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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.

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2 Answers 2

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

enter image description here

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  • $\begingroup$ 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? $\endgroup$ Jan 13, 2023 at 20:24
  • $\begingroup$ Through $f = F^{\mathrm{T}}w$ the weights are applied to the factor loadings, $F$, to get the factor exposures, $f$. $\endgroup$
    – Lisa Ann
    Jan 13, 2023 at 20:28
  • $\begingroup$ Ah, that makes sense. And then I assume the factor loadings and returns are already present in the term $\mu^T$ $\endgroup$ Jan 13, 2023 at 20:35
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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.

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

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