Analysis of CAPM and Three Factor Model

Introduction

Capital Asset Pricing Model (CAPM) is the backbone of modern portfolio theory. According to CAPM, the expected return on stock is a function of its relationship with the market portfolio defined by its beta. However, Eugene Fama and Kenneth French (1992) brought together two more factors and found that stock return is based on a combination of not just market beta but also firm size and value. They came up with a new model known as Three Factor Model as an alternative to CAPM.

What is Fama and French Three Factor Model?

Fama and French three factor model expands on the CAPM by adding size and value factors in addition to the market risk factor in CAPM. This model considers the fact that value and small cap stocks out-perform markets on a regular basis. Fama and French attempted to approach and measure equity returns in a different manner and found that value stocks outperform growth stocks and also small cap stocks tend to out perform large cap stocks. Thus, the performance of portfolios with a large number of small cap or value stocks is better than one with large cap and growth stocks but lower than the CAPM result. This is because the three factor model adjusts downward for small cap and value out-performance. However, the three factor model is considered to be a better model than its counterparts as by including two additional factors, the model adjusts for the out-performance tendency.

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The out-performance tendency of small cap stocks is a debated issue as out-performance may be considered as dependent on market efficiency or market inefficiency and may mean different things under each. In the case of market efficiency, the out-performance is generally explained by theexcess risk that value and small cap stocks face as a result of their higher cost of capital and greater business risk. However, in the case of market inefficiency, the out-performance can be explained by market participants arriving at the incorrect value of these companies. This results in excess return in the long run as the value adjusts.

Thus, according to Fama and French model, the expected return on a portfolio in excess of the risk free rate is explained by the sensitivity of its return to three factors. These are:

• excess return on a broad market portfolio
• Size factor represented as the difference between the return on a portfolio of small stocks and the return on a portfolio of large stocks (SMB) and
• Value factor represented as the difference between the return on a portfolio of high-book-to-market stocks and the return on a portfolio of low-book-to- market stocks (HML).

The model can be presented as follows:

(Rpt) = Rf + ßp[(Rmt) – Rf ] + sp(SMB) + hp(HML) + εpt

where:

• (Rpt) is the weighted return on portfolio p in period t.
• Rf is the risk-free rate;
• ßp is the coefficient loading for the excess return of the market portfolio over the risk-free rate
• sp is the coefficient loading for the excess average return of portfolios with small equity class over portfolios of big equity class.
• hp is the coefficient loading for the excess average returns of portfolios with high book-to-market equity class over those with low book-to-market equity class.
• εpt is the error term for portfolio p at time t.

Fama and French (1995) analysed the characteristics of firms with high book-to-market and those with low book-to-market equity. They found that firms with high book equity to market equity ratio tend to be steadily troubled in comparison to those with low book equity to market equity. In fact firms with low ratio have continued profitability conditions. This led them to conclude that high book equity to market equity stocks are riskier and the returns to holders of such equity is actually a compensation for holding less profitable and riskier stocks. They show that book-to-market equity ratio and slopes on HML in the three factor model is a compensation for relative distress. Weak firms with continuously low earnings tend to have high book equity to market equity ratio and positive slopes on HML. In contrast, strong firms with high earnings have low book equity to market equity and negative slopes on HML.

Fama and French Three Factor Model as a Performance Tool

Fama-French Three Factor Model is a highly useful tool for understanding portfolio performance. In recent times, this model has gained more relevance than CAPM. It is now taken as the most widely accepted explanation of stock price movements taken together and investor returns. Unlike CAPM which is a single factor model based on relationship between returns and market factor, the Fama-French model is based on stock return having its basis in not one but three separate risk factors: market, size and value or book to market based factor.

Fama and French reached this conclusion on the basis of two separate studies based on:

• Returns from a period from 1963 to 1990
• Returns from a period from 1929 to 1997.

Fama and French discovered that stock returns can best be explained when stocks are separated into portfolios based on size as measured by market capitalization and value-growth as measured by book/market ratios.

Thus, the model adopts a different approach to explain market pricing. These three factors, namely market, size and value collectively explain a significant part of the variation in mean returns. In other words, if we assume that stocks are priced rationally then systematic differences in average returns are explained by the market, size and value exposures of the risk factors in returns.

Though on one hand Fama and French model claims that the behaviour of stock returns in relation to market, size and value factors is consistent with the behaviour of earnings, on the other hand it admits to the weakness of the model especially in relation to the value factor. However, this weakness can be attributed to the measurement error problems in earnings data.

Objectives of the Present Study

The present study examines the applicability of the Fama-French three- factor model on the S&P 500 Stock Index for the period 1995-2006. It uses regression to test whether the three factors in the Fama-French model still hold for the period 1995-2006. It also attempts to compare its findings with the results of the earlier Fama-French study. Thus, the study attempts to:

• Test the one-factor linear pricing relationship implied by the CAPM and the three- factor linear pricing model of Fama and French.
• Analyze whether the market, size and value factors are all-encompassing in the cross-section of random stock returns.

About S&P 500 Index

The S&P 500 index is one of the indices maintained by Standard & Poor’s and forms part of the broader S&P 1500 and S&P Global 1200 stock market indices. The index is defined by the stocks of 500 Large-Cap corporations. Most of the companies included in this index are US based (Refer Appendix A). The index is used in reference not only to the index itself but also to the 500 actual companies whose stocks are included in the index.

All of the stocks in the index are those of large publicly held companies and trade on major US stock exchanges such as the New York Stock Exchange and Nasdaq

Risk-free Rate

The implied yield on the month-end auction of Treasury bills has been used as a risk-free rate on which the CAPM model is based.

Sample Stocks

There are 500 companies on the S&P 500 index. The sample for this analysis includes companies that have ranked in 10 top issues (in terms of market capitalization) on S&P 500 index during the period of study, that is, 1995-2006.

Ten companies have been selected as the sample of the study. These companies have appeared among first 10 more than one time in 10 years (1995-2006). These include:

• General Electric
• Microsoft
• Pfizer Inc
• Bank of America
• Exxon Mobil
• Procter & Gamble
• Wal-mart Stores
• Citigroup Inc
• AIG
• Johnson and Johnson

Methodology

Classifying companies on the Basis of Value

Each year, the sample companies are first classified on the basis of size (capitalization). Thereafter, the sample companies are classified on the basis of the value. Value is defined in terms of the ratio of book equity to market equity.

Book equity to market equity for each company on the sample is calculated each year by taking a ratio of book equity at the end of financial year and the market equity at the end of financial year. The companies are then ranked on the basis of the ratio. The top 30% companies H (3 companies) are taken to have a high value category as they have a higher book equity to market equity ratio. 40% companies are classified as M (4 companies in all) have a medium ratio while remaining 3 companies (30%) have a low ratio L.

For example in 2006, the sample companies have been grouped on the basis of book equity to market equity ratio as (Refer Appendix C for Book Equity to Market Equity Ratio)

Constructing and Sorting Portfolios Using Value and Size

As indicated earlier, the Fama and French model is based on three factors for explaining common stock returns:

• the market factor defined as market index return minus risk-free return proposed by the CAPM
• Factors relating to size
• Factors relating to value

On the basis of the value and size rankings, portfolios can be constructed for the sample companies. For example, in 2006 the portfolios can be as follows (Refer Appendix D for construction of portfolios methodology):

Monthly equally-weighted returns on the six portfolios are calculated for each year and the portfolios are re-formed in the following year. The six portfolios are constructed to be equally-weighted portfolios as compared with value-weighted portfolios.

The Factor Portfolios

The six portfolios can be classified into two factor portfolios (one representing size as a factor while the other representing value) as given below (Refer Appendix E):

• Size based factor portfolio: This can be represented in terms of Small Cap Minus Big Cap (SMB) and indicates the risk factor in returns related to size.

SMB = simple average of the returns of the three Small (Group 2) stock portfolios – average of the returns on the three big portfolios (Group 1).

SMB is the difference between the returns on small and big stock portfolios. SMB is clear of the value effects. It is focused on the differences in the behaviour of small and big stocks.

• Value based factor portfolio: This can be represented in terms of High Minus Low (HML). It indicates the risk factor in returns related to value.

HML = simple average of the returns on two high ratio portfolios – the average returns on two low ratio portfolios.

It is constructed to be relatively free of the size effect.

Analysing Returns

The following table highlights the mean or average return for each portfolio as well as standard deviation.

Statistics on Portfolio Returns

(Period Covered 1995- 2006)

Mean =  x / number of observations

The table above shows the relation between size and average return on one hand and value and average return on the other. The table clearly indicates that size and average return are positively correlated. The larger the size the higher the return. However, value and average return seems to be positive for small stocks, but negative for big stocks. This is similar to US findings of Fama-French which cite a strong value effect and a conditional size effect for the US market.

Correlation between Factor Portfolios

Correlation can be calculated as:

The SMB and HML also have a near perfect correlation. Thus size and the value factors seem to be closely correlated.

Statistical Analysis

In the discussion above correlation has been used to analyse the relationship between size and value of a company and how they impact returns. A more reliable analysis is possible by using multiple regression analysis.

Multiple regression analysis is an extension of correlation analysis, which involves the use of one independent variable, such as market risk, to measure corresponding changes in one dependent variable (the outcome, such as stock returns). This method, however, ignores the possible influence of other independent variables for example size and value. The independent variable is applied to the dependent variable say stock price returns.

Multiple regression is applied when a response variable (return in this case) may depend on more than one explanatory variable (market, size and value). Furthermore, these possible explanatory variables often co-vary with one another. This makes it impossible to subtract out the effects of the factors separately by performing successive linear regressions for each individual factor. In such cases it becomes necessary to perform multiple regression defined by an extended linear model.

Consider for example, a mutliple regression model having two explanatory factors. It can be given by

Through least squares approach the three  parameters can be estimated.

A good understanding of a multivariate model can be obtained by considering a bi-variate case with two factors. By solving the two normal equations, the best estimates for the beta parameters can easily be shown to be given by:

Where,

r₁₂ is the mutual correlation between the two x variables indicated as cor(x₁,x₂),

r_1y is the correlation between x₁ and y given by cor(x₁,y), and

r_2y is the correlation between x₂ and y given by =cor(x₂,y).

Correlations can be expressed in terms of beta parameters.

Thus, in such cases correlations consists of the sum of two parts: a direct effect and an indirect effect interceded by mutual correlation between the explanatory variables.

However, multiple regression analysis is different from the correlation analysis. It allows one to determine not just the relationship but the relative contribution from the direct and the indirect parts.

Regression Analysis on S &P 500 Index quoted shares in line with Fama-French Model

Multiple regression framework has been used in the tests of the Fama-French model. The multivariate regression model can be represented as:

R_at = aa + ba MKTt+ ca SMBt+ da HMLt+et, a=1,…,N ; t=1,…,T

Where

R_at= Excess return to portfolio ‘a’ in time period‘t’

M_t= Excess return to the market portfolio in time period‘t’

SMBt = return to the size factor portfolio in time period‘t’

HMLt = the return to the value factor portfolio in time period‘t’

aa = abnormal mean return of portfolio ‘a’

ba =market exposure of portfolio ‘a’

ca =size exposure of portfolio ‘a’

da =value exposure of portfolio ‘a’

et = mean- zero asset-specific return of portfolio ‘a’

The data from the sample can be tested using the above representation and taking some of the coefficients as zero. Doing so excludes them from regression and helps one estimate and test variants as in the case of Fama-French model.

Given rational pricing, the various factors must justify being used the asset pricing model only if they contribute substantially to the risk of well-diversified portfolios.

The market factor is responsible for largest fraction of common variation in stock returns. This can be seen in the case of the six size-value portfolios.

The table clearly indicates that the market factor alone accounts for almost 70% of the adjusted R². However, when the market factor is excluded and the other two factors, that is, value and size are used, the adjusted R² declines to below 20%. It can also be observed that the adjusted R² in the three-factor regression is higher than in the one-factor market model regression. It can also be observed that in certain cases, adding HML to the market model regression increases R² more than adding SMB to the market model regression. Thus, though all three factors impact returns on stock, the impact of market factor is much more than the other two factors. However, we cannot rank the remaining two factors in terms of their impact.

The factor exposure estimates in the regression applied to S&P index shares as also in the case of three- factor model indicate that the estimated size exposures increase with size ranking, and correspondingly for the estimated value exposures and value ranking. The market exposures of the portfolios are all slightly below one. When all three factors are considered together, it provides the most suitable description of all-included risk in these size and value-sorted portfolios.

The limitation of the study is that it is based on relatively small number of sorted portfolios, and the fact that the only sorting variables available to rely on the same characteristics of size and value are used to create the risk factors. Besides, it seems the sample size is too small to support any reliable conclusions, since there are virtually no statistically significant findings and the adjusted R²s are close to zero. In contrast, Fama and French study was based on much longer sample period and larger cross-section of earnings data. Even then the study found statistically weak relationships.

However, despite the limitations of the study, an interesting observation both in the case of the S&P shares as well as the Fama and French three factor model is that high returns are still taken as a reward for taking on high risk. This specifically implies that if returns increase with book/price, then stocks with a high book/price ratio must be more risky than average.

Conclusion

Fama and French findings that support their three- factor asset-pricing model are:

• The equity returns are impacted by the market, size and value
• The linear exposures of equities to these factors explains the cross-sectional dispersion of their mean returns.

This paper examines the two ce