The return on a stock (R) exhibits the following relationship with the market return (MR). $ R = -1.15 \times MR + 2\%; \; R^2 = 81\% $ Compute the ratio of the standard deviation of stock return to standard deviation of the market return. | Financial Risk Manager Part 1 Quiz

The return on a stock (R) exhibits the following relationship with the market return (MR).

$R = -1.15 \times MR + 2\%; \; R^2 = 81\%$

Compute the ratio of the standard deviation of stock return to standard deviation of the market return.

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Explanation:

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.

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