Formulating Mean-Variance Optimization with Factor Models

Summary

The document asks how to adapt the traditional mean-variance portfolio objective when expected returns and covariance are specified for factors rather than directly for assets. It proposes combining factor returns, asset weights, factor exposures, and the factor covariance matrix, but does not establish that the proposed expression is correctly dimensioned or mathematically equivalent to standard portfolio optimization.

One answer points to a CVXPY example as a source for a factor-model formulation. Another suggests treating the factors as assets in a conventional mean-variance problem, arguing that their expected returns would then influence optimal allocations. The discussion is brief and supplies no derivation, numerical example, or comparison of formulations. Readers should consult the referenced example and verify how exposures, residual asset risk, constraints, and expected returns are represented before applying a model; the replies do not resolve those details.

Key ideas

• A factor-model portfolio objective must map factor returns and risk through asset exposures.
• The proposed objective is not validated in the discussion.
• A response points to a CVXPY example for a factor-based formulation.
• Another suggestion is to include factors as assets in a standard mean-variance optimization.
• The short exchange does not address residual risk, constraints, or provide a worked derivation.

Tags

# Mean-variance optimization - objective function formation with factor models


# Mean-variance optimization - objective function formation with factor models












Tradition mean-variance optimization uses the following objective function in optimization:

$$ \mu w^T - \lambda w^T \Sigma w $$

Which I'm trying to adapt to a factor model. I've come up with:

$$ f \mu w^T - \lambda w^T \Sigma w f f^T $$

where:

- $f$ is the factor loadings (exposures)

- $\lambda$ is the risk aversion parameter

- $\mu$ is the factor returns

- $\Sigma$ is the factor variance-covariance matrix

- $w$ are the asset weights

Is this correct? I've tried to find literature detailing this adjustment but have not found anything. Thanks.

## Answer by Lisa Ann (score 1, accepted)

https://quant.stackexchange.com/a/74350

This summary can be found among the CVXPY examples, here.

## Answer by phdstudent (score 0)

https://quant.stackexchange.com/a/74349

That's not ideal. Why don't you just do the standard MV problem but including the factors themselves as assets? Since they generate positive alpha, the mean-variance problem will give them $w>0$, and your tangency portfolio would be higher.

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.