*Master student at the Faculty of Economics and Business, University of Groningen, The Netherlands
E-mail: tonpostema@gmail.com, student number: 2249324.
Fama and French Three-factor Model and Risk-return
Predictions with the Model Betas
Ton Postema*
Master’s Thesis Finance University of Groningen
Supervisor: Dr. L. Dam January 2014
Abstract
This paper examines the CAPM and Fama and French three-factor and four-fact model using U.K. data for the period 1986-2012. Remarkably, little research is done to test the ability of the three-factor model to build portfolios which are in line with the models predictions. This paper provides a more direct test of the FFM by forming portfolios based on estimated stock returns, predicted by historical factors. This study finds that for FFM there is a highly significant and positive correlation of 0.939 between expected and realized returns and a positive and significant correlation between expected and actual risk of 0.661 for the CAPM. Hence, FFM is better is estimating future return but the CAPM is a better risk estimator.
Keywords: capital asset pricing model, Fama and French three-factor model, four-factor model, beta.
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I. Introduction
“Essentially, all models are wrong, but some are useful” (Box and Draper, 1987). Since every model uses assumptions and is a simplification of reality, models are usually flawed. Nevertheless, they can be useful in explaining, predicting and understanding market movements. According to Box and Draper (1987) there is no model that can fully and complete accurate predict future returns. But how good are the models we use now? This paper will test two of the most used and taught models in finance; the Capital Asset Pricing Model (CAPM) and the Fama and French three-factor Model (FFM), on how accurate these models are and the power of these models to predict return variation using the models betas. For comparison purposes the four-factor model is also included.
The concept of risk is widely spread within the financial sector, hence everybody will include risk when making an investment analysis. One of the most widely accepted and used measures of risk is the beta coefficient. The beta coefficient is used in most asset pricing models, i.e. CAPM and the Fama and French three- and four-factor models. These asset pricing models use historical data to estimate the price of securities and portfolios, therefore these models rely heavenly on the estimated betas of the securities. Due to this simplifying assumption the CAPM is flawed. This flaw is well known and recognized, but despite this fact it is still one of the most popular models which is still frequently used and taught in finance textbooks and courses. Mostly due to it simplifying assumptions the CAPM make it easy to understand fundamental concepts of asset pricing.
The past twenty years of research shows that beta, or systematic risk of security alone, is not sufficient to explain the return behavior of the security. Research shows that more parameters should be included in the asset pricing models. Fama and French (1992) document that on average 70% of a portfolio return can be explained by the beta and the other 30% by other factors. The inclusion of firm specific parameters must increase the explanatory power of the asset pricing model. Fama and French (1993) elaborate on the use of firm specific characteristics by explaining the return behavior of portfolios by adding extra variables to the existing CAPM for size and value. They find that these two factors combined have the power to explain the behavior of stock returns. Furthermore the later addition of momentum by Carhart (1997) is also included.
three-3
factor model. The three-factor model has replaced the CAPM in relation to risk-adjusting. Remarkably, little research is done to test the ability of the three-factor model to build portfolios which are in line with the models predictions. Especially since the lack of this predicting ability is one of the limitations of the CAPM and according to some, let to the end of CAPM era. This paper provides a more direct test of the FFM by forming portfolios based on estimated stock returns, predicted by historical factors. The goal is to check whether portfolios formed on this basis will follow the predicted path by the model of risk and return. In addition the CAPM and four-factor model receive the same kind of testing. The dataset used in this research consist of all U.K. quoted firms for the period 1986-2012. The section about risk return prediction differs from previous studies in two ways. First it focusses on the U.K. rather than the U.S. market, secondly it includes the momentum factor.
In sum, this paper test the accuracy of the CAPM and FFM, with and without momentum, on U.K. data and the power of the models to predict return variation using the betas of these models.
The remainder of this paper is structured as follows. The following section reviews the existing literature and research on this topic. Section three describes the methodology used in this paper. The fourth section presents the data and descriptive statistics of the dataset. Section five describes the results and the six and final section is a summary and conclusion of this research.
II. Literature review
Preliminary
In the olden days, investments were generally made on expectations or intuition. Investors made their investment decisions based on news, sentiment and advices. Graham and Dodd (1937) argue that the prices at which stocks are traded often don’t reflect their true intrinsic value. They introduced the idea of value investing; buying stocks below their intrinsic value. Using value investing techniques can limit the downside risk of an investment. In 1952 Markowitz changed the game by introducing the idea of portfolios. Markowitz is the founder of modern portfolio theory (MPT), he made investors aware that risk can be managed by diversification, i.e. forming portfolios. Markowitz (1952) argues that it is possible to construct an efficient frontier which gives the maximum possible expected return for a predetermined level of risk.
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in finance textbooks and for evaluating performance of portfolios or cost of capital estimations. According to Fama and French (1993) the attraction of the CAPM is that it offers powerful and intuitively pleasing predictions about the relation between risk, expected return and how to measure risk. But the empirical record of the CAPM is poor. This empirical shortcomings may result in theoretical failings, mostly due to the simplifying assumptions.
CAPM
The CAPM starts with the assumption that there are two types of risk; systematic and unsystematic risk. Systematic risk cannot be diversified away, while unsystematic risk, or specific risk, can be diversified away. Specific risk represents the part of a stocks return that is not correlated with general market movements. Through diversification specific risk can be removed from a portfolio, however systematic risk remains a problem. Since systematic risk cannot be eliminated, Lintner (1965), Mossin (1966) and Sharpe (1964), developed a measurement for the systematic risk; beta. Beta measures the volatility of a stock in comparison to the market. According to the CAPM, beta is the only relevant measure of a stocks risk. This assumption is one of the major shortcoming of the model. Fama and French (1992) find no support for the positive relation of stock returns related to the market beta. Mainly due to its simplifying assumptions, which work perfect on paper but are too unrealistic in the real world and the fact that the CAPM ignores too many other factors that influence the models predictions, the model is far from perfect. Despite that the CAPM still lead as one of the most studied, taught and adapted pricing models.
Fama and French
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ratio E/P book to market equity and leverage. The main difference between the 1992 and 1993 papers of Fama and French is that their 1992 paper uses cross-section regression, like Fama and MacBeth (1973) and in their 1993 paper they use time-series regression.
The existing CAPM in combination with the above mentioned findings resulted in the creation of the Fama and French three-factor model. Fama and French (1993) construct two new variables to test if size and book-to-market ratios of equity affect the returns, namely SMB and HML. The SMB factor measures the size effect, small firms tend to be riskier than large firms and therefore investors in small firms require a larger risk premium than investors in large firms. The HML factor accounts for the difference in return between growth and value stocks. According to Fama and French value stocks outperform growth stocks due to high book-to-market ratios.
Fama and French conduct their research using monthly stock returns of the NASDAQ, NYSE and Amex for the period 1963-1990. Their first main finding is that average returns are negatively related to size, hence small cap firms denote higher returns than large cap firms. The second finding in their 1993 paper is that average returns are positively related to book-to-market equity, hence higher expected returns for value stocks.
Chung, Johnson and Schill (2006) find that stocks with low price to book values are typically companies that recently had some underperforming results compared to their forecast. This makes that these stocks are temporally less favorable resulting in a lower price. They also find that it is the other way around for companies with high price to book ratios; these stock are in the growth stage and stock prices of these companies are temporarily high. Hence sorting firms on metrics like price to book ratios tend be extremely negative in bad times and extremely positive in good times. Another finding by Chung, Johnson and Schill (2006) is that investors tend to over-forecast past performance of stocks, which leads to overvaluation of growth stocks and undervaluation of value stocks. Resulting in lower returns for growth stocks and higher returns for value stocks at the end of the cycle.
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higher returns (Dimson, Marsh and Staunton, 2004). The reason for this reversal is that small firms lose more when the economy is in a downstate and gain mere when the economy recovers. There are two contradicting explanations for the found value effect, one behavioral and one according to the efficient-market hypothesis. The behavioral explanation is that new information can lead to over- or underreaction in stock prices. Lakonishok, Schleifer and Vishny (1994) state that some investors have the tendency to extrapolate past performance into the future. Another explanation the way investors perceive the value of stocks. Book-to-market ratio can be used as a measure of how much a stock is really worth. Since growth stocks have a low book-to-market ratio and can therefore be described as overvalued. This is the other way around for value stocks, these have high book-to-market ratios and are therefore cheap. Eventually prices will go to the “true” value of the stock, resulting in an outperformance by the value stocks.
According to the efficient-market hypothesis value stocks have poor growth prospects, only few investment opportunities resulting in a higher risk premium. Papers by Daniel and Titman (1997), Griffin and Lemmon (2002) and Lakonishok, Schleifer and Vishny (1994) find that high to-market firms are more risky than low to-market firms because high book-to-market firms face financial distress risk. The possible risk of financial distress makes that the company’s stocks trade at a lower price. Fama and French (1998) and Dimson, Marsh and Staunton (2004) find that the value effect persisted while the size effect diminished.
Momentum
The benefits of the two extra factors of the FFM are acknowledged, but the model has been subjected to further improvement. Jegadeesh and Titman (1993) find that stock returns show short-term continuance, stocks that are performing well will continue to do so in the near future. This is now known as momentum. In 2001 Jegadeesh and Titman did further research on momentum and find that the momentum strategies also hold for the 1990s. In addition Rouwenhorst (1998) provided international evidence of momentum returns by showing that it also holds for non-US markets. Brennan, Chordia and Subrahmanyam (1998) and Grundy and Martin (2001) find that the CAPM and FFM fail to fully capture momentum.
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behavioral theories make that investors process stock information differently than one would expect based on market efficiency theory.
In an attempt to capture momentum, Carhart (1997) adds momentum as a fourth factor to the FFM. The momentum factor is the difference between return of month t on portfolios formed on portfolios formed on last year’s winners and losers. Fama and French (2010) add the momentum factor to their research when they research mutual fund performance. In 2012 Fama and French (2012) add the momentum factor again when they examine the size, value and momentum effect in four different regions (North America, Europe, Japan and Asia Pacific). They find strong momentum returns in three regions, only in Japan they did not find momentum returns. One of their findings is that momentum returns vary with firm size, and adding the momentum factor creates a better fit of the returns.
Avramov and Cordia (2006) find that the four-factor model introduced by Carhart fails to absorb all the momentum. They find that none of the models capture the impact of momentum on the cross-section of individual stock returns, even when returns are risk adjusted by momentum factors. They also state that momentum profits are inconsistent with the asset pricing misspecification that varies with the business cycle. This could point to a systematic risk rather than idiosyncratic source of momentum factors.
U.K. Stock Markets
8 Risk prediction with model betas
Koch and Westheide (2010) study the relationship between the betas of three-factor model and future returns. For 25 portfolios formed on size and book-to-market ratio Koch and Westheide argue that all three factors have significant predictive ability. They find evidence for a systematic relationship between the three-factor model betas and future returns. They state that the use of the three-factor model betas, estimated from historical data, is an appropriate method for forming portfolios. Hence, the realized returns are in accordance to what one would expect. Pettengill, Chang and Hueng (2013) study the relationship between systematic risk and return by forming portfolios based on expected risk measured by the three-factor model and comparing this to the risk and realized return of the portfolios. They find that the three-factor model accurately predicts future returns. Further they find that when applying the same procedure on the CAPM it gives almost similar results. Furthermore they find deficiencies in the three-factor model; portfolios with a low expected risk are predicted to have negative excess returns and the portfolios with high expected risk denote very high expected returns. Because the three-factor model is not a theoretical model, it appears to be an open question why this model should be used to predict research and return.
III. Methodology
Since the introduction in 1993 the FFM has been the dominant model in research studies for risk-adjusted returns. The more empirical nature of the FFM makes it a lot harder to perform the same type of testing than the more theoretical CAPM. The goal of this research is to test whether the FFM is able to account for different risk factors that influence stock returns and how well the FFM can make the risk-return tradeoff for portfolios. To check whether the FFM is truly a better risk prediction model than the CAPM and whether the addition of the fourth factor momentum, is a good addition to the model, the three models will be compared using identical sampling procedures. The FFM returns are estimated using Eq. (1):
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 (𝑅𝑚− 𝑅𝑓) + 𝑠𝑖 𝑆𝑀𝐵 + ℎ𝑖𝐻𝑀𝐿 + 𝜀𝑖 (1) Where Ri –Rf is the value-weighted monthly excess portfolio return; (Rm – Rf) the excess market
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For estimating the CAPM returns Eq. (2) is used:
𝑅𝑖 = 𝑅𝑓 + 𝛽𝑖 (𝑅𝑚− 𝑅𝑓) + 𝜀𝑖 (2) In addition to these two models the same procedures will also be applied on the four-factor model. For this estimation Eq. (3) is used:
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 (𝑅𝑚− 𝑅𝑓) + 𝑠𝑖 𝑆𝑀𝐵 + ℎ𝑖𝐻𝑀𝐿 + 𝑚𝑖𝑀𝑂𝑀 + 𝜀𝑖 (3) Where Ri –Rf,, SMB and HML are the same as in Eq. (1) and MOM is the factor for capturing
momentum.
Explanatory variables
The SMB and HML factors are calculated using the same method as Fama and French (1993). For the SMB factor, each year all stocks are ranked on size measured in June of year t. The median size is then used to split the stocks into two groups; Small (S) and Big (B). The HML factor is calculated by sorting the stocks on their book-to-market (BE/ME) ratio. Instead of two groups, as with the SMB factor, the stocks are now split into three groups based on percentiles. The top 30% is group high (value), middle 40% is the Medium (neutral) group and the bottom 30% is the Low (growth) group. Likewise as Fama and French (1993) negative BE firms are excluded.
These five groups are used to construct six portfolios; (S/L, S/M, S/H, B/L, B/M, B/H), see also Fig. 1 from the website of French.
Figure 11
Independent Portfolio Construction
Median ME
Small Value Big Value 70th BE/ME percentile
Small Neutral Big Neutral 30th BE/ME percentile
Small Growth Big Growth
These six portfolios can be used to construct the explanatory variables; small minus big (SMB) and high minus low (HML). The SMB factor is meant to mimic the risk factor in returns related to size. SMB is calculated by subtracting the simple average of the three big size portfolios (B/L, B/M, and B/H) from the simple average of the three small-stock portfolios (S/L, S/M, and S/H). HML is the difference, calculated each month, between the simple average of the returns of the two high BE/ME portfolios (S/H and B/H) and the simple average of the returns of the
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two low BE/ME portfolios (S/L and B/L). Fama and French (1993) state that difference between the small and big stocks is of portfolios with about the same BE/ME ratios the difference should be free of BE/ME influence, and vice versa the HML factor should be free from size effect. The third factor in the FFM model is the market factor, denoted as Rm-Rf.. Rm is in this research the return on the FTSE ALL Share Index, Rf is the monthly UK Treasury Bill rate (calculated as monthly rate).
The momentum factor, first introduced by Carhart (1997), is now also recognized and used by Fama and French. For the calculation of the momentum factor there are also six value-weighted portfolios formed; two portfolios based on size, and three portfolios based on prior monthly returns. The two size portfolios are formed in a similar fashion as with the SMB and HML factor. The other three portfolios split based on their last month performance, using the same 30th and 70th percentile breakpoints as earlier. These five groups are used to form following six portfolios: (S/H, S/N, S/L, B/H, B/N, B/L). The monthly momentum variable can then be calculated as the average of the big and small high portfolios minus the average of the big and small low portfolio.
Explained variable
The dependent variables consist of 25 excess return portfolios, formed on size and BE/ME ratio. By forming these 25 portfolios on size and BE/ME Fama and French (1993) try to mimic the SMB and HML portfolios and capture common factors in stock return which are related to size and BE/ME.
The 25 dependent portfolios are formed much like the 6 independent variables. The stocks are sorted at June of year t by size and independently by BE/ME. Size is measured at the end of June, BE/ME is measured at the end of December of t-1. The stocks are sorted on five size quintiles and independently on five BE/ME quintiles. For each portfolio I calculate the monthly value-weighted excess returns. These value-weighted excess returns for the period 1986-2012 are the dependent variables in the OLS time-series regressions using Eq. (1).
11 Risk prediction with model betas
The goal of this part of the research is to test the risk return predicting power of the FFM. Pettengill, Chang and Hueng (2013) state that due to the more empirical nature of the FFM it did not receive the same type of testing of its predictive power as the more theoretical CAPM. This section examines if the FFM can create portfolios with a risk-return tradeoff which is consistent with the actual performance of the portfolio. To test whether the FFM can predict an accurate risk-return tradeoff 10 portfolios are formed using the FFM. Where portfolio 1 contains the stocks with the highest expected risk, and thus return, and portfolio 10 the stock with the lowest expected return. In addition the same procedure is also applied to the CAPM and four-factor model.
The methodology in this section will closely follow that of Fama and MacBeth (1973) as they applied it to the CAPM. The idea is to form 10 portfolios in a 3 year estimation period and then monitor the performance of these portfolios in a subsequent one-year period. This section will use the same dataset as described before. Portfolios are formed for the time period of 1986-2012 using overlapping periods; hence there are 24 estimation and holding periods.
Fama and MacBeth (1973) use the betas of stock to form the portfolios, the higher the beta, the higher the expected risk and return. When using the FFM you obtain three betas instead of one. Since there is no theoretical basis on how to combine these three betas in a single risk measure, the estimated betas will be used to estimate the expected returns for each portfolio and for each estimation period. Then the stocks are ranked based on their expected returns instead of the single beta. If the FFM can effectively measure risk, the stocks with the highest expected returns should also have the higher expected risk. The stocks with the highest estimated returns will be in portfolio 1 and the stocks with the lowest expected returns will be in portfolio 10.
For each stock the market, size and value betas are estimated using the average of each explanatory factor over the entire sample period (these are the same as mentioned before). At the end of each estimation period the excess returns are estimated using Eq. (4).
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖+ 𝛽𝑖 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅ + 𝑠𝑖 𝑆𝑀𝐵̅̅̅̅̅̅ + ℎ𝑖𝐻𝑀𝐿̅̅̅̅̅̅̅ + 𝜀𝑖 (4) Where 𝑅𝑖 – 𝑅𝑓, 𝛽𝑖, 𝑠𝑖 & ℎ𝑖 mean the same as described in Eq. (1) and 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅, 𝑆𝑀𝐵̅̅̅̅̅̅ & 𝐻𝑀𝐿̅̅̅̅̅̅̅ are the averages of the factors for the entire sample period. To be included in an estimation period a stock needs to have return data for all 36 months.
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since only systematic risk should be added to the portfolio. But when measuring risk on an ex post basis, which is what this paper does, the standard deviation of monthly returns is an appropriate measure of realized risk. In line with the study by Pettengill, Chang and Hueng (2013) the Spearman rank order coefficient is used to provide a statistical measurement for the ability of creating portfolios which follow the expected risk and return tradeoff. Rank 1 of expected returns goes to portfolio 10, rank 2 is for portfolio 9, etc. Finally portfolio 1 receives a rank of 10. To calculate the spearman rank coefficient these ranks are linked to the realized ranks of actual returns. This is also done for the standard deviation, which is used as measure of actual risk, and the expected risk levels
For comparison, the same tests will also be applied to the CAPM and four-factor model. For the CAPM estimation the ten portfolios are formed on a single market beta, where portfolio 1 consist of the stocks with the highest betas and portfolio 10 consist of the stock with the lowest betas. For the four-factor model the same methodology as with the three-factor model is used, but the estimation or conducted using four betas in Eq. (5).
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅ + 𝑠𝑖 𝑆𝑀𝐵̅̅̅̅̅̅ + ℎ𝑖𝐻𝑀𝐿̅̅̅̅̅̅̅ + 𝑚𝑖𝑀𝑂𝑀̅̅̅̅̅̅̅ + 𝜀𝑖 (5) Where 𝑅𝑖 – 𝑅𝑓, 𝛽𝑖, 𝑠𝑖 & ℎ𝑖 mean the same as described in Eq. (1) and 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅, 𝑆𝑀𝐵̅̅̅̅̅̅ , 𝐻𝑀𝐿̅̅̅̅̅̅̅ & 𝑀𝑂𝑀̅̅̅̅̅̅̅ are the averages of the factors for the entire sample period. See the results section for the outcome of this part of the research.
Data
The dataset used in this paper is constructed using monthly stock returns of all U.K. quoted firms over the years 1986-2012. All data used is gathered from Thomson Reuters Datastream. The dataset is constructed using the FTSE All Share Index constituents list, plus, to overcome the survivorship bias, all the delisted (dead) stock during 1986-2012 period. The following data is collected for all firms: i. RI; return indexes including dividends and capital gains, ii. MV; Market value, measured by the share price multiplied by the number of ordinary shares in issue, iii. Book equity; this variable is constructed by the adding the common equity and deferred taxes. Finally book-to-market ratio (BE/ME) is constructed by dividing the book equity by market value.
Descriptive statistics
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increased significantly over the last 20 years. The number of firms per portfolio confirms the shift from small to big firms. In the 1963-1991 sample of Fama and French (1993) a total of 57% of the firms are in the five small-size quintiles versus 16% in the sample of this paper, for the five big-size quintiles it goes from 8.5% for the Fama and French (1993) dataset to almost 21% in the 1986-2012 dataset. The B/E ratios are all in line with Fama and French (1993) except for the five high-BE/ME portfolios, these are on average 50% higher.
Table 1
Descriptive statistics for 25 portfolios formed on size and book-to-market equity: 1986-2012, 26 yearsa
Size
quintile Low 2 3 Book-to-market equity (BE/ME) quintiles 4 High Low 2 3 4 High Average of annual number of firms in portfolio Average of annual averages of firm size
(in millions) Small 25.7 24.6 32.5 41.0 71.1 7.1 7.8 7.6 7.5 6.7 2 40.8 40.7 46.6 47.3 50.9 23.7 25.2 25.5 26.9 32.6 3 49.0 51.1 53.4 59.9 48.3 84.9 90.6 92.3 89.6 110.5 4 62.5 61.6 51.5 56.6 42.9 384.6 369.1 394.3 396.0 450.0 Big 63.8 63.0 57.1 36.2 28.2 148,219.9 93,962.1 47,239.6 24,465.9 8,493.3
Average of annual percent of market value in portfolio
Average of annual B/E ratios for portfolio Small 0.002 0.002 0.002 0.002 0.002 0.552 1.444 0.786 1.059 2.699 2 0.007 0.008 0.008 0.008 0.010 0.425 0.569 0.855 1.098 1.997 3 0.026 0.028 0.028 0.028 0.034 0.394 0.576 0.792 1.089 1.699 4 0.118 0.114 0.121 0.122 0.138 0.364 0.492 0.738 1.083 2.161 Big 45.604 28.910 14.535 7.528 2.613 0.237 0.518 0.710 1.050 3.066 aFor each year from 2012 the 25 size-BE/ME portfolios are formed as follows. Each year t from
1986-2012 size is measured at the end of June to compute the 5 size quintile breakpoints. Similarly the 5 quintile breakpoints for BE/ME are computed at the end of year t-1.
Summary statistics
Table 2 depicts the summary statistics for respectively the dependent and independent variables of the dataset used. Fama and French (1993) argue that value stocks; companies with high book-to-market ratio, outperform growth stocks; stocks with low book-book-to-market ratios. The means of the 25 portfolios, see Table 2 panel A, show that this assumption still holds; all the high BE/ME portfolios have significant higher average excess monthly returns than the low BE/ME portfolios. The other assumption of Fama French (1993) is that small firms tend to outperform large firms does not hold for this sample. Table 2 shows that the large firms have outperformed the small firms. The negative average SMB factor in Table 2 panel B, reflects the same contradicting finding. This is in line with findings of Dimson, Marsh and Staunton (2004), they find that the SMB effect reversed after it was first noticed and the HML effect has persisted and still exists.
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minus 1.043%, this confirms the fact that small firm effect reversed. The HML effect still exist; the average monthly returns for this factor is 0.991%. See appendix A for the descriptive statistics four-factor model and momentum factor.
Table 2
Panel A: Summary statistics for the monthly dependent and explanatory returns (in percent) Excess returns on 25 stock portfolios formed on ME and BE/ME
Size quintile
Book-to-market equity (BE/ME) quintiles
Low 2 3 4 High Low 2 3 4 High
Mean (in percent) Median (in percent)
Small -2.728 -1.473 -1.036 -0.700 0.812 -0.065 -0.056 -0.033 -0.018 -0.006
2 -0.183 -0.150 0.210 0.504 1.354 -0.031 -0.020 -0.013 -0.011 -0.006 3 0.616 0.713 0.946 0.762 1.532 -0.005 -0.005 -0.004 0.001 -0.006
4 0.926 0.966 0.968 0.763 1.362 0.003 0.006 0.006 0.007 0.014
Big 0.431 0.239 0.600 0.903 1.661 0.000 0.000 0.001 0.002 0.010
Standard deviation (in percent) Autocorrelation
Small 1.460 1.189 0.771 0.848 0.494 0.253 0.251 0.221 0.347 0.249
2 0.781 0.607 0.459 0.421 0.467 0.339 0.199 0.378 0.326 0.330
3 0.375 0.305 0.292 0.202 0.296 0.356 0.283 0.322 0.276 0.267
4 0.258 0.221 0.260 0.257 0.344 0.205 0.190 0.193 0.237 0.220
Big 0.880 0.595 0.677 0.840 0.846 0.113 0.054 0.081 -0.010 0.126
Min (in percent) Max (in percent)
Small -56.503 -14.294 -11.830 -34.642 -9.874 34.980 47.240 33.914 17.984 19.896 2 -14.777 -9.237 -5.527 -7.359 -4.906 43.423 30.381 15.859 14.578 18.861 3 -4.382 -2.354 -2.921 -2.319 -2.188 6.925 8.109 17.519 4.113 7.858 4 -3.858 -4.110 -3.615 -6.813 -3.035 6.121 3.273 3.836 9.703 14.346 Big -31.264 -16.527 -12.437 -22.633 -14.536 35.361 11.818 25.279 24.973 16.958 Skewness Kurtosis Small 4.026 2.200 -1.215 -1.754 4.797 104.533 104.907 211.021 249.693 246.769 2 4.894 7.134 2.372 -2.816 5.535 224.019 156.516 138.525 281.543 201.005 3 4.192 4.995 1.665 2.071 1.996 71.114 96.101 20.376 33.296 37.574 4 4.492 2.503 6.915 0.509 2.553 109.293 54.105 321.180 203.580 76.473 Big 6.840 1.268 -0.197 -1.186 -1.225 353.884 278.012 169.301 141.950 153.747
Panel B:Summary statistics for the monthly explanatory returns
in percent Correlation
Variable Mean Median St.Dev Min Max Skew Kurt Auto RM-RF SMB HML
RM-RF 0.395 0.864 4.588 -27.241 12.962 -0.979 3.865 0.074 1
SMB -1.043 -1.101 2.903 -9.756 11.973 0.581 1.886 0.109 -0.299 1
HML 0.991 1.046 2.919 -15.269 17.552 -0.178 7.803 0.246 -0.042 -0.395 1 Rm is the monthly return on the FTSE All Share Index. Rf is the one-month UK treasury bill rate. SMB is the
return on the portfolio for the factor size and HML is the return for the portfolio of the value factor.
15
IV. Results
This section describes and discusses the results of the estimation of the three models. The first part deals with the results of the regression of the models and the calculated GRS statistic. The second part discusses how well these models can make the risk-return tradeoff for portfolios using the models betas.
Table 3 shows the output of the FFM regression for all 25 portfolios. The intercepts are on average low and close to zero, this suggest that most of the returns are explained by the three factors. There are however eight alphas that are not significant at a 5% level; the four small and four big stocks in BE/ME portfolio 1-4. The market beta has high t-values and all the coefficients are significant. For the SMB coefficients the four big portfolios in BE/ME quintile 1-4 are not significant. The final coefficient, HML, has the poorest fit to the model, low t-values for the two bigger portfolios and nine out of ten portfolios in the 2 and 3 BE/ME quintiles are not significant. -40% -30% -20% -10% 0% 10% 20% 30% 40% 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12
Figure 2
Yearly Excess Market, SMB & HML Returns
16
Table 3
Regressions results of the monthly excess returns and the mimicking returns for size (SMB) and BE/ME (HML)
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 (𝑅𝑚− 𝑅𝑓) + 𝑠𝑖 𝑆𝑀𝐵 + ℎ𝑖𝐻𝑀𝐿 + 𝜀𝑖 Size
quintile
Book-to-market equity (BE/ME) quintiles
Low 2 3 4 High Low 2 3 4 High
α t(α) Small -0.008 -0.004 -0.002 -0.001 0.012 -1.690 -1.178 -0.706 -0.421 5.662 2 0.013 0.010 0.009 0.010 0.016 4.318 3.820 3.852 4.995 7.721 3 0.015 0.013 0.014 0.007 0.013 6.685 6.153 6.975 4.718 6.689 4 0.013 0.009 0.007 0.005 0.008 6.792 5.139 3.774 3.051 3.723 Big 0.003 -0.001 0.004 0.005 0.012 0.763 -0.225 1.018 1.694 4.613 β t(β) Small 1.084 0.989 0.888 0.993 0.932 11.322 13.652 15.560 15.945 20.018 2 1.113 1.049 1.061 1.001 1.106 17.798 18.922 22.084 23.977 24.922 3 1.088 1.112 1.073 1.120 1.083 22.087 24.622 25.846 33.238 25.156 4 1.096 1.103 1.126 0.985 1.118 27.056 30.856 29.524 28.541 23.421 Big 0.786 0.733 0.784 0.848 0.985 8.296 10.316 10.543 13.142 17.811 s t(s) Small 1.683 1.443 1.133 1.280 1.506 10.210 11.536 11.502 11.927 18.817 2 1.286 1.282 1.144 1.181 1.216 11.957 13.447 13.852 16.443 15.929 3 0.689 0.841 0.783 0.643 0.650 8.135 10.830 10.960 11.108 8.790 4 0.176 0.290 0.235 0.201 0.351 2.528 4.715 3.591 3.379 4.273 Big 0.088 -0.040 0.181 0.215 0.210 0.542 -0.325 1.414 1.938 2.207 h t(h) Small -0.634 0.028 -0.026 0.366 0.795 -4.058 -5.007 -7.906 -9.218 -0.771 2 -0.511 -0.220 0.141 0.350 0.614 0.233 -2.436 -1.761 -0.131 -0.078 3 -0.635 -0.130 -0.010 0.264 0.454 -0.277 1.792 -0.147 1.680 1.042 4 -0.609 -0.008 0.105 0.108 0.466 3.600 5.133 4.806 1.909 2.721 Big -0.119 -0.009 0.127 0.287 0.291 10.467 8.474 6.466 5.980 3.230 Adjusted R2 Small 0.450 0.604 0.673 0.744 0.187 GRS 6.003 2 0.459 0.609 0.676 0.756 0.272 p-value 0.000a 3 0.504 0.633 0.687 0.740 0.260 4 0.487 0.670 0.774 0.727 0.357 Big 0.636 0.678 0.662 0.640 0.509
Rm is the monthly return on the FTSE All Share Index. Rf is the one-month UK treasury bill rate. SMB is the
return on the portfolio for the factor size and HML is the return for the portfolio of the value factor.
a This p-value is nonzero but smaller than 0.0005
17
Table 4
Regressions results CAPM
𝑅𝑖= 𝑅𝑓 + 𝛽𝑖 (𝑅𝑚− 𝑅𝑓) + 𝜀𝑖 Size
quintile
Book-to-market equity (BE/ME) quintiles
Low 2 3 4 High Low 2 3 4 High
α t(α) Small -0.030 -0.017 -0.013 -0.010 0.006 -5.840 -4.503 -4.208 -3.027 1.929 2 -0.005 -0.005 -0.001 0.002 0.010 -1.367 -1.479 -0.398 0.725 4.028 3 0.002 0.004 0.006 0.004 0.012 0.838 1.461 2.691 2.236 5.588 4 0.005 0.006 0.005 0.004 0.010 2.416 3.541 3.270 2.673 4.386 Big 0.001 0.000 0.003 0.006 0.013 0.324 -0.148 0.947 2.091 5.278 β t(β) Small 0.785 0.717 0.676 0.746 0.626 6.934 8.610 10.244 10.600 9.848 2 0.883 0.812 0.841 0.769 0.860 11.138 11.608 14.500 14.432 15.421 3 0.974 0.956 0.925 0.991 0.948 16.212 18.347 19.689 26.719 20.892 4 1.079 1.048 1.079 0.945 1.039 23.853 30.076 29.649 28.734 21.967 Big 0.773 0.741 0.747 0.800 0.937 8.697 11.146 10.695 13.086 17.816 Adjusted R2 Small 0.128 0.188 0.247 0.258 0.229 GRS 7.867 2 0.276 0.293 0.393 0.391 0.423 p-value 0.000a 3 0.448 0.510 0.545 0.688 0.574 4 0.637 0.737 0.731 0.719 0.599 Big 0.188 0.276 0.260 0.345 0.495
Rm is the monthly return on the FTSE All Share Index. Rf is the one-month UK treasury bill rate. a This p-value is nonzero but smaller than 0.0005
Table 5 shows the regression results using the four-factor model. The alphas are higher in comparison to the three-factor model and CAPM. Furthermore the GRS statistic is the highest of all three tested models. The adjusted R2 is higher, but overall the model is a worse fit for the data. For this dataset the momentum factor does not give the model a better fit.
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Table 5
Regression results Four-factor Model
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖 + 𝛽𝑖 (𝑅𝑚− 𝑅𝑓) + 𝑠𝑖 𝑆𝑀𝐵 + ℎ𝑖𝐻𝑀𝐿 + 𝑚𝑖𝑀𝑂𝑀 + 𝜀𝑖 Size
quintile
Monthly prior (2-12) quintiles
Low 2 3 4 High Low 2 3 4 High
α t(α) Small 0.001 -0.001 -0.006 0.000 -0.013 0.610 -0.324 -0.905 -0.073 -1.436 2 0.018 0.006 0.010 0.013 0.021 7.073 3.250 5.413 6.104 3.306 3 0.025 0.007 0.008 0.012 0.022 8.002 4.040 5.670 7.777 9.724 4 0.023 0.010 0.006 0.008 0.017 6.761 6.027 4.701 5.416 3.844 Big 0.018 0.007 0.004 0.006 0.011 6.369 5.071 3.087 4.514 4.199 β t(β) Small 1.042 0.809 0.914 0.938 1.250 20.306 13.495 6.499 10.506 6.545 2 1.042 0.809 0.914 0.938 1.250 20.306 13.495 6.499 10.506 6.545 3 1.190 0.984 0.901 0.920 1.197 22.154 24.681 22.426 21.257 9.140 4 1.156 1.070 0.975 0.998 1.234 16.296 30.056 34.277 33.571 13.210 Big 1.023 0.956 0.899 0.842 0.923 17.406 31.171 30.537 32.430 17.196 s t(s) Small 1.667 1.223 1.161 1.190 1.152 19.202 12.061 4.877 7.893 3.566 2 1.342 1.021 1.062 1.044 1.515 14.780 15.136 15.619 14.247 6.841 3 0.801 0.695 0.605 0.683 0.839 7.299 10.984 11.433 12.705 10.621 4 0.054 0.232 0.250 0.269 0.610 0.452 3.850 5.191 5.352 3.860 Big -0.160 -0.018 -0.090 -0.010 -0.134 -1.607 -0.351 -1.798 -0.237 -1.473 h t(h) Small 0.172 0.107 0.463 0.291 0.532 2.041 1.090 1.977 1.975 1.698 2 -0.111 0.163 0.008 0.137 0.130 -1.254 2.483 0.118 1.921 0.606 3 -0.053 0.049 0.095 0.063 -0.220 -0.499 0.796 1.852 1.203 -2.863 4 -0.162 -0.048 0.044 -0.029 -0.035 -1.389 -0.817 0.939 -0.588 -0.228 Big -0.704 -0.381 -0.095 0.071 0.340 -12.667 -13.138 -3.403 2.902 6.689 m t(m) Small -0.304 -0.202 0.238 0.045 1.341 -6.256 -3.559 1.786 0.525 7.423 2 -0.514 -0.177 0.015 0.061 0.805 -10.116 -4.685 0.407 1.480 6.500 3 -0.766 -0.300 -0.091 0.028 0.273 -12.484 -8.482 -3.061 0.941 6.175 4 -0.952 -0.414 -0.155 0.026 0.596 -14.190 -12.284 -5.763 0.926 6.753 Big -0.704 -0.381 -0.095 0.071 0.340 -12.667 -13.138 -3.403 2.902 6.689 Adjusted R2 Small 0.697 0.486 0.126 0.294 0.198 GRS 9.840 2 0.722 0.708 0.670 0.628 0.289 p-value 0.000a 3 0.663 0.746 0.766 0.756 0.683 4 0.656 0.803 0.813 0.793 0.387 Big 0.665 0.824 0.789 0.792 0.553
Rm is the monthly return on the FTSE All Share Index. Rf is the one-month UK treasury bill rate. SMB is the
return on the portfolio for the factor size, HML is the return for the portfolio of the value factor and MOM is the return for momentum portfolio.
19 Risk prediction with model betas
For this part of the research 10 portfolios are formed based on expected return and risk using the betas of the three models. Portfolio 1 contains the stocks with the highest expected return, and thus also the highest risk. Thereafter nine additional portfolios are formed with a declining
11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 51 535254 55 -3 -2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 1 Act ua l R et urns Predicted Returns Figure 3A
Actual vs Predicted returns for the 25 FFM portfolios
11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 51 5253 54 55 -3 -2 -1 0 1 2 0.225 0.275 0.325 0.375 0.425 A ct ua l R et urn s Predicted Returns Figure 3b
Actual vs Predicted returns for the 25 CAPM portfolios
11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 51 52 53 54 55 -2 -1 0 1 2 -2.1 -1.6 -1.1 -0.6 -0.1 0.4 0.9 Act ua l R et urns Predicted Returns Figure 3c
20
amount of expecting return and risk until portfolio 10, which has the lowest expected return and risk. These 10 portfolios are formed for all 24 estimation periods during 1986-2012, hence a total of 240 portfolios. The goal here is to investigate if it is possible to reliable measure risk using the betas of the models. Table 5 shows the predicted and realized excess monthly returns for the 10 portfolios, for both FFM, CAPM and the four-factor models. The Spearman rank order coefficient is used to provide a statistical measurement for the ability of creating portfolios which follow the expected risk and return tradeoff. Rank 1 of expected returns goes to portfolio 10, rank 2 is for portfolio 9, etc. Finally portfolio 1 receives a rank of 10. To calculate the spearman rank coefficient these ranks are linked to the realized ranks of actual returns. This is also done for the standard deviation, which is used as measure of actual risk, and the expected risk levels. When this correlation is positive and significant it would imply that the model is able to predict returns using the betas.
Table 5 panel A shows a highly significant and positive correlation of 0.939 between expected and realized returns. Hence the FFM is able to create portfolios that follow the expected risk-return tradeoff. There is however no evidence for a positive correlation between expected and actual risk; correlation coefficient is negative and not significant. The results for the estimation with the CAPM beta are in panel B of Table 5. There is a positive correlation between expected and actual risk of 0.515, but not significant. There is however a positive and significant correlation between expected and actual risk of 0.661. For the four-factor model the correlations are smaller than with the estimations of the three-factor model, both are however not significant. In conclusion: the FFM does best when estimating future return but the CAPM is a better risk estimator.
21
Table 5
Expected & realized return per portfolio Panel A: Three-factor model
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖+ 𝛽𝑖 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅ + 𝑠𝑖 𝑆𝑀𝐵̅̅̅̅̅̅ + ℎ𝑖𝐻𝑀𝐿̅̅̅̅̅̅̅ + 𝜀𝑖 1 2 3 4 5 6 7 8 9 10 ρ p-value Expected return 0.017 0.009 0.006 0.004 0.002 0.000 -0.003 -0.005 -0.010 -0.020 - - Realized return 0.008 0.007 0.006 0.006 0.005 0.006 0.005 0.005 0.003 0.003 0.939 0.000 Standard deviation 0.124 0.099 0.091 0.088 0.090 0.091 0.096 0.104 0.119 0.151 -0.345 0.328 Panel B: CAPM 𝑅𝑖 – 𝑅𝑓 = 𝛽𝑖 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅ + 𝜀𝑖 Expected return 0.006 0.005 0.004 0.004 0.003 0.003 0.002 0.002 0.001 -0.001 - - Realized return 0.007 0.006 0.005 0.005 0.005 0.007 0.005 0.006 0.004 0.005 0.515 0.128 Standard deviation 0.152 0.120 0.105 0.100 0.098 0.095 0.093 0.092 0.091 0.107 0.661 0.038 Panel C: Four-factor model
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖+ 𝛽𝑖 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅ + 𝑠𝑖 𝑆𝑀𝐵̅̅̅̅̅̅ + ℎ𝑖𝐻𝑀𝐿̅̅̅̅̅̅̅ + 𝑚𝑖𝑀𝑂𝑀̅̅̅̅̅̅̅ + 𝜀𝑖
Expected return 0.019 0.010 0.006 0.004 0.001 -0.002 -0.004 -0.008 -0.013 -0.025 - - Realized return 0.006 0.007 0.005 0.006 0.006 0.005 0.006 0.003 0.005 0.005 0.503 0.138 Standard deviation 0.121 0.096 0.90 0.089 0.089 0.093 0.097 0.107 0.119 0.151 -0.406 0.244 The reported returns is the average of the monthly portfolio returns for the 24 holding periods. Where 𝑅𝑖 – 𝑅𝑓,
𝛽𝑖, 𝑠𝑖 & ℎ𝑖 mean the same as described in Eq. (1) and 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅, 𝑆𝑀𝐵̅̅̅̅̅̅, 𝐻𝑀𝐿̅̅̅̅̅̅̅ & 𝑀𝑂𝑀̅̅̅̅̅̅̅ are the averages of the factors for
the entire sample period. ρ is the spearman rank correlation coefficient.
Fig. 4 plots the average expected return of the ten portfolios against the realized returns. Especially for the three and four-factor model the extreme outliers are moderated when looking at realized returns. For the three-factor models the range of the expected returns is 3.69%, for the CAPM 0.73% and for the four-factor model it is 4.44%. The range for the realized returns is much smaller for all three model these are, respectively, 0.52%, 0.26% and 0.42%. These inconsistencies between expected and realized returns may suggest that these models are inefficient when it comes to realized and estimated betas.
-3.0% -2.0% -1.0% 0.0% 1.0% 2.0% 1 2 3 4 5 6 7 8 9 10 A v er ag e Mo n th ly E x ce ss R etu rn Figure 4a
Expected and realized monthly returns for the 10 portfolios using Three-factor model estimation
22
The lines in Fig. 5 represent the average yearly excess returns on the ten portfolios for the estimation period and the five years after the formation. For all three models the returns for the high expected return portfolios diminish immediately the year after the estimation period. From Fig. 5 it is clear that the high returns on the top portfolios are short-lived, the one-year performance is mostly eliminated after one year. For the portfolios with low expected returns this is exactly the other way around, they increase in the year after the formation period. For the three-factor and four-factor model the return for portfolio ten is even higher than the return of portfolio one. Fig. 5a shows the yearly excess returns using the FFM model. After two years the differences in returns is diminished. After five years portfolio one has the sixth highest returns of all portfolios and portfolio two the fifth. The portfolios with low expected returns, except portfolio nine who does remarkable well, do perform in line with expectations; portfolios eight, nine and ten have the lowest realized average yearly return over the five years.
Fig. 5b plots the yearly excess returns when using the CAPM for estimation of expected returns. Here the return of portfolio one declines for the first year but increases again after year two.
-0.2% 0.0% 0.2% 0.4% 0.6% 0.8% 1 2 3 4 5 6 7 8 9 10 A v er ag e Mo n th ly E x ce ss R etu rn Figure 4b
Expected and realized monthly returns for the 10 portfolios using CAPM estimation
Expected return Realized return
-3.0% -2.0% -1.0% 0.0% 1.0% 2.0% 3.0% 1 2 3 4 5 6 7 8 9 10 A v er ag e Mo n th ly E x ce ss R etu rn Figure 4c
Expected and realized monthly returns for the 10 portfolios using Four-factor model estimation
23
Portfolio one gets the highest average yearly returns in these five years. When using the CAPM for estimation the portfolios are most in line with expectations.
When using the four-factor model for the estimation portfolio ten has the highest average yearly returns over the five year period, and portfolio one the eight highest realized return. See, Fig. 5c. This is complete the other way around than expected. See Appendix B for all yearly returns per portfolio. -5% 0% 5% 10% 15% 20% 25% 30% Formation Period
+1 Year +2Year +3 Year +4 Year +5 Year
A v er ag e Y ea rly E x ce s retu rn s Figure 5a
Post-formation yearly excess returns using Three-factor model estimation Portfolio Portfolio 10 5% 7% 9% 11% 13% 15% Formation Period
+1 Year +2Year +3 Year +4 Year +5 Year
A v er ag e Yea rly E x ce s retu rn s Figure 5b
Post-formation yearly excess returns using CAPM estimation
Portfolio 1 Portfolio 10 -5% 0% 5% 10% 15% 20% 25% 30% Formation Period
+1 Year +2Year +3 Year +4 Year +5 Year
A v er ag e Yea rly E x ce s retu rn s Figure 5c
Post-formation yearly excess returns using Four-factor model estimation
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V. Conclusion
This paper provides a test of the most commonly used models that are, even they are flawed, still in the center of financial literature and courses. Is this righteous or should they find a newer model. The goal of this research is to test whether the FFM, CAPM and four-factor model are able to account for different risk factors that influence stock returns and on how well these model can make the risk-return tradeoff for portfolios using the models betas.
The overall positive returns on the HML portfolio indicates that the value effect persisted. For the SMB portfolio the majority of the yearly returns is negative, meaning that the size effect does not longer hold. This is in line the finding of Dimson, Marsh and Staunton (2004) who argue that small firms lose more when the economy is a downstate and gain more when the economy recovers.
In the test of the CAPM, three-factor and four-factor models the GRS-test rejects the hypotheses that the true intercepts are jointly zero. This suggest that the models do not succeeds in explaining all average returns, the GRS statistics are much higher than the rejection rate, suggesting that it leaves a lot of returns unexplained. This research shows that the three-factor model is indeed a better model for explaining and measuring returns than the CAPM. For the dataset in this paper the addition of a fourth factor, momentum, does not attribute in explaining the returns.
Since the FFM doesn’t have a single measure for risk, as the CAPM does, the relationship between systematic risk and returns are more difficult to study. For the FFM there is a highly significant and positive correlation of 0.939 between expected and realized returns. Hence the FFM is able to create portfolios that follow the expected risk-return tradeoff. There is however no evidence for a positive correlation between expected and actual risk; correlation coefficient is negative and insignificant. For the CAPM the coefficient between expected and actual risk of 0.515, but insignificant. There is a positive and significant correlation between expected and actual risk of 0.661. For the four-factor model the correlations are smaller than with the estimations of the three-factor model, but both are not significant, hence the four-factor model is not an improvement. In conclusion: the FFM does best when estimating future return but the CAPM is a better risk estimator.
25
basis of asset pricing. CAPM offers powerful and intuitively pleasing predictions about the relation between risk and expected return and how to measure risk.
Further research and limitations
26
Appendix A
Panel A: Summary statistics for the monthly dependent and explanatory returns (in percent) Excess returns on 25 stock portfolios formed on ME and prior returns
Size quintile
Monthly prior (2-12) returns
Low 2 3 4 High Low 2 3 4 High
Mean (in percent) Median (in percent)
Small -1.348 -1.149 -1.266 -0.792 -1.327 -1.076 -0.789 -0.704 -0.641 -0.805
2 0.183 -0.110 0.295 0.729 1.425 -1.145 -0.725 -0.513 -0.417 -0.413
3 1.151 0.099 0.574 0.927 1.793 -0.541 -0.384 -0.024 0.220 0.114
4 1.442 0.683 0.625 0.869 1.830 -0.039 0.304 0.463 0.607 0.547
Big 1.369 0.653 0.812 0.948 1.668 0.696 0.372 0.552 0.549 0.695
Standard deviation (in percent) Autocorrelation
Small 27.224 18.789 15.848 19.291 27.670 0.289 0.236 0.247 0.223 0.290
2 22.716 15.069 16.109 13.653 18.147 0.309 0.352 0.310 0.323 0.342 3 19.691 12.238 10.156 10.613 16.766 0.340 0.327 0.301 0.328 0.394
4 18.023 11.679 9.384 9.732 14.459 0.275 0.208 0.217 0.199 0.274
Big 14.007 10.154 13.332 8.933 12.507 0.177 0.156 0.088 0.111 0.049
Min (in percent) Max (in percent)
Small -25.451 -29.898 -37.896 -27.582 -28.895 29.188 28.957 19.766 22.199 52.989 2 -28.338 -21.538 -23.917 -24.215 -31.739 35.793 21.187 22.485 18.803 29.529 3 -30.771 -24.411 -22.510 -23.141 -27.411 57.101 31.208 20.273 14.836 31.325 4 -29.885 -27.368 -26.211 -30.116 -35.176 67.118 33.724 18.091 14.480 36.765 Big -27.131 -23.302 -29.510 -25.794 -35.497 36.220 29.316 17.802 12.401 37.690 Skewness Kurtosis Small 10.201 16.819 4.349 8.507 33.846 1.935 4.235 5.568 2.579 7.760 2 7.359 5.427 21.970 4.932 7.405 3.419 1.897 2.814 3.016 4.683 3 3.131 3.089 1.401 1.804 11.136 8.323 5.027 3.480 2.818 3.581 4 4.537 3.824 0.474 1.063 5.267 10.333 4.857 3.782 5.064 6.965 Big 1.084 0.307 -1.036 0.543 3.351 3.786 4.323 5.801 4.512 9.949 Panel B: Summary statistics for the monthly explanatory returns
in percent Correlation
Variable Mean Median St.Dev Min Max Skew Kurt Auto RM-RF SMB HML
27
Appendix B
Realized yearly returns per portfolio Panel A: Three-factor model
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖+ 𝛽𝑖 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅ + 𝑠𝑖 𝑆𝑀𝐵̅̅̅̅̅̅ + ℎ𝑖𝐻𝑀𝐿̅̅̅̅̅̅̅ + 𝜀𝑖 1 2 3 4 5 6 7 8 9 10 ρ p-value Estimation period 0.263 0.157 0.125 0.109 0.098 0.083 0.068 0.049 0.021 -0.041 1.000 0.000 Year +1 0.114 0.103 0.096 0.075 0.074 0.087 0.088 0.074 0.058 0.063 0.842 0.002 Year +2 0.080 0.093 0.078 0.091 0.095 0.101 0.092 0.088 0.080 0.090 0.030 0.934 Year +3 0.097 0.120 0.106 0.118 0.093 0.098 0.116 0.090 0.097 0.091 0.600 0.067 Year +4 0.117 0.097 0.123 0.113 0.100 0.119 0.095 0.098 0.095 0.108 0.442 0.200 Year +5 0.106 0.109 0.122 0.116 0.104 0.131 0.103 0.095 0.118 0.100 0.358 0.310 Average 5 years 0.103 0.104 0.105 0.103 0.093 0.107 0.099 0.089 0.090 0.090 0.697 0.025 Panel B: CAPM 𝑅𝑖 – 𝑅𝑓 = 𝛽𝑖 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅ + 𝜀𝑖 Estimation period 0.126 0.108 0.098 0.103 0.091 0.092 0.092 0.083 0.084 0.059 0.000 0.000 Year +1 0.091 0.094 0.074 0.078 0.077 0.092 0.080 0.089 0.075 0.080 0.212 0.556 Year +2 0.091 0.076 0.094 0.075 0.094 0.081 0.085 0.094 0.088 0.094 -0.164 0.651 Year +3 0.118 0.113 0.085 0.081 0.119 0.104 0.090 0.114 0.116 0.108 -0.018 0.960 Year +4 0.145 0.103 0.095 0.106 0.115 0.095 0.118 0.096 0.105 0.098 0.224 0.533 Year +5 0.126 0.123 0.108 0.110 0.106 0.127 0.095 0.082 0.113 0.103 0.527 0.117 Average 5 years 0.114 0.102 0.091 0.090 0.102 0.100 0.093 0.095 0.099 0.097 0.273 0.446 Panel C: Four-factor model
𝑅𝑖 – 𝑅𝑓 = 𝛼𝑖+ 𝛽𝑖 𝑅𝑚 − 𝑅𝑓̅̅̅̅̅̅̅̅̅̅̅̅ + 𝑠𝑖 𝑆𝑀𝐵̅̅̅̅̅̅ + ℎ𝑖𝐻𝑀𝐿̅̅̅̅̅̅̅ + 𝑚𝑖𝑀𝑂𝑀̅̅̅̅̅̅̅ + 𝜀𝑖 Estimation period 0.304 0.181 0.142 0.118 0.091 0.073 0.051 0.027 0.004 -0.062 0.000 0.000 Year +1 0.086 0.109 0.074 0.088 0.087 0.075 0.086 0.062 0.079 0.087 0.200 0.580 Year +2 0.064 0.092 0.074 0.099 0.090 0.074 0.091 0.087 0.097 0.110 -0.527 0.117 Year +3 0.083 0.130 0.098 0.104 0.105 0.086 0.117 0.114 0.108 0.087 -0.152 0.676 Year +4 0.110 0.091 0.104 0.103 0.094 0.119 0.120 0.087 0.108 0.146 -0.333 0.347 Year +5 0.120 0.097 0.111 0.108 0.095 0.121 0.115 0.102 0.120 0.122 -0.345 0.328 Average 5 years 0.093 0.103 0.092 0.100 0.094 0.095 0.106 0.090 0.103 0.110 -0.370 0.293 References
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