macrocephalopod

Now instead of estimating the ~2 million parameters of a 2000 x 2000 stock covariance matrix, you just need to estimate ~800 parameters of a 40 x 40 factor covariance matrix -- your risk model just got a whole lot simpler.

If you're doing this you would normally treat some factors as "risk" factors which have zero expectation (i.e. you want to hedge them to minimise risk) and some factors as "alpha" factors which have positive expectation, as well as risk --

you want exposure to the alpha factors, but you want *more* exposure to the ones which have higher expected return given the risk, and your factor model gives you a structured way to express that.

The vector of alphas, and the stock covariance matrix, are the key inputs to a portfolio optimizer (along with transaction costs, financing costs, position and turnover constraints, risk constraints etc). One particularly important fact is that portfolio optimizers are

known to behave *really* badly (recommending nonsense portfolios) when there is noise in the covariance matrix. You get noise when you try to estimate too many parameters from too little data, so using a factor model to reduce the number of parameters to estimate