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The Excess Returns of the S&p 500

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Question 1

In this section, we regress the excess returns of the S&P 500 on the market risk premium of the S&P 500 between January 1964 and January 1993. The mean returns of the 25 FF portfolios, as well as their betas and alphas, are presented in Tables 1 to 3. All the variables in the following regressions are retrieved from Kenneth R. French’s database using value weighted monthly average returns. The regressions are executed in STATA.

Table 1. Mean Excess Returns

 

Low

2

3

4

High

Small

0.29%

0.74%

0.76%

0.96%

1.09%

2

0.40%

0.67%

0.90%

0.96%

1.07%

3

0.44%

0.74%

0.68%

0.89%

1.01%

4

0.46%

0.38%

0.64%

0.79%

0.91%

Big

0.33%

0.34%

0.35%

0.50%

0.61%

Table 2. CAPM Betas

 

Low

2

3

4

High

Small

1.427

1.253

1.160

1.075

1.108

2

1.436

1.233

1.113

1.041

1.125

3

1.362

1.167

1.035

0.979

1.076

4

1.230

1.136

1.045

0.972

1.091

Big

1.005

0.987

0.868

0.853

0.869

Table 3. Alphas

 

Low

2

3

4

High

Small

-0.300

0.218

0.272

0.516

0.624

2

-0.196

0.153

0.432

0.530

0.603

3

-0.128

0.253

0.250

0.477

0.559

4

-0.057

-0.089

0.207

0.383

0.456

Big

-0.092

-0.067

-0.016

0.147

0.248

Question 2

As presented in Table 1, the mean excess returns tend to grow larger in the north-east direction. More specifically, as firms’ market capitalization decreases and book-to-market ratio increases, their mean excess return grows. This is consistent with the small-firm effect and the book-to-market ratio anomalies that are left unexplained while applying the CAPM.

Question 3

As presented in Figure 1, the mean excess returns of the 25 FF portfolios tend to grow larger as firm size decreases and the book-to-market ratio increases. However, for all observed portfolios, this relationship is not systematic. For instance, within the range of low book-to-market portfolios, there is no clear trend of increasing excess returns, given a decreasing firm size.

[pic 1]   [pic 2]

Figure 1. Mean Excess Returns                         Figure 2. Betas

The model, however, exhibits a more compelling, positive relationship between betas of the 25 FF portfolios, given a decreasing firm size and an increasing book-to-market ratio (as shown in Figure 2). If we imagine the anomaly as a ratio between the excess return and the given beta for a certain portfolio, it becomes clear that the portfolios within the lowest valuation quintile exhibits the lowest trade-off between mean excess return and beta. Therefore, the puzzle lies in the fact that CAPM overestimates betas for firms within the first and second quintile of the book-to-market ratio.

Question 4

The alpha intercepts capture the anomalies that the CAPM model fails to explain when regressing the excess return of the portfolios on the market risk premium. If the p-value of an alpha indicates significance, we fail to reject the null hypothesis and therefore, we have to question the reliability of CAPM.

         Overall, the alphas tend to be significant at higher book-to-market values whereas the CAPM is less impaired by the size effect. This is due to a negative correlation between the variation in average returns and the variation in market betas due to the variation in BE/ME. On the other hand, differences in size of the firm lead to a variation in average returns that is positively correlated to the variation in market betas.

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