Assuming that we have k factors and n stocks

Then Barra provides a k x k factor covariance matrix F, and a k x n factor exposure matrix E.

At runtime, we also observed a n x 1 vector r of individual stock return

My question is, how should we calculate the k x 1 vector of factor return f?

r =  E^t * f


r = E^t * F * f

basically, we need to solve a regression problem to get f . But I am not sure which of these 2 formula should I use.

IIRC it should be the second form, but I am not so sure.

Can someone please help to clarify?



1 Answer 1


IMHO, without additional risk sources, this looks like an exercise in linear algebra.

Given observed $r$ of dimension $n\times 1$, known $E$ of dimension $k \times n $ and unknown $f$ of dimension $k \times 1$, where $k\leq n$, we find:

$$ \begin{align} r&=E^Tf\\ \Rightarrow\left(EE^T\right)^{-1}Er&=f \end{align} $$

  • $\begingroup$ thanks. my question was actually, which of the following 2 functions is correct? it was r = E^t * F * f or r = E^t * F * f should we include the factor covariance F ? $\endgroup$
    – eight3
    Commented Jul 9 at 11:35
  • $\begingroup$ If you add the covariance matrix to the mix, you obtain orthonormalized factor returns. I do not have good knowledge about what is "right" there. $\endgroup$ Commented Jul 9 at 11:50
  • $\begingroup$ if we do not include F, we will have raw factor return as in bps, right ? $\endgroup$
    – eight3
    Commented Jul 9 at 11:54
  • $\begingroup$ Hi: you need to multiply the $k \times n$ factor exposure matrix by the factor returns estimated from the previous month which is a $1 \times k$ vector. Call it $FR$. Barra used to provide them and probably still does ? Then $FR \times E$ will give you an estimate of the total systematic factor return due to each stock. $\endgroup$
    – mark leeds
    Commented Jul 12 at 16:45

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