The third family refuses to name anything. Give it a covariance matrix of asset returns and it extracts the linear combinations that explain the most variance, in order, each uncorrelated with the ones before. Those are the principal components, and using them as factors is a statistical factor model.
The mathematics is exact and, on a well-chosen matrix, hand-computable. Take five assets that are all correlated with each other at 0.55 and nothing else. The correlation matrix has eigenvalues 1 + (n-1)ρ = 3.20 once and 1 - ρ = 0.45 four times, summing to 5 as they must. The first component explains 3.20 of 5, or 64.0% of the variance, and its loadings are equal on all five names. The equal-weighted market portfolio was not assumed anywhere; it fell out of the algebra. This is the strongest thing statistical models do. They find structure without being told what to look for.
What are the limits of principal components as factors?
The four remaining eigenvalues are all 0.45, and that tie is the weakest thing statistical models do. When eigenvalues are equal, any rotation of that subspace is an equally valid set of components. The second, third and fourth "factors" are not identified at all; the software will print four vectors and any orthogonal rotation of them would have been just as correct.
Break the symmetry and the problem changes shape without going away. Take four assets in two pairs, correlated at 0.80 within a pair and 0.40 across pairs. The eigenvalues are 2.60, 1.00, 0.20 and 0.20, so the first component takes 65.0% and the second 25.0%, ninety percent between them. The second component's loadings are (+0.5, +0.5, -0.5, -0.5): it is long one pair and short the other. That is a real, stable, interpretable spread.
The mathematics will not tell you whether it is value against growth, domestic against exporter, or two sectors that happened to move apart in this sample. You supply the name, from outside the model, and your name might be wrong.
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When is a statistical factor model worth using?
Statistical models earn their place in two situations. They are the right tool when you genuinely do not know the factor structure, as in a new asset class or a cross-market book. And they are a useful audit on a fundamental model: if your named factors explain far less variance than the leading principal components do, something systematic is moving your portfolio that you have not named.
Three costs come with them. The components drift as the sample changes, they cannot be traded directly without a construction step, and, worst for a risk report, a factor with no name cannot be discussed, hedged with intent or explained to a client.