So the question asks:
Consider three uncorrelated stocks in the market. Each stock has variance 1. The expected returns are given by $2, 3 $ and $ 5$ respectively. Find the optimal mean-variance portfolio min ($wCw^T-wm^T$) under the feasible portfolio constraint $w1 + w2 + w3 = 1$ and the fixed risk constraint $σ^2_V = 2$.
So so far I have : Let ri be the random variable associated with the rate of return for asset i, for i = 1, 2,3, and define the random vector $z=(2, 3, 5)^T$
Set µi = E(ri) and µ1 = 2, µ2 = 3, µ3 = 5
$m = (2,3,5) ^T$ , and cov(z) = Σ.
If w = (w1, w2,w3) then T is a set of weights associated with a portfolio, then the rate of return of this portfolio
r = Σ from i=1 to n $r_i w_i$
is also a random variable with mean $m^Tw$ and variance $w^ TΣw$. In the Markowitz theory an optimal portfolio is any portfolio solving the following quadratic program:
M minimize $1 /2w^ TΣw $
subject to $m^Tw ≥ µb$, and $e^Tw = 1$
where e always denotes the vector of ones, i.e., each of the components of e is the number 1.
The KKT conditions for this quadratic program are :
0 = Σw − λm − γe (1)
µb ≤ $m ^Tw,$ $e^ T w = 1$, 0 ≤ λ (2)
$λ^ T (m^Tw − µb) = 0 $ (3)
Are so far my solution right? But I really have no idea how to solve the system, I think the answer will be ($w,λ,γ$), right?