Review Multi-Factor Models
Recall that in a multi-factor model, returns,
\mathbf{r}
, are expressed in terms of factor exposures,
\mathbf{B}
, and factor returns
\mathbf{f}
. The part of returns not attributable to factors is called the idiosyncratic return,
\mathbf{s}
.
\mathbf{r} = \mathbf{B}\mathbf{f} + \mathbf{s}
.
If
\mathbf{r}, \mathbf{f},
and
\mathbf{s}
represent data from time
t
, then
\mathbf{B}
is determined from data from before time
t
.
If we have data from multiple time periods, the matrices look like this:
\scriptsize \left( \begin{array}{cccc}
r_{1, 1}&\cdots&\cdots &r_{1, T}\\
\vdots&\ddots&\ddots&\vdots\\
\vdots&\ddots&\ddots&\vdots\\
\vdots&\ddots&\ddots&\vdots\\
r_{N,1}&\cdots&\cdots &r_{N,T}
\end{array}\right)
=
\left( \begin{array}{ccc}
B_{1, 1}&\cdots &B_{1, K}\\
\vdots&\ddots&\vdots\\
\vdots&\ddots&\vdots\\
\vdots&\ddots&\vdots\\
B_{N,1}&\cdots &B_{N,K}
\end{array}\right)
\left( \begin{array}{cccc}
f_{1, 1}&\cdots&\cdots &f_{1, T}\\
\vdots&\ddots&\ddots&\vdots\\
f_{K,1}&\cdots&\cdots &f_{K,T}
\end{array}\right)
+
\left( \begin{array}{cccc}
s_{1, 1}&\cdots&\cdots &s_{1, T}\\
\vdots&\ddots&\ddots&\vdots\\
\vdots&\ddots&\ddots&\vdots\\
\vdots&\ddots&\ddots&\vdots\\
s_{N,1}&\cdots&\cdots &s_{N,T}
\end{array}\right)
where
N
is the number of companies,
K
the number of factors, and
T
the number of time points.