# How do you optimize an all equity portfolio while getting around the multicollinearity issue?

I am trying to optimize a portfolio of domestic and international equity funds. However being that they are very highly correlated it doesn’t really help. Is there a way to optimize and find an allocation which essentially provides the maximum diversification while minimizing risk? Could one translate these funds into factor and diversify across the factors then translate that back to weights for the portfolio? Perhaps using PCR?

• I have a suspicion that if you start with 100% allocation to a broad global portfolio like VT ( investor.vanguard.com/etf/profile/VT ) you already have a well diversified solution. You may want to see if adding any one other fund to the mix improves the situation. The "mean variance spanning test" of Huberman and Kandel 1987 offers a way to examine this question. Oct 23, 2020 at 13:13

This is a matter of your preference aka utility function. Perhaps the simplest reasonable approach would be the constrained Markowitz allocation $$w=argmax(w^T\mu - kw^T\Sigma w)\quad s.t.\quad w_i\ge 0,$$ where $$\mu$$ is the vector of funds' performances, $$\Sigma$$ is their covariance, and $$w$$ is the vector of your allocations. The weights will be inversely proportional to the risk aversion coefficient $$k$$. To make this quadratic programming problem more stable, you may want to add a sufficiently generous constant to the $$\Sigma$$'s diagonal (ridge or Tikhonov regularization). If the correlations are less than 1.0, you will get a modest improvement relative to an equal-weight allocation. I expect a PCA regression will give a noisier allocation unless you regularize it somehow.