LO 64.5: Apply a factor regression to construct a benchmark with multiple factors, measure a portfolio s sensitivity to those factors, and measure alpha against that benchmark.
Consider the CAPM formula, where E(R) is the expected return for asset i for a given level of beta exposure, and E(RM) is the expected return on the market:
E(Ri) = RF+(3[E(RM) – R F
If an investment has a beta of 1.3, then the following formulas demonstrate the algebraic evolution of this expression:
E(Ri) Rp +1.3[E(Rm) Rp] E(Rj) = Rp +1.3E(RM) 1.3(Rp) E(Ri) = 0.3Rp + 1.3E(RM)
In this example, the expected return on a $ 1 investment in asset i should be equal to a portfolio with a short position in the risk-free rate of $0.30 and a long position in the market of $1.30. Any return earned in excess of this unique blend will exceed our expectations and is, therefore, considered to be alpha. Using regression, the alpha is approximated as:
Ri,t Rf o l + (3(RM Rf ) + ijt
This exact process was conducted on Berkshire Hathaway stock over the period of January 1990 to May 2012 relative to S&P 500 Index. The results are shown in Figure 1. The monthly alpha coefficient is statistically significant at a 95% confidence level due to the absolute value of the /-statistic being greater than 2.0. Most regressions do not produce a statistically significant alpha.
2018 Kaplan, Inc.
Page 35
Topic 64 Cross Reference to GARP Assigned Reading – Ang, Chapter 10
Figure 1: Regression of Excess Returns
Alpha Beta Adjusted R2
Coefficient 0.72% 0.51 0.14
t-Statistic
2.02 6.51
This regression implies the following CAPM equation:
Rg 0.49Rp +0.51Rm
According to these regression results, a customized benchmark of 49% in the risk-free asset and 51% in the market would produce an expected alpha of 0.72% per month for Berkshire Hathaway. That is 8.6% (0.72% x 12) of annualized expected alpha! Since alpha is the excess return above the actual return, Ri5 you can think of alpha using the following formula:
a = R; – [0.49R f + 0.51E(RM)]
Professors Note: For Berkshire, it is important to note that their market capitalization has grown from less than $10 billion in the early 1990s to over $220 billion in 2012. In his Annual Letter to Shareholders for 2010, Warren Buffet told shareholders that the bountiful years, we want to emphasize, will never return. The huge sums of
capital we currently manage eliminate any chance o f exceptional performance.2 Thus, Berkshire Hathaway has acknowledged the law o f declining marginal returns due to scale.
In 1993, Eugene Fama and Kenneth French extended the traditional CAPM-based regression to include additional factors. They controlled for the size effect (small companies tend to outperform large companies) and for the value/growth effect (value stocks tend to perform better than growth stocks). They formally labeled the size premium as SMB, which stands for small minus big (the return on small stocks minus the return on big stocks), and they represented the value premium with HML, which stands for high minus low (high book-to-market stocks minus low book-to-market stocks). The factors for SMB and HML are long-short factors. The small minus big factor can be visualized as:
SMB = $ 1 in small caps (long position) – $ 1 in large caps (short position)
In a similar manner, we can visualize high minus low as:
HML = $ 1 in value stocks (long position) $ 1 in growth stocks (short position)
2. Berkshire Hathaway Annual Letter to Shareholders, 2010.
Page 36
2018 Kaplan, Inc.
Topic 64 Cross Reference to GARP Assigned Reading – Ang, Chapter 10
The Fama and French three-factor model is constructed as follows:
Ri – RF – a + (3i)MKX x (Rm Rp) + Pi,SMB x (SMB) + Pi5HML x (HML)
The SMB beta will be positive if there is co-movement with small stocks, and it will be negative if there is co-movement with large stocks. If a given asset does not co-move with either small or large companies (i.e., a medium company focus), then its beta coefficient will be zero. Likewise, the HML beta will be positive if the assets have a value focus, and it will be negative if the assets have a growth focus. Applying the Fama-French model to Berkshire Hathaway over the time period of January 1990-May 2012 yields the results displayed in Figure 2.
Figure 2: Fama-French Three-Factor Model Results
Alpha (a) Market beta ((3i MKT) SMB beta ((3i SMB) HML beta (P^h m i) Adjusted R2
Coefficient 0.65% 0.67 -0.50 0.38 0.27
t-Statistic
1.96 8.94 -4.92 3.52
The results in Figure 2 show several interesting aspects. First, the alpha declined slightly but is still very high. Second, the market beta rose from 0.51 to 0.67. Third, the SMB beta is negative, which suggests a large company bias. Fourth, the HML beta is positive, which suggests a value focus for the fund. The adjusted R2 also rose from 0.14 to 0.27, which suggests that SMB and HML do add value. Based on the results, the custom benchmark implied by the Fama-French three-factor model for Berkshire Hathaway is shown as follows:
Rg = 0.33(T-bills) + 0.67 x (market portfolio)
0.5(small caps) + 0.5(large caps)
+ 0.38(value stocks) 0.38(growth stocks)
All of the factor weights in this formula sum to 1.0, but adding the SMB and HML factors add explanatory ability to the regression equation. A test could also be added to account for the momentum effect, which is the theory that upward trending stocks will continue their upward movement while downward moving stocks will continue their downward trend. Thus, a fourth factor can be added to the Fama-French model. This fourth factor could be labeled as UMD, which stands for up minus down (upward trending stocks minus downward trending stocks). A positive UMD beta would suggest a focus on upward trending stocks, while a negative UMD beta would suggest a focus on downward trending stocks. As with the SMB and HML betas, a beta of zero suggests no relationship. Figure 3 shows the UMD factor added to the previous results. Using this data, it can be discerned that Berkshire Hathaway does not have exposure to momentum investing.
2018 Kaplan, Inc.
Page 37
Topic 64 Cross Reference to GARP Assigned Reading – Ang, Chapter 10
Figure 3: Fama-French Three-Factor Model Results With UMD Factor
Alpha (a) Market beta (Pi MKT) SMB beta (0i SMB) HML beta (Pi HML) UMD beta (Pi UMD) Adjusted R2
Coefficient 0.68% 0.66 -0.50 0.36 -0.04 0.27
t-Statistic
2.05 8.26 -4.86 3.33
0.66
One core challenge with using the Fama-French model is replication of indices. Fama and French have created an SMB index and an HML index to increase explanatory power, but there is no way to directly trade an SMB or HML portfolio. These indices are conceptual and not directly tradeable. It is important to include only tradeable factors because the factors chosen will greatly influence the calculated alpha.
M e a s u r e m e n t o f Tim e -Va r y in g Fa c t o r s
Ken’s Python in Finance | Quantitative Analysis | Risk Management | Project Management | Photographer | Designer