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Unformatted text preview: premium of the asset, so: RPA/β A = RPB/β B We can rearrange this equation to get: β B/β A = RPB/RPA If the rewardtorisk ratios are the same, the ratio of the betas of the assets is equal to the ratio of the risk premiums of the assets. 23. a. We 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) = .4(.20) + .4(.35) + .2(.60) = .3400 or 34.00% Normal: E(Rp) = .4(.15) + .4(.12) + .2(.05) = .1180 or 11.80% Bust: E(Rp) = .4(.01) + .4(–.25) + .2(–.50) = –.1960 or –19.60% And the expected return of the portfolio is: E(Rp) = .35(.34) + .40(.118) + .25(–.196) = .1172 or 11.72% 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, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is: σ 2p = .35(.34 – .1172)2 + .40(.118 – .1172)2 + .25(–.196 – .1172)2 σ 2p = .04190 σ p = (.04190)1/2 = .2047 or 20.47% 279 b. The risk premium is the return of a risky asset, minus the riskfree rate. Tbills are often used as the riskfree rate, so: RPi = E(Rp) – Rf = .1172 – .038 = .0792 or 7.92% c. The approximate expected real return is the expected nominal return minus the inflation rate, so: Approximate expected real return = .1172 – .035 = .0822 or 8.22% To find the exact real return, we will use the Fisher equation. Doing so, we get: 1 + E(Ri) = (1 + h)[1 + e(ri)] 1.1172 = (1.0350)[1 + e(ri)] e(ri) = (1.1172/1.035) – 1 = .0794 or 7.94% The approximate real risk premium is the expected return minus the inflation rate, so: Approximate expected real risk premium = .0792 – .035 = .0442 or 4.42% To find the exact expected real risk premium we use the Fisher effect. Doing do, we find: Exact expected real risk premium = (1.0792/1.035) – 1 = .0427 or 4.27% 24. We know the total portfolio value and the investment of two stocks in the portfolio, so we can find the weight of these two stocks. The weights of Stock A and Stock B are: wA = $180,000 / $1,000,000 = .18 wB = $290,000/$1,000,000 = .29 Since the portfolio is as risky as the market, the β of the portfolio must be equal to one. We also know the β of the riskfree asset is zero. We can use the equation for the β of a portfolio to find the weight of the third stock. Doing so, we find: β p = 1.0 = wA(.75) + wB(1.30) + wC(1.45) + wRf(0) Solving for the weight of Stock C, we find: wC = .33655172 So, the dollar investment in Stock C must be: Invest in Stock C = .33655172($1,000,000) = $336,551.72 280 We also know the total portfolio weight must be one, so the weight of the riskfree asset must be one minus the asset weight we know, or: 1 = wA + wB + wC + wRf 1 = .18 + .29 + .33655172 + wRf wRf = .19344828 So, the dollar investment in the riskfree asset must be: Invest in riskfree asset = .19344828($1,000,000) = $193,448.28 25. We are given the expected return and β of a portfolio and the expected return and β of assets in the portfolio. We know the β of the riskfree asset is zero. We also know the sum of the weights of each asset must be equal to one. So, the weight of the riskfree asset is one minus the weight of Stock X and the weight of Stock Y. Using this relationship, we can express the expected return of the portfolio as: E(Rp) = .1070 = wX(.172) + wY(.0875) + (1 – wX – wY)(.055) And the β of the portfolio is: β p = .8 = wX(1.8) + wY(0.50) + (1 – wX – wY)(0) We have two equations and two unknowns. Solving these equations, we find that: wX = –0.11111 wY = 2.00000 wRf = –0.88889 The amount to invest in Stock X is: Investment in stock X = –0.11111($100,000) = –$11,111.11 A negative portfolio weight means that you short sell the stock. If you are not familiar with short selling, it means you borrow a stock today and sell it. You must then purchase the stock at a later date to repay the borrowed stock. If you short sell a stock, you make a profit if the stock decreases in value. The negative weight on the riskfree asset means that we borrow money to invest. 26. 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) = .33(.082) + .33(.095) + .33(.063) = .0800 or 8.00% E(RB) = .33(–.065) + .33(.124) + .33(.185) = .0813 or 8.13% 281 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 of Stock A are...
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This note was uploaded on 07/10/2010 for the course FIN 6301 taught by Professor Eshmalwi during the Spring '10 term at University of TexasTyler.
 Spring '10
 eshmalwi
 Finance, Corporate Finance

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