# Project #18922 - Optimal Portfolio Choice

Please see problem 4.2 in the attachment

We consider optimal portfolio choice in a financial market with one risk-free asset B and two risky assets S1 and S2. The return on the risk-free asset B equals Rf and we denote by Ri, i = 1,2, the return on the risky asset Si, i = 1,2. 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 individual investor who optimizes her port- σ1 σ2

folio P according to the mean-variance criterion and denote by wi, i = 1, 2, her portfolio weights on the two risky assets.

1. What are the expected value and the variance of the return RP on the in- vestor’sportfolioP?(hint:computetheportfolio’sexpectedreturnμP and varianceσP2 asfunctionsofRf,wi,μi,σiandρ(i=1,2))

2. The investor is trying to solve the following optimal portfolio choice prob- lem: “choose a portfolio P which minimizes the variance var(RP ) for a given level of expected return μ.” Write this problem in mathematical terms and show that it is an optimization problem under constraints.

3. The solution to the optimization problem in 2.2 depends on the same con- stants a, b, c defined in class, and the variance of any portfolio on the mini- mum variance frontier takes the form

P (μRf)2 var(RMV)= a2bRf +cRf2.

4.3. OPTIMALPORTFOLIOCHOICEII 21 Plot the above frontier in the space of standard deviations and mean returns,

(σ(RP ), E(RP )).

1. Assume that no short sales on the risky assets S1 and S2 are allowed. What is
the minimum variance frontier of portfolios composed of the two risky assets
S1 and S2 only? Show that this minimum variance frontier can be represented by a hyper-
bola in the space of standard deviations and mean returns,
(σ(RP ), E(RP )).
Given that no short sales are allowed, which parts of the previous hyperbola
correspond to feasible portfolios?

2. Consider a portfolio P t whose expected return equals E(RtP ) = a bRf .

bcRf

Show that this portfolio lies both on the minimum variance frontier com- puter in 2.3 as well as on the minimum variance frontier computed in 2.4. What is the weight that the portfolio P t puts on the risk-free asset B?

3. Intheaboveframework,assumethatμ1 = 2,μ2 = 1,σ1 = 2andσ2 = 1. Plot the minimum variance frontiers of risky assets only (i.e. the one computed in 2.4) 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?

4. We now assume that ρ = 1/2 and that the risk-free interest rate Rf equals 20%. Compute and plot the minimum variance frontier with risk-free and risky assets derived in 2.3. Compute the expected return E(RtP ) and the variance var(RtP ) of the portfolio P t

 Subject Business Due By (Pacific Time) 12/06/2013 12:00 am
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