I'm reading the Barra risk model handbook (2004) available online and trying to understand the methodology. I've read a few materials on portfolio theory, so I can get at least the theoretical ideas behind a multi-factor risk model. But when you try to design one in reality, how do you test its predictions of the covariance between the assets? For an alpha model, you'll observe the subsequent realized alphas, and it's no big deal to compare your predictions with them. But isn't the realized covariance matrix practically impossible to compute fully (which is one of the reasons for decomposing the returns to a smaller number of factors in a risk model)? If so, how can we test the predictions of a risk model?
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$\begingroup$ Personally if BARRA showed me that for a sample of 25 representative stocks that I choose, the model risk is close to the historical covariance for those pairs of stocks I would be satisfied (without needing to look at all pairs out of 2500 stocks). And I would send the same 25 names to Axioma and Northfield as well and compare results. TBH I dont know if BARRA tests all pairs or not... $\endgroup$– nbbo2Jun 15, 2022 at 10:31
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$\begingroup$ @nbbo2 Thank you! I was counting this as a practical possibility too. But I feel that there's gotta be some robust test method, at least behind the scene, since in that case their models would certainly be more convincing, both for their clients and for themselves. Not sure though. $\endgroup$– whoknowsnotJun 15, 2022 at 10:38
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$\begingroup$ You can make @nbbo2 's comment more price and compare the implied multivariate distribution to the observed multivariate distribution. If there's a stastically significant difference in distributions, either the model is wrong (i.e. its assumptions are rejected), or the calibration is not representative for the observed sample. $\endgroup$– KermittfrogJun 15, 2022 at 10:56
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