Fama-French Model

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Project A: Estimating the Fama-French Model

Abstract Our paper estimates the Fama-French model and CAPM for portfolios with different

characteristics, including variation in stock size and book-to-market ratios. We find that the

Fama-French three factor model is more useful for estimating portfolio returns, whether for

weekly and monthly portfolio data in the US between 1926 and 2014, or for monthly portfolio

data for 22 non-US countries between 1990 and 2014. We test the stability of the Fama-

French factors over time, finding a structural break exists at 1963 for all portfolios. We

propose further research should consider including additional factors in the Fama-French

model, and whether developing countries with high economic growth will have different

factors or coefficients.

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1. Introduction Our research project will investigate the Fama-French model of asset pricing, by estimating

the coefficients of this three factor model, and comparing it to the simpler Capital Asset

Pricing Model (CAPM). Investigating the Fama-French model is important, as the CAPM

continues to be popular in financial analysis, despite the possibility that more than one factor

affects a stock's excess returns.

We will perform our analysis for both weekly and monthly US portfolio data, using portfolios

formed with stocks of different sizes and book-to-market ratios. The stability of the Fama-

French factors will be tested across time, as Fama-French (1992) and other authors have

concluded that they vary. Finally, we will consider non-US portfolio data, for developed

countries around the globe, to investigate whether the Fama-French model is specific to the

US, or if it can applied more generally. In our conclusion we will propose possible research

questions arising from our work.

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2. Theory and Literature Review of the Fama-French Model The Fama-French three-factor model was created by Fama and French (1992), to describe

expected stock returns in asset pricing and portfolio management, but its main predecessor

was the CAPM, which comprises only one variable to explain the expected returns of a

portfolio. This variable is the market risk, or non-diversifiable risk, β1, which is the slope of an

asset's excess return regressed on the market's excess return (Fama and French, 2004,

p.2), while there is no constant (Gujarati and Porter, 2009, p.147):

Ri - Rf = β1(Km-Rf) + ɛ The model is popular because of its simplicity and perceived ease of use to calculate

expected returns of both individual stocks and portfolios. However, Fama and French (1992)

found CAPM is only capable of explaining the expected returns generated in earlier periods,

prior to the 1960's in the US (Fama and French, 1992, p.450). Fama and French (1992)

concluded that CAPM faced too many setbacks in terms of empirical evidence, and in their

further research argue this may be because of “too many simplifying assumptions” (Fama

and French, 2004, p.25).

The Fama-French model instead utilizes three variables to describe expected stock return

(Cuthberston and Nitzsche, 2008, p.658):

Ri - Rf = α + β1(Km-Rf) + β2SMB + β3HML + ɛ In this Fama-French model, where Rf is the risk-free rate: Ri - Rf is the dependent variable, of

expected excess portfolio returns; α is the constant, known as Jensen's alpha; (Km-Rf) is the

excess market return; SMB is the size factor, and measures the difference in returns for

portfolios that comprise small stocks compared to those that comprise big stocks; and HML

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is the book-to-market factor, and measures the difference in returns between stocks with

high book-to-market ratios and those with low book-to-market ratios (Cuthberston and

Nitzsche, 2008, p.657).

Fama and French (1992, p.428) came to the conclusion of a three factor model after

observing that two classes of stocks did better than the market as a whole, with these being

stocks with small market capitalization and stocks with low a book-to-market ratio. Fama and

French (1992) had set out to determine whether the market risk factor, β1, helps explain the

cross-section of expected stock returns, and whether the new combination of stock size and

book-to-market ratio absorbs the effect of leverage and earnings-price ratios in stock returns

from 1963-1990 (Fama and French, 1992).

Portfolios used by Fama and French (1992) were formed on data from the NASDAQ, NYSE

and AMEX, excluding financial firms from the dataset. The reasoning behind this was to

eliminate the high leverage associated with financial firms (Fama and French, 1992, p.429).

Different portfolios were created by ranking the securities by size, market beta and book-to-

market value. In previous academic work, the CAPM was tested using cross-section

regressions of average portfolio returns to estimate betas, and other variables. Black (1972,

cited in Fama, 2014 p.1478) criticized this approach, as the results were too accurate given

the high volatility of market returns.

Regression methodology that improved accuracy was proposed by Fama and Macbeth

(1973), considering regression of average stock returns period-by-period, commonly month-

by-month. Fama and French (1992, p.438) relied on this methodology to estimate the cross

section return depending on log of the market size of the firm, it’s book-to-market ratio

captured at the beginning of each month. Fama and Frenchs' (1992) concluded that excess

portfolio returns were attributed not just to excess portfolio returns, but also to the size of

stocks and their book-to-market ratios. Fama and French (1992, p.450) initially included

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other parameters, including earnings-to-price and leverage, but these were found to be

“scaled versions of firm’s stock price”, and their effect was sufficiently covered by stock size

and book-to-market ratio. Fama and French (1992) discuss if their assumptions are correct,

and if market risk β1 is irrelevant to the Assest Pricing Model, or if there are other

explanatory variables correlated with β1. However, these theories are dismissed by Fama

and French (1992, p.438), due to the statisical significance of Fama and Frenchs' (1992)

findings.

The findings of Fama and French (1992) started a new era in asset pricing, but even two

decades on, the usefulness of the three factor model is still being debated compared to the

CAPM model. Bartholdy and Peare (2005, p.409-410) considered whether Fama-French or

CAPM is best to estimate the excess returns of an individual stock, finding Fama-French

only explains around 5%, and CAPM only explains around 3% of variation. Bartholdy and

Peare (2005, p.426) also found that excluding dividend from returns did not have a

significant impact.

Utilisation of the Fama-French model in non-US markets has also been explored, including

by Malin and Veeraraghavan (2004), using market data from France, Germany and the UK.

The methodology they used was similar to that of Fama and French (1992), using monthly

stock returns for stocks covering the period from 1992 to 2001. Malin and Veeraraghavan

(2004) found that: excess market returns, measuring systemic risk, was a significant variable

across al three countries; the size factor was significant in France and Germany, but not in

the UK; and that the book-to-market factor was also significant in all three countries.

The Fama-French model has also been tested over further time periods, including by Kothari

et al. (1995) post-1926 and post-1940, to see if excess market returns remained significant,

and also for 1947 and 1987 to test if the book-to-market ratio remains significant. Kothari et

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al. (1995) found that market and size factors held in all periods tested, but that the book-to

market factor had a statistically insignificant relationship.

Most recently, Fama (2014) published a paper summarising the research into CAPM and

Fama-French from the mid 1960s to the present. Fama (2014, p.1480) describes the Fama-

French model as one which looks backwards to explain returns, and while it is capable of

capturing the risks for portfolios through size, growth and book-to-market factors, he states

that other parameters may affect excess portfolio returns.

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3. Data The data used in this research project is from a data library provided by French (2014),

which gives data for portfolios containing stocks with different characteristics, as well as the

Fama-French factors. We will be using time series data, as our data is available for different

time periods, including weekly and monthly portfolio data.

We will begin by estimating the Fama-French model for US weekly portfolios, before moving

on to US monthly portfolios, and finally non-US monthly portfolios, using equal weighted

portfolio returns. Our US data runs from July 1926 until October 2014. Our non-US data runs

from November 1990 to November 2014, and includes the following 22 countries:

Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Hong

Kong, Republic of Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal,

Singapore, Spain, Switzerland, Sweden, UK.

This will allow us to test to see if there is any significant difference in the Fama-French

model depending on how regularly portfolio returns are measured, as well as if the Fama-

French model works best in or outside of the US. We will compare the Fama-French model

to the CAPM model in each case, to see which is more useful for explaining excess portfolio

returns, examining whether all factors of the Fama-French model remain statistically

significant. We will also perform tests for the stability of the Fama-French factors over time,

to see if different models are needed for different time periods.

Below we present our summary statistics tables for our weekly and monthly data for the US,

and our non-US monthly data. Our excess portfolio returns variables were constructed my

taking away the risk-free rate, RF, from the original portfolio returns, while excess market

returns was given by MktRF, size factor by SMB, and book-to-market factor by HML. We

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plotted our data to check for outliers, but none were found, and we checked for missing

values, leading to the deletion of four months at the beginning of our non-US monthly data.

The different characteristics of portfolios used as dependent variables are given by the

following data key:

(EW = equal weighted portfolio)

plus

(S = small stock size) or (B = big stock size)

plus

(G=low book-to-market ratio) or (N= neutral book-to-market) or (V= high book-to-market ratio)

30th book-to-market percentile 70th book-to-market percentile

US Weekly Returns Variable Observations Mean Std. Dev. Min Max

MktRF 4609 0.142 2.463 -18.79 17.02

SMB 4609 0.0344 1.229 -9.4 9.38

HML 4609 0.0857 1.410 -11.37 14.48

RF 4609 0.0710 0.0632 -0.016 0.335

weekly_EWSG 4609 0.178 3.114 -23.235 24.37

weekly_EWSN 4609 0.272 2.903 -26.265 28.446

weekly_EWSV 4609 0.387 3.175 -26.995 39.418

weekly_EWBG 4609 0.148 2.663 -19.385 18.63

weekly_EWBN 4609 0.195 2.720 -24.835 21.829

weekly_EWBV 4609 0.240 3.316 -26.625 27.442

US Monthly Returns Variable Observations Mean Std. Dev. Min Max

MktRF 1060 0.651 5.391 -29.07 37.93

SMB 1060 0.224 3.240 -16.4 37.47

HML 1060 0.394 3.490 -13.03 33.82

RF 1060 0.284 0.254 -0.06 1.35

monthly_EWSN 1060 1.132 7.786 -30.36 77.75

monthly_EWBN 1060 0.870 6.371 -30.41 57.67

Monthly Returns Excluding the US Variable Observations Mean Std. Dev. Min Max

MktRF 289 0.361 4.769 -21.14 13.7

SMB 289 0.0122 2.117 -5.73 7.34

HML 289 0.446 2.171 -11.4 9.42

RF 289 0.180 0.180 0 0.6

monthly_EWSN 289 0.607 5.191 -26.53 21.54

monthly_EWBN 289 0.496 4.800 -23.87 17.76

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4. Empirical Results We begin by discussing our first estimation of the Fama-French model, estimated for weekly

US portfolio returns between 1926 and 2014, comparing our estimations to the CAPM. We

then considered whether monthly data affects our results, before looking finally at non-US

monthly data, to investigate whether the Fama-French model is specific to the US.

i). US Weekly Our first estimations of the Fama-French model are given in Table 1, using excess returns

for two equal weighted portfolios, containing different stock sizes, but neutral book-to-market

ratios. In Model(1a) uses a portfolio containing stocks with small market equity, while

Model(2a) uses a portfolio containing stocks with big market equity.

Table 1

Model(1a) Fama-French

Model(1b) CAPM


Model(2b) CAPM

VARIABLES MktRF

SMB

HML

Constant

weekly_EWSN 0.941*** (0.010) 0.940*** (0.021) 0.401*** (0.023) 0.0718*** (0.0099)

weekly_EWSN 1.020*** (0.0087)

0.127*** (0.0215)

weekly_EWBN 1.014*** (0.00898) 0.202*** (0.0201) 0.335*** (0.0181) 0.0155* (0.0088)

weekly_EWBN 1.055*** (0.0048)

0.0454*** (0.0119)

Observations 4,609 4,609 4,609 4,609 R-squared 0.9395 0.7493 0.9494 0.9125

Adj. R-squared

0.9394

0.7493

0.9493

0.9125

Standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1

In Model(1a), we find the Fama-French coefficients for market return, size, and book-to-

market are significant at the 1% level, while the constant, Jensen's alpha, is also significant

at the 1% level. The R-squared value is also high, meaning that Model(1a) explains 93.95%

of the variation in excess portfolio returns, while the Adjusted R-squared value, which allows

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regressions with different numbers of explanatory variables to be compared, was slightly

lower at 0.9394.

To ensure the standard errors and significance of our coefficients is correct, Model(1a) was

tested for heteroscedasticity, using White's test. Our null hypothesis was for

homoscedasticity, and our alternative hypothesis was for heteroscedasticity. We found that

heteroscedasticity was present in the residuals, as the resulting p-value of our White test

was less than 0.05, meaning the variance of the error term was not constant. To correct this

for the rest of our analysis, we used robust standard errors in all our regressions, ensuring

our t and F statistics are correct.

Model(1a) shows all Fama-French factors have a positive effect on excess portfolio returns,

with both market return and size factors having a similar effect. Holding all other variables

constant: a 1 unit increase in market return in a 0.941 increase in excess portfolio returns;

and a 1 unit increase in the size factor results in a 0.940 increase in excess portfolio returns;

a 1 unit increase in the book-to-market factor results in only a 0.401 increase in excess

portfolio returns. Importantly for investigating the Fama-French model, the constant in

Model(1a) is significant at the 1% level, suggesting that Jensen's alpha does exist, albeit

with a small effect of only 0.0718.

We repeated the same process for Model(2a), with the dependent variable as excess returns

for portfolios that included stocks with big market equity. This gave different results, with the

coefficients for the size factor and constant both changing significantly compared to

Model(1a). In Model(2a), holding all other variables constant: a 1 unit change in market

return results in a 1.014 unit increase in excess portfolio returns; a 1 unit change in size

factor results in a 0.202 unit increase in excess portfolio returns; and a 1 unit change in

book-to-market results in a 0.335 increase in excess portfolio returns. The most interesting

result from Model(2a) is that the constant is only significant at the 10% level of significance,

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not reaching the 5% level that we would need to accept the Fama-French model, where a

constant is used, as opposed to CAPM where it is not.

To test whether CAPM explains excess portfolio returns more effectively than Fama-French,

we regressed the same dependent variables from Models (1a) and (2a) in Models (1b) and

(2b) respectively, but using only one explanatory factor, market return, as suggested by the

CAPM model. If CAPM is correct we would expect to see excess market returns is significant

at the 5% level, while the constant should not be significant at the 5% level. For both

Model(1b) and Model(2b), we find that excess market returns have very similar coefficient

coefficients, which are significant at the 1% level, and are also of a similar size to those

found in Model(1a) and Model(2a). However, the constant in both CAPM models is

significant at the 1% level, suggesting the Fama-French model is more useful than CAPM,

which predicts there would be no constant.

To confirm this result, we conducted an F-test for Models (1a) and (2a), with the null

hypothesis that the coefficients on the size and book-to-market factors are jointly equal to

zero, and an alternative hypothesis that at least one of the variables is not equal to zero. Our

F-tests, with F(2, 4605), both rejected the null hypothesis at the 5% level, as the p-values for

both tests were less than 0.05, meaning we can accept the alternative hypothesis for both

Models (1a) and (2a). This again suggests that the Fama-French model is better than CAPM

at estimating excess portfolio returns.

Consulting the Adjusted R-squared values for Models (1) and (2) confirms this finding, as

they are much higher in the Fama-French models than for the CAPM models. Model(1a)'s

Adjusted R-squared is 0.9394, while Model(1b)'s Adjusted R-squared is much lower, at only

0.7493. This is repeated for Model(2a)'s Adjusted R-squared, which is 0.9493, while

Model(2b)'s Adjusted R-squared is 0.9125. It is interesting to note for portfolios that contain

big stocks, CAPM performs much better than a portfolio containing small stocks, with an

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Adjusted R-squared that is not too far away from its Fama-French counterpart. This means

that while the size and book-to-market factors have a statistically significant effect in the

Fama-French estimation in Model(2a), the size of their explanatory power is relatively small

compared to the market return factor.

Given our analysis of portfolios that contain stocks with neutral book-to-market ratios, and

containing stocks with both big and small market equity, we find that the Fama-French model

is better than the CAPM model at explaining excess portfolio returns. However, it is

important to assess the stability of the Fama-French model and its parameters over time. To

do this, we undertook a Chow Test for both the portfolios in Models (1) and (2). We decided

to test our portfolios with a break point of 1963, as Fama and French (1992, p.450) found

that the Fama-French model held for the time period 1963 to 1990.

Our null hypothesis was that the parameters of the Fama-French model were constant

across these two periods, and our alternative hypothesis was that the parameters varied. For

Model(1a) our F-test, F(4, 4601), our F-critical was 2.45 at the 5% level, and our F-

test=61.16, meaning our F-test>F-critical, and so we rejected the null hypothesis, and

instead accepted the alternative hypothesis that the parameters varied in Model(1a). We

repeated this Chow Test for Model(2a), finding F-test=34.55, meaning our F-test>F-critical,

and so we again rejected the null, and accepted the alternative. This confirms Fama and

French (1992, p.450) that there is variation in factors over time. Table 2 shows the

coefficients for the Fama-French factors for both Models (1) and (2) across the two time

periods.

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Table 2

Model(1a) Model(1a) Model(2a) Model(2a) 1926-1962 1963-2014 1926-1962 1963-2014

VARIABLES weekly_EWSN weekly_EWSN weekly_EWBN weekly_EWBN MktRF 1.014*** 0.868*** 1.043*** 1.027***

(0.01455) (0.01063) (0.01326) (0.005525) SMB 0.992*** 0.886*** 0.234*** 0.189***

(0.03594) (0.02058) (0.02760) (0.009936) HML 0.341*** 0.342*** 0.248*** 0.426***

(0.03711) (0.02267) (0.02243) (0.01029) Constant 0.0813*** 0.0733*** 0.0288** -0.00165

(0.01709) (0.01168) (0.01330) (0.01159)

Observations 1904 2705 1904 2705 R-squared 0.9521 0.9260 0.9660 0.9290 Adj. R-squared 0.9520 0.9259 0.9660 0.9289


For Model(1a), the coefficients across the Fama-French factors remain relatively constant,

with the main change only being for the coefficient of excess market returns, which changes

from 1.014 for the period 1926-1962, to 0.868 for the period 1963-2014, thus moving further

away from a one-to-one relationship between excess market returns and excess portfolio

returns in the later period. All coefficients in Model(1a) are significant at the 1% level for both

periods, confirming that the addition of size and book-to-market factors are a good addition

to the model, and that the Fama-French model is more useful than the CAPM model.

For Model(2a) the coefficients also remain similar, albeit the book-to-market factor increases

from 0.248 in the period 1926-1962, to 0.426 in the second period 1963-2014. All the

coefficients reach the 5% level of significance, apart from the constant for Model(2a) in the

later period of 1963-2014, which is not significantly different from zero. However, as the

other coefficients are all significantly different from zero at the 1% level, we can conclude

that the Fama-French factors are more useful than the simpler CAPM model.

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So far we have considered portfolios only where the size of stocks has varied between small

and big, while the book-to-market ratio has remained neutral. In Table 3 we calculated the

Fama-French and CAPM models of portfolios that include small stocks, but have different

book-to-market ratios, ranging from growth stocks, which have low book-to-market ratios, to

value stocks, which have high book-to-market ratios.

Table 3


Model(3b) CAPM


Model(4b) CAPM

VARIABLES weekly_EWSG weekly_EWSG weekly_EWSV weekly_EWSV MktRF 1.0298*** 1.077*** 0.914*** 1.033***

(0.0102) (0.0181) (0.0111) (0.0262) SMB 1.094*** 1.077***

(0.0237) (0.0295) HML -0.00516 0.758***

(0.0186) (0.0250) Constant -0.00467 0.0258 0.155*** 0.240***

(0.0134) (0.0239) (0.0125) (0.0272)

Observations 4,609 4,609 4,609 4,609 R-squared 0.9108 0.7256 0.9180 0.6423 Adj. R- squared 0.9107 0.7255 0.9179 0.6422 Standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1

Model(3a) uses excess portfolio returns where the book-to-market ratio is low. The

coefficients on market return and size factors are both significant at the 1% level, but neither

the coefficient on the book-to-market factor, nor the constant, which both have a negative

effect on excess portfolio returns, reach the 10% level of significance. However, an F-test of

the null hypothesis that both the size and book-to-market factors are jointly equal to zero

returns a p-value less than 0.05, and so we accept the alternative hypothesis that at least

one of the factors is significantly different from zero.

To test whether CAPM would be a better estimation, Model(3b) uses the same dependent

variable, but only including market return as an explanatory variable. This gives a broadly

similar coefficient for market return of just over 1, which is significant at the 1% level.

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Importantly, the constant in this model does not have a significant coefficient, failing to reach

the 10% level of significance. This suggests CAPM is a suitable model for excess portfolio

returns where the book-to-market ratio is low and the stocks are small. However, its

Adjusted R-squared is 0.7255, which is lower than the Adjusted R-squared for Model(3a) of

0.9107. The fact that in Model(3a) the size factor was significant at the 1% level, and the

Adjusted R-squared was higher, tells us that CAPM is not the best possible estimation of

excess portfolio returns in this case, and the Fama-French model is still more useful.

Model(4a) and (4b) repeats this process for excess portfolio returns where stocks are still

small, but the book-to-market ratio is high. All the coefficients in Model(4a) are significant at

the 1% level, with and Adjusted R-squared of 0.9179, while the coefficients in Model(4a)

have similar effects to Model(1a), although the book-to-market factor has a coefficient closer

to 1 here.

Model(4b) estimates the CAPM model using the same dependent variable, with the

coefficient for market return being significant at the 1% level, and having a similar effect to

Model(4a). Importantly, the constant is significant at the 1% level also, which suggests we

can reject the CAPM model as a suitable estimation of excess portfolio returns in this case.

The Adjusted R-squared in Model(4b) is also much lower than Model(4a), meaning

Model(4b). Finally, a joint hypothesis test of the size and book-to-market factors in Model(4a)

finds they are significantly different from zero, so we can conclude that the Fama-French

model is more useful than the CAPM model in this case.

Table 4 continues our analysis by considering excess portfolio returns where the size of

stocks in the portfolio is big, and stock book-to-market ratio varies between high and low.

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Table 4


Model(5b) CAPM


Model(6b) CAPM

VARIABLES weekly_EWBG weekly_EWBG weekly_EWBV weekly_EWBV MktRF 1.051*** 1.0439*** 1.124*** 1.215***

(0.00785) (0.00897) (0.02060) SMB 0.223*** 0.232***

(0.0172) (0.0180) HML -0.180*** 0.851***

(0.0152) (0.0169) Constant 0.00699 0.000299 -0.000183 0.0678***

(0.008502) (0.0102) (0.0108) (0.02095)

Observations 4,609 4,609 4,609 4,609 R-squared 0.9521 0.9323 0.9476 0.8147 Adj. R- squared 0.9521 0.9323 0.9476 0.8146 Standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1

Model(5a) is a Fama-French model for big stocks with low book-to-market ratios. The

coefficients for market return, size and book-to-market factors are all individually significant

at the 1% level, which compares to Model3(a), where the book-to-market factor was not

individually significant.

Model(5a) has an Adjusted R-squared of 0.9521, however the constant is not significantly

different from zero, with an extremely low coefficient of 0.00699. Thus a CAPM model is

tested in Model(5b), where the coefficient for market return is significant at the 1% level, with

a similar effect as in Model(5a), with an extremely small constant of 0.000299, which is not

significantly different from zero. The Adjusted R-squared of 0.9323 for Model(5b) is also very

close to that of Model(5a). This suggests the size and book-to-market ratio factors are

adding very little explanatory power, but this small effect is significant at the 1% level for both

factors. Thus while there may be a zero intercept, in this case the Fama-French model is

more useful than the simpler CAPM model.

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An important difference between Model(5a) and Model(2a), which used stocks with neutral

book-to-market ratios, is the book-to-market factor in Model(5a) has a negative effect on

excess portfolio returns at -0.180 for a 1 unit increase in the book-to-market factor, while in

Model(2a) it had a positive explanatory effect of 0.335 for a 1 unit increase in the book-to-

market factor.

Model(6a) explores a Fama-French model for big stocks, but where the book-to-market ratio

is high. All the coefficients on market return, size and book-to-market reach the 1% level of

significance, while, as with Model(5a), the constant is not significantly different from zero.

Compared to Model(5a), where the book-to-market ratio was low, the coefficient of book-to-

market factor has a positive sign, and so holding other variables constant, a 1 unit increase

in the book-to-market factor results in a 0.851 increase in excess portfolio returns, where

Model(5a) saw a decrease of 0.180.

Comparing the Fama-French model of Model(6a) to the CAPM model of Model(6b), we find

that the Fama-French model has a higher Adjusted R-squared, meaning it explains more of

the variation in excess portfolio returns than Model(6b). The constant in Model(6b) is also

significant at the 1% level, and so we can conclude that the Fama-French model is more

useful than the CAPM model for estimating the excess returns of big stocks with high book-

to-market ratios.

To finish our analysis of weekly excess returns, having found that the Fama-French model is

more useful than the CAPM model for stocks of different sizes and with different book-to-

market ratios, we tested whether the Fama-French models had stable parameters over time.

We did this by conducting a Chow Test for Models (3a), (4a), (5a) and (6a), with a break-

point of 1963, inline with the work of Fama and French (1992, p.450). We found that for all

four models we could reject the null hypothesis, and instead parameters varied across the

two periods, confirming Fama-French parameters are not stable over time.

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ii). US Monthly We now consider monthly excess portfolio returns, focusing on portfolios that have neutral

book-to-market ratios, but with variation in stock size between big and small. By considering

monthly data, it allows us to see whether the Fama-French model continues to be a better

model of excess portfolio returns, or whether CAPM becomes the better model. Our

regression analysis is presented in Table 5.

Table 5


Model(7b) CAPM


Model(8b) CAPM

VARIABLES monthly_EWSN monthly_EWSN monthly_EWBN monthly_EWBN MktRF 0.993*** 1.258*** 1.048*** 1.128***

(0.0158) (0.0528) (0.0177) (0.0288) SMB 1.043*** 0.160***

(0.0500) (0.0326) HML 0.370*** 0.342***

(0.0321) (0.0268) Constant 0.105*** 0.312*** 0.0163 0.135***

(0.0464) (0.1038) (0.0407) (0.0550)

Observations 1,060 1,060 1,060 1,060 R-squared 0.9585 0.7585 0.9524 0.9115 Adj. R-squared 0.9584 0.7583 0.9523 0.9114


Model(7a) is a Fama-French model of excess portfolio returns, where the portfolio contains

small stocks. All coefficients in Model(9a) are significant at the 1% level, with an Adjusted R-

squared of 0.9584. This is a significantly higher Adjusted R-Squared than for the CAPM

variant in Model(7b), which had a lower Adjusted R-squared of 0.7583. While the coefficients

are significant at the 1% level in the CAPM, the fact further variables also reach this level of

significance in the Fama-French model shows that if we simply accepted the CAPM, we

would have underspecified the model, with omitted variable bias, giving incorrect confidence

intervals.

Page 20 of 23

Model(8a) repeats this analysis for a portfolio containing big stocks. All coefficients are

significant at the 1% level, apart from the constant, which is not significantly different from

zero. However, when comparing this Fama-French model to the CAPM in Model(8b), we find

that the Adjusted R-squared is higher for the Fama-French model, while the coefficient in the

CAPM is also significantly different from zero, so we can reject CAPM being better than

Fama-French at estimating excess portfolio returns in this case.

Comparing Models (7a) an (8a) with the equivalent weekly portfolios of Models (1a) and

(1b), we find that the Fama-French factors do not vary substantially, and in both cases the

Fama-French model is more useful at estimating excess portfolio returns than the CAPM.

We also found the Fama-French factors did vary across our two time periods for Models (7a)

and (8a), when we tested for a structural break at 1963.

iii). Monthly excluding the US

We finish by looking at portfolios formed from stocks excluding the US, to see if our result of

the Fama-French model being continues to hold.

Table 6


Model(9b) CAPM


Model(10b) CAPM

monthlyexUS monthlyexUS monthlyexUS monthlyexUS VARIABLES _EWSN _EWSN _EWBN _EWBN MktRF 1.046*** 0.965*** 1.002*** 0.972***

(0.0250) (0.0418) (0.0182) (0.0219) SMB 0.947*** 0.274***

(0.0486) (0.0331) HML 0.0356 0.291***

(0.0436) (0.0413) Constant 0.202** 0.259* 0.00164 0.145**

(0.0844) (0.144) (0.0566) (0.0734)

Observations 289 289 289 289 R-squared 0.9288 0.7855 0.9617 0.9324 Adj. R- squared 0.9280 0.7848 0.9613 0.9322


Page 21 of 23

Model(9a) uses Fama-French factors for excess portfolio returns that contain small stocks

with neutral book-to-market ratios. The coefficients for market return and size are significant

at the 1% level, and the constant is significant at the 5% level, but the book-to-market factor

is not significantly different from zero. Model(9b) is the CAPM variant, where excess market

returns is significant at the 1% level, but the constant is only significant at the 10% level. As

the constant does not reach the 5% level, it suggests the CAPM criteria of no constant is

fulfilled. However, Model(9a) has a much higher Adjusted R-squared of 0.9280 compared to

0.7848, and using an F-test on size and book-to-market factors, we can not reject the

hypothesis that they are jointly significant. Model(10a) repeats this process, only for

portfolios with big stocks, with the same conclusion drawn that the Fama-French model is

more useful than the CAPM, as it has a higher Adjusted R-squared, with all the Fama-

French factors being significant at the 1% level.

Comparing the models for monthly excess portfolio returns in the US, in Models (7a) and

(8a) with those excluding the US in Models (9a) and (10a), we find that for portfolios with the

same characteristics, the Fama-French factors do not substantially vary. The most

substantial change is the book-to-market factor, which is significant at the 1% level with a

coefficient of 0.370, for US portfolios with small stocks, and changes to 0.0356 in non-US

portfolios, but this coefficient is not significant. Comparing the portfolios with big stocks, the

variations in coefficients are extremely small, while the constant in each is not significant.

Finally, a Chow Test for the non-US Fama-French factors shows they have a structural

break too, as was the case for US models. As the non-US data is over a shorter time period,

the Chow Test was for variability in the parameters of the Fama-French model for the period

prior to November 2000, and from November 2000 onwards.

Page 22 of 23

5. Conclusion In conclusion, we have found that the Fama-French model using three factors, representing

excess market returns, stock size and stock book-to-market ratio, is a better and more useful

model of portfolio returns than the CAPM. We found this to be the case for both weekly and

monthly US portfolio returns between 1926 and 2014, while also finding that the Fama-

French factors are not stable over time, with a Chow Test showing a structural break either

side of 1963, confirming the work of Fama and French (1992).

We also considered non-US monthly returns, for 22 developed countries across a 24 year

period between 1990 and 2014, and found that the Fama-French model remains more useful

than CAPM, with the coefficients of factors not substantially varying compared to US monthly

returns. A Chow Test was again performed to check for parameter stability over time, finding

a structural break either side of November 2000.

Our work has confirmed Fama and Frenchs' (1992) findings, that the Fama-French three

factor model is preferable to CAPM for estimating excess portfolio returns, even when those

portfolios contain stocks with varying sizes and book-to-market ratios. However, there

continues to be debate over whether more factors are required in the Fama-French model,

as suggested by Fama (2014), and a Ramsey RESET test for omitted variables in our

weekly portfolios shows this could be the case, as we could not reject the null hypothesis of

further explanatory factors.

Thus further work could explore whether there are other factors that are significant in

explaining excess portfolio returns, which if proven to be the case, would mean our Fama-

French models are suffering from omitted variable bias. Further work could also be

undertaken to examine what factors are significant for economies that display significantly

different economic characteristics from the US, such as developing countries that are seeing

high rates of economics growth, which may affect returns.

Page 23 of 23

6. References Bartholdy, J. and Peare, P. (2005) 'Estimation of expected return: CAPM vs. Fama and French', International Review of Financial Analysis, 14(4), pp. 407–427.

Cuthbertson, K. and Nitzsche, D. (2008) Investments, 2nd edn., US: John Wiley & Sons, Ltd.

Fama, E. & MacBeth, J., (1973) 'Tests of the multi-period two-parameter model', Journal of Financial Economics, 1(1), p. 43– 66.

Fama, E.F. and French, K.R. (1992) 'The cross-section of expected stock returns', The Journal of Finance, 47(2), pp. 427-465.

Fama, E.F. and French K.R. (2004) 'The capital asset pricing model: theory and evidence', Journal Of Economic Perspectives, 18(2), pp. 25-46.

Fama, E.F. (2014) 'Two pillars of asset pricing', American Economic Review, 104(6), pp. 1467-1485.

French (2014) Data Library, Available at:http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html (Accessed: 19th November 2014).

Gujarati, D.N. and Porter, D.C. (2009) Basic Econometrics, 5th edn., US: McGraw-Hill.

Kothari, S., Shanken, J. and Sloan, R. G., (1995) 'Another look at the cross-section of expected stock returns', Journal of Finance, 50(1), pp. 185-224.

Malin, M. and Veeraraghavan, M. (2004), 'On the robustness of the Fama and French multifactor model: evidence from France, Germany, and the United Kingdom', International Journal of Business and Economics, 3(2), pp. 155-176.

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