# Project #18648 - portfolio anaysis

4.5 Optimal Portfolio Choice IV

Consider a financial market with only two risky assets S1 and S2, whose returns we denote by R1 and R2. For i = 1, 2, let μi denote the expected return on the risky asset Si, σi2 its variance and ρ the correlation between the two assets. In

other words, μi = E(Ri), σi2 = var(Ri) and ρ = cov(Ri,Rj). Consider an σ1 σ2

individual investor who optimizes her portfolio P according to the mean-variance criterion and denote by wi, i = 1, 2, her portfolio weights on the two risky assets. We shall assume that short sales are allowed, so wi (i = 1, 2) can be positive, zero or negative.

1. What are the expected value μP and the variance σP2 of the return RP on the investor’s portfolio P? (hint: compute the portfolio’s expected return μP andvarianceσP2 asfunctionsofwi,μi,σiandρ(i=1,2)).

2. What is the equation of the minimum variance frontier of portfolios com- posed of the two risky assets S1 and S2 only? Show that this minimum vari- ance frontier can be represented by a hyperbola in a (σ(R), E(R)) space. (hint: compute the variance σP2 of a portfolio composed of the risky assets only, as function of its expected return μP and of μi, σi and ρ (i = 1, 2)).

3. Intheaboveframework,assumethatμ1 =4,μ2 =1,σ1 =4,andσ2 =1. Compute the variance of the minimum variance portfolio obtained when ρ = 0 and ρ = 0.5.

4. What happens to the minimum variance portfolio when the two assets are perfectly negatively correlated, i.e. ρ = 1? What is its variance in this case?

5. Intheaboveframework,assumethatμ1 = 2,μ2 = 1,σ1 = 2andσ2 = 1. Plot the minimum variance frontiers corresponding to the cases ρ = 1, ρ = 1/2, ρ = 0 and ρ = 1 (use the same graph for the four cases). What happens in the limiting cases ρ = 1 and ρ = 1

 Subject Business Due By (Pacific Time) 12/03/2013 12:00 am
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