Lab — pedagogical demos

The signal is collective

Reproducible demos behind the article: how a tiny shared edge invisible at the strategy level becomes a confident signal at the portfolio level.

The mathematics

Consider N strategies whose per-period excess returns share a common signal ε and a one-factor noise structure:

x_{i, t} = ε + ρ F_{t} + 1 - ρ Z_{i, t}, F_{t}, Z_{i, t} \sim iid N (0, 1) .

Each x_i,t has mean ε, variance 1, and pairwise correlation ρ. The Sharpe ratio of any single strategy is mean / std = ε; over T periods, the t-statistic is

t_{strategy} = T ε .

Now form the equal-weight portfolio

\overset{x}{ˉ}_{t} = \frac{1}{N} i = 1 \sum N x_{i, t} = ε + ρ F_{t} + 1 - ρ \overset{ˉ}{Z}_{t},

where $\overset{ˉ}{Z}_{t} = \frac{1}{N} \sum_{i} Z_{i, t}$ has variance 1/N. The portfolio mean is still ε. Its variance, however, has shrunk:

Var (\overset{x}{ˉ}_{t}) = ρ + \frac{1 - ρ}{N} = \frac{1 + ( N - 1 ) ρ}{N} .

Hence the portfolio t-statistic over T periods is

t_{portfolio} = \frac{T ε}{( 1 + ( N - 1 ) ρ ) / N} = T ε \frac{N}{1 + ( N - 1 ) ρ} .

The diversification factor √(N / (1 + (N−1)ρ)) is what does the work. As ρ → 0 it grows like √N; as ρ → 1 it saturates at 1. For modest ρ = 0.05 and N = 50 the factor is ≈ 5.0, so the portfolio t-stat is five times the per-strategy t-stat.

Why this is the article’s central claim, mathematically

The strategy and the portfolio are computed from the same data. They sample the same noise. The signal ε is identical in both. What changes between them is purely the variance of the noise — the signal-to-noise ratio scales by the diversification factor. This is the toy demonstration of the empirical claim in Edge is in the Process: at strategy level, ε is undetectable; at portfolio level, the same ε is a statistically reliable edge.

Worked example

Take N = 50, T = 252, ε = 0.04, ρ = 0.05. Plug into the formulas:

E[t_strategy] = √252 · 0.04 ≈ 0.635. The empirical histogram of strategy t-stats peaks near this value with standard deviation ≈ 1; positive-fraction is ~74%, > 2 fraction is < 5%.
E[t_portfolio] = 0.635 · √(50 / (1 + 49·0.05)) = 0.635 · √(50 / 3.45) ≈ 2.42. Empirical positive-fraction is > 99%; > 2 fraction is around 65%.

The interactive demo below confirms these numbers in real time. Sweep ε to zero and watch both distributions collapse onto the standard normal — the diversification advantage vanishes when there is no shared signal to extract.

Demo — The signal is collective

Per-period model: x_i = ε + √ρ · F + √(1−ρ) · Z_i. Same ε goes into every strategy; portfolio averages dilute idiosyncratic noise.

N strategies50

T periods252

ε (signal)0.040

ρ (cross-corr)0.05

trials2000

seed=9

E[t] strategy

0.63

E[t] portfolio

2.42

mean t̂ — strat

0.66

mean t̂ — port

2.43

P(t > 0) strat

75%

P(t > 0) port

99%

P(t > 2) strat

8.9%

P(t > 2) port

66.6%

Gray: distribution of t̂ across 2000 draws of a single random strategy. Amber: distribution of t̂ across 2000 draws of an N=50 equal-weight portfolio of strategies, same data-generating process. The mean of the amber distribution exceeds the gray by exactly the diversification factor √(N / (1 + (N−1)ρ)).

Figures

39 BTC strategies and the equal-weight portfolio cumulative trajectory — Fig. 1 —The toy model made empirical on real data: 39 BTC strategies (top quartile by mean per-day P&L, scaled by per-strategy std) plotted as cumulative trajectories (light grey spaghetti) alongside the equal-weight portfolio (amber). Individual strategies are noisy and many spend years underwater; the population mean compounds the small shared positive drift into a clean upward trajectory.

Empirical portfolio t-statistic as a function of N with the diversification-factor fit — Fig. 2 —Empirical portfolio t-statistic as N grows from 1 to 39, averaged over 80 random sub-samples per N (band: 10-90% across draws). The dashed teal curve is the theoretical scaling t₁ · √(N / (1 + (N−1)ρ̄)) with the empirical mean pairwise correlation ρ̄ ≈ 0.08. Single-strategy t ≈ 0.47 — a strategy you would dismiss as noise — lifts to t ≈ 1.4 at N = 39, exactly tracking the diversification-factor prediction.

Why this matters for systematic strategies

The toy model is an oversimplification — real strategy returns are not jointly Gaussian with constant ρ — but the qualitative claim is robust. As long as a strategy population has any shared, mean-positive signal that is not perfectly correlated across strategies, equal-weighting (or any sensible weighting) amplifies it. The corollary is that strategy selection by individual t-statistics throws away the amplification: you keep only strategies that are individually significant, but the strategies that carry the most signal are typically not individually significant — they are individually shaped to extract a small piece of the shared edge.

This is why the firm’s production pipeline never selects strategies on individual statistical significance. It selects on filter-conditioned population statistics — Daru Finance’s proprietary regime-confidence diagnostic, the regime-conditional Sharpe, the FDR-controlled subset under M/02. The signal lives in the collective.

Reproducibility

DaruFinance / signal-is-collective

Python — open source reference implementation

Minimal invocation

import numpy as np
from signal_is_collective import simulate

results = simulate(N=50, T=252, eps=0.04, rho=0.05, trials=2000, seed=9)
print(f"Mean strategy t = {results.strategy_t.mean():.2f}")
print(f"Mean portfolio t = {results.portfolio_t.mean():.2f}")
# Reproduces the article's headline numbers exactly.

References

[1]Gatto, D. V. (2026). Edge is in the Process. daru-finance.com / SSRN companion article.
[2]Sharpe, W. F. (1994). The Sharpe ratio. Journal of Portfolio Management 21(1), 49–58.
[3]Fama, E. F. (1976). Foundations of Finance. Basic Books, New York.

All projects View on GitHub