How to Maximize Portfolio Sharpe Ratio using Lagrange Multipliers in a Factor Model

Question

I've come across the notes of the 2003 lecture "Advanced Lecture on Mathematical Science and Information Science I: Optimization in Finance" by Reha H. Tutuncu.

It describes on page 62 in section 5.2 a way to reformulate the tangency portfolio to the efficient frontier as a quadratic optimization problem:

$$ \min_{y,\kappa} y^T Q y \qquad \text{where} \quad (\mu-r_f)^T y = 1,\; \kappa > 0 $$

I'm wondering if anyone has seen an adaptation or similar work to incorporate a factor model. I believe an adaptation for the $y$ vector will need to take place but I'm unsure what that would be.

Answer 1

I think I figured it out. If:

• $\Sigma$ is the factor VCV matrix (m assets by m assets)
• $f$ is the factor exposures (n factors by m assets)

Then you can re-create the security level VCV matrix with just:

$$ f\Sigma f^T $$

Similarly if:

• $\mu$ is the factor returns (n factors by m assets)

You can recreate the security level returns by just:

$$ \mu f^T $$

Now that you have security level risk and return parameters you can apply the technique mentioned above in the linked paper.