CAPM. The Fama French three factor model cross section and time series test

Table of Contents

1. Introduction

2. Literature review

3. Data

4. Methodology

5. Results
5.1.1 Times Series Test of the CAPM
5.1.2 Cross Sectional Test of CAPM
5.2 F ama-French Three F actor Model
5.2.1 Times Series Test of Fama-French Three Factor Model
5.2.2 Cross Sectional Test of Fama-French Three Factor Model
5.4 Carhart Four Factor Model with Sentiment Index
5.4.1 Times Series Test of Carhart Four Factor Model with Sentiment Index
5.4.2 Cross Sectional Test of Carhart Four Factor Model with Sentiment Index

6. Conclusion

7. References

8. Appendix
8.1 - Table 1 Time-Series Regression of CAPM
8.2 - Table 2 Cross-Sectional Regression of CAPM
8.3 - Figure 1
8.4 - Table 3 Time Series Regression of FAMA fidi period.
8.5 - Table 4 Cross-Sectional Regression of Fama-French Three Factor Model
8.6 - Table 5 Time Series Regression Carhart full period.

1. Introduction

The САРМ model was developed by Sharpe (1964) and tries to give insight into the relation of risk and return characteristics of assets, in particular how risk adjusted excess returns of securities are influenced by the market. Fama and French (1996) further developed the CAPM to a three-factor model. Their aim was to enhance the explanatory power of the CAPM, thereby including the size (SMB) and book to market (HML) effect to achieve more explanatory insight of what drives returns. Carhart (1997) even included a fourth factor, namely the momentum anomaly (WML) as found out by Jagadeesh and Titman (1993), to further resolve the CAPM pricing error of not fully predicting returns, and add explanatory power. Additionally, we retrieved the sentiment index from Datastream to test a fifth explanatory factor. This research paper tests these four different models based on historical European data from the Kenneth R. French website and 50 European stocks and one European real estate index from Datastream. The structure of the research is closely tied to the set up used by Wang.

The paper continues with a short literature review on the CAPM, the three-factor model, the four-factor model, and the sentiment index. Next, a description of the data and methodology is given. Then first the CAPM is tested, followed by the three-factor model, four-factor model and lastly the sentiment index is included. The results are discussed individually in each section. Finally, we draw an overall conclusion and include some limitations.

2. Literature review

Several studies in the past find mixed results regarding the validity of the CAPM model (Nartea and Ward, 2009). Fama and French (1992) find very little variation of stock returns with the market as predicted by the CAPM and included the book-to-market (HML) and size (SMB) effect. These additional risk factors should help to explain and predict the returns better and produce a model with better explanatory power than the CAPM. The book to market effect was first introduced by Rosenberg et al. (1985), and says that stocks with high book-to-market ratios return a premium. The reason is that strong firms with high earnings tend to have higher stock prices and lower В/M ratios, while firms with high В/M ratios have lower stock prices due to earnings volatility or bad earnings (Fama and French, 1996). The size effect shows that smaller stocks trade at a premium to larger stocks. This is due to the higher riskiness (higher volatility) and lower liquidity. The momentum anomaly describes that stocks with high returns in the past continue to generate high returns (WML). Stambaugh, Yu and Yuan (2011) use the sentiment index as a good proxy of stock market returns which captures investor irrationality, since one can measure the psychological state of investors to a certain degree. This happens because investors are influenced by irrationality (feelings and overall impressions of the market), even though this is in violation with one of the CAPM assumptions that investors are assumed to act rational. The economic sentiment index, is determined by surveying companies, consumers, the construction industry and the retail industry about their economic expectations within the next six months (Eurostat, 2013).

Nartea and Ward (2009) give a very comprehensive survey of the findings of these effects all around the world and find mixed evidence for the different effects in different regions. For example, Mahlin and Veeraraghavan (2004) do not find evidence of the book-to-market effect in Germany, France and the UK, while in New Zealand the effect is strongly pronounced (Narea et ak, 2009). However, Mahlin et al. (2004) find a statistically significant relationship for the small firm effect in France and Germany and the big firm effect in the United Kingdom. Nevertheless, overall it appears that more studies provide evidence for the explanatory power of the size, b/m ratio and momentum risk factor than against it (Narea et ak, 2009).

3. Data

The data for the analysis is European monthly data, retrieved from Kenneth R. French’s website for six portfolio returns (French, 2013). French’s data library provides the European risk factors for the market, the risk free rate, HML, SMB and momentum. All returns are measured in US dollars and are total returns. Therefore, the returns include the price return and the dividend return. The market return is the European value weighted return minus the one-month risk free rate. The data ranges from July 1990 until September 2013 for a total of 278 observations.

The portfolios for the HML and SMB are sorted into two market capitalization classifications and three book-to-market categories at the end of each June (French, 2013). The stocks are once ranked for market size as well as for В/M ratio for the other portfolio construction. Large stocks are the top 90% of market capitalization and small stocks are the bottom 10%. High book to market ratios are above 70th percentile, low book to market ratios are below the 30th percentile and everything in between is considered neutral. The momentum portfolio (WML) is found by using the equal-weighted average for the returns for the two winner portfolios from the large capitalization and low capitalization category. The same was done for the loser portfolio. The classification into winner and losers was made by using the cumulative return for the last 10 months.

The European stock, index and sentiment data are retrieved from Datastream for the same monthly periods using the operator RI to get the total return data. We decided to retrieve single stock data for the Eurostoxx 50, MSCI Europe Real Estate index and the European sentiment indicator.

4. Methodology

To account for time-variability of asset pricing theory, for example due to a crisis, changing tastes, sentiment, mindset or other unforeseen circumstances, we divided the sample into three categories (Wang). The first category uses the data from 1990 until the end of 2002 (150 months), the second time period from January of 2003 until September 2013 (129 months) and the third time period spans the overall sample from January 1990 till September 2013 (279 months). The research of Wang serves as a starting point. The first two analyses from the Wang paper are replicated on European data. Next, we extend the models to test additional risk factors.

5. Results

The Capital Asset Pricing Model (CAPM) gives the relationship of an asset’s risk-adjusted excess return, given its correlation to the market. Therefore, as the formula below shows, the asset’s excess return is a function of the risk free rate and the asset’s sensitivity to the market. Consequently, there exists a relationship that more risky assets have higher volatility and, as a result, a higher expected return.

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5.1.1 Times Series Test of the CAPM

The time-series regression model to test the CAPM, regresses the CAPM with the data outlined in section 3 above. The equation, following this paragraph, is used to specify an assets return over the risk free rate. The right side of the equation shows that alpha gives the assets abnormal return over the return predicted by the beta coefficient above the market. If Jenson’s alpha, aj, is dramatically positive different from zero and significant, then one can conclude that the respective asset was able to generate abnormal returns for its investors above what is expected by the model and given the asset’s correlation to the market. However, with the assumption that the CAPM model holds, the hypothesis is that the beta coefficient sufficiently explains the assets abnormal returns. Therefore, the null hypothesis states that aj = 0. Thus, a two-sided test is performed, and the alternative is that alpha is different from zero. If there is sufficient evidence, meaning that the t-statistic is high enough, the null hypothesis can be rejected and it can be concluded that the CAPM is not able to predict stock returns for this sample and period. The MSCI Europe Real Estate index is included as stock number 51 in the time-series regression results, but in line with the other 50 European stocks, the CAPM is not able to explain the returns very well.

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The regression results are displayed in table 1 in the appendix. As one can see, the CAPM model is not a very good predictor of the stock return. All of the alphas are different from zero, but only two observations are significant at a five percent level. As a result, our output fails to reject the null hypothesis that alpha is equal to zero. Additionally, not a single beta is significant, indicating weak explanatory power of the market factor. Moreover, the explanatory power of the model is not too strong, since the R2 statistic is most of the time very low with an average of 0.0094, while not a single R2 is larger than 0.027, which means that the model only explains at most 2.7% of the variation in the asset return. The MSCI Europe Real Estate index does either exhibit an alpha or a beta significantly different from zero. Moreover, the explanatory power as measured with an R2 of 0.003 is also very low. In conclusion, in all of the observations, the CAPM does not sufficiently explain the average excess returns of the assets.

5.1.2 Cross Sectional Test of CAPM

The cross sectional regression tests if y0is zero, thus no abnormal return, and if У1 is positive. If У1 is positive, beta, the slope coefficient, is positively related to the excess return. This would show the hypothesized relationship between risk and return. The following formula specifies the model:

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Table 2 in the appendix gives the cross sectional results for the different time-periods that are tested. The R2 for the overall period and for the first period from 1990 till 2002 are low in a 0­5% range. The R2 for the period from 2003 till 2013 is much larger at 15%. This means that the cross sectional test comes with little explanatory power. One can see that the intercept is positive and statistically significant different from zero for the second sub period and the overall period, this is different from the time-series test where it was found that there was no abnormal return. Moreover, y 1=0 cannot be rejected, except for the second period. However, У1 is negative for all periods. Figure 1 plots the expected return and beta and fits a regression line. The downward sloping relation is exact the opposite of what we would expect. The higher the risk, the higher the beta, the higher the expected return. Unfortunately, this relationship cannot be observed in our case. As a result, the CAPM is not sufficiently able to explain the returns and we should add other factors next to see if there exists a better model.

5.2 Fama-French Three Factor Model

The three-factor Fama-French model takes, as already outlined in detail in the data sample section, two additional risk factors into account, namely the SMB and HML factors.

[illustration not visible in this excerpt] 5.2.1 Times Series Test of Fama-French Three Factor Model

If the three-factor model of Fama and French is valid, the intercept, namely alpha should be equal to zero and the risk factors, such as the market, SMB and HML should be positively different from zero and significant to explain the returns of an asset.

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Table 3 in the appendix shows the results for the time-series regression for the three-factor model over the whole period.

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