An interesting side note in linear factor models

19 May, 2026

Linear factor models

Linear factor models of stock returns are commonly used by practitioners and academics. The topic is well covered (see Elements of Quantitative Investing, or this post), so I won't repeat the basics, but instead dig a little more into an interesting result that I've seen mentioned a few times but never fully explored.

Pure factor portfolios as estimators

Our model for stock returns is

where $r$ is an $N$-dim vector of observed stock returns, $X$ is an $(N,k)$ matrix of known factor loadings, $f$ is an unobservable $k$-dim vector of latent factor values and $ϵ$ is an $N$-dim vector of idiosyncratic stock residuals.

Given observed $r$, known $X$, and $D$, an ex-ante diagonal covariance matrix for $ϵ$, we can estimate values for $f$ using weighted-OLS, i.e.

where $W=D^{-1}$ is a diagonal matrix of weights inversely proportional to the variances of the idio-residuals.

The solution to this is the WOLS estimator for $f$:

which we can also write as

where $Q=WX(X^{T}WX{)}^{-1}$. These estimated factor values are given by the returns of portfolios with holdings equal to the columns of $Q$, which we call pure factor portfolios. They are "pure" in the sense that the pure factor portfolio corresponding to factor $j$ has exposure 1 to factor $j$ and zero exposure to all other factors:

Pure factor portfolios as minimum risk portfolios

The interesting equivalence, often mentioned in passing, is that these pure factor portfolios can also be constructed as the minimum risk portfolios with exposure 1 to a given factor and zero exposure to all other factors, i.e. we can find the $j$-th factor's pure portfolio from

Firstly let's prove this result, then discuss it in more detail.

Using the standard method of Lagrange multipliers we first form the Lagrangian

the FOCs are

and

solving for $\lambda$ gives

which gives us

which is just the $j$-th column of $Q$ - simple.

Why the equivalence?

To me this equivalence is surprising, and I haven't yet seen a simple explanation. The two routes to the pure factor portfolios seem completely disconnected - firstly as WOLS estimators, and secondly as minimum risk portfolios.

BLUE Portfolios

One route to understand the equivalence is to think about what properties we want our factor value estimator to have. Let's say that we are generally interested in a linear estimator, i.e. we can write our estimator as

where $A$ is some $(k,N)$ matrix.

We almost certainly want $\tilde{f}$ to be "unbiased", in that the expected value is equal to the true factor values, or equivalently in the limit of zero residual noise we get the correct result. If we write

we can see that if $ϵ=0$ then in order for $\tilde{f}=f$ we need

which is the same as saying that row $j$ of $A$ must have unit exposure to factor $j$ and zero exposure to all other factors.

We would also like $\tilde{f}$ to give us the "best" estimate of $f$, in the sense that our estimates have the minimum possible variance. We can write this as

and therefore saying that we want our estimator to have the lowest variance is equivalent to

which is exactly equivalent to our minimum risk formulation above. Of course, the solution to this is given by

where $W=D^{-1}$.

Summary

Looking at this now the result is actually quite understandable. Saying that the pure factor portfolios are minimum risk portfolios is equivalent to saying that they are the Best Linear Unbiased Estimators (BLUE) for our factor values, and minimum risk equates to minimum variance in our estimates. This is essentially just a rederivation of Gauss-Markov through a portfolio and risk based lens.