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Unformatted text preview: ult is the variance. So, the expected return and standard deviation of each stock are: Asset 1: E(R1) = .15(.25) + .35(.20) + .35(.15) + .15(.10) = .1750 or 17.50%
2 σ 1 =.15(.25 – .1750)2 + .35(.20 – .1750)2 + .35(.15 – .1750)2 + .15(.10 – .1750)2 = .00213 σ 1 = (.00213)1/2 = .0461 or 4.61% Asset 2: E(R2) = .15(.25) + .35(.15) + .35(.20) + .15(.10) = .1750 or 17.50% σ 2 =.15(.25 – .1750)2 + .35(.15 – .1750)2 + .35(.20 – .1750)2 + .15(.10 – .1750)2 = .00213 2 σ 2 = (.00213)1/2 = .0461 or 4.61% Asset 3: E(R3) = .15(.10) + .35(.15) + .35(.20) + .15(.25) = .1750 or 17.50%
2 σ 3 =.15(.10 – .1750)2 + .35(.15 – .1750)2 + .35(.20 – .1750)2 + .15(.25 – .1750)2 = .00213 σ 3 = (.00213)1/2 = .0461 or 4.61% 288 b. To find the covariance, we multiply each possible state times the product of each assets’ deviation from the mean in that state. The sum of these products is the covariance. The correlation is the covariance divided by the product of the two standard deviations. So, the covariance and correlation between each possible set of assets are: Asset 1 and Asset 2: Cov(1,2) = .15(.25 – .1750)(.25 – .1750) + .35(.20 – .1750)(.15 – .1750) + .35(.15 – .1750)(.20 – .1750) + .15(.10 – .1750)(.10 – .1750) Cov(1,2) = .000125 ρ 1,2 = Cov(1,2) / σ 1 σ 2 ρ 1,2 = .000125 / (.0461)(.0461) ρ 1,2 = .5882 Asset 1 and Asset 3: Cov(1,3) = .15(.25 – .1750)(.10 – .1750) + .35(.20 – .1750)(.15 – .1750) + .35(.15 – .1750)(.20 – .1750) + .15(.10 – .1750)(.25 – .1750) Cov(1,3) = –.002125 ρ 1,3 = Cov(1,3) / σ 1 σ 3 ρ 1,3 = –.002125 / (.0461)(.0461) ρ 1,3 = –1 Asset 2 and Asset 3: Cov(2,3) = .15(.25 – .1750)(.10 – .1750) + .35(.15 – .1750)(.15 – .1750) + .35(.20 – .1750)(.20 – .1750) + .15(.10 – .1750)(.25 – .1750) Cov(2,3) = –.000125 ρ 2,3 = Cov(2,3) / σ 2 σ 3 ρ 2,3 = –.000125 / (.0461)(.0461) ρ 2,3 = –.5882 c. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so, for a portfolio of Asset 1 and Asset 2: E(RP) = w1E(R1) + w2E(R2) E(RP) = .50(.1750) + .50(.1750) E(RP) = .1750 or 17.50% The variance of a portfolio of two assets can be expressed as:
2 2 σ 2 = w 1 σ 1 + w 2 σ 2 + 2w1w2σ 1σ 2ρ 1,2 P 2 2 σ 2 = .502(.04612) + .502(.04612) + 2(.50)(.50)(.0461)(.0461)(.5882) P σ 2 = .001688 P And the standard deviation of the portfolio is: σ P = (.001688)1/2
289 σ P = .0411 or 4.11% d. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so, for a portfolio of Asset 1 and Asset 3: E(RP) = w1E(R1) + w3E(R3) E(RP) = .50(.1750) + .50(.1750) E(RP) = .1750 or 17.50% The variance of a portfolio of two assets can be expressed as:
2 2 2 2 σ 2 = w 1 σ 1 + w 3 σ 3 + 2w1w3σ 1σ 3ρ 1,3 P σ 2 = .502(.04612) + .502(.04612) + 2(.50)(.50)(.0461)(.0461)(–1) P σ 2 = .000000 P Since the variance is zero, the standard deviation is also zero. e. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so, for a portfolio of Asset 1 and Asset 3: E(RP) = w2E(R2) + w3E(R3) E(RP) = .50(.1750) + .50(.1750) E(RP) = .1750 or 17.50% The variance of a portfolio of two assets can be expressed as:
2 2 σ 2 = w 2 σ 2 + w 3 σ 3 + 2w2w3σ 2σ 3ρ 1,3 P 2 2 σ 2 = .502(.04612) + .502(.04612) + 2(.50)(.50)(.0461)(.0461)(–.5882) P σ 2 = .000438 P And the standard deviation of the portfolio is: σ P = (.000438)1/2 σ P = .0209 or 2.09% f. As long as the correlation between the returns on two securities is below 1, there is a benefit to diversification. A portfolio with negatively correlated stocks can achieve greater risk reduction than a portfolio with positively correlated stocks, holding the expected return on each stock constant. Applying proper weights on perfectly negatively correlated stocks can reduce portfolio variance to 0. 290 37. a. The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock is: E(RA) = .15(–.08) + .70(.13) + .15(.48) = .1510 or 15.10% E(RB) = .15(–.05) + .70(.14) + .15(.29) = .1340 or 13.40% b. We can use the expected returns we calculated to find the slope of the Security Market Line. We know that the beta of Stock A is .25 greater than the beta of Stock B. Therefore, as beta increases by .25, the expected return on a security increases by .017 (= .1510 – .1340). The slope of the security market line (SML) equals: SlopeSML = Rise / Run SlopeSML = Increase in expected return / Increase in beta SlopeSML = (.1510 – .1340) / .25 SlopeSML = .0680 or 6.80% Since the market’s beta is 1 and the riskfree rate has a beta of zero, the slope of the Security Market Line equals the expected market risk premium. So, the expected market risk premium must be 6.8 percent. We could also solve this problem using CAPM. The equations for the expected returns of the two stocks are: .151 = Rf +...
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 Spring '10
 eshmalwi
 Finance, Corporate Finance

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