TD - Risk free and risky assets and the market price of risk-Quiz & sol

Question
Consider a portfolio Π which is composed of a proportion λ < 1 of a risk free
asset S0 with associated return R0 and a proportion 1−λ of a risky portfolio
S1 with associated return R1 and variance σ
2
1
. Show that as λ varies, Π
lies along a straight line in the risk/reward diagram, the line having slope
θ = (R1 − R0)/σ1. Explain briefly why this implies that the problem of
finding the capital market line reduces to that of maximizing θ over all risky
portfolios.
Now consider a scenario where there are three risky assets S1, S2 and S3 with
respective expected returns
R1 = 0.08, R2 = 0.10, R3 = 0.12.
The variances and covariances between the assets are given by
σ
2
1 = 0.008
σ12 = 0.004
σ13 = 0
σ
2
2 = 0.006
σ23 = 0.002
σ
2
3 = 0.008
and the risk free rate is 0.05. Short selling and borrowing are allowed. Show
that the optimal portfolio of risky assets consists of investing proportions
1/11, 4/11 and 6/11 of one’s total wealth in S1, S2 and S3 respectively. Show
that the associated risk and return are q
520/121 ∼ 2.07% and 120/11 ∼
10.91% respectively, and the market price of risk is
θ =
√
130
4
∼ 2.85