Testing the CAPM, Fama-French Three-FactorModel and Carhart’s Four-Factor-Model on 25 portfolios formed on Size and BE/ME Literature and empirical study

Bachelor Thesis Finance Author: P.R. Kleij (ANR: s520105) Supervisor: M.F. Boons

Study program: Bachelor Bedrijfseconomie Tilburg University, May 18th, 2012

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Abstract Previous work showed the performance of several asset pricing models on several portfolios; size and BE/ME, E/P, C/P, Industries and so on. Since those studies all use data from periods before 2000, I want to test if the results found in those studies still hold nowadays. In this paper, I study the performance of the CAPM, Three-factor-model and the Four-factor-model on 25 portfolios formed on Size and BE/ME for the period 1980-2011. I found that, in accordance with previous work, the three-factor-model performs significantly better than the CAPM. Furthermore, the concept of momentum does not seem to have further improvement on the explanatory power of the model.

With the capital asset pricing model (CAPM), a theory for asset pricing is introduced. Nowadays, almost four decades later, it is still the most widely used model to explain the relationship between risk and expected return (Fama and French, 2004). The base for this model was formed by Markowitz (1952) and further developed by Sharpe (1964) and Lintner (1965) to what we now know as the CAPM. Markowitz (1952) introduced the concept of mean-variance efficient portfolios, which means that investors choose portfolios which maximizes expected returns for a given level variance and minimizes variance for a given level of expected return. In addition to Markowitz, Tobin (1958) introduced the separation theorem, where he introduced the concept of borrowing and lending when forming portfolios. Furthermore, Sharpe and Lintner (1964, 1965) introduced the assumption that investors have a joint probability distribution function, which is contrary to the assumptions of Markowitz, who stated that every investor has its own probability distribution. Moreover, they introduced the CAPM. The CAPM states that the return of any asset is explained by its market beta, and the market beta only is sufficient to explain expected return. In addition to the CAPM, Merton (1973) developed the intertemporal capital asset pricing model (ICAPM), which differs from the CAPM in the basic assumptions. In the ICAPM, investors not only care about what they earned at the end of the period, they also care about opportunities to invest in with their previously obtained earnings. Furthermore, Jagannathan and Wang (1996) introduced the conditional CAPM, they introduced the assumption that betas and market premiums can shift over time, so the beta is no longer a constant factor as in the CAPM. In the eighties, several empirical studies found anomalies. Basu (1977) was one of the first; he found that low price-earnings portfolios tend to have higher returns than high price-earnings stocks. Furthermore, Banz (1981) introduced the size-effect; he found smaller firms have had higher risk adjusted returns than larger firms. Even so, Statman (1980) and Rosenberg, Reid and Lanstein (1985) found that stocks with high book-to-market ratios had higher average returns than low book-to-market ratios. Again, Basu (1983) found that E/P ratios help to explain the cross-section of expected return. Bhandari (1988) found stock returns are positively 2

related to the debt-to-equity ratio, so leverage helps to explain the cross-section of returns. Fama and French (1992, 1993, 1996), summarizes these anomalies in the three-factor-model. According to Fama and French (1992, 1993, 1996), the market beta is not sufficient to explain the cross-section of expected returns. They found two more factors, in addition to the market beta, can help to explain the cross-section of expected returns. These two factors are the size of the firm and the book-tomarket ratio of its assets, and together with the market beta, according to Fama and French, they can proxy for risk and explain the cross-section of expected returns. Furthermore, they found that the combination of size and book-to-market equity seems to absorb the roles of leverage and E/P (Fama and French, 1992). Fama and French (1996) noticed their model was not sufficient to explain the continuation of short-term returns. Jegadeesh and Titman (1993) discovered the momentum effect, which can explain the continuation of short-term expected return. Previous research by De Bondt and Thaler (1985, 1987), showed stock prices overreact to information, and as a consequence, investors are able to obtain abnormal returns by buying past losers and selling pas winners. Later, Jegadeesh (1990) and Lehman (1990) found more evidence for these contrarian strategies. They found that selecting stocks based on their performance in the previous month have had abnormal returns. Finally, Jegadeesh and Titman (1993) shows that abnormal returns can be realized by buying past winners and selling past losers, in a three- to twelve-month period. A few years later, Carhart (1997) was the first who introduced the concept of momentum in his four-factor model, in addition to the three-factor-model by Fama and French. He found that individual mutual funds who follows the momentum strategy of buying stocks on past performance, obtain lower abnormal average returns when expenses are deducted. In the years to follow, several empirical studies found evidence for the effect of SMB, HML and MOM, but the relationship between those factors and future economic growth never have been investigated. Liew and Vassalou (2000) were the first who attempted to find this relationship and found that there was a relationship between SMB, HML and future economic growth. Even so, the predictive power of SMB and HML for economic growth does not changed when other variables were included, so those factors have a large degree of independency (Liew and Vassalou, 2000). Vassalou (2003) continued studying this relationship and attempted to add a factor to the CAPM that predicts future economic growth. She found that, after this factor was implemented, the performance of the model significantly improved, and the SMB and HML lost their function in explaining the cross-section of average returns.

This paper will test the three capital asset pricing models, the CAPM, Fama-French three-factor model and Carhart’s four-factor model, and test which of these models predicts the patterns of stock returns most accurately. To start, 25 portfolios based on size and book-to-market equity are used from 1980-2011, which are formed by Kenneth French. 3

First, time-series regression will be run to obtain estimates of alphas, betas, R2 and t-statistics. To find the model that explains the cross-section of average returns best, several aspects need to be evaluated. First, when the intercepts (alphas) are close to zero, the model describes the expected return. So the model with the lowest intercepts performs better than models with higher intercepts. Another important factor are the R2, when those are high; most of the variation in expected return is captured by the model. So the model with the highest R2 performs better than models with lower ones. Furthermore, a GRS test will be performed to test if all alphas are equal to zero. When the time-series regressions are finished, cross-sectional regressions will be run. The earlier obtained beta estimates from the time-series regression will be used as independent variable. Running these cross-sectional regressions gives estimates of averages for each month; even so, they provide standard deviations, t-statistics and p-values. The latter two are used to check for significance. When the time-series and cross-sectional regressions for the whole period (1980-2011) are done, I continue testing it on three subsamples; i. 1980-1990, ii. 1991-2000 and iii. 2001-2011. I choose three subsamples of each 10 years to look if there are any differences in comparison with the 30 year sample. Most of the anomalies were found in the eighties, so you might expect that, for example, the size- and book-to-market effects become weaker because investors are following new strategies of portfolio formation. Numbers of empirical papers have studied the performance of the several asset pricing models. However, most of those studies are performed on data from the past, Fama and French (1996) used data from the period 1963-1993, where Carhart used data from the period 1962-1993. My goal is to test if the results found for those periods, still holds for the period 1980-2011. Due to changing conditions, like investment behavior, economic climate, new empirical findings and so on, you might expect changes in the explanatory power of these models. I found that, in accordance with Fama and French (1996), the Fama-French three-factor-model explains the cross-section of average returns better than the CAPM does. Furthermore, when the momentum factor is included in the model, the four-factor-model does not improve its explanatory power. Furthermore, as stated by Fama and French, the beta does not seem to be sufficient to explain the cross-section of average returns. When additional factors, SMB and HML, are included, the model explains more of the cross-section of average returns. Furthermore, when cross-sectional regressions are run on the three asset pricing model, all three models are rejected. So, when testing with cross-sectional regressions, all three models lose its explanatory power. When testing on subsamples from several periods (1980-1990, 1991-2000 and 2001-2011), no more interesting new patterns were found that were significantly different from patterns of the whole sample (1980-2011).

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This thesis consist of several sections, the first one provides a review of where the literature currently stands. Section II provides empirical results of time-series regressions on the several asset pricing models. Section III provides empirical results of cross-sectional regressions on the asset pricing models. To continue, section IV provides the results of time-series on the different subsamples, period 1 (1980-1990), period 2 (1991-2000) and period 3 (2001-2011). Finally, section V consist the conclusion.

I. Literature Review Fundament of capital asset pricing The base for capital asset pricing was formed by Markowitz (1952), he introduced a completely new view on asset pricing. He rejected the assumption that investors only maximize discounted returns; every investor only invests in securities with the greatest discounted value. When all investors only choose portfolios which maximize expected returns, than there never is a diversified portfolio that is more preferable than a non-diversified portfolio (Markowitz, 1952). So, investors should not only maximize discounted returns, they even so need to diversify. Here, Markowitz firstly introduces the concept of diversification, which is nowadays broadly carried. Diversification must be used in the right way, and with the right reasons according to Markowitz. Portfolios created with assets that are all perfectly correlated, like a portfolio consisting of assets from one industry, are not well diversified because when the industry performs poorly, the whole portfolio will perform poorly. When a portfolio is held with assets from multiple industries, this effect is much less likely to occur, and as a consequence, this portfolio is well more diversified than a portfolio with assets from the same industry. But not all variance could be diversified. Due to the law of large numbers, not all the variance can be diversified because of the inter-correlation of the securities. This is one of the first important findings of Markowitz, which stated that not all the risk in a portfolio could be eliminated; there will always remain risk, systematic risk. The risk that can be eliminated, the unsystematic risk, could be eliminated by building a portfolio with both correlated and uncorrelated assets. The following figure, figure 1, graphically presents the effect of diversification, Figure 1

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Markowitz developed a model where he assumes that investors are risk-averse, and they only care about their mean and variance of their one-period investment return when choosing among portfolios. So, investors are looking for portfolios that minimizes variance using a given level of expected return and portfolios that maximizes expected return for a given level of variance, this is well known as mean-variance efficient (Markowitz, 1952). According to this theory, investors want to have portfolios with high means (expected returns) and low variance (risk). After Markowitz formed the base for capital asset pricing, many people made additions to it. Tobin (1958) introduced the “separation theorem”. When investors are able to borrow and lend at the same rate, the process of portfolio formation can be broken down into steps. First, the investor chooses the optimal combination of risky assets; and secondly he has to choose the allocation of funds between this optimal combination of risky assets and risk-free assets, depending on the investors’ risk profile (Tolbin, 1958). Assuming every investor faces the same efficient frontier, and the market is efficient, everyone is forming the same portfolio. As a result, there is only one portfolio of risky assets and risk-free assets, which has to be the market portfolio. The following figure, figure 2, summarizes this story. Figure 2

The figure above graphically presents the mean-variance model introduced by Markowitz. The minimum variance frontier, the curve abc, represents portfolios of risky assets which minimize variance for different levels of expected return. Clearly, there is a relationship between expected return and risk. For example, investors who require a high level of expected return bear, as a consequence, higher levels of risk. Point b. on the minimum variance curve is often called the minimum variance portfolio, since it has the lowest possible variance. When the risk-free borrowing 6

and lending rate is introduced, the efficient set turns into a straight line. For example, consider a portfolio with a share in risk-free assets and a share in risky assets, like point g. When an investor invests all his funds in risk-free assets, like treasury bills, he will bear no risk and have a return equal to the risk-free interest rate. To obtain the optimal portfolios, one must draw a line from Rf up, and to the far left as possible, to obtain the tangency portfolio, as shown in point T. This line is called the capital market line, and is well known as the mean-variance efficient frontier. When following Tolbin’s separation theorem, all investors will have the same opportunity set, and as a consequence, combine the same risky tangency portfolio. When all investors hold the same tangency portfolio, the tangency portfolio can be considered as the value-weight market portfolio. The weight of each risky asset in the tangency portfolio equals the total market value of all outstanding assets divided by the total market value of all risky assets (Fama and French, 2004). The only differences, when investors are forming portfolios, are cause to the risk profiles of individual investors. Investors that are riskaverse will form portfolios which have a larger share of risk-free assets and a smaller share of risky assets. On the contrary, riskier investors will hold a larger share of risky assets in comparison to riskfree assets. To summarize, the assumptions of Markowitz and Tolbin states that the market portfolio must be on the minimum variance frontier, so the market portfolio overlaps the tangency portfolio. This implies that the relationship that holds for the minimum variance portfolio must also hold for the market portfolio.

Sharpe-Lintner CAPM Sharpe (1964) and Lintner (1965) attempted to further improve the foundation of capital asset pricing. The assumption of Markowitz and Tolbin that security prices are given and each investor has its own probability distribution among rates of returns, is rejected by Sharpe and Lintner. They assume that all investors have the same distribution function, so there is a joint probability distribution. Furthermore, they assume that all investors assign the same means, variances and covariances to the distribution of these returns (Lintner, 1965), which is in line with the theory of joint probability distribution. Another important assumption is that each investor can invest, as many as he requires, in risk-free assets. He can do this by borrowing and lending at the risk-free rate, and invests the proceeds in risky assets. Here they break with an assumption of Tolbin, who assumed borrowing was not permitted. With the assumptions of borrowing and lending, an important conclusion emerged. Sharpe and Lintner found that, when following the assumptions of borrowing and lending at the risk-free rate, every individual investor will make identical decisions when forming portfolios. As a consequence, only one point on the mean-variance-efficient frontier is important; the tangency portfolio. This tangency portfolio, point T in figure 1, is the same for every investor, due to the key assumptions of Sharpe and Lintner. Because it is the same for every investor, the tangency 7

portfolio must be the market portfolio. The only difference among portfolios for individual investors is caused by the investors’ risk profile. Due to this risk profile, portfolios only differ in combinations of risky and risk-free assets, but it always is on the tangency portfolio.

To illustrate these findings, Sharpe and Lintner introduced the Sharpe-Lintner CAPM model: (1) Where

is the excess return of any asset i,

is the market beta of asset i and

is the market risk premium. According to this model, investors are only compensated for the systematic risk. The extent to which investors are compensated depends on two factors, market beta and the market risk premium. Because the unsystematic risk can be eliminated by diversification, investors are not compensated for it. The market beta can also be denoted as the covariance of its return with the market, divided by the variance of the market return (Fama and French, 2004): (2) The market beta measures the sensitivity of the return of any asset i to the market return. In common words, if the market beta is 1, one euro invested in asset i contributes 1 euro to the market portfolio. Furthermore it can be seen as a risk factor, when the beta is higher the expected return of asset i will be higher. So, this implies the expected return of any asset i is linearly related to its market beta. Figure 3 graphically represents this relationship: Figure 3

When a security has a beta of zero, it is equal to the risk-free rate. When a security has a beta of 1, the expected return of asset i is equal to the expected return of the market. This linear slope,

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denoted as the security market line (SML), has an upward slope as long as the expected return of the market is larger than the risk-free rate. Even though the CAPM was a major breakthrough in capital asset pricing, it was not perfect and several empirical studies argued the simplifying assumptions of the CAPM. Black (1972) was one of the first who argue the logic of the CAPM. According to Black, unrestricted borrowing and lending is not realistic. So, Black rejected the assumption of unrestricted borrowing and lending and introduced unrestricted short selling of risky assets. Black stated that the market portfolio, which is meanvariance-efficient, can be acquired by permitting unrestricted short-selling of risky assets. Furthermore, he had a slightly other view on the relationship between the market beta and expected return than Sharpe and Lintner. They both agreed there was a linear relationship between beta and expected return, but there view was different on the expected return of assets that did not correlate with the market. According to Sharpe and Lintner, uncorrelated assets have betas of zero and therefore contribute nothing to the market portfolio. Because assets with betas of zero are uncorrelated with the market, they are riskless and, as a consequence, must equal the risk-free interest rate. The premium for beta risk is the difference between the expected return of the market and the risk-free rate. In contrast, Black stated that expected returns of uncorrelated assets, must be less than the expected return of the market, and as a consequence, the premium for beta is positive.

ICAPM and Conditional CAPM In addition to the CAPM, Merton (1973) developed the intertemporal capital asset pricing model (ICAPM). The main difference between the CAPM and the ICAPM can be found in the assumptions of both models. In the CAPM, investors only care about what they earned with their portfolio at the end of the period. In the ICAPM, investors not only care about what they earned at the end of the period, they also care about opportunities to invest in with their previously obtained earnings. Important factors introduced by Merton are state variables, for example, consumption goods prices, labor income, investment opportunities. So, when investors are forming portfolios using the ICAPM, they pay attention to changes in the state variables because fluctuations in state variable may alter future investment opportunities. Another important factor in the ICAPM is the covariance between the portfolio return and the state variables. So, portfolios are formed on having the highest expected return with a given level of variance and covariances with the state-variables, and as a consequence, portfolios formed are multifactor-efficient (Merton, 1973). Several empirical studies concluded that the static version of the CAPM is unable to explain the cross-section of expected return (Banz, 1981; Basu (1983); Bhandaari (1988)). The static CAPM by Sharpe and Lintner stated that the betas remain constant over time. Jagannathan and Wang (1996) argue this assumption; they state that the risk of a firm will vary over time, like in periods of 9

recession. In a recession, firms in distress will be riskier and, as a consequence, their beta will be higher. They introduced the assumption that betas and market premiums can shift over time, so the beta is no longer a constant factor. They found that the conditional CAPM explains almost 30 percent of the cross-sectional variation in average returns, while the static CAPM only explains 1 percent (Jagannathan and Wang, 1996).

Anomalies in the eighties During the next years, the basics of the CAPM got criticized more and more. One of the key critics was that the market beta was unrelated to the variation in the expected return. Basu (1977) was one of the first who argue the validity of asset pricing models. He found that low price-earnings portfolios tend to have higher returns than high price-earnings stocks. Due to this finding, the efficient market hypothesis, which states that stock prices fully reflect information so excess returns could not be acquired, is violated. An explanation for this difference can be found in the processing of information for stock prices. Because of delay in adjustments, important information, like P/E ratios, could not be fully reflected by stock prices (Basu, 1977). Banz (1981) introduced the size-effect; he found smaller firms have had higher risk adjusted returns than larger firms (Banz, 1981). Banz stated that because of this size-effect, the CAPM is not well specified. Furthermore, he found that this size effect occurs mainly for very small firms and in much lesser extent for average-sized and large-sized firms. A clear explanation for this effect was not found by Banz, he stated that size might be one of the factor that proxy for this effect, but he did not exclude that there are other, unknown, factors that correlates with size to create this effect. Another empirical study by Basu (1983) shows the effect of earnings-price ratios on stock returns. Basu found that E/P ratios help to explain the cross-section of expected return, while market beta and size were included in his testing. Furthermore Statman (1980) and Rosenberg, Reid and Lanstein (1985) found that stocks with high book-to-market ratios had higher average returns than low book-to-market ratios. Bhandari (1988) found stock returns are positively related to the debt-to-equity ratio, so leverage helps to explain the cross-section of returns. In the CAPM model, this leverage effect should be captured by its market beta, because the market beta is the only factor that describse the crosssection of expected returns (Bhandari, 1988). However, Bhandari not only includes the market beta in his test but also size of the firm, and found that leverage helps to explain the cross-section of average returns.

Three-factor model With these empirical studies in mind, Fama and French (1982) tested the relationship between expected return and size, book-to-market equity, leverage and E/P ratios. One of their key results 10

after multivariate testing was that the market beta does not helps to explain the cross-section of return and the combination of size and book-to-market equity seems to absorb the roles of leverage and E/P (Fama and French, 1992). Furthermore, they found that stock risks are multidimensional. The first dimension of risk is captured by size; the second dimension is captured by book-to-market equity. So, Fama and French (1992, 1993) concluded the market beta was not sufficient anymore to explain the cross section in expected returns. Therefore, they developed a three-factor-model as a complement on the CAPM. Fama and French introduced two additional factors to describe the crosssection of expected returns, the size of a stock (small minus big) and book-to-market equity (high minus low). So, the expected excess return is, (3) Where

is the excess return of any asset i,

is the market beta and

is the

market risk premium. SMB, Small Minus Big, measures the differences between the returns on small stock portfolios and big stock portfolios. High Minus Low, HML, measures the differences between the returns on high book-to-market equity firms and low book-to-market equity firms. Fama and French find that the three-factor-model was a good model when testing it on portfolios formed on book-to-market equity and size. Also, the three-factor-model seems to explain average returns when testing it on portfolios formed on cashflow-price ratios (C/P), sales growth, earningsprice (E/P) and long-term past returns. Moreover, Fama and French found that portfolios formed on C/P, sales growth, long-term past returns and E/P does not reveal any other dimensions of risk that are not captured by size and book-to-market equity. So, in the three-factor-model, the SMB and HML capture systematic risk, and for that reason they are priced. After the introduction of the CAPM they were called anomalies because the effect of leverage, size, book-to-market equity and E/P ratios were not captured by the CAPM. Since the three-factor-model captures these effects, they are no longer anomalies and, as a consequence, explain most of the variation in average returns. So, in the three-factor model, expected returns are explained, first, by the excess return on the market. Second, by the difference between small stocks and big stocks, and finally by the difference between low book-to-market stocks and high book-to-market stocks. Fama and French (1995) found that HML slopes and book-to-market equity proxy for relative distress. They found that weak firms with low earnings tend to have positive slopes on HML and have high book-to-market equity. On the contrary, strong firms with high earnings have negative slopes on HML and low book-to-market equity (Fama and French, 1995). The second factor introduced by Fama and French, size, also relates to profitability. Small firms tend to have lower earnings on assets than big firms (Fama and French, 1992).

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The concept of momentum As concluded by Fama and French (1996), the three-factor-model does not cover expected return on all securities and portfolios; it does not explain the continuation of short-term expected returns. Therefore, Jegadeesh and Titman (1993) discovered the momentum effect, which can explain the continuation of short-term expected return. This momentum anomaly is a market imperfection, because new information is slowly processed (Jegadeesh and Titman, 1993). According to De Bondt and Thaler (1985, 1987), stock prices overreact to information, and as a consequence, investors are able to obtain abnormal returns by buying past losers and selling pas winners. De Bondt and Thaler (1985, 1987) found that holding stocks that performed poorly in the past (three to five years) have had 25% higher average returns than stocks that performs well in the past. A very remarkable result was that, the so called, loser portfolio had very large positive excess returns in January. Even more remarkable was that the effect occurs 5 years after the portfolios were formed. A clear explanation for this effect was not found. However, a lot of people argued the study of De Bondt and Thaler; some argued that the findings can be explained by the size effect and systematic risk (Chan, 1988). Even so, because the remarkable results of loser portfolios in January, some argued the results not necessarily relate to overreaction. Jegadeesh (1990) and Lehman (1990) found more evidence for these contrarian strategies; when investors select stocks based on the past performance of these stocks. They found that selecting stocks based on their performance in the previous month have had abnormal returns. However, it is hard to explain this effect because their studies are very short-term orientated (months) and therefore, it is not necessarily related to overreaction. Jegadeesh and Titman (1993) not totally agreed with the periods for testing this effect. They argued the very short-term period of Jegadeesh (1990) and Lehman (1990), which was one week or one month. They also disagreed with the very long-term period of De Bondt and Thaler (1985, 1987) which was three to five years. So, Jegadeesh and Titman used a period between those two, they thought a three to twelve-months would be more appropriate for testing. They found that; when buying past winners and selling past losers, abnormal returns can be acquired. When selecting stocks on their previous six-month return, and holding them for another 6 months, they found that these stocks obtained an excess return of 12% (Jegadeesh and Titman, 1993). Furthermore, they found that past winners realized higher returns than past losers in the first seven months. After these seven months, past losers realized higher returns than past winners. A clear explanation for this effect was not given, according to Jegadeesh and Titman, a more sophisticated model of investor behavior is needed. However, the interpretation of Jegadeesh and Titman (1993); when investors are buying past winners and selling past losers, prices move away from their long-run values and, as a consequence, cause prices to overreact. 12

Furthermore, Grinblatt, Titman and Wermers ( 1995), studied the trading patterns of mutual funds. They examined the portfolio choices of these mutual funds and tried to make a separation in this choice. They want to examine the extent to which they form portfolios on herding and to which extent the form portfolios based on the past performance of stocks. They found that 77 percent of the mutual funds bought stocks that performed well in the past. But on the contrary, they did not systematically sell past losers (Grinblatt, Titman and Wermers, 1995). Furthermore, they found that momentum investors, buying stocks on past performance, performed significantly better than other mutual funds. Moreover, they found small evidence for the extent of herding.

Four-factor-model Carhart (1997) further studied the effect of momentum, and find that individual mutual funds who follows the momentum strategy of buying stocks on past performance, obtained lower abnormal average returns when expenses are deducted. So, he concludes that these expenses, transaction costs, offset the gains obtained following the momentum strategy. With the concept of momentum in mind, Carhart (1997) further altered the three-factor model with a fourth, momentum factor. Therefore, the expected excess return is, (4)

Where

is the excess return of any asset i,

is the market beta and

is the

market risk-premium. SMB measures the differences between the returns on small stock portfolios and big stock portfolios. HML measures the differences between the returns on high book-to-market equity firms and low book-to-market equity firms. And finally, MOM measures the differences between the returns of past winners and past losers. According to Carhart (1997), buying past winners and selling past losers, realize a return of 8 percent per year. Furthermore, he found significant negative relationships between performance and expense ratios, load fees and portfolio turnover. Finally, three important rules-of-thumb can be derived from his study. First, avoid funds with continually poor performance. Second, funds with high past returns have above-average expected returns next year. And finally, the investment costs of expense ratios, load fees and transaction costs have a direct, negative impact on performance (Carhart, 1997)

New insights on asset pricing models Since the introduction of SMB, HML and MOM, several empirical studies found evidence for those factors, but the relationship between those three factors and economic risk factors never have been

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studied. Liew and Vassalou (2000) were the first who attempted to find this relationship, the core of the study was to investigate if there was a relationship between SMB, HML, MOM and future economic growth. The key result of this study was that there is a relationship between SMB, HML and future economic growth, and this relationship holds for several countries in even good and bad states. Even so, the predictive power of SMB and HML for economic growth does not change when other variables are included, like market factors and other business variables. So, SMB and HML are, for a large extent, independent of any other factors. Also they found that the regression coefficients of the SMB, HML and the market are positive and of the same magnitude (Liew and Vassalou, 2000). Because of the finding that SMB and HML can predict future economic growth, the assertions by Fama and French (1992, 1993 and 1995) that these factors act as state variables, is confirmed. Furthermore, no relationship was found between MOM and economic growth. A few years later, Vassalou (2003) continued the study on this relationship. Vassalou (2003) attempted to include a factor to the CAPM that predicts future economic growth. He found that, after this factor was implemented, the performance of the model significantly improved. Even so, the pricing errors have the same size and patterns as the Fama-French three-factor-model (Vassalou, 2003). But when the factor is added to a model with SMB and HML, the latter two factors lose their value in explaining the cross-section of returns (Vassalou, 2003). This is in line with the theory above. Since SMB and HML can predict future economic growth, and a factor that contains news about economic growth is implemented, the SMB and HML lose its function because the new factor captures this.

II. Emprical results for time-series regressions CAPM To test the CAPM, the Fama-French 3 factor model and Carhart’s four factor model, 25 portfolios were formed on size and BE/ME. These portfolios are formed by Kenneth French1 and include all stocks from the NYSE, AMEX and NASDAQ index for which market equity and book equity data was available. There are 5 portfolios formed on size and 5 on BE/ME, which is 25 in total. To form these portfolios, breakpoints are needed. So, to create these portfolios, Kenneth French used market equity quintiles of the NYSE, at the end of June for every year. To start, the first table of Table 1 shows the average excess returns and standard deviations of the 25 portfolios. The table shows that small stocks (0.72), on average, tend to have higher excess returns than big stocks (0.59). The same effect holds for the difference between average excess returns of low book-to-market stocks and high book-to-market stocks. On average, high book-to-market stocks tend to have higher excess returns (0.89) than low book-to-market stocks (0.43). Another conclusion,

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Kenneth French Data Library: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

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based on table 1, is that small stocks (6.37) are, on average, riskier than big stocks(4.78). Furthermore, low book-to-market stocks (6.63) are riskier, on average, than high book-to-market stocks (5.49). There are large differences between the number of firms in portfolios, the small stocks have, on average, 545 firms in their portfolio while big stocks only have 63 firms, on average. The difference between low book-to-market portfolios (249) and high book-to-market portfolios (189) is much smaller. The same effect is true for average firm size, but only in the opposite direction. The average size of small stocks (78) is much lower than the average size of big stocks (15071), while the difference between low and high book-to-market stocks (4696 vs. 3051) is much smaller. Table 1 Summary statistics on 25 portfolios formed on size and BE/ME for the period 1980-2011, 384 months 25 portfolios, formed on size and BE/ME for the period 1980-2011. These portfolios are formed by Kenneth French and include all stocks from the NYSE, AMEX and NASDAQ index for which market equity and book equity data was available. There are 5 portfolios formed on size and 5 on BE/ME, which is 25 in total. To form this portfolios, breakpoints are needed. So, to create these portfolios, Kenneth French used market equity quintiles of the NYSE, at the end of June for every year. The * stand for different levels of significance. *, **, *** are significant at levels of 90%, 95% and 99%, respectively. Size

Small 2 3 4 Big

Small 2 3 4 Big

Small 2 3 4 Big

To

Low

-0.03 0.40 0.52 0.71 0.57

-0.06 1.06 1.49 2.26** 2.3**

0.76 0.73 0.79 0.70 0.66

2.17** 2.43** 2.81*** 2.57** 2.72***

83 400 935 2378 19683

continue,

Book-to-Market Equity (BE/ME) Quintiles 3 4 High Summary Statistics Means

2

0.85 0.94 0.81 0.69 0.54

0.97 0.92 0.82 0.78 0.56

t-statistics means 2.91*** 3.55*** 3.47*** 3.46*** 3.17*** 3.25*** 2.58** 3.21*** 2.26** 2.43**

1.03 0.90 1.08 0.80 0.64

time

series

regressions

2

3

4

High

Standard Deviations 8.09 7.31 6.78 6.13 4.85

3.5*** 2.97*** 4*** 2.97*** 2.43**

Average firm size 89 86 75 406 412 405 945 943 955 2306 2267 2291 16998 14670 12475

Low

671 203 143 113 114

6.85 5.92 5.52 5.32 4.73

5.72 5.29 5.01 5.27 4.70

5.37 5.19 4.97 4.79 4.51

5.80 5.93 5.27 5.30 5.13

Number of firms in portfolio 408 417 492 142 140 115 101 88 71 79 65 57 70 54 48

736 85 51 42 30

55 398 974 2300 11527

are

run

for

each

portfolio.

The

Regression,

, is run for every month. This time-series regression gives estimates for betas for each portfolio. Table 2 represents the results of these time-series regressions for the CAPM. When the intercepts (alpha) are close to zero, the CAPM describes the expected return. For the CAPM, only the biggest stocks (0.10) approach zero, for all other stocks the average 15

excess returns are unexplained. The difference in intercepts between small stocks (0.12) and big stocks are not very different (0.1), but for low book-to-market (-0.25) and high book-to-market (0.36) there is a large difference. When higher book-to-market stocks are included in the portfolios, the larger the intercepts become and more excess returns are unexplained. Important to notice is the large negative intercept (-0.78) of the small, low book-to-market portfolio. This effect is called the small-growth anomaly, which is nowadays still a point of discussion in literature. There are large differences between the betas of the portfolios. Small stocks tend to be more volatile to the market (1.09) than big stocks (0.9). Furthermore, stocks with low book-to-market values (1.24) tend to be more volatile than stocks with high book-to-market values (0.96). So, small stocks are riskier than big stocks, and low book-to-market stocks are riskier than high book-to-market stocks. The average of the R² for the CAPM is 0.74, so 74% of the variation in expected returns is captured by this model. This also implies that 25% of the variation in expected returns is not explained by this model and there must be other factors that explain this variation. According to this table, big stocks with low book-to-market tends to have higher R² than other stocks and so they tend to explain variation in expected return better than other stocks. Table 2 Time-series regressions on the CAPM on 25 portfolios formed on size and BE/ME for the period 1980-2011, 384 months. Rf is the risk-free interest rate, or the return on one-month treasury bills. Rm is the market return, the value weighted return on all stocks in the several portfolios. The Rf, Rm and the portfolios are obtained from the data library of Kenneth French. The * stand for different levels of significance. *, **, *** are significant at levels of 90%, 95% and 99%, respectively.

16

Another way to test the CAPM is using the GRS test. This multivariate test, introduced by Gibbons, Ross and Shanken (1989), tests if all the alphas are equal to zero. To test, first hypothesis must be formed. H0: αi=0 H1: αi≠0 After testing, GRS statistics of 5.05 are acquired with a p-value of 6.62e-13. We reject H0 when the pvalue