Certainly! Let's solve the given problem step-by-step:
### Part (a): Calculate Stock A's beta.
We are given the following information:
- Risk-free rate [tex]\(\operatorname{rRF} = 5\%\)[/tex]
- Market return [tex]\(\operatorname{rM} = 10\%\)[/tex]
- Return on Stock A [tex]\(\operatorname{rA} = 12\%\)[/tex]
We use the Capital Asset Pricing Model (CAPM) formula to calculate the beta of Stock A. The CAPM formula is:
[tex]\[
rA = rRF + \beta_A \times (rM - rRF)
\][/tex]
Solving for [tex]\(\beta_A\)[/tex]:
[tex]\[
12\% = 5\% + \beta_A \times (10\% - 5\%)
\][/tex]
Break it down step-by-step:
1. Subtract [tex]\(5\%\)[/tex] from [tex]\(12\%\)[/tex]:
[tex]\[
12\% - 5\% = 7\%
\][/tex]
2. Calculate the excess market return (market risk premium):
[tex]\[
10\% - 5\% = 5\%
\][/tex]
3. Now, we have:
[tex]\[
7\% = \beta_A \times 5\%
\][/tex]
4. Solve for [tex]\(\beta_A\)[/tex]:
[tex]\[
\beta_A = \frac{7\%}{5\%} = 1.4
\][/tex]
So, Stock A's beta ([tex]\(\beta_A\)[/tex]) is approximately 1.4.
### Part (b): Calculate A's new required rate of return if Stock A's beta were 2.0
Now, we are given the new beta value for Stock A:
- New [tex]\(\beta_A = 2.0\)[/tex]
We need to calculate the new required rate of return ([tex]\(rA_{new}\)[/tex]) using the CAPM formula again:
[tex]\[
rA_{new} = rRF + \beta_{new} \times (rM - rRF)
\][/tex]
Substitute the given values:
[tex]\[
rA_{new} = 5\% + 2.0 \times (10\% - 5\%)
\][/tex]
Break it down step-by-step:
1. Calculate the excess market return (market risk premium):
[tex]\[
10\% - 5\% = 5\%
\][/tex]
2. Multiply this with the new beta:
[tex]\[
2.0 \times 5\% = 10\%
\][/tex]
3. Add this to the risk-free rate:
[tex]\[
rA_{new} = 5\% + 10\% = 15\%
\][/tex]
So, the new required rate of return for Stock A ([tex]\(rA_{new}\)[/tex]) is approximately 15%.
### Summary of results:
- Stock A's beta ([tex]\(\beta_A\)[/tex]) is approximately 1.4.
- New required rate of return for Stock A ([tex]\(rA_{new}\)[/tex]) is approximately 15%.
