macrocephalopod

The return of stock i on day t is modeled as a linear combination of the unknown factor returns plus an unknown residual, where the dependence of stock i on factor j on day t is given by the *known* exposure X(i, j, t)pic.twitter.com/CbdvNRGura

If we understand that r, f and epsilon are vectors and X is a matrix then we can drop all the subscripts and write it in matrix notation which is much easier. It's important to understand that the exposures X are known in advance, and its the factor returns f that are unknownpic.twitter.com/uKK7PuGjuQ

Now you have a set of linear equations on each day, and you can solve the linear equations to get the vector of factor returns for each day using the normal equation -pic.twitter.com/YwVkUzSM69

What's the point of this? The factor models tells you how the covariance matrix of stocks is related to the covariance matrix of factors - it's the sum of the covariance due to factor exposures - Σ_f - and the covariance of the residuals Ω which is normally assumed to be diagonalpic.twitter.com/H7ruwNaEtR

Even more interestingly, if you are not just a risk manager but a quant equity pm, you can take expectations and get a model for the expected return (or alpha) of each stock in terms of the expected returns on each factor --pic.twitter.com/JflXE3Qj8V