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Unformatted text preview: Statistics: Var(x) = E[x – E(x)]2 In our case: Var(RP) = E[RP – E(RP)]2 301 Note however, to use this, first we must find RP and E(RP). So, using the assumption about equal weights and then substituting in the known equation for Ri: 1 N 1 RP = N RP = ∑R ∑ i (0.10 + β F + ε i) 1 N RP = 0.10 + β F + ∑ε i Also, recall from Statistics a property of expected value, that is:
~ ~~ If: Z = aX + Y ~ ~ where a is a constant, and ~ , X , and Y are random variables, then: Z ~ ~ ~ E(Z) = E(a )E(X) + E(Y) and E(a) = a Now use the above to find E(RP): 1 εi E(RP) = E 0.10 + βF + N 1 E(ε i ) E(RP) = 0.10 + β E(F) + N 1 0 E(RP) = 0.10 + β (0) + N E(RP) = 0.10 ∑ ∑ ∑ Next, substitute both of these results into the original equation for variance: Var(RP) = E[RP – E(RP)]2
1 Var(RP) = E 0.10 + βF + ∑ ε i  0.10 N 1 Var(RP) = E βF + N 2 ∑ ε 2 1 Var(RP) = E β 2 F 2 + 2βF N ∑ 1 ε+ 2 N (∑ε ) 2 2 2 1 1 Var(RP) = β 2 σ 2 + σ 2ε + 1  Cov(ε i , ε j ) N N 302 Finally, since we can have as many stocks in each market as we want, in the limit, as N → ∞ , 1 → 0, so we get: N Var(RP) = β 2σ 2 + Cov(ε i,ε j) and, since: Cov(ε i,ε j) = σ iσ jρ (ε i,ε j) and the problem states that σ 1 = σ 2 = 0.10, so: Var(RP) = β 2σ 2 + σ 1σ 2ρ (ε i,ε j) Var(RP) = β 2(0.01) + 0.04ρ (ε i,ε j) So now, summarize what we have so far: R1i = 0.10 + 1.5F + ε 1i R2i = 0.10 + 0.5F + ε 2i E(R1P) = E(R2P) = 0.10 Var(R1P) = 0.0225 + 0.04ρ (ε 1i,ε 1j) Var(R2P) = 0.0025 + 0.04ρ (ε 2i,ε 2j) Finally we can begin answering the questions a, b, & c for various values of the correlations: a. Substitute ρ (ε 1i,ε 1j) = ρ (ε 2i,ε 2j) = 0 into the respective variance formulas: Var(R1P) = 0.0225 Var(R2P) = 0.0025 Since Var(R1P) > Var(R2P), and expected returns are equal, a risk averse investor will prefer to invest in the second market. b. If we assume ρ (ε 1i,ε 1j) = 0.9, and ρ (ε 2i,ε 2j) = 0, the variance of each portfolio is: Var(R1P) = 0.0225 + 0.04ρ (ε 1i,ε 1j) Var(R1P) = 0.0225 + 0.04(0.9) Var(R1P) = 0.0585 Var(R2P) = 0.0025 + 0.04ρ (ε 2i,ε 2j) Var(R2P) = 0.0025 + 0.04(0) Var(R2P) = 0.0025 Since Var(R1P) > Var(R2P), and expected returns are equal, a risk averse investor will prefer to invest in the second market. 303 c. If we assume ρ (ε 1i,ε 1j) = 0, and ρ (ε 2i,ε 2j) = .5, the variance of each portfolio is: Var(R1P) = 0.0225 + 0.04ρ (ε 1i,ε 1j) Var(R1P) = 0.0225 + 0.04(0) Var(R1P) = 0.0225 Var(R2P) = 0.0025 + 0.04ρ (ε 2i,ε 2j) Var(R2P) = 0.0025 + 0.04(0.5) Var(R2P) = 0.0225 Since Var(R1P) = Var(R2P), and expected returns are equal, a risk averse investor will be indifferent between the two markets. d. Since the expected returns are equal, indifference implies that the variances of the portfolios in the two markets are also equal. So, set the variance equations equal, and solve for the correlation of one market in terms of the other: Var(R1P) = Var(R2P) 0.0225 + 0.04ρ (ε 1i,ε 1j) = 0.0025 + 0.04ρ (ε 2i,ε 2j) ρ (ε 2i,ε 2j) = ρ (ε 1i,ε 1j) + 0.5 Therefore, for any set of correlations that have this relationship (as found in part c), a risk adverse investor will be indifferent between the two markets. 9. a. In order to find standard deviation, σ , you must first find the Variance, since σ = from Statistics a property of Variance:
~~ If: ~ = aX + Y Z Var . Recall ~ ~ where a is a constant, and ~ , X , and Y are random variables, then: Z ~ ~ ~ Var(Z) = a 2 Var(X) + Var(Y) and: Var(a) = 0 The problem states that returngeneration can be described by: Ri,t = α i + β i(RM) + ε i,t 304 Realize that Ri,t, RM, and ε i,t are random variables, and α i and β i are constants. Then, applying the above properties to this model, we get: Var(Ri) = β i2 Var(RM) + Var(ε i) and now we can find the standard deviation for each asset: σ 2 = 0.72(0.0121) + 0.01 = 0.015929 A σA = 0.015929 = .1262 or 12.62% σ 2 = 1.22(0.0121) + 0.0144 = 0.031824 B σB = 0.031824 = .1784 or 17.84% 2 σ C = 1.52(0.0121) + 0.0225 = 0.049725 σC = b. 0.049725 = .2230 or 22.30%
Var(ε i ) → 0, so you get: N From the above formula for variance, note that as N → ∞ , Var(Ri) = β i2 Var(RM) So, the variances for the assets are: σ 2 = 0.72(.0121) = 0.005929 A σ 2 = 1.22(.0121) = 0.017424 B
2 σ C = 1.52(.0121) = 0.027225 c. We can use the model: R i = RF + β i( R M – RF)
which is the CAPM (or APT Model when there is one factor and that factor is the Market). So, the expected return of each asset is: R A = 3.3% + 0.7(10.6% – 3.3%) = 8.41% R B = 3.3% + 1.2(10.6% – 3.3%) = 12.06% R C = 3.3% + 1.5(10.6% – 3.3%) = 14.25%
We can compare these results for expected asset returns as per CAPM or APT with the expected returns given in the table. This shows that assets A & B are accurately priced, but asset C is overpriced (the model shows the return should be higher). Thus, rational investors will not hold asset C. 305 d. If short selling is allowed, rational investors will sell short asset C, causing the price of asse...
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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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