CAPM and Three Factor Model in Cost of Equity Measurement
Disclaimer: This dissertation has been submitted by a student. This is not an example of the work written by our professional dissertation writers. You can view samples of our professional work here.
Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of UK Essays.
1.0 INTRODUCTION AND OBJECTIVES
Central to many financial decisions such as those relating to investment, capital budgeting, portfolio management and performance evaluation is the estimation of the cost of equity or expected return. There exist several models for the valuation of equity returns, prominent among which are the dividend growth model, residual income model and its extension, free cash flow model, the capital asset pricing model, the Fama and French three factor model, the four factor model etc. Over the past few decades, two of the most common asset pricing models that have been used for this purpose are the Capital Asset Pricing Model (a single factor model by Sharpe 1964, Lintner 1965) and the three factor model suggested by Fama and French (1993). These two models have been very appealing to both practitioners and academicians due to their structural simplicity and are very easy to interpret. There have however been lots of debates and articles as to which of these two models should be used when estimating the cost of equity or expected returns. The question as to which of these two models is better in terms of their ability to explain variation in returns and forecast future returns is still an open one. While most practitioners favour a one factor model (CAPM) when estimating the cost of equity or expected return for a single stock or portfolio, academics however recommend the Fama and French three factor model (see eg. Bruner et al, 1998).
The CAPM depicts a linear relationship between the expected return on a stock or portfolio to the excess return on a market portfolio. It characterizes the degree to which an asset's return is correlated to the market, and indirectly how risky the asset is, as captured by beta. The three-factor model on the other hand is an extension of the CAPM with the introduction of two additional factors, which takes into account firm size (SMB) and book-to-market equity (HML). The question therefore is why practitioners prefer to use the single factor model (CAPM) when there exist some evidence in academics in favour of the Fama and French three factor model. Considering the number of years most academic concepts are adopted practically, can we conclude that the Fama and French three factor model is experiencing this so-called natural resistance or is it the case that the Fama and French model does not perform significantly better than the CAPM and so therefore not worth the time and cost?
The few questions I have posed above form the basis for this study. It is worth noting that while the huge academic studies on these models produce interesting results and new findings, the validity of the underlying models have not been rigorously verified. In this paper, while I aim to ascertain which of the two models better estimates the cost of equity for capital budgeting purposes using regression analysis, I also will like to test whether the data used satisfy the assumptions of the method most academicians adopt, i.e. the Ordinary Least Squares (OLS) method. I will in particular be testing for the existence or otherwise of heteroscedasticity, multicollinearity, normality of errors serial correlation and unit roots, which may result in inefficient coefficient estimates, wrong standard errors, and hence inflated adjusted R2 if present in the data. I will then correct these if they exist by adopting the Generalised Least Squares (GLS) approach instead of the widely used Ordinary Least Squares (OLS) before drawing any inference from the results obtained.
My conclusion as to which of the models is superior to the other will be based on which provides the best possible estimate for expected return or cost of equity for capital budgeting decision making. Since the cost of capital for capital budgeting is not observed, the objective here, therefore, is to find the model that is most effective in capturing the variations in stock returns as well as providing the best estimates for future returns. By running a cross sectional regression using stock or portfolio returns as the dependent variable and estimated factor(s) based on past returns as regressors, R2 measures how much of the differences in returns is explained by the estimation procedure. The model that produces the highest adjusted R2 will therefore be deemed the best.
The Fama-French (1993, 1996) claimed superiority of their model over CAPM in explaining variations in returns from regressions of 25 portfolios sorted by size and book-to-market value. Their conclusion was based on the fact that their model produced a lower mean absolute value of alpha which is much closer to the theoretical value of zero. Fama and French (2004, working paper) stated that if asset pricing theory holds either in the case of the CAPM (page 10), or the Fama and French three-factor model (page 21), then the value of their alphas should be zero, depicting that the asset pricing model and its factor or factors explain the variations in portfolio returns. Larger values of alpha in this case are not desirable, since this will imply that the model was poor in explaining variation in returns. In line with this postulation, the model that yields the lowest Mean Absolute Value of Alpha (MAVA) will therefore be considered the best. But since alpha is a random variable, I will proceed to test the null hypothesis H0: αi = 0 for all i, by employing the GRS F-statistic postulated by Gibbons, Ross and Shanken (1989).
My third and fourth testing measures are based on postulates by econometricians that, the statistical adequacy of a model in terms of its violations of the classical linear regression model assumptions is hugely irrelevant if the models predictive power is poor and that the accuracy of forecasts according to traditional statistical criteria such as the MSE may give little guide to the potential profitability of employing those forecasts in a market trading strategy or for capital budgeting purposes. I will therefore test the predictive power of the two models by observing the percentage of forecast signs predicted correctly and their Mean Square Errors (MSE).
One other motivation for this study is also to ascertain whether the results of prior studies are sample specific, that is, whether it is dependent on the period of study or the portfolio grouping used. Theoretically, the effectiveness of an asset pricing model in explaining variation in returns should not be influenced by how the data is grouped. Fama and French (1996) claimed superiority of their model over the CAPM using the July 1963 to December 1993 time period with data groupings based on size and book-to-market equity. I will be replicating this test on the same data grouping but covering a much longer period (from July 1926 to June 2006) and then on a different data grouping based on industry characteristics. Testing the models using the second grouping of industry portfolios will afford me the opportunity to ascertain whether the effectiveness of an asset pricing model is sample specific. I will also carry out the test by employing a much shorter period (5 years) and comparing it to the longer period and then using the one with the better estimate in terms of alpha and R2 to carry out out-of-sample forecasts.
The rest of this paper is structured as follows. Chapter 2 will review the various models available for the estimation of equity cost with particular emphasis on the two asset-pricing models and analysing some existing literature. Chapter 3 will give a description of the data, its source and transformations required, with Chapter 4 describing the methodology. Chapter 5 will involve the time series tests of hypothesis on the data and Chapter 6 will involve an empirical analysis of the results for the tests of the CAPM and the Fama and French three-factor model. Finally, Chapter 7 contains a summary of the major findings of my work and my recommendation as well as some limitations, if any, of the study and recommended areas for further studies.
2.0 RELEVANT LITERATURE
The estimation of the cost of equity for an industry involves estimation of what investors expect in return for their investment in that industry. That is, the cost of equity to an industry is equal to the expected return on investors' equity holdings in that industry. There are however a host of models available for the estimation of expected returns on an industry's equity capital including but not limited to estimates from fundamentals (dividends and earnings) and those from asset pricing models.
2.1 Estimations from Fundamentals
Estimation of expected returns or cost of equity in this case from fundamentals involves the use of dividends and earnings. Fama and French (2002) used this approach to estimate expected stock returns. They stated that, the expected return estimates from fundamentals help to judge whether the realised average return is high or low relative to the expected value (pp 1). The reasoning behind this approach lies in the fact that, the average stock return is the average dividend yield plus the average rate of capital gain:
A(Rt) = A(Dt/Pt-1) + A(GPt) (1)
where Dt is the dividend for year t, Pt-1 is the price at the end of year t - 1, GPt = (Pt - Pt-1)/Pt-1 is the rate of capital gain, and A( ) indicates an average value. Given in this situation that the dividend-price ratio, Dt/Pt , is stationary (mean reverting), an alternative estimate of the stock return from fundamentals is:
A(RDt) = A(Dt/Pt-1) + A(GDt) (2)
Where GDt = (Dt - Dt-1)/Dt-1is the growth rate of dividends and (2) is known as the dividend growth model which can be viewed as the expected stock return estimate of the Gordon (1962) model. Equation (2) in theory will only apply to variables that are cointegrated with the stock price and may not hold if the dividend-price ratio is non-stationary, which may be caused by firms decision to return earnings to stockholders by moving away from dividends to share repurchases (Fama and French 2002). But assuming that the ratio of earnings to price, (Yt/Pt), is stationary, then an alternative estimate of the expected rate of capital gain will be the average growth rate of earnings, A(GYt) = A((Yt - Yt-1)/Yt-1). In this case, the average dividend yield can be combined with the A(GYt) to produce a third method of estimating expected stock return, the earnings growth model given as:
A(RYt) = A(Dt/Pt-1) + A(GYt) (3)
It stands to reason from the model in Lettau and Ludvigson (2001) that the average growth rate of consumption can be an alternative mean of estimating the expected rate of capital gain if the ratio of consumption to stock market wealth is assumed stationary.
Fama and French (2002) in their analysis concluded that the dividend growth model has an advantage over the earnings growth model and the average stock return if the goal is to estimate the long-term expected growth of wealth. However, it is a more generally known fact that, dividends are a policy variable and so subject to changes in management policy, which raises problems when using the dividend growth model to estimate the expected stock returns. But this may not be a problem in the long run if there is stability in dividend policies and dividend-price ratio resumes its mean-reversion (although the reversion may be at a new mean level). Bagwell and Shoven (1989) and Dunsby (1995) have observed that share repurchases after 1983 has been on the ascendancy, while Fama and French (2001) have also observed that the proportion of firms who do not pay dividends have been increasing steadily since 1978. The Fama and French (2001) observation implies that in transition periods where firms who do not pay dividends increases steadily, the market dividend-price ratio may be non-stationary; overtime, it is likely to decrease, in which case the expected return will likely be underestimated when the dividend growth model is used.
The earnings growth model, although not superior to the dividend growth model (Fama and French (2002)), is not affected by possible changes in dividend policies over time. The earnings growth model however may also be affected by non-stationarity in earnings-price ratio since it ability to accurately estimate average expected return is based on the assumption that there are permanent shifts in the expected value of the earnings-price ratio.
2.2 Estimations from Asset-Pricing Models
One of the most fundamental concepts in the area of asset-pricing is that of risk versus reward. The pioneering work that addressed the risk and reward trade-off was done by Sharpe (1964)-Lintner (1965), in their introduction of the Capital Asset Pricing Model (CAPM). The Capital Asset Pricing Model postulates that the cross-sectional variation in expected stock or portfolio returns is captured only by the market beta. However, evidence from past literature (Fama and French (1992), Carhart (1997), Strong and Xu (1997), Jagannathan and Wang (1996), Lettau and Ludvigson (2001), and others) stipulates that the cross-section of stock returns is not fully captured by the one factor market beta. Past and present literature including studies by Banz (1981), Rosenberg et al (1985), Basu (1983) and Lakonishok et al (1994) have established that, in addition to the market beta, average returns on stocks are influenced by size, book-to-market equity, earnings/price and past sales growth respectively. Past studies have also revealed that stock returns tend to display short-term momentum (Jegadeesh and Titman (1993)) and long-term reversals (DeBondt and Thaler (1985)).
Growing research in this area by scholars to address these anomalies has led to the development of alternative models that better explain variations in stock returns. This led to the categorisation of asset pricing models into three: (1) multifactor models that add some factors to the market return, such as the Fama and French three factor model; (2) the arbitrage pricing theory postulated by Ross (1977) and (3) the nonparametric models that heavily criticized the linearity of the CAPM and therefore added moments, as evidenced in the work of Harvey and Siddique (2000) and Dittmar (2002). From this categorization, most of the asset-pricing models can be described as special cases of the four-factor model proposed by Carhart (1997). The four-factor model is given as:
E(Ri) - Rf = αi + [ E(RM) - Rf ]bi + si E(SMB) + hi E(HML) + wiE(WML) + εi (4)
where SMB, HML and WML are proxies for size, book-to-market equity and momentum respectively. There exist other variants of these models such as the three-moment CAPM and the four-moment CAPM (Dittmar, 2002) which add skewness and kurtosis to investor preferences, however the focus of this paper is to compare and test the effectiveness of the CAPM and the Fama and French three-factor model, the two premier asset-pricing models widely acknowledged among both practitioners and academicians.
2.3 Theoretical Background: CAPM and Fama & French Three-Factor Model
There exist quite a substantial amount of studies in the field of finance relating to these two prominent asset pricing models. The Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) has been the first most widely recognized theoretical explanation for the estimation of expected stock returns or cost of equity in this case. It is a single factor model that is widely used by Financial Economists and in industry. The CAPM being the first theoretical asset pricing model to address the risk and return concept and due to its simplicity and ease of interpretation, was quickly embraced when it was first introduced. The models attractiveness also lies in the fact that, it addressed difficult problems related to asset pricing using readily available time series data. The CAPM is based on the idea of the relationship that exists between the risk of an asset and the expected return with beta being the sole risk pricing factor. The Sharpe-Lintner CAPM equation which describes individual asset return is given as:
E(Ri) = Rf + [ E(RM) - Rf ]βiM i = 1,......,N (5)
where E(Ri) is the expected return on any asset i, Rf is the risk-free interest rate, E(RM) is the expected return on the value-weighted market portfolio, and βiM is the asset's market beta which measures the sensitivity of the asset's return to variations in the market returns and it is equivalent to Cov(Ri, RM)/Var(RM).
The equation for the time series regression can be written as:
E(Ri) - Rf = αi + [ E(RM) - Rf ]βiM + εi i = 1,......,N; (6)
showing that the excess return on portfolio i is dependent on excess market return with εi as the error term. The excess market return is also referred to as the market premium.
The model is based on several key assumptions, portraying a simplified world where: (1) there are no taxes or transaction costs or problems with indivisibilities of assets; (2) all investors have identical investment horizons; (3) all investors have identical opinions about expected returns, volatilities and correlations of available investments; (4) all assets have limited liability; (5) there exist sufficiently large number of investors with comparable wealth levels so that each investor believes that he/she can purchase and sell any amount of an asset as he or she deems fit in the market; (6) the capital market is in equilibrium; and (7) Trading in assets takes place continually over time. The merits of these assumptions have been discussed extensively in literature.
It is evident that most of these assumptions are the standard assumptions of a perfect market which does not exist in reality. It is a known fact that, in reality, indivisibilities and transaction costs do exist and one of the reasons assigned to the assumption of continual trading models is to implicitly give recognition to these costs. It is imperative to note however that, trading intervals are stochastic and of non-constant length and so making it unsatisfactory to assume no trading cost. As mentioned earlier, the assumptions made the model very simple to estimate (given a proxy for the market factor) and interpret, thus making it very attractive and this explains why it was easily embraced. The CAPM stipulates that, investors are only rewarded for the systematic or non-diversifiable risk (represented by beta) they bear in holding a portfolio of assets. Notwithstanding the models simplicity in estimation and interpretation, it has been criticized heavily over the past few decades.
Due to its many unrealistic assumptions and simple nature, academicians almost immediately began testing the implications of the CAPM. Studies by Black, Jensen and Scholes (1972) and Fama and MacBeth (1973) gave the first strong empirical support to the use of the model for determining the cost of capital. Black et al. (1972) in combining all the NYSE stocks into portfolio and using data between the periods of 1931 to 1965 found that the data are consistent with the predictions of the Capital Asset Pricing Model (CAPM). Using return data for NYSE stocks for the period between 1926 to 1968, Fama and MacBeth (1973) in examining whether other stock characteristics such as beta squared and idiosyncratic volatility of returns in addition to their betas would help in explaining the cross section of stock returns better found that knowledge of beta was sufficient.
There have however been several academic challenges to the validity of the model in relation to its practical application. Banz (1981) revealed the first major challenge to the model when he provided empirical evidence to show that stocks of smaller firms earned better returns than predicted by the CAPM. Banz's finding was not deemed economically important by most academicians in the light that, it is unreasonable to expect an abstract model such as the CAPM to hold exactly and that the proportion of small firms to total market capital is insignificant (under 5%). Other early empirical works by Blume and friend (1973), Basu (1977), Reinganum (1981), Gibbons (1982), Stambaugh (1982) and shanken (1985) could not offer any significant evidence in support of the CAPM.
In their paper, Fama and French (2004) noted that in regressing a cross section of average portfolio returns on portfolio beta estimates, the CAPM would predict an intercept which is equal to the risk free rate (Rf) and a beta coefficient equal to the market risk premium (E(Rm) - Rf). However, Black, Jensen and Scholes (1972), Blume and Friend (1973), Fama and MacBeth (1973) and Fama and French (1992) after running series of cross-sectional regressions found that the average risk-free rate, which is proxied by the one month T-bill, was always less that the realised intercept. Theory stipulates that, the three main components of the model (the risk free, beta and the market risk premium) must be forward-looking estimates. That is they must be estimates of their true future values. Empirical studies and survey results however show substantial disagreements as to how these components can be estimated. While most empirical researches use the one month T-bill rate as a proxy to the risk-free rate, interviews depicts that practitioners prefer to use either the 90-day T-bill or a 10-year T-bond (normally characterised by a flat yield curve). Survey results have revealed that practitioners have a strong preference for long-term bond yields with over 70% of financial advisors and corporations using Treasury-bond yields with maturities of ten 10 or more years. However, many corporations reveal that they match the tenor of the investment to the term of the risk free rate.
Finance theory postulates that the estimated beta should be forward looking, so as to reflect investor's uncertainty about future cash flows to equity. Practitioners are forced to use various kinds of proxies since forward-looking betas are unobservable. It is therefore a common practice to use beta estimates derived from historical data which are normally retrieved from Bloomberg, Standard & Poors and Value Line. However, the lack of consensus as to which of these three to use results in different betas for the same company. These differences in beta estimates could result in significantly different expected future returns or cost of equity for the company in question thereby yielding conflicting financial decisions especially in capital budgeting. In the work of Bruner et al. (1998), they found significant differences in beta estimates for a small sample of stocks, with Bloomberg providing a figure of 1.03 while Value Line beta was 1.24. The use of historical data however requires that one makes some practical compromises, each of which can adversely affect the quality of the results. Forinstance, the statistically reliability of the estimate may improve greatly by employing longer time series periods but this may include information that are stale or irrelevant. Empirical research over the years has shown that the precision of the beta estimates using the CAPM is greatly improved when working with well diversified portfolios compared to individual securities.
In relation to the equity risk premium, finance theory postulates that, the market premium should be equal to the difference between investors expected returns on the market portfolio and the risk-free rate. Most practitioners have to grapple with the problem of how to measure the market risk premium. Survey results have revealed that the equity market premium prompted the greatest diversity of responses among survey respondents. Since future expected returns are unobservable, most of the survey participants extrapolated historical returns in the future on the assumption that future expectations are heavily influenced by past experience. The survey participants however differed in their estimation of the average historical equity returns as well as their choice of proxy for the riskless asset. Some respondents preferred the geometric average historical equity returns to the arithmetic one while some also prefer the T-bonds to the T-bill as a proxy for the riskless asset.
Despite the numerous academic literatures which discuss how the CAPM should be implemented, there is no consensus in relation to the time frame and the data frequency that should be used for estimation. Bartholdy & Peare (2005) in their paper concluded that, for estimation of beta, five years of monthly data is the appropriate time period and data frequency. They also found that an equal weighted index, as opposed to the commonly recommended value-weighted index provides a better estimate. Their findings also revealed that it does not really matter whether dividends are included in the index or not or whether raw returns or excess returns are used in the regression equation.
The CAPM has over the years been said to have failed greatly in explaining accurate expected returns and this some researchers have attributed to its many unrealistic assumptions. One other major assumption of the CAPM is that there exists complete knowledge of the true market portfolio's composition or index to be used. This assumed index is to consist of all the assets in the world. However since only a small fraction of all assets in the world are traded on stock exchanges, it is impossible to construct such an index leading to the use of proxies such as the S&P500, resulting in ambiguities in tests.
The greatest challenge to the CAPM aside that of Banz (1981) came from Fama and French (1992). Using similar procedures as Fama and MacBeth (1973) and ten size classes and ten beta classes, Fama and French (1992) found that the cross section of average returns on stocks for the periods spanning 1960s to 1990 for US stocks is not fully explained by the CAPM beta and that stock risks are multidimensional. Their regression analysis suggest that company size and book-to-market equity ratio do perform better than beta in capturing cross-sectional variation in the cost of equity capital across firms. Their work was however preceded by Stattman (1980) who was the first to document a positive relation between book-to-market ratios and US stock returns. The findings of Fama and French could however not be dismissed as being economically insignificant as in the case of Banz.
Fama and French therefore in 1993 identified a model with three common risk factors in the stock return- an overall market factor, factors related to firm size (SMB) and those related to book-to-market equity (HML), as an alternative to the CAPM. The SMB factor is computed as the average return on three small portfolios (small cap portfolios) less the average return on three big portfolios (large cap portfolios). The HML factor on the other hand is computed as the average return on two value portfolios less the average return on two growth portfolios. The growth portfolio represents stocks with low Book Equity to Market Equity ratio (BE/ME) while the value portfolios represent stocks with high BE/ME ratio. Their three-factor model equation is described as follows:
E(Ri) - Rf = αi + [ E(RM) - Rf ]bi + si E(SMB) + hi E(HML) + εi (7)
Where E(RM) - Rf, , E(SMB) and E(HML) are the factor risk premiums and bi , si and hi are the factor sensitivities. It is however believed that the introduction of these two additional factors was motivated by the works of Stattman (1980) and Banz (1981).
The effectiveness of these two models in capturing variations in stock returns may be judged by the intercept (alpha) in equations (6) and (7) above. Theory postulates that if these models hold, then the value of the intercept or alpha must equal zero for all assets or portfolio of assets. Fama and French (1997) tested the ability of both the CAPM and their own three-factor model in estimating industry costs of equity. Their test considered 48 industries in which they found that their model outperformed the CAPM across all the industries considered. They however could not conclude that their model was better since their estimates of industry cost of equities were observed to be imprecise. Another disturbing outcome of their study is that both models displayed very large standard errors in the order of 3.0% per annum across all industries.
Connor and Senghal (2001) tested the effectiveness of these two models in predicting portfolio returns in india's stock market. They tested the models using 6 portfolio groupings formed from the intersection of two size and three book-to-market equity by examining and testing their intercepts. Connor and Senghal in this paper examined the values of the intercepts and their corresponding t-statistics and then tested the intercepts simultaneously by using the GRS statistic first introduced by Gibbons, Ross and Shanken (1989). Based on the evidence provided by the intercepts and the GRS tests, Connor and Senghal concluded generally that the three-factor model of Fama and French was superior to the CAPM.
There have been other several empirical papers ever since, to ascertain which of these models is better in the estimation of expected return or cost of equity, most producing contrasting results. Howard Qi (2004) concluded in his work that on the aggregate level, the two models behave fairly well in their predictive power but the CAPM appeared to be slightly better. Bartholdy and Peare (2002) in their work came to the conclusion that both models performed poorly with the CAPM being the poorest.
3.0 DATA SOURCES
The most complete set of data for this study is one on the US economy provided by Kenneth French on his online library. The data for this study is monthly time series returns from three basic factors Rm - Rf , SMB and HML which were used as independent variables, for two separate periods. The first period spans July 1926 to December 2007, a total of 978 observations and the second spans July 2001 to June 2006 a total of 60 observations with a third period (July 2006 to December 2007) being used for out-of-sample forecasts. The longer sample is an extension of the original Fama-French analysis to include earlier and recent data. Prior research (by Jan Bartholdy and Paula Peare, 2002) has shown that a shorter monthly sample of 5 years is appropriate for this kind of study.
As stated earlier, my sample consists of two different portfolio groupings which form the dependent variables in this study. The first being the excess returns of 25 portfolios formed by the intersection of both size (from small to big market cap) and book-to-market equity (from low to high). These 25 portfolios, which are constructed at the end of each June, are the intersections of 5 portfolios formed on size (market equity, ME) and 5 portfolios formed on the ratio of book equity to market equity (BE/ME). The size breakpoints for year t are the NYSE market equity quintiles at the end of June of t. BE/ME for June of year t is the book equity for the last fiscal year end in t-1 divided by ME for December of t-1. The BE/ME breakpoints are NYSE quintiles. The portfolios for July of year t to June of t+1 include all NYSE, AMEX, and NASDAQ stocks for which there exist market equity data for December of t-1 and June of t, and (positive) book equity data for t-1.
The second consists of excess returns of 17 industry portfolios which are used as dependent variables in the two models under study. This portfolio grouping includes: Food; Mining & Minerals; Oil & Petroleum Products; Textiles, Apparel & Footware; Consumer Durables; Chemicals; Drugs, Soap, Perfumes & Tobacco; Construction & Construction Materials; Steel Works; Fabricated Products; Machinery & Business Equipment; Automobiles; Transportation; Utilities; Retail Stores; Banks, Insurance Companies & Other Financials and Others. These portfolios are formed by assigning each NYSE, AMEX and NASDAQ stock to an industry portfolio at the end of June of a specified year t based on its four-digit SIC code at that time. The SIC codes used here are Compustat SIC codes for the fiscal year ending in a given calendar year t-1. CRSP SIC codes for June of year t are used in cases where Compustat SIC codes are unavailable. Returns are then computed from July of the current year t to June of year t+1. As mentioned earlier, the introduction of this portfolio grouping in the analysis is to ascertain whether the effectiveness of a model is sample specific or dependent on the way the data is grouped.
The Fama and French factors are constructed using 6 value-weight portfolios formed on size and book-to-market. SMB (Small Minus Big) is the average return on the three small portfolios minus the average return on the three big portfolios:
1/3 (Small Value + Small Neutral + Small Growth)
HML (High Minus Low) is the average return on two value portfolios minus the average return on the two growth portfolios,
1/2 (Small Value + Big Value)
Rm-Rf, the excess return on the market, is the value-weight return on all NYSE, AMEX, and NASDAQ stocks (from CRSP) minus the risk-free rate which in this paper is proxied by the one-month Treasury bill yield provided by Ibbotson Associates. Rm-Rf includes all NYSE, AMEX, and NASDAQ firms. SMB and HML for July of year t to June of t+1 include all NYSE, AMEX, and NASDAQ stocks for which there exists market equity data for December of t-1 and June of t, and (positive) book equity data for t-1. These dependent variables as well as their descriptions were also sourced from Kenneth French's online library.
I began my analysis by running a time series regression using Ordinary Least Squares (OLS) on the excess returns of one of the 17 industry portfolios (Food) as against the independent variable in the Capital Asset Pricing Model. I then proceeded to test for the presence or otherwise of heteroscedasticity, multicollinearity, serial correlation, normality and unit root on the sample data under the OLS method.
The Newey-West (1987) procedure, which is a consistent variance-covariance estimator in the presence of both heteroscedasticity and serial correlation was then used to run the time series regression on the excess returns of the 42 portfolios against the regressors in each model (6 and 7). This I carried out by automatically invoking the Newey-West standard error in RATS by setting the number of lags to 12. These regressions provided 25 and 17 separate intercepts or alphas and adjusted R2's which I used in my analysis to ascertain which model is most effective in explaining the variations in equity returns.
My methodology for the evaluation process initially involved a direct comparison of the adjusted R2's of the two models in each of the 42 distinct portfolios of the two portfolio groupings within the two sample periods. The model with the highest mean adjusted R2 in each portfolio was deemed superior. This paper also used three other methods to ascertain which of the models is effective. As mentioned earlier, I first examined the Mean Absolute Values of the Alphas (MAVA) and then ascertained their statistical significance by examining the t-statistics. Here the model with the lowest mean absolute value for the alphas is theoretically a better model at explaining the variations in portfolio returns.
The third evaluation method involving the alphas involves testing the null hypothesis H0: αi = 0, for all i, or simply put, a joint test on the intercepts using the GRS statistic propounded by Gibbons, Ross and Shanken (1989). This test is necessary because alpha is a random variable. I then carried out this test by running GLS regression and then testing whether the alphas are jointly zero. The decision rule here is that the GRS statistic should be zero if the alphas are jointly zero. This therefore implies that a larger value of the GRS statistic is not desirable in respect to the effectiveness of the asset pricing model. The GRS statistic is constructed using the intercepts and error terms described in both models. The test statistic for the CAPM is given as:
Ì´ F (N, T-N-1)
where T is the number of time series observations, N is the number of portfolios, m and σm are the average excess return and standard deviation of the market portfolio, αp is an N×1 matrix of estimated portfolio alphas and Σ is an N×N disturbance covariance matrix. The test statistic for the Fama and French three factor model is an extension of that of the CAPM to incorporate the k factors in the model which follows the concept introduced by Jobson and Korkie (1985). The modified test statistic is given as:
Ì´ F (N, T-N-K)
where K is the number of factors in the model, k is a K×1 matrix of factor means and Ω is the K×K covariance matrix of the factor returns.
My fourth method for testing model superiority is based on some suggestions by econometricians that, the statistical adequacy of a model in terms of its violations of the classical linear regression model assumptions is hugely irrelevant if the models predictive power is poor. I therefore used the July 2001 to June 2006 sample period estimates to forecast the ex-post sample period from July 2006 to December 2007 using the two portfolio groupings. This generated a series of 18 months forecast of expected excess returns for the various portfolios which I compared with the actual values to determine the accuracy of the forecasts. I then found the Mean Squared Errors by employing the following formulae:
Where N is the total sample size (in-sample + out-sample), T is the first out-of-sample forecast observation, Yt+s and Ft,s are the sth out-of-sample actual and forecast values respectively. Under this test, a larger value of the MSE is undesirable and the intuition is that the model with the lowest MSE has the highest predictive power. However, recent studies (Gerlow, Irwin and Liu, 1993) have shown that the accuracy of forecasts according to traditional statistical criteria such as the MSE may give little guide to the potential profitability of employing those forecasts in a market trading strategy or for capital budgeting purposes. Leitch and Tanner (1991) have found models that can accurately forecast the sign of future returns or can predict turning points in a series to be profitable. Based on this, I proceeded to use the formulae suggested by Pesaran and Timmerman (1992) to compute the proportion of correctly predicted signs for any given out-of-sample time t+s. The formula for the computation is given as:
% of Correct Sign Prediction
Where Zt+s = 1 if (Yt+s*Ft,s) > 0
Zt+s = 0 otherwise
The decision rule here is that the model with the highest mean percentage sign prediction under the two porfolio groupings will be deemed best.
In summary, my methodology for ascertaining which model is superior would depend on which:
Yields the lowest Mean Absolute Value of Alpha (MAVA).
Yields the highest average adjusted R2
Pass the GRS test on H0: αi = 0, for all i.
Has the highest predictive power in terms of producing the least Mean Square Error (MSE) and the highest average percentage of correct sign prediction
4.1 Forecasting Procedure
In economics and finance, decisions often involve committing scarce economic resources to long-term projects whose returns are often unpredictable. The returns on such projects often depend on future events in an economy and its financial markets. In this light, decisions made presently will reflect forecasts of expectations of future state of the world and the more accurate these forecasts are the more the financial gain or utility.
In an attempt to determine the predictive power of the two models under review, I employed the two main approaches to forecasting; the econometric (structural) and time series forecasting. Econometric or structural forecasting involve forecasting a dependent variable given one or more independent variables while time series forecasting involve forecasting the future values of a series given its previous values. The estimated models for the July 2001 to June 2006 are used to carry out forecast for the period spanning July 2006 to December 2007, producing 18 months out-of-sample forecasts.
The initial estimation procedure involved using time series forecast to estimate future values of the three independent variables (Rm - Rf, SMB and HML) used in the two models. I initially estimated a regression model of the form:
Yt = γt + λtYt-12 + εt
Where Y represents the independent variables, t is the current period and t-12 is the current period lagged 12 months. The proposed model is based on my believe that the present values of the series are correlated with the previous year's values. I proceeded by estimating the models for the three independent variables and then employed the estimated model to forecast the independent variables for the 18 months out-of-sample forecast period. These 18 time series forecasts for each of the independent variables are employed to forecast the excess returns for each of the 42 portfolio groupings using the estimated econometric models.
Some econometricians however prefer to use the previous year's values of the independent variables in the structural forecasts. That is, instead of carrying out a time series forecast of the independent variables as stated above, the prefer to use the previous 12 months values as estimate for the current values, depicted mathematically as:
SMBt = SMBt-12
To enable me draw inference as to which forecast procedure is better, I have in this paper produced forecast using both procedures under the CAPM and the Fama and French three factor model. As stated earlier, the criteria for ascertaining which model and estimation procedure is better depends on which produces the lowest Mean Square Error (MSE) and the highest percentage of correctly predicted signs (%CPS).
5.0 TIME SERIES TESTS
5.1 Test For Presence Of Heteroscedasticity
The Ordinary Least Squares (OLS) linear regression estimation method assumes that the variance of the errors is constant, that is homoscedasticity. A violation of this assumption leads to what is termed heteroscedasticity implying that the variances of the errors are not constant. The use of OLS in the presence of heteroscedasticity although produces estimates that are still unbiased and consistent, the standard errors of these estimates may be wrong and hence any inferences made could be misleading. The intercept which forms the major basis for making a decision as to which model is better, will have a large standard error hence making inferences from the t-static can be misleading. The test is carried out using the following hypothesis:
By applying the white test using the CAPM and Food portfolio from the industry grouping, it is evident there is the presence of heteroscedasticity in our data (White Chi-square Statistic = 19.71692 with a P-value = 1.9428 × 10-4), hence the null hypothesis is rejected.
5.2 Test For Presence Of Autocorrelation
Another assumption under the use of OLS is that the errors are uncorrelated with one another (i.e. cov(εi, εj) = 0). A violation of this assumption, where the errors or disturbance terms are correlated with each other is termed serial correlation or autocorrelation. The consequences of ignoring autocorrelation when present are similar to that of heteroscedasticity, although here the coefficient estimates are inefficient. Using OLS in the presence of autocorrelation could also result in inflated R2 relative to its true value and since my analysis is also based on the use of R2, any inferences I make will be misleading. Although there exist several tests for autocorrelation, I used the Breusch-Godfrey test, which allows for a joint test of εt and several of its lagged values at the same time, under the following hypothesis:
H0: No autocorrelation (¿½?1=0 and ¿½?2 = 0 and....and ¿½?r= 0)
H1: Autocorrelation (¿½?10 or ¿½?2 =0 or ......or ¿½?r = 0); where r is the number of lags
I conducted this test in RATS by regressing the residuals on a constant and twelve lags and testing the restriction that the coefficients on all the ten lags are jointly zero and the resulting F-statistic is given as: F(12, 952) = 2.80634 with Significance Level 0.00087927. We therefore reject the null hypothesis and conclude that the errors are autocorrelated.
5.3 Test For Presence Of Multicollinearity
One other implicit assumption made with the use of OLS estimation method is that the independent or explanatory variables are uncorrelated (or orthogonal) with one another. A violation of this assumption will result in significant changes in the coefficients of other variables (eg Rm-Rf) when some other variables (eg. SMB and HML) are added or removed. There is almost always some degree of correlation but this is expected to be very minimal. A higher degree of correlation between the explanatory variables is termed multicollinearity. The simplest method for this test is constructing a correlation matrix between the individual variables and observing their values. The result is depicted below:
Rm-Rf SMB HML
SMB 0.3265 1.0000
HML 0.2153 0.0905 1.0000
The highest correlation here (0.33) exists between Rm-Rf and SMB and this is relatively low so I have safely assume that multicollinearity is absent from explanatory variables.
5.4 Test for Normality of Errors
In order to perform single or joint tests of hypothesis on the model parameters, it is required that the disturbances are normally distributed. The Bera-Jacque test is one of the most commonly used tests for normality. The test applies the property of a normally distributed random variable that the total distribution is characterised by the first two moments, that is, the mean and the variance. Skewness and kurtosis, which are the standardised third and fourth moments of a distribution can be used to determine whether the errors are normally distributed or not. The distribution of the errors is normal when the distribution is not skewed and having a coefficient of kurtosis of 3 or excess kurtosis of zero. The Bera and Jarque (1981) test, tests whether the coefficients of skewness and excess kurtosis are jointly zero. The hypothesis of interest is given as:
H0: Normal Errors
H1: Non-Normal Errors
The test statistic under the null hypothesis asymptotically follows a Chi-Square distribution with 2 degrees of freedom. The test resulted in excess kurtosis of 4.77 and χ2(2) of 930.45 with a significance level of zero. This result implies that the null hypothesis for residual or error normality is rejected, meaning that any inferences I make about the coefficient estimates could be wrong. However since my sample is sufficiently large, there is no cause for much concern.
5.5 Test for Presence of Unit Root
The use of ordinary least squares (OLS) often relies on the assumption that the stochastic process is stationary, that is, it does not contain unit roots. In the case where the stochastic process is non-stationary (contains unit roots), the estimates produced by OLS may be invalid. Such invalid estimates are referred to as spurious regression (by Granger and Newbold, 1974), having high t-ratios and R-squared thus producing results with no economic interpretation. The pioneering work on the test for unit root involving time series data was done by Dickey (1976) and Fuller (1979). The Dickey-Fuller or Augmented Dickey-Fuller (ADF) test seeks to examine the null hypothesis that ψ = 0 in equations of the form:
âˆ†yt = ψyt-1 + ut
This implies that the relevant hypotheses are:
H0: Series contains a unit root
H1: Series is stationary
I proceeded with the test by programming a prebuilt command in RATS called DFUNIT.SRC, which I used to estimate the Dickey-Fuller (DF) statistic with no lag and the Augmented Dickey-Fuller (ADF) statistic with 12 lags on the differences. The results are as shown below:
1% = -3.44, 5% = -2.87, 10% = -2.57
Although the testing procedure follows that of the student t-test, the actual distribution under the null hypothesis of non-stationarity means that the test statistic does not follow the usual t-distribution. Fuller (1976) derived these critical values from simulation experiments. In comparison, it is evident that the test statistics under both tests are more negative than the critical values at each level of significance, implying a rejection of the null hypothesis of unit root in the returns.
6.0 EMPIRICAL ANALYSIS OF RESULTS
6.1 Results for the 25 Size and Book-to-Market Portfolios
The regression results depicted in Tables 1 to 4 in this section and subsequent ones were obtained using the Generalised Least Squares (GLS) method by employing the Newey-West standard errors procedure to deal with the problems of heteroscedasticity and autocorrelation using 12 lags.
The first result in Table 1 is based on the 25 size and book-to-market portfolios on the entire available data from July 1926 to December 2007, a total of 978 observations. This therefore is an extension of the analysis covering July 1963 to December 1993 by Fama and French in 1993. It is evident from table 1 that the Mean Absolute Value (MAV) of alpha for the Fama and French three factor model is less than that of the CAPM, thus demonstrating its superiority in line with the results of Fama and French. Whilst the CAPM displays a MAVA of 0.0022, that of the Fama and French three factor model is 0.0018. In terms of á¹œ2, the Fama and French three factor model again proofs its superiority over the CAPM across all the 25 portfolios. The average á¹œ2 for the CAPM is 0.75 as against the average of 0.88 for the Fama and French three factor model.
The GRS statistic as stated earlier is another useful tool for measuring the effectiveness of an asset-pricing model. The test suggests that, if the null hypothesis is true, that is the alphas are jointly zero, then the statistic should also be zero. The computed statistics for CAPM and Fama and French three factor models are approximately 3.78 and 3.40 respectively. It is evident that these figures are undesirable since they are further away from zero, depicting the ineffectiveness of the two models. The upper 5% critical values of the resultant F-distributions are approximately 1.50 in each case, hence we reject the null hypothesis that H0: αi = 0 for all i, since the values produced by the test statistic are greater than the critical values. This decision implies that the alphas are not jointly zero and hence the models are not effective in explaining variations in the returns of portfolios grouped according to size and book-to-market equity. It is important to note that even though the null hypothesis is rejected under both models the resultant values of the statistic are in line with those of the MAVA, where the CAPM underperforms the Fama and French three factor model.
The Fama and French three factor model produced 12 negative alphas and 13 positive with 5 alphas being statistically significant at the 95% level of confidence. The CAPM on the other hand showed 17 positive alphas with 8 negative and 4 significant at the 95% level of confidence.
Table 1: CAPM and Fama and French three-factor regressions for 25 Portfolios sorted on
Size, Book-to-Market Equity (BE/ME) l926:07 to 2007:12.
The following table displays the regression results for both the CAPM and Fama and French three-factor model for 25 portfolios. The 25 portfolios are constructed at the end of each June and represent the intersections of five portfolios formed on size (market equity, ME) and five portfolios formed on the ratio of book equity to market equity (BE/ME). The size breakpoints for year t are the NYSE market equity quintiles at the end of June of t (1926-2007). BE/ME for June of year t is the book equity for the last fiscal year end in t-1 divided by ME for December of t-1. The BE/ME breakpoints are NYSE quintiles. The portfolios for July of year t to June of t+l include all NYSE, AMEX, and NASDAQ stocks for which we have market equity data for December of t-1 and June of t, and (positive) book equity data for t-1. The data runs monthly from 1926:07 to 2007:12 for a total of 978 observations. MAV = Mean Absolute Value.
CAPM Fama and French
Ri - Rf = αi + βi(Rm - Rf) + εi Ri - Rf = αi + βi(Rm - Rf) + si SMB + hi HML + εi
Size, BE/ME α t (α) β á¹œ2 α t (α) β s h á¹œ2
Small, Low -0.0063 -2.71 1.66 0.51 -0.0087 -4.90 1.32 1.35 0.43 0.64
Small, 2 -0.0016 -0.80 1.48 0.55 -0.0040 -2.81 1.11 1.67 0.27 0.79
Small, 3 0.0011 0.58 1.40 0.66 -0.0012 -1.51 1.09 1.24 0.42 0.85
Small, 4 0.0033 1.62 1.32 0.66 0.0007 0.79 0.98 1.29 0.53 0.92
Small, High 0.0049 2.37 1.40 0.61 0.0014 1.95 0.99 1.42 0.83 0.92
2, Low -0.0026 -1.49 1.25 0.70 -0.0031 -3.70 1.07 1.09 -0.26 0.89
2, 2 0.0010 0.77 1.28 0.76 -0.0004 -0.54 1.05 1.04 0.16 0.93
Size, BE/ME α t (α) β á¹œ2 α t (α) β s h á¹œ2
2, 3 0.0025 1.87 1.19 0.76 0.0008 1.54 0.96 0.91 0.32 0.93
2, 4 0.0029 1.94 1.23 0.76 0.0007 1.41 0.98 0.86 0.51 0.94
2, High 0.0032 1.76 1.36 0.71 0.0003 0.43 1.06 0.97 0.78 0.94
3, Low -0.0017 -1.28 1.29 0.81 -0.0021 -3.20 1.15 0.82 -0.18 0.92
3, 2 0.0013 1.34 1.13 0.86 0.0006 1.14 1.01 0.55 0.07 0.92
3, 3 0.0021 2.00 1.15 0.85 0.0009 1.39 1.02 0.44 0.32 0.92
3, 4 0.0025 2.04 1.13 0.80 0.0009 1.60 0.96 0.50 0.46 0.92
3, High 0.0020 1.50 1.40 0.76 -0.0007 -0.92 1.17 0.53 0.87 0.92
4, Low -0.0003 -0.27 1.08 0.86 0.0003 0.43 1.07 0.29 -0.36 0.92
4, 2 0.0002 0.22 1.10 0.90 -0.0004 -0.48 1.03 0.27 0.12 0.92
4, 3 0.0015 1.49 1.10 0.87 0.0006 0.84 1.00 0.26 0.28 0.91
4, 4 0.0019 1.41 1.18 0.82 0.0003 0.35 1.06 0.21 0.55 0.91
4, High 0.0013 0.73 1.45 0.75 -0.0014 -1.44 1.25 0.33 0.94 0.90
Big, Low -0.0004 -0.69 0.98 0.92 0.0003 0.65 1.04 -0.15 -0.23 0.95
Big, 2 0.0000 0.08 0.93 0.91 0.0002 0.48 0.97 -0.17 0.00 0.92
Big, 3 0.0003 0.32 0.98 0.85 -0.0002 -0.27 0.98 -0.22 0.32 0.91
Big, 4 -0.0001 -0.09 1.14 0.79 -0.0016 -2.03 1.07 -0.16 0.68 0.91
Big, High -0.0108 -0.90 1.25 0.25 -0.0128 -1.08 1.12 0.03 0.83 0.30
MAV 0.0022 1.23 0.75 0.0018 1.06 0.67 0.43 0.88
GRS F-Test 3.7841 3.4002
Critical Values 1.50 1.50
The second result in Table 2 is also based on the 25 size and book-to-market portfolios but this time on a much shorter data set from July 2001 to June 2006, a total of 60 observations. This is in line with the general practice of using five years of monthly data in empirical analysis such as this to allow for parameter stability and eliminate non-synchronous trading. It is evident from table 2 that the Mean Absolute Value (MAV) of alpha for the Fama and French three factor model is still less than that of the CAPM, with just some slight increase in that of the Fama and French as against a significant increase in that of the CAPM over the previous figures. Whilst the CAPM displays a MAVA of 0.0051, an increase of 0.0029 over the previous figure, that of the Fama and French three factor model is 0.0021, recording an increase of 0.0003. In terms of á¹œ2, the Fama and French three factor model again proves its superiority over the CAPM across all the 25 portfolios. The average á¹œ2 for the CAPM is 0.78 as against the average of 0.92 for the Fama and French three factor model.
Despite the increase in the MAVA over the previous ones of the 1926 to 2007 period, there has been significant reduction in the resultant values of the GRS statistics in both models. It is evident from the table that the Fama and French three factor model outperformed the CAPM with a much lower figure of approximately 0.89 as against 1.32 by the CAPM. The upper 5% critical values of the resultant F-distributions are approximately 1.90 and 1.88 for the CAPM and the Fama and French models respectively. It is evident that the values of the statistics are less than the corresponding critical values, hence the lack of evidence to reject the null hypothesis that the alphas are jointly zero for both asset-pricing models. This result implies that under the much shorter period, the models explain better the variations in returns which are also evident in the high á¹œ2'srecorded within this period.
In terms of the signs and significance of the alphas, the Fama and French three factor model produced 17 negative alphas and 8 positive with 3 alphas being statistically significant at the 95% level of confidence. The CAPM on the other hand showed substantial weight towards positive intercepts producing 20 positive alphas with 5 negative and 9 significant at the 95% level of confidence.
Table 2: CAPM and Fama and French three-factor regressions for 25 Portfolios sorted on
Size, Book-to-Market Equity (BE/ME) 2001:07 to 2006:06.
The following table displays the regression results for both the CAPM and Fama and French three-factor model for 25 portfolios. The data runs monthly from 2001:07 to 2006:06 for a total of 60 observations. Please see Table 3 for descriptions of all regression variables. MAV = Mean Absolute Value.
CAPM Fama and French
Ri - Rf = αi + βi(Rm - Rf) + εi Ri - Rf = αi + βi(Rm - Rf) + si SMB + hi HML + εi
Size, BE/ME α t (α) β á¹œ2 α t (α) β s h á¹œ2
Small, Low -0.0054 -1.46 1.72 0.76
Cite This Dissertation
To export a reference to this article please select a referencing stye below: