Using Factor Covariance and Returns in Tangency Portfolio Optimization

Summary

The document asks how to adapt a quadratic formulation of the maximum-Sharpe tangency portfolio to a factor model. Its proposed approach is to first convert factor-level inputs into security-level inputs, then apply the existing portfolio optimization method. It describes reconstructing asset covariance from factor exposures and factor covariance, and reconstructing asset returns from factor returns and exposures.

This is a conceptual suggestion rather than a worked derivation. The note gives no numerical example, empirical test, or detailed treatment of the optimization constraints. Its notation and matrix dimensions are potentially unclear, and the proposal does not discuss residual or idiosyncratic asset risk, which can matter in factor covariance models. Readers should verify the dimensions and model assumptions before using the formulas in an implementation.

Key ideas

• A tangency portfolio can be framed as a quadratic optimization problem with a return constraint.
• The proposed adaptation maps factor-level covariance and returns into security-level inputs before optimization.
• The covariance reconstruction uses asset factor exposures and the factor covariance matrix.
• The note omits a worked example and does not address idiosyncratic risk.

Tags

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


# 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.

## Answer by LattePrincess (score 1, accepted)

https://quant.stackexchange.com/a/75704

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.