Financial Risk Manager Part 1

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An analyst is trying to establish the relationship between the return on stock X(Rx) and the return on stock S(Rs). Stock X is listed on the Bombay Stock Exchange (BSE). The analyst has assumed a linear relationship as follows.

R_x = a + b \times R_s + \epsilon_t

Furthermore, the analyst has gathered the following historical data.

Expected return on stock X	15%
Expected return on S	10%
Standard deviation of return on stock X	20%
Standard deviation of return on stock S	15%
Correlation between returns on stock X and S	0.3

Which of the following is the correct model that can be deduced using the ordinary least squares technique?_

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TTanishq

$E(R_x) = 0.40 + 0.40 \times E(R_s)$

$E(R_x) = 0.11 + 0.40 \times E(R_s)$

$E(R_x) = 0.40 + 0.11 \times E(R_s)$

None of the above

Explanation:

Explanation

For the linear regression model:

R_x = a + b \times R_s + \epsilon_t

The expected value equation becomes:

E(R_x) = a + b \times E(R_s)

To find the coefficients using ordinary least squares:

Step 1: Calculate the slope coefficient (b)

b = \frac{\text{Cov}(R_s, R_x)}{\text{Var}(R_s)} = \frac{[\text{Corr}(R_s, R_x) \times SD(R_s) \times SD(R_x)]}{\text{Var}(R_s)} = \text{Corr}(R_s, R_x) \times \frac{SD(R_x)}{SD(R_s)}

Given:

Correlation = 0.3
SD(R_x) = 20% = 0.20
SD(R_s) = 15% = 0.15

b = 0.3 \times \frac{0.20}{0.15} = 0.3 \times 1.333 = 0.40

Step 2: Calculate the intercept (a)

a = \bar{R_x} - b \times \bar{R_s} = 0.15 - 0.40 \times 0.10 = 0.15 - 0.04 = 0.11

Step 3: Final Model

E(R_x) = 0.11 + 0.40 \times E(R_s)

This matches option B exactly._

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