Thirty-One Signals, Six Bets · part 5 of 10

§ 3

The central result: 31 signals are 6 bets

What survived the neutral construction is the other half of Grinold's fundamental law. IR = IC × √breadth pays the square root only if the bets are independent. Thirty-one signals that all say "buy what went up" are one bet with thirty spare copies.

We built 31 signals across six declared families — momentum, reversal, volatility, liquidity, technical, shape — from the published anomaly literature. Each states the prior fixing its sign before seeing data, because choosing signs afterwards is 31 free parameters and would fit anything. Two pairs are deliberate near-duplicates, planted as positive controls for the correlation matrix.

Controls before conclusions

The instrument was tested against a signal that is the forward return (must measure IC ≈ 1) and against pure noise (must measure IC ≈ 0 and effective breadth ≈ the signal count). An instrument that cannot see a perfect edge cannot be trusted when it reports none, and one that finds structure in noise would flatter every result in this paper.

3.1No individual signal has a detectable edge

Figure 1 — Rank IC t-statistics, 31 signals, both universes Each row is one signal. Filled marks are 112 large caps; open marks are 110 small caps. The dashed lines are |t| = 2. Nothing crosses in large caps; one signal does in small caps. t-statistics are deflated by √(horizon/stride) to account for overlapping 21-session forward windows — without that correction they would be twice as large, which is the difference between "nothing here" and two apparent discoveries.
Signals at |t| ≥ 2, large
01.6 expected by chance across 31 tests
Signals at |t| ≥ 2, small
1illiquidity, t = 2.06
Signs matching prior
13 / 31binomial p = 0.24 — a coin flip

The one signal that crosses is illiquidity in small caps. Against 1.6 expected by chance, one hit is not a discovery — and it is the same trade §2.5 already ruled out, since the measured spread on that universe is 14.27 bps against an edge of about 3 bps. The single signal that shows up is the one that cannot be harvested.