Toyota Motor Corp Case

First of all, we chose Toyota Motor Corp. as the large-cap stock, and American Energy Industries Inc. as the small-cap stock. We then downloaded monthly returns including dividends on these stocks and returns on S&P 500 index for the past five years from database CRSP. We considered S&P 500 index as the proxy of market portfolio. We used monthly returns on three-month treasury bills as proxy of risk free rate, and downloaded the monthly returns for the same period.

We ran the first regression for two stocks using: . We used the least square method to get the results. are chosen to minimize deviations from observed values.

measures systematic risk of stock j, and measures its unique or diversifiable risk relative to S&P 500.

More directly, we ran the second regression using: . In this equation, risk-free rate is the average of last five-year monthly returns on 3-month treasury bills, so it is fixed.

In the first regression, is calculated by multiplying with (1- ), which varies with different betas. However, we use fixed risk free rate as in the second regression, which does not vary with different betas. All the regression output is presented in the following pages.

For the first regression, Toyota Motor's Beta is 0.74, and the Beta's standard deviation is 0.1451. While, for American energy industries' Inc., the Beta is 1.6647, and Beta's standard deviation is 0.3155. To construct a 95% confidence intervals, we know that beta deviated from the mean about 2 standard deviations, which in statistics terms is . From this, we could get intervals for Toyota Motor Corp. as following: . Therefore Beta lies between 0.4498 and 1.0302 with 95% confidence. The intervals for the American energy industries Inc. is . Hence, we are 95% sure that the true beta is between 1.0337 and 2.2957.

In the second regression, the beta and its standard deviation of Toyota Motor Corp. are 0.74 and 0.1451. And the beta and its standard deviation for American Energy Industries Inc. are 1.6647 and 0.3155. From this, we found out that the beta and standard deviation in two regressions are the same. Therefore, we could conclude with 95% confidence that the true beta for Toyota Motor Corp. lies between 0.4498 and 1.0302, and the true beta for American energy industries Inc. is between 1.0337 and 2.2957.

In first regression, the R square for Toyota Motor Corp. and American Energy Industries Inc. are 0.3097 and 0.3243. In the Second regression, The R square for Toyota Motor Corp. is also 0.3097 and for American Energy Industries Inc. is 0.3243. Based on these data, we could say in both regressions that for Toyota Motor Corp., only 30.97% variation in its returns can be explained by the changes in returns on S&P 500, and for American Energy Industries Inc., only 32.43% variation in its returns