Orthogonalising a Signal Against Known Factors

A new signal earns a decile spread of 40 bps a month. Before celebrating, ask the deflating question: how much of that spread is really value, or momentum, or size, riding along inside the new field because the new field happens to be correlated with names that are cheap, or have been rising, or are small? If a known, already-owned factor explains most of the return, the "new" signal adds nothing to a book that already holds that factor — it is the same bet, badly disguised.

What orthogonalisation actually does

Orthogonalising a signal means regressing it, cross-sectionally, on the known factors you want to strip out, and keeping only the residual — the part of the signal that known factors cannot explain. Concretely: each period, run a regression with your raw signal as the dependent variable and factor exposures (value, momentum, size, sector, whatever you already own) as the independent variables. The residual from that regression, name by name, is the orthogonalised signal.

Geometrically, think of the raw signal as an arrow, and each known factor as another arrow. Some of the raw signal's arrow points in the same direction as value; some points in the same direction as momentum. Orthogonalisation finds the piece of the arrow left over once you subtract away everything that points along a direction you already have exposure to — literally, the projection is removed, and what remains is perpendicular ("orthogonal") to every factor you regressed against.

$s_{\perp} = s - \sum_k \beta_k f_k$

In words: the orthogonalised signal $s_\perp$ equals the raw signal $s$ minus its estimated exposure $\beta_k$ to each known factor $f_k$, summed across all the factors you're stripping out. What's left is, by construction, uncorrelated with every $f_k$ in the regression.

A worked example

A researcher builds a "quality shock" signal from a sudden jump in return-on-equity. Raw decile spread: 35 bps/month, t-stat 2.6 — looks promising. Regressing the signal cross-sectionally each month against value (book-to-price) and momentum (12-1 month return) gives factor loadings of 0.31 on value and 0.18 on momentum: the quality-shock names tend to also be cheap and have been rising.

Take the residual from that regression as the new, orthogonalised signal, and re-run the same decile spread test. The spread drops to 14 bps/month, t-stat 1.4. Most of the original spread was value and momentum riding along; the piece that is genuinely new to a book that already holds those two factors is much smaller — real, plausibly, but a fraction of what the headline number suggested.

	Raw signal	Orthogonalised (vs. value, momentum)
Decile spread	35 bps/mo	14 bps/mo
t-stat	2.6	1.4
Correlation with value	0.31	0.00 (by construction)
Correlation with momentum	0.18	0.00 (by construction)
Marginal value to a book already long value + momentum	Overstated	The honest number

This is not a reason to discard the signal — 14 bps net of two major factors, if it holds up out of sample, may still be worth adding. It is a reason to size the expectation correctly and stop attributing someone else's factor return to your new idea.

Where it bites

Orthogonalisation is required whenever a PM asks "what does this add to what I already run," and it is standard practice before a signal enters a multi-factor blend, so that the combination step isn't secretly double-weighting the same underlying bet under two different names.