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Beta and required return

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The riskless return is currently 6%, and Chicago Gear has estimated the contingent returns given here.

a. Calculate the expected returns on the stock market and on Chicago Gear stock.

b. What is Chicago Gear's beta?

c. What is Chicago Gear's required according to the CAPM?

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This solution provides steps necessary to determine the beta and required return.

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1) if these are the only two investments in her portfolio, what is her portfolio's beta? 2) If the market required rate of return is 14 percent and the risk-free rate is 6 percent, what is the fund's required rate of return? 3) Use a spreadsheet or calculator with a linear regression function to determine stock X's beta coefficient. Plot the security market line. 4) Construct a scatter diagram showing the relationship between returns on stock Y and the market. Use a spreadsheet or a calculator with a linear regression function to estimate beta.

1. An individual has $35,000 invested in a stock which has a beta of 0.8 and $40,000 invested in a stock with a beta of 1.4, if these are the only two investments in her portfolio, what is her portfolio's beta?

2. Suppose you are the money manager of a $4 million investment fund. The fund consists of 4 stocks with the following investments and betas;

Stock Investment Beta
A $ 400,000 1.50
B 600,000 (0.50)
C 1,000,000 1.25
D 2,000,000 0.75

If the market required rate of return is 14 percent and the risk-free rate is 6 percent, what is the fund's required rate of return?

3. Given the following set of data;

Historical Rates of Return

Year NYSE Stock X
1 (26.5%) (14.0%)
2 37.2 23.0
3 23.8 17.5
4 (7.2) 2.0
5 6.6 8.1
6 20.5 19.4
7 30.6 18.2

a. Use a spreadsheet or calculator with a linear regression function to determine stock X's beta coefficient.

b. Determine the arithmetic average rates of return for stock X and the NYSE over the period given. Calculate the standard deviations of returns for both stock X and the NYSE.
c. Assuming (1) that the situation during years 1 to 7 is expected to hold true in the future (that is, Rx = Rx; Rm = Rm; and both Ox and bx in the future will equal their past values. (2) That stock X is in equilibrium (that is, it plots on the security market line), what is the risk-free rate?

d. Plot the security market line.

e. Suppose you hold a large, well-diversified portfolio and are considering adding to the portfolio and are considering adding to the portfolio either stock X or another Stock, stock Y, that has the same beta as stock X but a higher standard deviation of returns. Stocks X and Y have the same expected returns; that is
^ ^
RX = RY = 10.6%. Which stock should you choose?

4. Given the following set of data:

Historical Rates of Return

Year NYSE Stock Y
1 4.0% 3.0%
2 14.3 18.2
3 19.0 9.1
4 (14.7) (6.0)
5 (26.5) (15.3)
6 37.2 33.1
7 23.8 6.1
8 (7.2) 3.2
9 6.6 14.8
10 20.5 24.1
11 30.6 18.0
Mean = 9.8% 9.8%
o = 19.6% 13.8%

a. Construct a scatter diagram showing the relationship between returns on stock Y and the market. Use a spreadsheet or a calculator with a linear regression function to estimate beta.
b. Give a verbal interpretation of what the regression line and beta coefficient show about stock Y's volatility and relative riskiness as compared with those of other stocks.
c. Suppose the scatter of points had been more spread out, but the regression line was exactly where your present graph shows it. How would this affect (1) the firm's risk if the stock is held in a one-asset portfolio and (2) the actual risk premium on the stock if the CAPM holds exactly?
d. Suppose the regression line had been downward sloping and the beta coefficient had been negative. What would this imply about (1) stock Y's relative riskiness, (2) its correlation with the market, (3) its probable risk premium?

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