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Unformatted text preview: so the expected return of the portfolio in each state of the economy is: Boom: E(Rp) = (.07 + .15 + .33)/3 = .1833 or 18.33% Bust: E(Rp) = (.13 + .03 − .06)/3 = .0333 or 3.33% To find the expected return of the portfolio, we multiply the return in each state of the economy by the probability of that state occurring, and then sum. Doing this, we find: E(Rp) = .80(.1833) + .20(.0333) = .1533 or 15.33% b. This portfolio does not have an equal weight in each asset. We still need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: E(Rp)=.20(.07) +.20(.15) + .60(.33) =.2420 or 24.20% Bust: E(Rp) =.20(.13) +.20(.03) + .60(− .06) = –.0040 or –0.40% And the expected return of the portfolio is: E(Rp) = .80(.2420) + .20(− .004) = .1928 or 19.28% To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance of the portfolio is: σ p2 = .80(.2420 – .1928)2 + .20(− .0040 – .1928)2 = .00968 10. a. This portfolio does not have an equal weight in each asset. We first need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: Good: Poor: Bust: E(Rp) = .30(.3) + .40(.45) + .30(.33) = .3690 or 36.90% E(Rp) = .30(.12) + .40(.10) + .30(.15) = .1210 or 12.10% E(Rp) = .30(.01) + .40(–.15) + .30(–.05) = –.0720 or –7.20% E(Rp) = .30(–.06) + .40(–.30) + .30(–.09) = –.1650 or –16.50% And the expected return of the portfolio is: E(Rp) = .20(.3690) + .35(.1210) + .30(–.0720) + .15(–.1650) = .0698 or 6.98% 273 b. To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation the portfolio is: σ p2 = .20(.3690 – .0698)2 + .35(.1210 – .0698)2 + .30(–.0720 – .0698)2 + .15(–.1650 – .0698)2 σ p2 = .03312 σ p = (.03312)1/2 = .1820 or 18.20% 11. The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. So, the beta of the portfolio is: β p = .25(.75) + .20(1.90) + .15(1.38) + .40(1.16) = 1.24 12. The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. If the portfolio is as risky as the market it must have the same beta as the market. Since the beta of the market is one, we know the beta of our portfolio is one. We also need to remember that the beta of the riskfree asset is zero. It has to be zero since the asset has no risk. Setting up the equation for the beta of our portfolio, we get: β p = 1.0 = 1/3(0) + 1/3(1.85) + 1/3(β X) Solving for the beta of Stock X, we get: β X = 1.15 13. CAPM states the relationship between the risk of an asset and its expected return. CAPM is: E(Ri) = Rf + [E(RM) – Rf] × β i Substituting the values we are given, we find: E(Ri) = .05 + (.12 – .05)(1.25) = .1375 or 13.75% 14. We are given the values for the CAPM except for the β of the stock. We need to substitute these values into the CAPM, and solve for the β of the stock. One important thing we need to realize is that we are given the market risk premium. The market risk premium is the expected return of the market minus the riskfree rate. We must be careful not to use this value as the expected return of the market. Using the CAPM, we find: E(Ri) = .142 = .04 + .07β i β i = 1.46 274 15. Here we need to find the expected return of the market using the CAPM. Substituting the values given, and solving for the expected return of the market, we find: E(Ri) = .105 = .055 + [E(RM) – .055](.73) E(RM) = .1235 or 12.35% 16. Here we need to find the riskfree rate using the CAPM. Substituting the values given, and solving for the riskfree rate, we find: E(Ri) = .162 = Rf + (.11 – Rf)(1.75) .162 = Rf + .1925 – 1.75Rf Rf = .0407 or 4.07% 17. a. Again, we have a special case where the portfolio is equally weighted, so we can sum the returns of each asset and divide by the number of assets. The expected return of the portfolio is: E(Rp) = (.103 + .05)/2 = .0765 or 7.65% b. We need to find the portfolio weights that result in a portfolio with a β of 0.50. We know the β of the riskfree asset is zero. We also know the weight of the riskfree asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So: β p = 0.50 = wS(.92) + (1 – wS)(0) 0.50 = .92wS + 0 – 0wS wS = 0.50/.92 wS = ....
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 Spring '10
 eshmalwi
 Finance, Corporate Finance

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