Explanation

The coefficient of determination ($R^2$) measures the fraction of the total variation in the dependent variable that is explained by the independent variable. The return on R explains approximately 81% of the variation from the return on MR.

The correlation coefficient is given by:

$r = [\text{Sign of estimated slope coefficient}] \sqrt{R^2} = -\sqrt{0.81} = -0.9$

From OLS, the slope coefficient ($b$) is given by:

$$b = \frac{\text{Cov}(MR, R)}{\text{Var}(MR)} = \frac{[\text{Corr}(MR, R) * SD(MR) * SD(R)]}{\text{Var}(MR)} = \text{Corr}(MR, R) * \frac{SD(R)}{SD(MR)}$$ $$-1.15 = -0.9 * \frac{SD(R)}{SD(MR)}$$ $$\frac{SD(R)}{SD(MR)} = \frac{1.15}{0.9} = 1.28$$

Note: Given a regression $Y = a + bX$, the sign of $r$ depends on the sign of the estimated slope coefficient $b$:

• If $b$ is negative, then $r$ takes a negative sign.
• If $b$ is positive, then $r$ takes a positive sign.