Answer: 1.28

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.

Author: Nikitesh Somanthe