312

PortfolioNumber

*whereβ1istheexcessreturnonthemarketindex

1 2 3 4 5t-stat 2.6455 4.7132 4.8809 2.6935 4.1440p(t-stat) 0.0086*** 0.0000*** 0.0000*** 0.0075*** 0.0000***f-stat 6.9985 22.2146 23.8227 7.2547 17.1727p(f-stat) 0.0086*** 0.0000*** 0.0000*** 0.0075*** 0.0000***chi-square 6.9985 22.2146 23.8227 7.2547 17.1727p(chi-square) 0.0082 0.0000 0.0000 0.0071 0.0000

WaldTestRestrictionStatisticsforCAPMTimeSeriesTests,EntirePeriod(January,1990-January,2016)

Table2


H0:α=0

PortfolioNumber



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hypothesis. This indicates that there is significant evidence to reject the null

hypothesis that α is equal to zero.

Fama-French 3-Factor Results

Similar to the CAPM results, we can note that the coefficient associated to the market

(β1) is positive across all portfolios and is between 1.44 and 0.61 (see Table 3). In

addition, exposure to size (β2) appears to be negative across all portfolios except for

portfolio 1 which has a positive value of 0.3666 meaning that for a 1% move in the

coefficient, the dependent variable (excess return on portfolio) moves upwards by

0.37%. The p-value associated with the coefficient seems to be highly significant at a

significance level of 0.01. We can see that the p-values for portfolios 2 and 3 are not

significant when exposed to size while portfolios 4 and 5 indicate a fall in the value of

the dependent variable. Exposure to value (β3) shows positive results across all

portfolios however when looking at the p-values most portfolios are not significant.

Similar to the CAPM results, the p-values for the F-statistic in the 3-factor test are

also 0.0000 across all portfolios indicating that the model is significant.

Adjusted 𝑅! in our 3-Factor results ranges from 0.5078 to 0.7329 meaning our model

can explain between 50.78% and 73.29% of the total variability of the dependent

variable. This is slightly more than our CAPM results, which could only explain

between 42.88% and 72.90% meaning that adding size and value coefficients helps

explain the total variability of the dependent variable even better.

In our regression, the DW statistic ranges from between 1.8518 and 2.1117. Since it is

close to 2, we can conclude that we have evidence of no serial correlation in the

residuals.



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The results in Table 4 show the Wald restriction tests results. Again, we can see that

the p-value associated to the F-statistic is 0.0000 across all portfolios meaning that we

have significant evidence to reject the null hypothesis.

Fama-French 5-Factor Results The risk coefficients range from 1.2673 for portfolio 1 to 0.7677 for portfolio 5 for

the Fama-French 5 factor model.

We can note that the coefficient associated with market exposure (β1) is positive

across all portfolios and is between 1.26 and 0.76 (see Table 5). In contrast to both the

CAPM and 3-factor results, we can observe that the variability of the coefficients are

significantly reduced across all five portfolios on the 5-factor model and are more

concentrated around 1.0000 (which is the beta of the market portfolio). An inference

can be made that by adding these extra factors; the systematic risk (market risk) can

be slightly reduced.

1 2 3 4 5

f-stat 8.2839 9.6929 8.2113 33.5281 20.7468

p(f-stat) 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***

chi-square 24.8517 29.0786 24.6387 100.5842 62.2403

p(chi-square) 0.0000 0.0000 0.0000 0.0000 0.0000

Table4

WaldTestRestrictionStatisticsfor3-FactorTimeSeriesTests,EntirePeriod(January,1990-January,2016)


H0:α=0andβ2=0,β3=0

PortfolioNumber

1 2 3 4 5 R_mtβ1* 1.4190 1.1662 1.0340 0.7188 0.6451 1.0000β2* 0.3666 -0.0085 -0.0138 -0.4672 -0.2974β3* 0.0635 0.1362 0.0642 0.0358 0.0367α 0.6887 0.7613 1.0910 0.5534 0.7142

p(β1) 0.0000 0.0000 0.0000 0.0000 0.0000p(β2) 0.0000 0.8743 0.8494 0.0000 0.0000p(β3) 0.5050 0.0182 0.4112 0.5271 0.4933p(α) 0.0137 0.0000 0.0000 0.0009 0.0000SE(β1) 0.0665 0.0401 0.0545 0.0395 0.0374SE(β2) 0.0887 0.0535 0.0727 0.0528 0.0499SE(β3) 0.0952 0.0574 0.0780 0.0566 0.0535t-stat(β1) 21.3516 29.1029 18.9830 18.1818 17.2674t-stat(β2) 4.1306 -0.1583 -0.1900 -8.8494 -5.9616t-stat(β3) 0.6674 2.3732 0.8229 0.6331 0.6859f-stat 168.3305 285.4454 122.4673 131.0488 107.9514p(f-stat) 0.0000 0.0000 0.0000 0.0000 0.0000R^2 0.6212 0.7355 0.5440 0.5607 0.5125DW

Table3SummaryofStatisticsforTimeSeriesTests,EntirePeriod(January,1990-January,2016)

SampleSizeForEachRegression=312PortfolioNumber

*whereβ1istheexcessreturnonthemarketindex,β2"isthedifferencebetweenthereturnsondiversifiedportfoliosofsmallandbigstocks(SMB)"andβ3"isthedifferencebetweenthe

returnsondiversifiedportfoliosofhighandlowB/Mstocks(HML)"(French,2016b)



21

In addition exposure to size (β2) appears to be positive across most portfolios except

for portfolio 4 and 5, which reflect the portfolios with the smallest betas. Similar to

previous results, the p-value associated with the β2 coefficient seems to be highly

significant at a significance level of 0.01. Exposure to value (β3) shows positive

results across only portfolios 1 and 2, which represent the companies with the highest

betas. Subsequently, looking at the p-values, portfolio 1, 4 and 5 are significant at

0.01 confidence level, portfolio 2 is significant at 0.05 and portfolio 3 is not

significant with a p-value of 0.6844. Exposure to profitability (β4) is positive across

portfolios 2 to 5. Portfolio 1 seems to show a negative value of -0.3242. The p-values

associated with the coefficient are all significant (with portfolio 1 being significant to

a 0.05 confidence level). Exposure to investment (β5) is negative for the majority of

the portfolios (1, 2 and 3) and positive for portfolios 4 and 5. The p-values associated

with the coefficient seem significant with portfolio 2 being significant to a 0.10

confidence level and portfolio 3 not being significant (0.5607).

Similar to the CAPM and 3-Factor results, the p-values for the F-statistic in the 5-

factor test are also 0.0000 across all portfolios indicating that the model is significant.

1 2 3 4 5 R_mtβ1* 1.2673 1.1648 1.1224 0.8399 0.7677 1.0000β2* 0.2990 0.0623 0.2671 -0.3821 -0.1514β3* 0.5313 0.1898 -0.0421 -0.3160 -0.2786β4* -0.3242 0.1627 0.7495 0.3311 0.4741β5* -0.9098 -0.1907 -0.0858 0.6465 0.5025α 1.0014 0.7281 0.7865 0.2881 0.4151

p(β1) 0.0000 0.0000 0.0000 0.0000 0.0000p(β2) 0.0029 0.3124 0.0008 0.0000 0.0052p(β3) 0.0001 0.0197 0.6844 0.0000 0.0001p(β4) 0.0224 0.0639 0.0000 0.0001 0.0000p(β5) 0.0000 0.0993 0.5607 0.0000 0.0000p(α) 0.0040 0.0000 0.0004 0.0754 0.0063t-stat(β1) 17.6715 26.2272 19.7929 20.2138 19.7895t-stat(β2) 3.0073 1.0119 3.3976 -6.6338 -2.8158t-stat(β3) 4.0631 2.3436 -0.4068 -4.1708 -3.9381t-stat(β4) -2.2953 1.8593 6.7097 4.0455 6.2042t-stat(β5) -4.8843 -1.6533 -0.5825 5.9906 4.9875f-stat 113.6849 176.2483 94.8814 97.8398 85.5882p(f-stat) 0.0000 0.0000 0.0000 0.0000 0.0000R^2 0.6501 0.7423 0.6079 0.6152 0.5831DW 2.0916 1.9586 1.9401 2.2881 2.0207

*whereβ1istheexcessreturnonthemarketindex,"β2isthedifferencebetweenthereturnsondiversifiedportfoliosofsmallandbigstocks(SMB),β3isthedifferencebetweenthereturnsondiversifiedportfoliosofhighandlowB/Mstocks(HML),β4istheaveragereturnonthetworobustoperatingprofitabilityportfoliosminustheaveragereturnonthetwoweakoperatingprofitabilityportfolios(RMW),β5istheaveragereturnonthetwoconservativeinvestmentportfoliosminustheaveragereturnonthetwoaggressiveinvestmentportfolios(CMA)"

(French,2016b)

SummaryofStatisticsforTimeSeriesTests,EntirePeriod(January,1990-January,2016)SampleSizeForEachRegression=312

PortfolioNumber

Table5



22

Adjusted 𝑅! in our 5-Factor results ranges from 57.63% to 73.80%, which has a

significantly greater explanatory power of the returns of stocks relative to both the

CAPM and 3-Factor model, which could only explain, between 42.88% and 73.29%

of the total variability of the dependent variable.

In our regression, the DW statistic ranges from between 1.9401 and 2.2881. Similar to

both the CAPM and 3-Factor results, since it is close to 2, we can conclude that we

have evidence of no serial correlation in the residuals.

The results in Table 6 show the Wald restriction 5-Factor tests results. We can see

that the p-value associated to the F-statistic is 0.0000 across all portfolios meaning

that there is significant evidence to reject the null hypothesis that the coefficients are

equal to zero.

4.3.1 Further Interpretation of the Results Overall, the results from the conducted tests show an overwhelming rejection of the

null hypothesis. The inference that can be made from this is that the tested data

showed significant signs that a stock (or portfolio of stocks) was able to produce

abnormal returns given exposure to various factors such as market, size, value,

profitability and investment; thus arbitrage opportunities seem to exist. In addition, α

and the corresponding p-values all showed positive values across all portfolios and

were all within a 0.10 confidence level. It could be argued that the beta alone was not

sufficient to determine the expected returns on stocks/portfolios. The results also

showed that the expected returns on a portfolio were more sensitive to one factor

1 2 3 4 5

f-stat 10.4007 7.5419 15.6708 31.4784 24.8117

p(f-stat) 0.0000*** 0.0000*** 0.0000*** 0.0000*** 0.0000***

chi-square 52.0034 37.7097 78.3541 157.3922 124.0587

p(chi-square) 0.0000 0.0000 0.0000 0.0000 0.0000

Table6

WaldTestRestrictionStatisticsfor5-FactorTimeSeriesTests,EntirePeriod(January,1990-January,2016)


H0:α=0andβ2=0,β3=0,β4=0,β5=0,β6=0

PortfolioNumber



23

more than another, which gives credence to the concept of diversification in modern

portfolio theory as a way of potentially reducing risk.

4.3.2 Limitations of the Data Analysis The limitations of this analysis are as follows:

Due to time constraints, only a small selection of firms were chosen from the S&P

500 index and as such the coefficient results calculated in this paper may not fully

reflect the expected performance of other stocks (and portfolios) on the respective

index.

The tested period of 26 years is a relatively short time span to truly explain the

variability of excess returns on stocks (and portfolios) and better results may be

achieved by testing over a longer period.



24

5. Conclusion

It is difficult to determine exactly why the returns on risky assets (stocks) are much

higher than theoretically risk-free assets in US markets or indeed global markets as

observed by Dimson et al (2011). The research outlined in this paper suggests that

various forms of assumptions for preferences such as risk aversion and market

inefficiency may be underlying factors for higher asset prices, which subsequently

lead to a higher equity risk premium. While some of the models outlined in this paper

have potential to resolve the equity premium anomaly, Cochrane (1997) points out,

“the most promising of them involve deep modifications to standard models necessary

to explain asset pricing phenomena” and that “every quantitatively successful current

story still requires astonishingly high risk aversion”. This leads us back to the second

plausible resolution to the “puzzle”, specifically that the historical premium may be

unreliable. As Cochrane (1997) puts it, perhaps it was simply “100 years of good

luck” and that US equity investors have enjoyed good fortune.

The findings of this study seem to contradict the hypothesis of contemporary asset

pricing models such as the CAPM in that stock returns should not be able to generate

abnormal returns (alpha) and that the expected return should be equal to the excess

returns on the market portfolio. The results showed positive alpha values (abnormal

returns) across all tested portfolios, with the majority within a 0.01 significance level

indicating a high level of confidence for rejection of the null hypothesis. In addition,

introducing more coefficients by using Fama-French’s 3 and 5-factor models proved

to yield similar results with alpha being observed across all portfolios tested. While

arbitrage opportunities do seem to exist, it is important to note, however, that in the

presence of transaction and taxation costs, alpha may be significantly reduced or

completely eliminated.

The results obtained also provide credibility to the linear structure of the CAPM

equation being a good explanation of stock returns. The 3 and 5-factor models proved

to be even more useful in explaining the percentage of the total variation in the

dependent variable (excess return on portfolio). In view of the above results, it can be



25

concluded that beta alone is insufficient in determining the expected returns on

stocks/portfolios.

5.1 Recommendations for Future Research Further research on preference assumptions such as risk aversion, aggregate

consumption, and market inefficiency may help advance debates on the equity

premium puzzle. In particular, modern financial theories, which present implications

towards market inefficiency, such as the efficient market hypothesis (Fama, 1970) and

the random walk hypothesis (Malkiel, 1973) could provide a better understanding

behind such high equity risk premia. Two key questions should be asked: If asset

prices were to encapsulate all available information including insider information,

implying strong form market efficiency (Fama, 1970), would there be smaller asset

price fluctuations which in turn lead to a smaller equity risk premium. Also, would

asset price movements be more predictable and thus reduce the risk element in asset

prices, if they followed a non-random walk as pointed out by Lo and MacKinlay

(2002).



26

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Appendix A

The list of firms used in this paper are detailed below:

S&P 500 Sample List No. COMPANY TICKERSYMBOL

1 Apple APPL2 AlphabetInc. GOOGL3 Microsoft MSFT4 ExxonMobilCorp. XOM5 BerkshireHathaway BRK.B6 Amazon AMZN7 Facebook FB8 Johnson&Johnson JNJ9 GeneralElectric GE10 AT&TInc. T11 WellsFargo&Co. WFC12 JPMorgan JPM13 ProcterandGamble PG14 VerizonCommunicationsInc. VZ15 PfizerInc. PFE16 Coca-ColaCompany KO17 ChevronCorporation CVX18 VisaInc. V19 HomeDepotInc. HD20 Merck&Co.Inc. MRK21 PhillipMorrisInternationalInc. PM22 ComcastCorporation CMCSA23 IntelCorporation INTC24 BankofAmericaCorporation BAC25 Disney DIS

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