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# Capital Asset Pricing Model of a Portfolio Share

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1. Compute the expected return and volatility of return for a portfolio that has a portfolio share of 0.9 in the S&P 500 and 0.1 in an emerging market index. The S&P 500 has a volatility of return of 15 percent and an expected return of 12 percent. The emerging market has a return volatility of 30 percent and an expected return of 10 percent. The correlation between the emerging market index return and the S&P 500 is 0.1.

2. If the S&P 500 is a good proxy for the market portfolio in the CAPM, and the CAPM applies to the emerging market index, use the information in previous question to compute the beta and risk premium for emerging market index.

3. A firm has an expected cash flow of \$500 million in one year. The beta of the common stock of the firm is 0.8 and this cash flow has the same risk as the firm as a whole. Using a risk-free rate of 5 percent and a risk premium on the market portfolio of 6 percent, what is the present value of the cash flow? If the beta of the firm doubles, what happens to the present value of the cash flow?

4. Using the data in the previous question, consider how hedging the cash flow against systematic risk might affect the firm. If management wants to eliminate the systematic risk of the cash flow completely, how could it do so? How much would the firm have to pay investors to bear the systematic risk of the cash flow?

5. Consider problem 3. To hedge the firm's systematic risk, management has to pay investors to bear this risk. Why is it that the value of the firm for shareholders does not fall when the firm pays other investors to bear the cash flow's systematic risk?

#### Solution Preview

1. Compute the expected return and volatility of return for a portfolio that has a portfolio share of 0.9 in the S&P 500 and 0.1 in an emerging market index. The S&P 500 has a volatility of return of 15 percent and an expected return of 12 percent. The emerging market has a return volatility of 30 percent and an expected return of 10 percent. The correlation between the emerging market index return and the S&P 500 is 0.1.

Volatility of return = standard deviation of return

Security Portfolio Share (x) Expected return ( r ) Standard deviation (s) Correlation (rho)
S&P 500 (A) 0.9 12% 15.00% 0.1
Emerging Market Index (B) 0.1 10% 30.00%
Total= 1

Expected return = xA rA + xB rB= 11.80% =0.9*0.12+0.1*0.1
Standard deviation =square root of (xA^2sA^2 + xB^2sB^2+ 2 xAxBrhosAsB)= 14.12% =square root of (0.9^2*0.15^2+0.1^2*0.3^2+2*0.9*0.1*0.1*0.15*0.3)

Expected return of the portfolio= 11.80%
Volatility (standard deviation) of return of the portfolio= 14.12%

2. If the S&P 500 is a good proxy for the market portfolio in the CAPM, and the CAPM applies to the emerging market index, use the information in previous question to compute the beta and risk premium for emerging market index.

Beta = correlation between S&P 500 return and Emerging Market Index return x Standard Deviation of return for Emerging Market Index / Standard Deviation of return for S&P 500

Standard deviation of return for Emerging Market Index= 30.00%
Standard ...

#### Solution Summary

Calculates
1) expected return and volatility of a portfolio.
2) beta and risk premium of an index
3) present value of cash flow
Discusees how hedging the cash flow against systematic risk might affect the firm.

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