# Markowitz implied returns

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. Oct 13, 2014 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.)

If you know or can estimate the correlation structure of your assets, you can use Kritzman et. al. 2008 and the closed form solution mentioned therein for the calculation of the implied returns and thus inverting Markowitz:

Given a mean-variance optimized fully invested portfolio X, a risk aversion parameter L and a var-covar matrix C. the closed form solution for the vector E that depends on the "expected returns" M is:

E = L C X^T + (-L+ 1 C^{-1} M^{T})/(1 C^{-1} 1^T} 1^T)

Note that E, M and X (your portfolio weights) are vectors, C is a matrix and L a scalar. You can use the matrix functions in excel to calculate it given your covariance matrix and portfolio weights vector.