In trying to evaluate a portfolio allocation, I want to do a reverse markowitz, that is find the implied returns that would result in the current allocation to be optimal. This results in some of the implied returns to be negative, which i have a hard time grasping since all the weights are positive. Am I missing something in my calculation or could this in fact be the case?

Classical markowitz is w=E[r]*E[O] where r is a 1*n vector of returns, and O is the NN covariance matrix. Im simply isolating for E[r] instead by E[r]=wE[O]^-1.

  • This might be a better question for the Quantitative Finance exchange. – rhaskett Oct 13 '14 at 17:35

The optimal portfolio will be optimised for high return and low variance and/or covariance. So, in order to select negatively correlated assets to lower the covariance, sometimes an asset with some negative returns may be included.

To make a simplified example. If assets A & B are perfectly negatively correlated their combined volatility will be zero. That's the same volatility as cash. Even though asset B has a negative average return, since the combined holding returns more than cash, in theory you could invest in both A and B, and even leverage the investment by borrowing cash, and the investment would be low/zero risk (based on the historical performance).

(Perfect negative correlation doesn't typically happen.)

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