Measuring a signal: IC, breadth and the fundamental law

SIG · Chapter 312 min readAsked at Two Sigma, Citadel, AQR, DE Shaw

Assumes OLS from three angles.

After this lesson you should be able to

  • Define the information coefficient and the information ratio.
  • Use the fundamental law to trade off signal quality against breadth.
  • Explain why an IC of 0.030.03 can be a very good signal.

A signal is judged by how well it ranks what comes next and by how many independent bets it lets you take. The fundamental law of active management ties those two into one number, and it explains most of what separates a research desk from a stock picker.

Definition 3.1

The information coefficient

IC, IC=Corr(signalt, returnt+1)\mathrm{IC} = \mathrm{Corr}\big(\text{signal}_t,\ \text{return}_{t+1}\big) — The cross-sectional correlation between your forecast and the realised forward return, usually computed per period and then averaged. Rank correlation is the common choice, since it is robust to the outliers that dominate a Pearson estimate on returns.

Proposition 3.2

What a good IC looks like

An IC of 0.020.02 to 0.050.05 is a real, tradeable equity signal. Anything above 0.100.10 on a liquid universe should be assumed to be a bug — look-ahead, survivorship, or a return that is inside the signal — until you have proved otherwise. The corresponding R2R^2 is IC2\mathrm{IC}^2, so a 0.050.05 signal explains a quarter of one per cent of the variance, and that is fine.

Holds when

  • The bar depends on horizon: intraday signals can carry higher ICs because there is less noise to fight per unit time.
  • ICs are noisy. The standard error of a mean IC over TT periods is roughly 1/T⋅N1/\sqrt{T \cdot N}, so judge it with an error bar.

Equation 3.3

The fundamental law of active management

The information ratio you can achieve is the quality of each forecast times the square root of how many independent forecasts you make.

IR≈IC×BR\mathrm{IR} \approx \mathrm{IC} \times \sqrt{\mathrm{BR}}
IR\mathrm{IR}
Information ratio: active return divided by tracking error.
BR\mathrm{BR}
Breadth: the number of *independent* bets per year.

Example 3.4

One researcher has an IC of 0.050.05 on 500 names rebalanced monthly. Another has an IC of 0.300.30 on four macro calls a year. Who has the better information ratio?

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Worked solution

  1. Formula
    IR=ICBR\mathrm{IR} = \mathrm{IC}\sqrt{\mathrm{BR}}
  2. Substitute
    BR1=500×12=6000,BR2=4\mathrm{BR}_1 = 500 \times 12 = 6000, \qquad \mathrm{BR}_2 = 4
  3. Solve
    IR1=0.056000≈0.05×77.5≈3.9\mathrm{IR}_1 = 0.05\sqrt{6000} \approx 0.05 \times 77.5 \approx 3.9
  4. IR2=0.30×2=0.6\mathrm{IR}_2 = 0.30 \times 2 = 0.6
  5. Answer
    the cross-sectional signal, by a factor of about six\text{the cross-sectional signal, by a factor of about six}

Sanity check. The 6,000 figure is optimistic — names within a sector are far from independent — but even at a tenth of that breadth the conclusion holds. Breadth is why systematic research exists.

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