Risk constraint expression for factor risk as percentage of total risk

[Yubo Bao]

Hi all,

I'm trying to implement a optimizer for multi-factor model that constraint factor risk as percentage of total risk (which is a combination of factor and specifici risk). 

    eg. constraint: factor risk % < 45% (of total risk)

What's the expression for this constraint?

The example I see only shows constraint expression for total risk in terms of absolute value:

https://nbviewer.org/github/MOSEK/PortfolioOptimization/blob/main/python/notebooks/ch5_factor_model_large.ipynb

Thanks,

Yubo

[Michal Adamaszek]

Hi,

It is not possible in full generality because an inequality between two quadratic forms (or norms) like that is not convex. 

Perhaps you can heuristically optimize for some tradeoff combination of

factor_risk + lambda*spec_risk

and depending on the other constraints it may lead to a desired balance, but directly comparing the difference or quotient of those is not possible.

Best,

Michal