It is standard in the literature to use ten portfolios when sorting (e.g., Fama & French, 1996;
Leong et al., 2009), and so we chose to use that as our benchmark. However, it would be interesting to see how varying the number of portfolios affects our results. We therefore also look at the results from using 5 and 15 portfolios.
8.3.1 5 portfolios
First, we look at how the results change when we only sort companies in 5 portfolios. The monotone tendencies are apparent, and the resulting long-short strategy gives an annual return of about 6.18% as shown in Table 5, with a higher risk-adjusted semi-annual return than in the 10 portfolio case, shown in Table 23. First, see that the monthly return generated by the (5-1) spread portfolio is lower than the (10-1), however with lower standard deviation as well.
This results in a return which, even though lower, has a higher Sharpe-ratio. See Table 20 and Table 23 for complete results.
5 Portfolio Returns, Value-Weighted
Portfolio 1 2 3 4 5 (5-1)
Mean 0.15 0.37 0.41 0.37 0.66** 0.50***
t-stat 0.455 1.234 1.322 1.150 2.064 2.629
t-stat* 0.431 1.286 1.234 1.054 2.179 2.870
SE* 0.004 0.003 0.003 0.004 0.003 0.002
Skew -2.59 -1.59 -0.56 -1.74 -1.49 0.17
Kurt 18.73 11.21 9.83 10.19 12.26 14.20
Companies 387 405 412 411 398
Average market cap $26.56 $25.68 $27.66 $27.31 $25.41
N 247
Mean and standard deviation of returns in percentage.
Average market cap in millions.
We calculate standard errors using a constant regression. *The standard errors are heteroskedasticity and auto-correlastion robust (HAC) up to 6 lags.
Table 5: Value-Weighted with News - Monthly.
8.3.2 15 portfolios
Here we will look at the case where we increase the number of portfolios used in the sorting to 15. Statistical evidence on monthly frequency as well as performance on semi-annual frequency, are reported in Table 21 and Table 24 respectively. The results indicate that the value-weighted spread portfolio (15-1) performs much worse than the (10-1) and (5-1), with an annual return of only 3.79% as shown in Table 15. Even though there is a tendency of the positive relationship between the similarity score and return, the increased number of portfolios appears to have introduced significant noise, see Figure 22.
9 Fama-French Multifactor Models
Now that we have established that we can generate returns by sorting portfolios according to a similarity score we want to check whether these returns are abnormal, that is, if it is not accounted for by well-known risk factors. We will test this by running both the Fama-French three factor model, as well as the Fama-French five-factor model. The Fama-French three-factor (Fama & French, 1993) in all its simplicity is an extension of the CAPM, that aims to describe stock returns through three factors. The first is the market risk (MKT), which is the difference between the expected return of the market and the risk-free rate. In other words, it is the excess return required as compensation by the investor for the additional volatility of returns above the risk-free rate. The second is the outperformance of companies with a small market capitalization relative to companies with a large market capitalization (SMB). The justification of SMB is that in the long-term, companies that have a small market capitalization is more likely to experience higher returns than companies with large market capitalization. The third factor is the outperformance of high book-to-market companies versus low book-to-market companies (HML). The factor rationalization of the HML is that value companies, i.e., high book-to-market ratio companies tend to have higher returns than growth companies, i.e., low book-to-market ratio companies.
The Fama-French three-factor model was further extended by Fama and French in 2015 when they introduced the Fama-French five-factor model. The Fama-French five-factor model proposed two new factors to the model. The fourth factor they proposed is the difference between returns on diversified portfolios of stocks with robust and weak profitability (RMW), and the difference between the returns on diversified portfolios of the stocks of low and high investment firms, which
they refer to as conservative and aggressive (CMA).
According to the Fama-French setup, abnormal returns are indicated by an intercept which is statistically significantly different from zero. We will report all estimates in separate tables, and use a significance levelα= 5% when referring to statistical significance.
9.1 Fama-French Three-Factor Model
We use data on the three factors and the risk-free rate from Kenneth French’s data library2.The Fama-French 3 factor model is specified as
Ri=α+biRmt +SiSM Bt+hiHM Lt,
where Rm is the market excess return, SM B is the small minus big factor, andHM Lt is the high minus low factor. Finally,Ri is the excess return of asset i and will be the excess return of the 5 and 10 portfolios long-short strategy. First, we run the regression on the 10 portfolio case, and then on the 5 portfolio case.
9.1.1 10 portfolios We specify the model as
R10=α+biRtm+SiSM Bt+hiHM Lt,
where R10 is the excess return of the 10 portfolios long-short strategy. To calculate the excess return, we subtract the risk-free rate we got from Kenneth French’s web site from the returns we calculated earlier. The results are summarized in Table 6. As stated in Table 6, the three-factor estimates for the 10 portfolio long-short strategy is not jointly statistically different from zero. We notice that the intercept is statistically different from zero, which implies that we can generate returns in excess of what the asset pricing model would imply.
9.1.2 5 portfolios We specify the model as
R5=α+biRtm+SiSM Bt+hiHM Lt,
where R is the excess return of the 5 portfolios long-short strategy. To calculate the excess return, we subtract the risk-free rate we got from Kenneth French’s web site from the returns
2https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
we calculated earlier. The results are summarized in Table 6. As stated in Table 6, the three-factor estimates for the 5 portfolio long-short strategy is jointly significantly different from zero.
Moreover, as the intercept is statistically different from zero, it implies that we are also able to generate abnormal returns with the 5 portfolios long-short strategy.
Table 6: Fama-French Three-Factor - monthly frequency 5 Portfolios
Standard errors are autocorrelation and heteroskedasticity robust (HAC)