I came up with a conjecture, any ideas to prove or disprove it?
I have 5 different stocks: A, B, C, D, E
I have the correlation values for all combinations in the set. I need to make sure that the "net correlation" of the system is minimized.
To do that I will solve the following system:
A*B*CORREL(A,B) = A*C*CORREL(A,C) = A*D*CORREL(A,D) = B*C*CORREL(B,C) = B*D*CORREL(B,D) = C*D*CORREL(C,D)
Is that correct?
Also, is "net correlation" a thing?
No, it's wrong. I think what you mean is something like this. You will buy $x_i$ worth of stock $i$, with $x_1 + \ldots + x_5 = 1$ (say). The return per dollar on stock $i$ is $R_i$, with covariances $\text{Cov}(R_i, R_j)$. You want to minimize the variance in the total return, namely
$$\text{Var}\left( \sum_i x_i R_i \right) = \sum_{i,j} x_i x_j \text{Cov}(R_i, R_j)$$
Assume you have a solution where all $x_i > 0$ (or that you can buy negative amounts). Using a Lagrange multiplier we look at $$ F(x_1, \ldots, x_5, \lambda) = \sum_{i,j} x_i x_j \text{Cov}(R_i, R_j) + \lambda \left(1 - \sum_i x_i\right)$$
EDIT: Taking derivative with respect to $x_i$ gives you the conditions
$$ 2 \sum_j x_j \text{Cov}(R_i, R_j) - \lambda = 0 $$
Thus if ${\bf x}^T = (x_1, \ldots, x_5)$, ${\bf e}^T = (1, \ldots, 1)$, and $C$ is the covariance matrix with entries $\text{Cov}(R_i, R_j)$, $$ {\bf x} = \frac{\lambda}{2} C^{-1} {\bf e} $$