# 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.

## 2 Answers

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

• In this example, the asset weights are only applied to the idiosyncratic component: $w^TDw$ Shouldn't the asset weights be applied to the systematic risk (factor loadings) as well? Commented Jan 13, 2023 at 20:24
• Through $f = F^{\mathrm{T}}w$ the weights are applied to the factor loadings, $F$, to get the factor exposures, $f$. Commented Jan 13, 2023 at 20:28
• Ah, that makes sense. And then I assume the factor loadings and returns are already present in the term $\mu^T$ Commented Jan 13, 2023 at 20:35

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.

• If I used factors as assets I would be solving for factor weights. I'm trying to solve for asset weights. Commented Jan 13, 2023 at 19:35