To summarize the relationship between the market and Lincoln’s return, a regression model was fit. The fitted regression is begin{equation*} widehat{LINCOLN}=-0.00214+0.973 MARKET. end{equation*} The resulting estimated standard error, s = 0.0696 is lower than the standard deviation of Lincoln’s returns, (s_y)=0.0859. Thus, the regression model explains some of the variability of Lincoln’s returns. Further, the (t)-statistic associated with the slope (b_1) turns out to be (t(b_1))=5.64, which is significantly large. One disappointing aspect is that the statistic (R^2=35.4%) can be interpreted as saying that the market explains only a little over a third of the variability. Thus, even though the market is clearly an important determinant, as evidenced by the high t-statistic, it provides only a partial explanation of the performance of the Lincoln’s returns.
In the context of the market model, we may interpret the standard deviation of the market, (s_x), as non-diversifiable risk. Thus, the risk of a security can be decomposed into two components, the diversifiable component and the market component, which is non-diversifiable. The idea here is that by combining several securities we can create a portfolio of securities that, in most instances, will reduce the riskiness of our holdings when compared with a single security. Again, the rationale for holding a security is that we are compensated through higher expected returns by holding a security with higher riskiness. To quantify the relative riskiness, it is not hard to show that begin{equation} s_y^2 = b_1^2 s_x^2 + s^2 frac{n-2}{n-1}. end{equation}
The riskiness of a security is due to the riskiness due to the market plus the riskiness due to a diversifiable component. Note that the riskiness due to the market component, (s_x^2), is larger for securities with larger slopes. For this reason, investors think of securities with slopes (b_1) greater than one as “aggressive” and slopes less than one as “defensive.”