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Unformatted text preview: Firm C’s stock is underpriced, and you should buy it. 31. Because a well-diversified portfolio has no unsystematic risk, this portfolio should lie on the Capital Market Line (CML). The slope of the CML equals: SlopeCML = [E(RM) – Rf] / σ M SlopeCML = (0.12 – 0.05) / 0.19 SlopeCML = 0.36842 a. The expected return on the portfolio equals: E(RP) = Rf + SlopeCML(σ P) E(RP) = .05 + .36842(.07) E(RP) = .0758 or 7.58% b. The expected return on the portfolio equals: E(RP) = Rf + SlopeCML(σ P) .20 = .05 + .36842(σ P) σ P = .4071 or 40.71% 32. First, we can calculate the standard deviation of the market portfolio using the Capital Market Line (CML). We know that the risk-free rate asset has a return of 5 percent and a standard deviation of zero and the portfolio has an expected return of 9 percent and a standard deviation of 13 percent. These two points must lie on the Capital Market Line. The slope of the Capital Market Line equals: SlopeCML = Rise / Run SlopeCML = Increase in expected return / Increase in standard deviation SlopeCML = (.09 – .05) / (.13 – 0) SlopeCML = .31 285 According to the Capital Market Line: E(RI) = Rf + SlopeCML(σ I) Since we know the expected return on the market portfolio, the risk-free rate, and the slope of the Capital Market Line, we can solve for the standard deviation of the market portfolio which is: E(RM) = Rf + SlopeCML(σ M) .12 = .05 + (.31)(σ M) σ M = (.12 – .05) / .31 σ M = .2275 or 22.75% Next, we can use the standard deviation of the market portfolio to solve for the beta of a security using the beta equation. Doing so, we find the beta of the security is: β I = (ρ I,M)(σ I) / σ M β I = (.45)(.40) / .2275 β I = 0.79 Now we can use the beta of the security in the CAPM to find its expected return, which is: E(RI) = Rf + β I[E(RM) – Rf] E(RI) = 0.05 + 0.79(.12 – 0.05) E(RI) = .1054 or 10.54% 33. First, we need to find the standard deviation of the market and the portfolio, which are: σ M = (.0429)1/2 σ M = .2071 or 20.71% σ Z = (.1783)1/2 σ Z = .4223 or 42.23% Now we can use the equation for beta to find the beta of the portfolio, which is: β Z = (ρ Z,M)(σ Z) / σ M β Z = (.39)(.4223) / .2071 β Z = .80 Now, we can use the CAPM to find the expected return of the portfolio, which is: E(RZ) = Rf + β Z[E(RM) – Rf] E(RZ) = .048 + .80(.114 – .048) E(RZ) = .1005 or 10.05% 286 Challenge 34. The amount of systematic risk is measured by the β of an asset. Since we know the market risk premium and the risk-free rate, if we know the expected return of the asset we can use the CAPM to solve for the β of the asset. The expected return of Stock I is: E(RI) = .15(.09) + .55(.42) + .30(.26) = .3225 or 32.25% Using the CAPM to find the β of Stock I, we find: .3225 = .04 + .075β I β I = 3.77 The total risk of the asset is measured by its standard deviation, so we need to calculate the standard deviation of Stock I. Beginning with the calculation of the stock’s variance, we find:
σ I2 = .15(.09 – .3225)2 + .55(.42 – .3225)2 + .30(.26 – .3225)2 σ I2 = .01451 σ I = (.01451)1/2 = .1205 or 12.05% Using the same procedure for Stock II, we find the expected return to be: E(RII) = .15(–.30) + .55(.12) + .30(.44) = .1530 Using the CAPM to find the β of Stock II, we find: .1530 = .04 + .075β II β II = 1.51 And the standard deviation of Stock II is: σ II2 = .15(–.30 – .1530)2 + .55(.12 – .1530)2 + .30(.44 – .1530)2 σ II2 = .05609 σ II = (.05609)1/2 = .2368 or 23.68% Although Stock II has more total risk than I, it has much less systematic risk, since its beta is much smaller than I’s. Thus, I has more systematic risk, and II has more unsystematic and more total risk. Since unsystematic risk can be diversified away, I is actually the “riskier” stock despite the lack of volatility in its returns. Stock I will have a higher risk premium and a greater expected return. 287 35. Here we have the expected return and beta for two assets. We can express the returns of the two assets using CAPM. If the CAPM is true, then the security market line holds as well, which means all assets have the same risk premium. Setting the reward-to-risk ratios of the assets equal to each other and solving for the risk-free rate, we find: (.15 – Rf)/1.4 = (.115 – Rf)/.90 .90(.15 – Rf) = 1.4(.115 – Rf) .135 – .9Rf = .161 – 1.4Rf .5Rf = .026 Rf = .052 or 5.20% Now using CAPM to find the expected return on the market with both stocks, we find: .15 = .0520 + 1.4(RM – .0520) RM = .1220 or 12.20% 36. a. .115 = .0520 + .9(RM – .0520) RM = .1220 or 12.20% The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. 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 res...
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