How I do a reweight a portfolio, given a particular correlation matrix? I have a portfolio of 14 distinct instruments.
Some of the instruments returns are correlated, which means the portfolio is not as diversified as it seems.
Is there a general way to down-weight instruments which are highly correlated, and vice-versa, to make a portfolio as diversified as it can be?
 A: A classic question in portfolio theory is what's the minimum variance portfolio of risky assets?
Let $\Sigma$ be the covariance matrix for the returns of your set of assets. Let $\mathbf{w}$ be a vector denoting the portfolio weights. The minimum variance portfolio is the solution $\mathbf{w}^*$ to the optimization problem:
$$
\begin{equation}
 \begin{array}{*2{>{\displaystyle}r}}
 \mbox{minimize (over $w_i$)} & \mathbf{w}'\Sigma \mathbf{w} \\
 \mbox{subject to} & \sum_i w_i = 1
 \end{array}
\end{equation}
$$
Or if you have a no short selling constraint:
$$
\begin{equation}
 \begin{array}{*2{>{\displaystyle}r}}
 \mbox{minimize (over $w_i$)} & \mathbf{w}'\Sigma \mathbf{w} \\
 \mbox{subject to} & \sum_i w_i = 1\\
    & w_i \geq 0
 \end{array}
\end{equation}
$$
The main problem in this exercise is that a poorly estimated covariance matrix $\Sigma$ of risky assets or a $\Sigma$ that changes over time (which it almost certainly the case) may yield garbage estimates of $\mathbf{w}^*$  going forward. Estimates of $\Sigma$ aren't very precise, and if you treat $\Sigma$ as if it were estimated without any error, you may get garbage results.
Also be aware that the returns of financial assets tend to have huge cross-sectional correlation. Diversification averages out idiosyncratic risk but it cannot eliminate systemic risk. As you increase the number of assets, the variance of the minimum variance portfolio does not go to zero because of economy-wide, systemic risk: $ \lim_{n\rightarrow \infty} \min \mathbf{w'}\Sigma\mathbf{w} \neq 0$.
