# Sign retention in mean variance optimization

The mean variance optimization to the objective: $$h^T\alpha - \lambda h^T V h$$ results in the solution:

$$h = \frac{V^{-1} \alpha}{2 \lambda}$$

Would a positive value for an asset in $$\alpha$$ result in a positive value in the weights vector $$h$$ ?

• what is $\alpha$? – develarist Nov 22 '19 at 21:39
• Expected returns. This is the signal generated for tomorrow's returns. – whisperer Nov 25 '19 at 19:35

The sign of the portfolio weight $$h_n$$ assigned to one asset cannot be solely determined by the sign of that asset's expected return, $$\alpha_n$$, due to the model having to also take as an input that asset's dependence structure with the other $$N-1$$ assets being considered, captured within the covariance matrix $$V$$. Those other assets are also vying for inclusion in the portfolio.
• Mathematically the weights are determined by $h=\frac{1}{2\lambda} V^{−1} \alpha$. Knowing that a particular element $\alpha_i$ of the vector $\alpha$ is >0 does not guarantee that the corresponding element $h_i$ is >0. That is just not how matrix multiplication works. $h_i$ depends on the entire row $i$ from the matrix and all the elements of $\alpha$. – noob2 Apr 24 at 2:27