What it shows: the Fundamental Law of Active Management says your Information Ratio ≈ skill × √breadth — IR ≈ IC × √BR. Combining many signals is a way to buy skill and breadth, but only independent breadth counts. This simulator equal-weights N signals (each with its own IC) at a chosen average pairwise correlation and shows the closed-form composite IC = c·√N ⁄ √(1 + (N−1)ρ), its ceiling c/√ρ, and the effective (independent-equivalent) breadth N ⁄ (1 + (N−1)ρ).
All figures are illustrative — nothing here is a performance claim or investment advice. The equal-weight, single-common-factor model is a teaching simplification; production signal combination uses covariance-aware weighting and shrinkage. Related tools: Information Coefficient Calculator · Backtest Overfitting Simulator · Cointegration & Pairs Trading Simulator · Signal Decay Calculator.
Combine some signals
Set how many weak signals you're stacking, how predictive each one is on its own, and how correlated they are with one another. The combined Information Coefficient updates live — watch it run into the ceiling as you add more correlated signals.
how many candidate signals you equal-weight into one
each signal's own correlation with the forward return (0.02–0.05 is typical)
0 = the signals are independent · high = they all say the same thing
for the implied Information Ratio: IR ≈ composite IC × √breadth
You've hit the correlation ceiling
Each signal on its own has an IC of 0.030. Equal-weighting 25 of them gives a composite IC of 0.049 — 1.63× a single signal. But at a pairwise correlation of 0.35, your 25 signals only carry the information of 2.7 independent ones — so the composite is stuck near its ceiling of 0.051 (96% of the way there). Adding a tenth, a hundredth, a thousandth correlated signal barely moves it. Independent signals would have reached 0.150 instead.
What the combination buys you
Composite IC
0.049
1.63× a single signal
Single-signal IC
0.030
what you started with
If independent (ρ=0)
0.150
c·√N — the breadth ideal
Ceiling (∞ signals)
0.051
c/√ρ — more never beats this
Effective breadth
2.7
of 25 nominal
Diversification ratio
1.63×
composite ÷ single
Implied IR
1.09
IC × √500
IR if independent
3.35
the breadth you left behind
Reading this
The Fundamental Law of Active Management says IR ≈ IC × √breadth. Stacking signals is a way to buy breadth — but only independent breadth counts. Correlated signals repeat the same bet, so 25 of them at ρ=0.35 act like just 2.7. The real edge in signal research isn't finding more signals; it's finding signals that are different from the ones you already have. Chase orthogonality, not count.
More signals, less and less payoff
If the signals were independent (grey dashed) the composite IC would keep climbing as c·√N. At your ρ (blue), it bends over and flattens into the ceiling c/√ρ — the wall that more correlated signals can't break.
Why it caps: correlation eats your breadth
Effective breadth is how many independent signals your correlated set is worth. Independent signals sit on the 45° line (N signals = N bets); correlated ones flatten at 1/ρ no matter how many you add.
And it's hard to even measure
The composite's true IC is 0.049 (teal line), but any single backtest over ~1000 observations measures it with huge noise: across 160 samples the readings average 0.000 and 0% came out negative. A real edge this small is easy to lose in sampling noise — which is the whole problem the overfitting and IC significance tools exist to police.
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