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

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.

• If there is any hope of someone answering ( I don't have time to read the section of the paper right now but that doesn't mean I could answer if I did. I have a feeling that others don't have time either ), you should be more specific about the factor model that you are referring to. There are variations and the variation probably matters to the answer. Nevertheless, it's an interesting pdf at a quick glance. Thanks for posting. Mar 29 at 10:45
• The factor model is irrelevant. Whether you use a cross-sectional, time series, or statistical model at the end of the day you will end up with asset-level factor exposures, forward-looking factor returns, and a factor VCV matrix. The question is around optimization once those variables have been determined. Mar 29 at 12:47
• okay. hopefully I can find a chance to read the relevant pages. thanks. Mar 30 at 2:40

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.