ECO4323

TESTING CAPM — Project Report Instruction

2024 Fall

1 Logistics

Congratulations! You have finished two midterm tests and all the quizzes, i.e., most of the coursework. This project report is the last piece for your evaluation. You have one week to finish the report. The deadline for submission is Dec 12, at 10:30 exactly. Be sure to submit your report on time. As we have agreed, late submissions will be penalized.

Treat this take-home project as an essay with a bit of guidance. Try to solve the problem using the concepts and methodologies you learned in class, and present your work as much as you can. Data is provided to you, and detailed introductions with regard to how to understand and handle the data are provided as well. Read the instructions very carefully before you proceed. Discussions between students are encouraged, but individual report writing is required. Copy-and-paste work will be assigned zero points.

2 Exercise

2.1 Estimate Betas

Financial economists and financial market participants typically estimate CAPM betas for individual stocks using 5 years of monthly data. The 5-year data sample is long enough to provide statistically precise estimates. But, by going back only 5 years, these analysts avoid using data from the more distant past that may not be relevant for predicting stock returns in the future.

We will use monthly data and annual data in this exercise, both available on Moodle. In this section, we use only monthly data. The monthly data set mentioned above contains data on monthly returns running from January 2019 through December 2023 on the market as a whole and on six individual stocks: AT&T, Exxon-Mobil, JetBlue, Nvidia, PepsiCo, and Tesla.

Question 1. Use these data to estimate the CAPM betas for each of the six individual stocks.

There are at least three ways of doing this. Choose whichever is easiest or most convenient for you. One way is to load the data into a statistics or econometrics software package like Stata or R, and regress each individual stock return on a constant and the market return. The slope coefficient from this regression equals the CAPM beta: the covariance between the dependent variable (the individual stock return) and the independent variable (the market return) divided by the variance of the independent variable (the market return).

It is also possible to estimate the slope coefficient from the regression described above using Google Sheets. To do this, download the Excel spreadsheet and open it instead with Google Sheets. Then, to estimate the beta for AT&T’s stock, use the command “=slope(C12:C71,B12:B71)”.

For the other five stocks, keep the range of the independent variable B12:B71 as above but change the range of the dependent variable to match the columns containing data for the other stock returns.

A third way is to use Excel, Google Sheets, or some other computer program to calculate the covariance between the individual stock return and the market return and the variance of the market return, then to divide the covariance by the variance. Now you know how to estimate CAPM betas!

2.2 CAPM Betas and Expected Returns

Question 2. Use the CAPM betas that you estimated above to determine the expected return on each of the six individual stocks.

To do this, you also need to estimate the risk-free rate and the expected return on the market portfolio. For this task, it is useful to find data on a slightly longer sample of data since stock returns, in particular, have been unusually high over the past five years alone. The annual data set mentioned above contains data on annual returns, running all the way back to 1927 and extending through 2023. This period includes the Great Depression of 1929 through 1933 and the era of very high inflation during the 1970s. Therefore, it may be too long to reliably predict stock returns in the near future.

Instead, let’s use the annual data but focus on just the 24 years from 2000 through 2023. Over that period, the average return on the stock market has been 8.96 percent – approximately 9 percent – which suggests setting ❊[r˜M] = 9 in the CAPM formula.

Meanwhile, although the Federal Reserve is presently targeting interest rates in a range between 5.25 and 5.50 percent, most financial market participants expect that the Fed will begin reducing interest rates later this year. Based on these considerations, let’s take rf = 5 as our best estimate for the risk-free rate in the CAPM formula.

Now, you are ready to compute the CAPM estimates of the expected return on each of the six stocks.

2.3 Value Stocks and the CAPM

Many successful investors have learned that it is possible to “beat the market,” that is, earn average returns above those provided by the market as a whole, by investing in “value stocks,” that is, shares in old-fashioned companies that most other investors ignore and whose prices are beaten down and by avoiding “growth stocks,” that is, shares in newer and more popular companies whose prices have already been bid up. Warren Buffett, most famously, made his fortune partly through “value investing,” but many others have profited from the strategy as well.

The superior performance of value stocks over growth stocks can be quantified using the “HML” measure developed by Eugene Fama and Kenneth French and described in more detail below. According to this measure, value stocks provided annual returns that were, on average, 5.9 percentage points higher than returns on growth stocks over a long period extending from 1927 through 2006. Interestingly, growth stocks have done much better recently: from 2007 through 2023, they have returned, on average, 3.4 percentage points more than value stocks. Even after accounting for this recent reversal, over the full period from 1927 through 2023, the annual return on value stocks remains 4.3 percentage points higher, on average, than the return on growth stocks.

Financial economists will always be quick to point out that higher average or expected returns on value compared to growth stocks do not necessarily violate the implications of the CAPM. Suppose, in particular, that value stocks have higher CAPM betas than growth stocks. Then, the CAPM predicts that value stocks will have higher expected returns as well. The interpretation of Warren Buffett’s success would then be: he earned higher returns, but only because he was willing to take on more aggregate risk.

It turns out, though, that two famous research articles, one by Barr Rosenberg, Kenneth Reid, and Ronald Lanstein (“Persuasive Evidence of Market Inefficiency,” Journal of Portfolio Management, Spring 1985) and the other by Eugene Fama and Kenneth French (“Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics, February 1993), find that higher average returns on value stocks compared to growth do violate the CAPM. That is, value stocks offer higher average returns than growth stocks, even after accounting for differences in their CAPM betas.

By answering this question, you will see whether or not the earlier results presented by Rosenberg, Reid, and Lanstein and by Fama and French continue to hold in the longer set of annual data, running from 1927 through 2023, mentioned above. This data set runs well beyond those used in the original studies. This exercise is both interesting and important because, as noted above, value investing strategies have performed quite poorly, in recent years, while growth stocks like Apple, Facebook, and Nvidia have provided far superior returns. If you include this recent data in your sample, will it still appear that value stocks outperform. growth stocks on average? Are the differences in expected returns still larger than what is predicted by the CAPM? And are the deviations from the CAPM’s predictions still “statistically significant?” These are the broader issues you will address here.

Rosenberg, Reid, and Lanstein and Fama and French distinguish value from growth stocks by comparing each company’s “book value” to its “market value.” Book value measures the accounting value of the firm’s assets per share. Market value is just the market price of each share of stock. According to Rosenberg and Fama and French, value stocks are those with high book compared to market values, and growth stocks are those with low book-to-market values.

Let r˜H be the random return on a portfolio of high book-to-market value stocks, and let r˜L be the random return on a portfolio of low book-to-market growth stocks. As usual, let r˜M be the random return on the market portfolio, and let rf be the risk-free rate. Then, as we know, the CAPM implies

E[r˜H] = rf + βH(E[r˜M] − rf )                      (1-1)

E[r˜L] = rf + βL(E[r˜M] − rf )                       (1-2)

where βH and βL are the CAPM betas on the portfolios of value and growth stocks.

To concentrate specifically on the returns from a value investing strategy, Fama and French define an HML (“high-minus-low”) portfolio that takes long positions in value stocks and short positions of equal value in growth stocks. The random return on this HML portfolio is therefore

r˜HML = r˜H − r˜L                                    (1-3)

and, according to the CAPM

E[r˜HML] = (βH − βL)(E[r˜M] − rf )                              (1-4)

This implication of (1-4) will form. the basis of our statistical test of the CAPM and, specifically, of the continued profitability of the value investing strategy embodied in Fama and French’s HML portfolio.

Consider, in particular, a regression of the return on the HML portfolio on a constant and the difference between the market return and the risk-free rate:

r˜HML,t = α + β(r˜M,t − rf,t) + ϵt                               (1-5)

We can see that the slope coefficient β from the regression equation (1-5) provides an estimate of the HML portfolio’s CAPM beta. More importantly for our purposes, (1-4) implies that according to the CAPM, the intercept term α in the regression equation (1-5) should equal zero. We can test the CAPM’s implications, therefore, by seeing whether the estimated value of α in (1-5) is statistically different from zero. The annual data set mentioned above contains data on the HML return rHML,t and the excess return on the market that you can use to run this statistical test. Since, for this question, we are interested in interpreting the past as opposed to predicting the future, it makes sense to use the full range of annual data running from 1927 through 2023.

Question 3. Use these data to estimate the regression in (1-5).

Then, if you are interested or have learned a bit of econometrics, think about the following optional questions.

Question 4 (optional). Use the t-statistic to test the null hypothesis, implied by the CAPM, that the intercept is equal to zero. Discuss your findings.

Question 5 (optional). Check whether your answer to the above question is robust or not by restricting the sample to different time periods. Discuss your findings.