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In: Finance

Suppose there are three assets: A, B, and C. Asset A’s expected return and standard deviation...

Suppose there are three assets: A, B, and C. Asset A’s expected return and standard deviation are 1 percent and 1 percent. Asset B has the same expected return and standard deviation as Asset A. However, the correlation coefficient of Assets A and B is −0.25. Asset C’s return is independent of the other two assets. The expected return and standard deviation of Asset C are 0.5 percent and 1 percent.
(a) Find a portfolio of the three assets that has the smallest variance among all portfolios that yields the expected return of 0.9 percent.
(b) Find a portfolio of the three assets that has the smallest variance among all portfolios that yields the expected return of rp percent. Find the variance of the portfolio.
(c) Suppose the risk-free rate is zero. Find the tangency portfolio.
(d) Suppose an investor’s mean-variance utility function is E(r)−0.005·A·σ2, where A = 500. Find the investor’s optimal portfolio of the three risky assets and the risk-free asset.

Solutions

Expert Solution

Expected Return of assetA=R1 1%
Expected Return of assetB=R2 1%
S1=Standard Deviation of assetA 1%
S2=Standard Deviation of assetB 1%
Variance of asset A=V1                          1 %%
Variance of asset B=V2                          1 %%
Correlation between asset A and B=Corr(1,2) -0.25
Covariance(1,2)=Cov(1,2)=Corr(1,2)*S1*S2=0 -0.25 %%
w1=Investment in asset A
w2=Investment in asset B
Portfolio Return=Rp(Percentage)
w1*R1+w2*R2=w1*1+w2*1=w1+w2 ……..Equation (1)
Vp=Portfolio Variance=(w1^2)*V1+(w2^2)*1+2*w1*w2*(-0.25)
Vp=Portfolio Variance=(w1^2)*+(w2^2)-0.5*w1*w2….....Equation(2)
Sp=Portfolio Standard Deviation=Square root of Variance=SQRT(Vp)
ALL POSSIBLE PORTFOLIOS
w1 w2 Rp=w1+w2 Vp(Using Equation (2)
Weight of Weight of
AssetA AssetB Portfolio Return(%) Portfolio Variance(%%)
0 1 1 1.0000
0.2 0.8 1 0.6000
0.35 0.65 1 0.4313
0.4 0.6 1 0.4000
0.5 0.5 1 0.3750
0.51 0.49 1 0.3753
0.6 0.4 1 0.4000
0.8 0.2 1 0.6000
1 0 1 1.0000
BEST FEASIBLE CAL: Minimum Variance Portfolio
Variance 0.3750 %%
Weight of Asset 1(Stock Fund (S) 50%
Weight of Asset 2(Bond Fund (B) 50%
Portfolio Return 1%

jeff jeffy answered 5 months ago
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