Return Horizon Choice and Signal-to-Noise

Suppose you have a signal — say, a valuation ratio — and you want to know how good it is at predicting returns. The first decision you make, often without thinking about it, is what return window to test it against: tomorrow's return? Next week's? Next month's? That single choice can make an identical signal look useless or look excellent, because the "true" relationship between signal and return unfolds over its own natural horizon, and testing at the wrong one buries the signal in noise it never had to fight.

The core idea: matching the window to the decay

A signal that predicts returns has a real, but not infinite, useful life. A short-term order-flow imbalance might only say something meaningful about the next few minutes; a valuation ratio might say something meaningful about the next several months. Test the order-flow signal against next month's return and its genuine one-minute effect is drowned out by weeks of unrelated market movement — signal-to-noise collapses even though the relationship is real at its native horizon. Test a slow valuation signal against tomorrow's return, and you throw away almost all of its predictive content, because one day of price action has little to do with a multi-month reversion story.

Signal-to-noise here means the ratio of variance explained by the signal to the total variance of the return being predicted. Each extra period added to the window contributes roughly the same amount of unrelated noise variance, while the signal's genuine explanatory power only accumulates over periods where it's actually still informative. Stretch the window past the signal's natural decay, and every additional period is pure noise added to a numerator that has stopped growing — so the ratio falls.

What this looks like in practice

A common workflow is to test the same signal against a grid of horizons — 1 day, 5 days, 21 days, 63 days — and plot a statistic like the information coefficient at each one. The horizon where that relationship peaks is a rough estimate of the signal's natural decay: not a free parameter to tune until a result looks good, but itself a piece of information about how the signal works. An IC that peaks at 5 days and decays smoothly on either side looks like a real process; an IC that only looks good at one oddly specific horizon among many tested, with no coherent pattern around it, is a warning sign of having searched until noise produced a good-looking number.

Worked example. A signal's information coefficient (IC) measured at four horizons: 1 day, IC = 0.01; 5 days, IC = 0.05; 21 days, IC = 0.03; 63 days, IC = 0.01. The peak at 5 days, with a smooth rise and fall on either side, is a coherent picture of a signal whose real predictive content decays over roughly a week — testing it at 1 day nearly misses the effect, and testing it at 63 days has already mostly decayed away. This is a stronger basis for trusting the signal than the raw 5-day number alone, since the pattern around it looks like a real decay curve, not an isolated lucky number.

Horizon choice also interacts with trading costs: a signal that only "works" at a horizon you can't actually trade at — because rebalancing that often would be eaten by costs — isn't tradeable regardless of its statistical strength, so the practical horizon is the best statistical one intersected with what your cost structure allows.