Suppose that the S&P 500, with a beta of 1.0, has an expected return of 13% and T-bills provide a risk-free return of 4%. a. What would be the expected return and beta of portfolios constructed from these two assets with weights in the S&P 500 of (i) 0; (ii) 0.25; (iii) 0.50; (iv) 0.75; (v) 1.0

a. The answers are as follows:

(i) Expected of Return of Portfolio = 4%; and Beta of Portfolio = 0

(ii) Expected of Return of Portfolio = 6.25%; and Beta of Portfolio = 0.25

(iii) Expected of Return of Portfolio = 8.50%; and Beta of Portfolio = 0.50

(iv) Expected of Return of Portfolio = 10.75%; and Beta of Portfolio = 0.75

(v) Expected of Return of Portfolio = 13%; and Beta of Portfolio = 1.0

b. Change in expected return = 9% increase

Explanation:

Note: This question is not complete as part b of it is omitted. The complete question is therefore provided before answering the question as follows:

Suppose that the S&P 500, with a beta of 1.0, has an expected return of 13% and T-bills provide a risk-free return of 4%.

a. What would be the expected return and beta of portfolios constructed from these two assets with weights in the S&P 500 of (i) 0; (ii) 0.25; (iii) 0.50; (iv) 0.75; (v) 1.0

b. How does expected return vary with beta? (Do not round intermediate calculations.)

The explanation to the answers are now provided as follows:

a. What would be the expected return and beta of portfolios constructed from these two assets with weights in the S&P 500 of (i) 0; (ii) 0.25; (iii) 0.50; (iv) 0.75; (v) 1.0

To calculate these, we use the following formula:

Expected of Return of Portfolio = (WS&P * RS&P) + (WT * RT) ………… (1)

Beta of Portfolio = (WS&P * BS&P) + (WT * BT) ………………..………………. (2)

Where;

WS&P = Weight of S&P = (1) – (1v)

RS&P = Return of S&P = 13%, or 0.13

WT = Weight of T-bills = 1 – WS&P

RT = Return of T-bills = 4%, or 0.04

BS&P = 1.0

BT = 0

After substituting the values into equation (1) & (2), we therefore have:

(i) Expected return and beta of portfolios with weights in the S&P 500 of 0 (i.e. WS&P = 0)

Using equation (1), we have:

Expected of Return of Portfolio = (0 * 0.13) + ((1 - 0) * 0.04) = 0.04, or 4%

Using equation (2), we have:

Beta of Portfolio = (0 * 1.0) + ((1 - 0) * 0) = 0

(ii) Expected return and beta of portfolios with weights in the S&P 500 of 0.25 (i.e. WS&P = 0.25)

Using equation (1), we have:

Expected of Return of Portfolio = (0.25 * 0.13) + ((1 - 0.25) * 0.04) = 0.0625, or 6.25%

Using equation (2), we have:

Beta of Portfolio = (0.25 * 1.0) + ((1 - 0.25) * 0) = 0.25

(iii) Expected return and beta of portfolios with weights in the S&P 500 of 0.50 (i.e. WS&P = 0.50)

Using equation (1), we have:

Expected of Return of Portfolio = (0.50 * 0.13) + ((1 - 0.50) * 0.04) = 0.0850, or 8.50%

Using equation (2), we have:

Beta of Portfolio = (0.50 * 1.0) + ((1 - 0.50) * 0) = 0.50

(iv) Expected return and beta of portfolios with weights in the S&P 500 of 0.75 (i.e. WS&P = 0.75)

Using equation (1), we have:

Expected of Return of Portfolio = (0.75 * 0.13) + ((1 - 0.75) * 0.04) = 0.1075, or 10.75%

Using equation (2), we have:

Beta of Portfolio = (0.75 * 1.0) + ((1 - 0.75) * 0) = 0.75

(v) Expected return and beta of portfolios with weights in the S&P 500 of 1.0 (i.e. WS&P = 1.0)

Using equation (1), we have:

Expected of Return of Portfolio = (1.0 * 0.13) + ((1 – 1.0) * 0.04) = 0.13, or 13%

Using equation (2), we have:

Beta of Portfolio = (1.0 * 1.0) + (1 – 1.0) * 0) = 1.0

b. How does expected return vary with beta? (Do not round intermediate calculations.)

There expected return will increase by the percentage of the difference between Expected Return and Risk free rate. That is;

Change in expected return = Expected Return - Risk free rate = 13% - 4% = 9% increase

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