1. The famous capital asset pricing model or CAPM is a single factor model that characterizes the portion of an asset’s total return that results from systematic movements in the broader market. It is given by the equation:

r_S = r_RF + β_S × (r_M – r_RF)

where r_S is the total return on the asset (stock) of interest, r_RF is the risk-free rate of return available to investors and r_M is the total return on a well-diversified portfolio of stocks. The difference between the market and risk-free rates of return or (r_M – r_RF) is called the market or equity risk premium. β_S or the asset’s beta value is a measure of the asset’s systematic risk. If greater than one, the stock is viewed as riskier than the market. If less than one, the stock is viewed as less risky than the market. If equal to one, the stock is viewed as having market risk.

The CAPM is estimated using historical returns fitted to the following linear regression model:

(r_S – r_RF) = α_S + β_S × (r_M – r_RF) + ε

where α_S is a measure of any arbitrage opportunity (i.e., the stock’s systematic return, positive or negative. in absence of market movement) and ε is the unsystematic portion of the stock’s excess return, (r_S – r_RF), due to firm- and-or industry-specific risk. In an efficient capital market, there should be no systematic arbitrage opportunity and so α_S or alpha is expected to be zero.

Now, consider the following regression analysis.

This analysis extends the capital asset pricing model (CAPM) for Apple, Inc. Specifically, the model below is a Fama-French three factor model for Apple in which two other variables are added to the regression equation.

The three factor model regresses the excess return on a stock, in this case Apple, on the excess return for the overall market, an excess return (called “small minus big” or SMB) measuring the difference between market returns on small market cap stocks and large market cap stocks, and an excess return (called “high minus low” or HML) measuring the difference between market returns on companies with high book-to-market ratios and companies with low book-to-market ratios.

Recall that an excess return is the difference between an asset’s total return (dividends plus capital gains) and the risk free rate of return. Recall also that market cap is determined by multiplying a firm’s share price by its shares outstanding. The excess market return variable in the CAPM and three factor model provides an estimate of the firm’s beta value which indicates any premium associated with systematic market risk. The SMB variable is included to see if there is any risk premium associated with size (i.e., small market cap stocks generally provide higher returns than large cap stocks but also higher price risk). In periods when small cap stocks outperform large cap stocks, SMB will be a positive number and vice versa. The HML variable is included to see if there is any premium associated with book-to-market risk (i.e., value stocks tend to have higher book-to-market ratios while growth stocks tend to have lower book-to-market ratios). In periods when value stocks outperform growth stocks, HML will be a positive number and vice versa.

Here’s another explanation of the CAPM and the three factor model: http://www.forbes.com/sites/frankarmstrong/2013/05/23/fama-french-three-factor-model/

For this model, Apple’s monthly total returns were developed from data obtained from Yahoo! Finance. The excess market return and the SMB and HML excess returns were obtained from Professor Kenneth R. French’s Data Library website at Dartmouth. The monthly Treasury bill return (i.e., the risk-free return on investment) also was obtained from the French Data Library. The equation is estimated from monthly time series data covering the period from October 1984 through December 2013. The notation is as follows:

RAAPL-RF Apple’s monthly excess return which equals RAAPL minus RF

RAAPL Apple’s monthly total return (dividends plus capital gains adjusted for any splits)

RF Monthly return on the Treasury bill (its annual yield divided by 12)

Mkt-RF Monthly excess return in the overall market which equals Mkt minus RF

Mkt The market’s monthly total return (dividends plus capital gains adjusted for any splits)

SMB The monthly small minus big excess return

HML The monthly high minus low excess return

t Time period t

The formal three factor model is given as follows:

(RAAPL-RF)_t = α + β*(Mkt-RF)_t + β_SMB*SMB_t + β_HML*HML_t + ε_t

From the parameter estimation, we have the following:

Variable - Parameter	Coefficients	Standard Errors	t-Statistics
Intercept - α	1.5237	0.6321	2.4108
(Mkt-RF)t - β	1.1462	0.1450	7.9023
SMBt - βSMB	0.3164	0.2107	1.5015
HMLt - βHML	-0.8347	0.2236	-3.7323

Adjusted R² 0.2557 F-ratio 39.7390

Number of observations 351 Durbin-Watson 1.8941

a) Interpret the model from a financial perspective. That is, what are these four estimated coefficients telling you about Apple’s systematic stock market performance? Does each make theoretical and intuitive sense? Explain your answers. (Note that a non-zero intercept (alpha (α)) denotes the presence of arbitrage opportunities.)

b) Comment briefly on the statistical significance of the intercept and each of the independent variables as indicated by their t-ratios and on the statistical significance of the model as a whole as indicated by the F-ratio. What is the interpretation of R²?

c) The graph below shows the residual plot for the estimated three factor equation. Considering both auto- or serial correlation and heteroskedasticity, what, if any, potential problems do you see?

d) The Durbin-Watson statistic can be used to test for serial autocorrelation as a visual inspection of the residuals may only be indicative, not conclusive. Auto-correlation means that the errors are correlated across time. So, for instance, if there is first-order serial autocorrelation that means that knowing the error in time period n, would give you information about the likely error in time n+1. In 12^th order, serial correlation (for instance in monthly data where there is a yearly cycle such as retail sales), means that the errors of time n and time n+12 (or n-12) are correlated.

This model has 351 observations (n=351) and three independent variables (K=3), four if you include the intercept (alpha) term. At a 5% significance level, the lower and upper boundaries for the Durbin-Watson statistic are approximately 1.8074 and 1.8419, respectively. At a 1% significance level, the lower and upper boundaries for the Durbin-Watson statistic are approximately 1.7353 and 1.7697, respectively.

What can be said about first-order autocorrelation given the Durbin-Watson value of 1.8941 calculated from the three-factor model residuals? (In other words, interpret the Durbin-Watson test using the critical test values given. More info can be found in Chapter 9 of your Defusco text.)

How does this relate to your answer to part c? (Again, in other words, what feature of the graph is the Durbin-Watson test testing for?)

e) Examine the following two tables. The first shows correlations among different lagged residuals, running from zero lags (this period) to 12 lags (12 periods (months) ago). So, for instance, you will see that every lag is perfectly correlated with itself while other lags are both positively and negatively correlated with each other.

Practically speaking, this means that knowing something about the residual (error) five periods ago may give you some information about the residual today. In this particular case that correlation is 0.1190. But is a particular value of residual correlation enough to indicate residual autocorrelation? That depends on the sample size and the degree of confidence we want to achieve.

The second table gives the associated t-statistics for each value in the residual correlation matrix (except for correlations with themselves). It also gives critical t-values for several levels of significance.

Please use the following residual correlations analysis of the three factor model residuals to make a statement (determination) of whether or not first- and higher-order autocorrelation is present? How do the statistics below relate to the visual information given in the graph from part c?

f) Please examine the model computed below. In this model we are using the three factors of the three-factor model, the squares of each of those factors, the interaction terms between those factors, and an intercept to explain excess returns of Apple stock. Critical t-statistics and f-statistics are also provided.

Please explain:

1) In a general sense, what does the F-test test for? What do you conclude from the F-statistic and critical F-values given below.

2) In general what do t-tests test for? What do you conclude from the t-statistics and critical t-values given below?

3) Is the relationship between the three factors of the three factor model and the stock’s returns linear or non-linear? Please explain

4) Finally, assume you had concluded that the model is non-linear (whether you actually did or not). How does this relate to the presence or absence of heteroskedasticity? Please explain.

g) Compared with the three factor model, the Capital Asset Pricing Model (CAPM) for Apple, only uses and intercept (known as alpha (α )) and the market risk premium (Mkt-RF)_t – Beta (β) to try to explain movements in Apple’s stock price. See the explanation at the top of this document for more information.

Running this regression model results in a beta (β) value of 1.3596, an R² of 0.2106 and an F-ratio of 93.1237. 1) Has there been any appreciable gain in predictive power with the three factor model? (In other words, did the three factor model do a better job than CAPM of predicting Apple’s stock price?)

2) Conduct a t-test on the null hypothesis that beta (β) truly equals 1.3596 given the above estimate of 1.1462 and its standard error of 0.1450. (This is basically asking whether or not the Beta estimated by each model are statistically different from one another.)

3) Finally, comment on the CAPM intercept term of 1.2041 which is statistically different from zero. What are the financial or practical implications for investors?

Variable - Parameter	Coefficients	Standard Errors	t-Statistics
Intercept – α	1.2041	0.6432	1.8720
(Mkt-RF)t – β	1.3596	0.1409	9.6501

h) For January 2014, the actual value of (Mkt-RF) was minus 3.26% (-3.26), SMB was plus 0.85% (+0.85) and HML was minus 1.83% (-1.83). Given the three factor regression output at the beginning of this document and the actual January values for the variables here, what does the model predict for Apple’s excess return (RAAPL-RF) in January? How does that compare to the actual value for (RAAPL-RF) which was minus 10.77% (-10.77) for January 2014? Is the difference statistically significant?