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: rS = rRF + ?S � (rM � rRF) where rS is the total return on the asset (stock) of interest, rRF is the risk-free rate of return available to investors and rM is the total return on a well-diversified portfolio of stocks. The difference between the market and risk-free rates of return or (rM � rRF) is called the market or equity risk premium. ?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: (rS � rRF) = ?S + ?S � (rM � rRF) + ? where ?S is a measure of any arbitrage opportunity (i.e., the stock�s systematic return, positive or negative. in absence of market movement) and ? is the unsystematic portion of the stock�s excess return, (rS � rRF), due to firm- and-or industry-specific risk. In an efficient capital market, there should be no systematic arbitrage opportunity and so ?S or alpha is expected to be zero. Now, consider the following regression analysis. This analysis extends the 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. For this model, Apple�s monthly total returns and those for the 13-week Treasury Bill (i.e., the risk free investment) were developed from data obtained from Yahoo! Finance. The excess market return and the SMB and HML excess returns were obtained from Professor French�s Data Library website at Dartmouth. The equation is estimated from monthly time series data covering the period from October 1984 through July 2012. The notation is as follows: XRAAPL Apple�s monthly excess return which equals RAAPL minus RTBILL RAAPL Apple�s monthly total return (dividends plus capital gains adjusted for any splits) RTBILL Monthly return on the 13-week Treasury bill (annual yield divided by 12) XRMKT Monthly excess return in the overall market which equals RMKT minus RTBILL RMKT 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: XRAAPLt = ? + ?*XRMKTt + ?SMB*SMBt + ?HML*HMLt + ?t From the parameter estimation, we have the following: Variable - Parameter Coefficients Standard Errors t-Statistics Intercept - ? 1.7953 0.6557 2.7379 XRMKTt - ? 1.2689 0.1471 8.6230 SMBt - ?SMB 0.4674 0.2283 2.0474 HMLt - ?HML -0.4300 0.1816 -2.3679 Adjusted R2 0.2376 F-ratio 35.6016 Number of observations 334 Durbin-Watson 1.9168 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 or alpha 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 the adjusted R2? c) The graph below shows the residual plot for the estimated equation. What, if any, potential problems do you see? d) The model has 334 observations (n=334) and 3 independent variables (K=3). Even though the number of observations exceeds those reported in most Durbin-Watson tables, what can be said about autocorrelation given the value of 1.9168 calculated from the model residuals? How does this relate to your answer to part c? e) What does the following information from a residual correlations analysis tell you about autocorrelation? How does this relate to your answer to part c? f) An F-ratio of 2.36 is determined for a model in which the squared residuals from the three factor model above are regressed on the independent variables, the squares of the independent variables and the cross-products of the independent variables. This model is a full White test with coefficient details shown below. What do these tell you and how does this relate to your answer to part c? g) The CAPM for Apple, which excludes the SMB and HML variables, has a beta (?) value of 1.3835, an adjusted R2 of 0.2152 and an F-ratio of 92.3279. Has there been any appreciable gain in predictive power with this three factor model? Conduct a t-test on the null hypothesis that beta (?) truly equals 1.3835 given the above estimate of 1.2689 and its standard error of 0.1471. Also, comment on the intercept term of 1.7167 and its implications for investors given its statistical significance. Variable - Parameter Coefficients Standard Errors t-Statistics Intercept - ? 1.7167 0.6642 2.5845 XRMKTt - ? 1.3835 0.1440 9.6087 h) For August 2012, the actual value of XRMKT is 2.58, SMB is 0.50 and HML is 0.79. The actual value for XRAAPL is 9.38 for August 2012. Given the three factor model above, what is the predicted value of XRAAPL for August 2012 in comparison to its actual value?