Question

A stock has a beta of 1.37 and an expected return of 13.5 percent. A risk-free asset currently earns 4.65 percent. a. What is the expected return on a portfolio that is equally invested in the two assets? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) b. If a portfolio of the two assets has a beta of .97, what are the portfolio weights? (Do not round intermediate calculations and round your answers to 4 decimal places, e.g., .1616.) If a portfolio of the two assets has an expected return of 12.7 percent, what is its beta? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) d. If a portfolio of the two assets has a beta of 2.57, what are the portfolio weights? (A negative answer should be indicated by a minus sign. Do not round intermediate calculations and round your answers to 4 decimal places, e.g., 1616.) a. b Expected return Weight of stock Weight of risk-free asset Beta Weight of stock Weight of risk-free

Answer 1

Stock

Beta of stock = β_S = 1.37

Expected return on stock = R_S = 13.5%

Risk-free asset

Risk-free assets have zero beta

Beta of risk-free asset = β_F = 0

Return on risk-free asset = R_F = 4.65%

Part a

Equally invested in two assets means that the weight of the stock and the risk-free asset is the same and equal to 0.5 or 50%

Weight of stock in the portfolio = W_S = 0.5

Weight of risk-free asset in the portfolio = W_F = 0.5

Expected return on the portfolio is calculated using the formula:

E[R_P] = W_S*R_S + W_F*R_F = 0.5*13.5% + 0.5*4.65% = 9.075% ~ 9.08% (Rounded to two-decimal places)

Answer a

Expected return = 9.08%

Part b

Beta of the portfolio = β_P = 0.97

Weight of stock in the portfolio = W_S, Weight of risk-free asset in the portfolio = W_F

The sum of weights of two assets in the portfolio is 1 or 100%

W_S + W_F = 1

W_F = 1 - W_S

Now, Beta of the portfolio is calculated using the formula:

Beta of the portfolio = β_P = W_S*β_S + W_F*β_F

β_S = 1.37, β_F = 0

β_P = W_S*β_S + W_F*β_F

0.97 = W_S*1.37 + W_F*0

W_S = 0.97/1.37 = 0.708029197080292 ~ 0.7080

W_F = 1 - W_S = 1 - 0.7080 = 0.2920

Answer b

Weight of stock = 0.7080

Weight of risk-free asset = 0.2920

Part c

Expected return on stock = R_S = 13.5%

Return on risk-free asset = R_F = 4.65%

Expected return of the portfolio = E[R_P] = 12.7%

We need to first calculate the weights of the two assets in the portfolio and then calculate the beta of the portfolio

Weight of stock in the portfolio = W_S, Weight of risk-free asset in the portfolio = W_F

W_S + W_F = 1

W_F = 1 - W_S

The expected return of the portfolio is calculated using the formula:

E[R_P] = W_S*R_S + W_F*R_F

E[R_P] = W_S*R_S + (1-W_S)*R_F

12.7% = W_S*13.5% + (1-W_S)*4.65%

12.7% = W_S*13.5% + 4.65% - W_S*4.65%

12.7% - 4.65% = W_S*8.85%

8.05% = W_S*8.85%

W_S = 8.05%/8.85% = 0.909604519774011

W_F = 1 - W_S = 1 - 0.909604519774011 = 0.0903954802259888

Now, Beta of the portolio is calculated using the formula

Beta of the portfolio = β_P = W_S* β_S + W_F*β_F

Where,  β_S = 1.37, W_S = 0.909604519774011, β_F = 0, W_F = 0.0903954802259888

β_P = 0.909604519774011*1.37 + W_F*0 = 0.909604519774011*1.37 = 1.25

Answer c

Beta = 1.25

Part d

Portfolio beta = β_P = 2.57

Portfolio beta is calculated using the formula:

Portfolio beta = β_P = W_S*β_S + W_F*β_F

Where, β_S = 1.37, β_F = 0, Weight of stock = W_S, Weight of risk-free asset = W_F

2.57 = W_S*1.37 + W_F*0

2.57 = W_S*1.37 + 0

W_S = 2.57/1.37 = 1.8759

W_F = 1 - W_S = 1 - 1.8759 = -0.8759

Answer d

Weight of stock = 1.8759

Weight of risk-free = -0.8759

Answers

a.	Expected Return	9.08	%
b.	Weight of stock	0.7080
	Weight if risk-free asset	0.2920
c.	Beta	1.25
d.	Weight of stock	1.8759
	Weight of risk-free	-0.8759