The Theory Behind The Capital Asset Pricing Model Finance Essay
The Capital Asset Pricing Model (CAPM) is being used since the 1960s to measure portfolio performance and to calculate the cost of capital. In the 1990s Eugene Fama and Kenneth French tried to improve the performance of the CAPM by adding two factors to the model. The first factor is the book-to-market ratio of stocks in the portfolio and the second factor is the stock's underlying company size. This model was to be superior to the CAPM. However, Graham and Harvey (2001) proved that in 2001 the CAPM was still used by 73,5% of the U.S. CFO's to calculate the cost of capital and portfolio performance. In Europe, Brounen, de Jong and Koedijk (2004) showed that this percentage was still 45%. Why do CFO's still rely on an inferior model, or isn't the Fama French Three Factor Model superior to the CAPM? The goal of this paper is to determine whether the CAPM or the Fama French Three Factor Model is superior to one another in size and book-to-market portfolios.
2.1 The Capital Asset Pricing Model
The Capital Asset Pricing Model is a model that describes the relationship between risk and expected return. It is mainly used by investors to value assets and to determine expected returns. The main idea behind the model is that investors need to be compensated for their exposure to systematic risk, because not all of the investments are truly risky. By diversifying a portfolio, it is possible to reduce risk. The risk reduced is called the Risk Free Rate. In other words, the expected return of a security or a portfolio is formed by the risk free rate plus a risk premium.
The CAPM by Sharpe (1964) and Lindtner (1965) is in fact an extension of the one period mean variance portfolio models of Tobin (1958) and Markowitz (1959). It builds on the model of portfolio choice, where an investor selects a portfolio that minimizes the variance of portfolio return and maximizes the expected return, given this variance. This is often referred to as the "mean variance model". The CAPM is in fact a prediction test about the coherence between risk and expected return and identifies a portfolio that is efficient if asset prices are to clear the market of all assets. The two added assumptions to this mean variance model identify a mean-variance-efficient portfolio are "complete agreement" and "borrowing and lending at risk free rate". Complete agreement stands for the agreement amongst investors on the joint distribution of asset returns from which the returns we use to test the model are drawn. Borrowing and lending at a risk free rate exists and is equal for all investors and does not depend on the amount borrowed or lent. The relationship between risk and the expected return for portfolios is apparent. The higher the risk, the higher the payoff and vice versa according to Fama and French (2004).
The Sharpe-Lindtner CAPM equation:
E(Ri) = Rf + ï¢im * (E(Rm) - Rf)
Where E(Ri) is the expected return on asset i, Rf is the risk-free rate, ï¢im is the market beta of asset i, E(Rm) is the expected return on the market. Where ï¢im consists of the following factors:
ï¢im = cov(Ri, Rm)/ï³2(Rm)
Where cov(Ri, Rm) is the covariance risk of asset i in m and ï³2(Rm) is the variance of the market return. In words, the expected return on asset i is the risk-free rate (Rf) times the unit premium of beta risk, E(Rm) - Rf.
The expected return on an investment portfolio is equal to the weighted average of all of the assets' expected returns in the portfolio, thus is linearly combined. The standard deviation of a portfolio is nonlinearly combined, because of the diversification of risk that takes place when a portfolio is formed as seen in Figure 1  . For example, when a portfolio of two equally risky assets is formed, with equally expected returns, the expected return on the portfolio will be equal to the expected return of one of the assets, though the standard deviation of the portfolio will be lower than the standard deviation of each of the underlying assets because of the diversification effect. In other words, diversification leads to a risk reduction without diminishing the expected return.
Figure 1 describes the various portfolio possibilities and explains the CAPM further. On the horizontal axis portfolio risk is set and on the vertical axis expected return is set. The curve is called the minimum variance frontier, and is a line that describes the minimum variance at different risk levels of multiple risky portfolios, when there is no opportunity of risk free borrowing. For example, when an investor is looking for a high expected return, this automatically brings a high risk along with it. The optimal choice of a portfolio for an investor lies on the minimum variance frontier, since it maximizes the expected return for a given volatility. By adding the opportunity to borrow at a risk free rate, the mean variance efficient frontier comes into being. This is now the efficient set, in stead of the minimum variance frontier according to Fama and French (2004).
Graph 1 thesis.jpg
2.2 The Pro's and Con's of the Capital Asset Pricing Model
The main advantage of the CAPM over any other pricing mode is its simplicity to use. However, there are some anomalies. During the 80's and 90's, several deviations in the CAPM were discovered which show anomalies in the CAPM and question it's correctness. According to Basu (1977) future returns on high Earnings to Price ratios are in reality higher than the ratios predicted by the CAPM. Banz (1981) finds that the low market value stocks actually had a higher return than predicted by the CAPM. Bhandari (1988) showed that leverage high debt equity stocks had returns that were too high relative to their betas according to the CAPM. He also shows that there is a positive relation between between leverage and average return in the CAPM. Leverage could be associated with risk and expected return, but according to Bhandari (1988), the leverage risk should be explained by the market ï¢.
2.3 The Fama French Three Factor Model.
Fama and French (1992) criticize the empirical adequacy of the CAPM and claim to improve the model by adding two empirically based factors. The factors are a product of empirical data research and there is no underlying theory to explain these factors. Of all the researched factors, these turn out to be the most effective. Fama and French (1992) used the cross-sectional regression approach of Fama and MacBeth (1973) to show that ï¢ doesn't suffice to explain average return. Size captures differences in average stock returns where ï¢ misses them. To improve the predictive value of the CAPM they added two factors to the model. According to the model the risk premium or excess return above the risk free rate is a composition of three factors, namely
The risk premium on the market portfolio (Rm - RF).
The difference in returns between the small stock portfolios and the big stock portfolios (SMB).
The difference in returns between the high book to market stock portfolios and the low book to market stock portfolios (HML).
The Fama and French Three Factor Model:
E(Ri) - Rf = bi * (E(Rm) - Rf) + si E(SMB) + hi E(HML)
E(Rm) is the expected return on the market, Rf is the risk free rate, E(SMB) and E(HML) are the expected premiums and bi, si and hi are the regression slopes.
The first factor is Small Minus Big (SMB) which is designed to measure the excess return investors have historically received by investing in company stocks with small market capitalization. This excess return is most commonly known as the 'size premium' . Fama and French (2006) compose six value weight portfolios, SG, SN, SV, BG, BN, and BV. They state that: "The portfolios are intersections stocks of NYSE, AMEX (after 1962) and Nasdaq (after 1972) into two size groups, Small and Big and three book to market (B/M) equity groups Growth (firms in the bottom 30% of NYSE B/M), Neutral (middle 40% of NYSE B/M) and Value (high 30% of NYSE B/M).  " The SMB factor is the average returns on the small stock portfolios minus the average returns on the big portfolios according to Fama and French (2006) and is computed as follows:
SMB =1/3 (Small Value + Small Neutral + Small Growth) Â - 1/3 (Big Value + Big Neutral + Big Growth)
Small Value are the firms with the June market cap below the NYSE median that are at the high 30% of NYSE B/M.
Small Neutral are the firms with the June market cap below the NYSE median that are at the middle 40% of NYSE B/M.
Small Growth are the firms with the June market cap below the NYSE median that are at the bottom 30% of the NYSE B/M.
Big Value are the firms with the June market cap value above NYSE median that are at the high 30% of NYSE B/M.
Big Neutral are the firms with the June market cap value above NYSE median that are at the middle 40% of NYSE B/M.
Big Growth are the firms with the June market cap value above NYSE median that are at the bottom 30% of NYSE B/M.
When this value is positive, the small caps have outperformed the large caps in the particular month and vice versa. The second is High Minus Low (HML) which is designed to measure the 'value premium' investors get for investing in high book-to-market companies. The HML factor is the average returns on value portfolios minus the average returns on growth portfolios according to Fama and French (2006) and is computed as follows:
HML =1/2 (Small Value + Big Value)Â - 1/2 (Small Growth + Big Growth)
When this value is positive, the growth stocks outperformed the value stocks in that particular month and vice versa.
The market risk premium (Rm-Rf) that is used is the value to weight return on all NYSE, AMEX and Nasdaq stocks diminished by the one-month Treasury bill rate.
2.4 The Pro's and Con's of the Three Factor Model
The additional two factors in the Fama and French Three factor model are purely empirical. There is no underlying theory as there exists in the CAPM. Though the three factor model needs additional date compared to the CAPM (the SMB factor and the HML factor) the higher costs in using the three factor model compared to the CAPM is not justified according to Bartholdy (2002), because the three factor model doesn't seem to outperform the CAPM significantly on individual stock returns estimation.
2.5 The Momentum factor
Carhart (1997) adds another factor to the equation, creating his Four Factor Model. This fourth factor describes the effect of Jegadeesh and Titman's (1993) one year momentum anomaly. This model is a market equilibrium-based four-risk factor model. Carhart's factor (PR1YR) brings the one-year momentum return to the equation, which enlarges the explanation of the model compared to the three factor model, so the fourth factor substantially improves the performance of the model according to Carhart (1997). In the three factor model, errors concerning last year's stock portfolios are observably reduced. Carhart managed to reduce most of the patterns in pricing errors. This indicates a well performing model on describing cross sectional variation in average stock returns.
The Carhart four factor model:
E(Ri) - Rf = bi * (E(Rm) - Rf) + si E(SMB) + hi E(HML) + pi E(PR1YR)
3. Research Questions and Hypotheses
Which of the two models, the CAPM or the Three Factor Model, is superior to one another for the different portfolios of size and book-to-market in the time period between 1993 and 2009?
The Three Factor model is superior in predicting the value of securities in the portfolio size in the time period 1993-2009.
The Three Factor model is superior in predicting the value of securities in the portfolio book-to-market in the time period 1993-2009.
Not all the factors of the three factor model are contributing to the adequacy of the model in valuing the portfolios size and book-to-market in the time period 1993-2009.
The momomentum factor helps explain returns.
4. Methodology and Data Collection
This paper uses the data gathered by Fama and French and published on the website  of Kenneth R. French. According to Fama and French (1992) this data consists of all non-financial firms in the intersection of the NYSE, AMEX and NASDAQ files and the merged COMPUSTAT annual industrial files of income statement and balance sheet data, maintained by the Center for Research in Security Prices (CSRP). They have excluded the financial firms from their dataset because high leverage in these firms is quite normal in opposition to high leverage in non-financial firms. In non-financial firms this high leverage could possibly indicate distress. In the Small-Minus-Big factor in the Three Factor Model, accounting variables play a role, and to ensure that these accounting variables are known before the returns they are used to explain, the accounting data for all fiscal year-ends t, are calculated after a minimum of 6 months has passed the fiscal year-end.
4.2 Estimating market ï¢
This paper uses the market ï¢ provided by Fama and French and published on the website of Kenneth R. French. The asset pricing tests performed in this paper use the cross sectional regression approach of Fama and MacBeth (1973). The market ï¢'s for portfolios are more precise than the individual ï¢'s, so the approach of Fama and MacBeth (1973) is to estimate the portfolio's ï¢ and then assign this ï¢ to each stock in the underlying portfolio. By using this method, the useage of individual stocks in the asset-pricing tests of Fama and MacBeth is enabled.
SPSS 15 will be used for the data analysis in this paper, which will consist of multivariate regression analysis to answer the research question and hypotheses. To determine whether the two additional factors of the Three Factor Model add extra predictive value to the CAPM, this paper uses multiple linear regression in SPSS to see if the R-squared value increases with these additional factors for each of the portfolios. . In the data library section on their site  , Fama and French provide data about portfolios formed on size and portfolios formed on book-to-market. To investigate the size and book-to-market factors in the Three factor model, the data of these portfolios are used in the regression analysis.
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