TD - Riskless and risky investments and the market price of risk

Question
Let O be the opportunity set, in risk/return space, for a set of risky assets
(none of which are perfectly negatively correlated). Assume that short selling is allowed and that there is also a riskless investment with return RF
available. Suppose that an investor choses to invest X ≥ 0 in a purely risky
portfolio P, with variance σ
2
P
and expected return RP, and 1−X in the riskless investment. Show that as X varies the investor’s total portfolio lies along
a line of slope (RP − RF )/σP and intercept RF . Hence or otherwise, deduce
that the problem of finding the capital market line reduces to maximizing
the slope of the line above all possible portfolios in O.
Consider a situation where there are three risky assets S1, S2 and S3 with
respective expected returns
R1 = 0.1, R2 = 0.12, R3 = 0.18,
variances and covariances given by
σ
2
1 = 0.0016, σ12 = 0.0016, σ13 = 0,
σ
2
2 = 0.01, σ23 = 0.0012, σ
2
3 = 0.0144.
Further, assume that the risk free rate is 0.06, short selling and borrowing
are allowed. Show that under these circumstances, the market price of risk
is
θ =
167
√
13861
∼ 1.418
and that the optimal portfolio of risky assets consists of the following proportions of total wealth invested in S1, S2 and S3, respectively,
237
331
∼ 0.716
12
331
∼ 0.036
82
331
∼ 0.247