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Unformatted text preview: bout the company. So, the unsystematic return of the stock is –2.6 percent. The total return is the expected return, plus the two components of unexpected return: the systematic risk portion of return and the unsystematic portion. So, the total return of the stock is: R= R +m+ε R = 10.80% + 2.81% – 2.6% R = 11.01% 3. a. If m is the systematic risk portion of return, then: m = β GNPΔ%GNP + β rΔInterest rates m = 2.04(2.6% – 1.8%) – 1.15(4.8% – 4.3%) m = 1.06% b. The unsystematic is the return that occurs because of a firm specific factor such as the increase in market share. If ε is the unsystematic risk portion of the return, then: ε = 0.45(27% – 23%) ε = 1.80% c. The total return is the expected return, plus the two components of unexpected return: the systematic risk portion of return and the unsystematic portion. So, the total return of the stock is: R= R +m+ε R = 10.50% + 1.06% + 1.80% R = 13.36% 4. The beta for a particular risk factor in a portfolio is the weighted average of the betas of the assets. This is true whether the betas are from a single factor model or a multifactor model. So, the betas of the portfolio are: F1 = .20(1.45) + .20(0.73) + .60(0.89) F1 = 0.97 F2 = .20(0.80) + .20(1.25) + .60(–0.14) F2 = 0.33 F3 = .20(0.05) + .20(–0.20) + .60(1.24) F3 = 0.71 So, the expression for the return of the portfolio is: Ri = 5% + 0.97F1 + 0.33F2 – 0.71F3 Which means the return of the portfolio is: Ri = 5% + 0.97(5.50%) + 0.33(4.20%) – 0.71(4.90%) Ri = 8.21% 298 Intermediate 5. We can express the multifactor model for each portfolio as: E(RP ) = RF + β 1F1 + β 2F2 where F1 and F2 are the respective risk premiums for each factor. Expressing the return equation for each portfolio, we get: 16% = 4% + 0.85F1 + 1.15F2 12% = 4% + 1.45F1 – 0.25F2 We can solve the system of two equations with two unknowns. Multiplying each equation by the respective F2 factor for the other equation, we get: 4.00% = 1.0% + .2125F1 + 0.2875F2 13.8% = 4.6% + 1.6675F1 – 0.2875F2 Summing the equations and solving F1 for gives us: 17.8% = 5.6% + 1.88 F1 F1 = 6.49% And now, using the equation for portfolio A, we can solve for F2, which is: 16% = 4% + 0.85(6.490%) + 1.15F2 F2 = 5.64% 6. a. The market model is specified by: R = R + β (RM – R M ) + ε so applying that to each Stock: Stock A: RA = R A + β A(RM – R M ) + ε A RA = 10.5% + 1.2(RM – 14.2%) + ε A Stock B: RB = R B + β B(RM – R M ) + ε B RB = 13.0% + 0.98(RM – 14.2%) + ε B Stock C: RC = R C + β C(RM – R M ) + ε C RC = 15.7% + 1.37(RM – 14.2%) + ε C 299 b. Since we don't have the actual market return or unsystematic risk, we will get a formula with those values as unknowns: RP = .30RA + .45RB + .25RC RP = .30[10.5% + 1.2(RM – 14.2%) + ε A] + .45[13.0% + 0.98(RM – 14.2%) + ε B] + .25[15.7% + 1.37(RM – 14.2%) + ε C] RP = .30(10.5%) + .45(13%) + .25(15.7%) + [.30(1.2) + .45(.98) + .25(1.37)](RM – 14.2%) + .30ε A + .45ε B + .25ε C RP = 12.925% + 1.1435(RM – 14.2%) + .30ε A + .45ε B + .25ε C c. Using the market model, if the return on the market is 15 percent and the systematic risk is zero, the return for each individual stock is: RA = 10.5% + 1.20(15% – 14.2%) RA = 11.46% RB = 13% + 0.98(15% – 14.2%) RB = 13.78% RC = 15.70% + 1.37(15% – 14.2%) RC = 16.80% To calculate the return on the portfolio, we can use the equation from part b, so: RP = 12.925% + 1.1435(15% – 14.2%) RP = 13.84% Alternatively, to find the portfolio return, we can use the return of each asset and its portfolio weight, or: RP = X1R1 + X2R2 + X3R3 RP = .30(11.46%) + .45(13.78%) + .25(16.80%) RP = 13.84% 7. a. Since five stocks have the same expected returns and the same betas, the portfolio also has the same expected return and beta. However, the unsystematic risks might be different, so the expected return of the portfolio is: R P = 11% + 0.84F1 + 1.69F2 + (1/5)(ε 1 + ε 2 + ε 3 + ε 4 + ε 5) 300 b. Consider the expected return equation of a portfolio of five assets we calculated in part a. Since we now have a very large number of stocks in the portfolio, as: N → ∞, 1 →0 N But, the ε js are infinite, so: (1/N)(ε 1 + ε 2 + ε 3 + ε 4 +…..+ ε N) → 0 Thus: R P = 11% + 0.84F1 + 1.69F2
Challenge 8. To determine which investment an investor would prefer, you must compute the variance of portfolios created by many stocks from either market. Because you know that diversification is good, it is reasonable to assume that once an investor has chosen the market in which she will invest, she will buy many stocks in that market. Known: EF = 0 and σ = 0.10 Eε = 0 and Sε i = 0.20 for all i If we assume the stocks in the portfolio are equallyweighted, the weight of each stock is Xi = 1 for all i N 1 , that is: N If a portfolio is composed of N stocks each forming 1/N proportion of the portfolio, the return on the portfolio is 1/N times the sum of the returns on the N stocks. To find the variance of the respective portfolios in the 2 markets, we need to use the definition of variance from...
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 Spring '10
 eshmalwi
 Finance, Corporate Finance

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