Signal Combination & Decay

Here is the dirty secret of quantitative investing: your best signal is barely correlated with what stocks actually do next. If you plotted “what the model predicted” against “what really happened,” you would see a shapeless cloud with the faintest upward tilt — a relationship so weak that a fundamental investor would laugh you out of the room. And yet that whisper of an edge, applied across thousands of bets and stacked alongside other faint whispers, is the entire business. The quant doesn’t find one loud signal; they find many quiet ones and add them up.

This lesson is about three numbers that govern that game: how strong a signal really is (the information coefficient), how to combine several weak ones into something tradable, and how fast each signal goes stale — because a signal you can’t act on before it decays, or can’t afford to trade often enough to capture, is worth exactly nothing.

The information coefficient (IC)

Analogy. Imagine a weather forecaster who is right 51% of the time about tomorrow’s rain. On any single day, useless — basically a coin flip. But let them bet a dollar on rain every day for thirty years, and that 1% edge quietly drains the casino. A trading signal is the same: the IC measures how much better than a coin flip your forecast is, and on any one stock that edge is invisible. The money comes from making the bet ten thousand times.

Definition. The information coefficient is the (rank) correlation between a signal’s predictions and the subsequently realized returns. Cross-sectionally, you rank every stock by your signal today, rank them by their actual return over the next period, and correlate the two rankings:

$\text{IC} = \operatorname{corr}\big(\text{signal}_t,\; r_{t+1}\big).$

IC ranges from $-1$ (perfectly wrong) through $0$ (no skill) to $+1$ (perfect foresight). Real equity signals live in a humblingly narrow band: a typical IC is 0.02 to 0.05. An IC above 0.10 is suspiciously good and usually means you’ve leaked future information into the backtest.

Why a ‘weak’ correlation is worth a fortune. Recall the Fundamental Law of Active Management from the first lesson: your information ratio is roughly your skill per bet times the square root of how many independent bets you make.

$\text{IR} \approx \text{IC}\,\sqrt{\text{breadth}}$

Breadth is the lever. A microscopic IC, swung across enough independent bets, produces a serious IR.

Worked example. Take a signal with $\text{IC} = 0.03$ and a strategy that makes 500 independent bets per year (say, ~500 names rebalanced roughly once a year, or fewer names traded more often — what counts is independent bets):

$\text{IR} \approx 0.03 \times \sqrt{500} = 0.03 \times 22.36 \approx 0.67.$

An IR of 0.67 is a perfectly respectable, fundable strategy — built entirely out of a 3% correlation. Step through the breadth ladder and watch the IR climb:

Read the last two rows together: bumping IC from 0.03 to 0.05 at fixed breadth lifts the IR from 0.67 to 1.12. Skill is precious — but so is breadth, and breadth is usually the cheaper lever to pull.

When to use it

IC is your signal’s report card — measure it out of sample before you ever trade. A stable, positive out-of-sample IC (even 0.02) is a green light; an IC that’s huge in-sample but evaporates out-of-sample is the classic fingerprint of overfitting. Use IC to rank and compare candidate signals, and use IC·√breadth to forecast whether the whole strategy can clear a fundable IR before you build it.

Combining signals

Analogy. One slightly-loaded die barely beats fair. But roll five independently-loaded dice and average them, and the law of large numbers sharpens your edge — the noise in each cancels while the bias reinforces. Combining alphas works the same way: each raw signal is mostly noise plus a sliver of truth, and averaging independent signals cancels the noise faster than it cancels the truth.

The magic of orthogonality. Suppose you have two signals, each with the same IC, and their errors are uncorrelated. Averaging them keeps the full signal but halves the independent noise, so the combined IC rises. For two equally-good, independent signals:

$\text{IC}_{\text{combined}} \approx \text{IC}\,\sqrt{2}.$

More generally, blending $n$ equally-good, mutually uncorrelated signals scales the combined IC by $\sqrt{n}$ — the exact same $\sqrt{\,\cdot\,}$ diversification math that governs bets in the Fundamental Law, now applied to signals instead of positions. Two signals at IC 0.03 combine to ≈ 0.042; four to ≈ 0.06; nine to ≈ 0.09.

Why a correlated signal adds almost nothing. If a new signal is highly correlated with what you already trade, it carries no new information — it just re-states a bet you’ve already made. The $\sqrt{n}$ bonus assumes independence; correlation eats it. Adding your tenth momentum variant is nearly free of value, while adding a genuinely different signal (a quality metric, a sentiment score, a positioning gauge) is where the IC actually jumps.

Combination methods, weakest temptation last.

Worked example. You hold three signals, each IC ≈ 0.03, pairwise uncorrelated. Equal-weight them:

$\text{IC}_{\text{combined}} \approx 0.03 \times \sqrt{3} = 0.03 \times 1.732 \approx 0.052.$

You nearly doubled your effective skill — from 0.03 to 0.052 — without finding a single new idea, just by combining what you had orthogonally. Now suppose those three were actually 0.6-correlated with each other. The diversification benefit shrinks dramatically; the effective $\sqrt{n}$ collapses toward $\sqrt{1}$, and the combined IC barely exceeds 0.03. Same three signals, wildly different payoff — orthogonality is the whole game.

When to use it

Reach for equal-weight as the default and the benchmark every fancier method must beat. Graduate to IC-weighting only when you have stable, out-of-sample IC estimates and several signals of genuinely different quality. Save regression/ML for when you have lots of data, strong cross-validation discipline, and a real prior reason to expect interactions — and even then, hold equal-weight in your back pocket as the honest baseline.

Alpha decay

Analogy. A signal is a melting ice cube. The moment it forms it’s at full size, and it predicts the very near future sharply — but every day you wait, it shrinks. A “fast” signal (a tiny cube) is intense but gone in days; a “slow” signal (a big cube) lasts for months but was never as cold to begin with. Wait too long to act and you’re holding a puddle.

Definition / formula. A signal’s predictive power fades geometrically as the holding horizon $h$ lengthens. Model the IC at horizon $h$ as an exponential decay:

$\text{IC}(h) = \text{IC}_0\, e^{-h/\tau}$

where $\text{IC}_0$ is the IC at the shortest horizon and $\tau$ (the decay constant, in days) sets how fast it bleeds away. A small $\tau$ is a fast signal — predictive for a few days, then noise; a large $\tau$ is a slow signal — predictive further out but typically weaker to start.

The headline summary statistic is the signal half-life — the horizon at which the IC has fallen to half its initial value:

$\text{half-life} = \tau \ln 2 \approx 0.693\,\tau.$

Worked example. A signal has $\tau = 8$ days. Its half-life is:

$\text{half-life} = 8 \times \ln 2 = 8 \times 0.693 = 5.5\ \text{days}.$

So after ~5.5 days the signal is half as predictive; after another 5.5 days, a quarter; and so on. Tabulating $\text{IC}(h)/\text{IC}_0 = e^{-h/8}$:

By day 20 the signal retains 8% of its bite — effectively noise. If your operational reality (capital, liquidity, compliance) means you can only act on this signal a week after it fires, you’ve already surrendered more than half its value before you place the trade.

When to use it

Estimate every signal’s decay curve, not just its peak IC. The half-life tells you the natural clock of the signal: a 2-day half-life demands intraday or daily rebalancing and tight execution; a 60-day half-life is a patient, low-turnover position. Match your trading infrastructure to the half-life — a fast signal in a slow execution pipeline is alpha thrown in the bin. This decay profile feeds directly into the trade-off below.

The decay-vs-turnover trade-off

This is the heart of the lesson. Alpha decay pushes you to trade fast — act on the signal while it’s still fresh. Transaction costs push you to trade slow — every rebalance pays the round-trip spread, and re-trading constantly bleeds you dry. These two forces fight, and the winner is a hump: net alpha per unit time is maximized at an optimal holding horizon in the middle.

Analogy. Picking fruit. Pick too early (trade too fast) and you’re back at the orchard every day paying the entrance fee — turnover cost — before the fruit has ripened. Wait too long (trade too slow) and the fruit rots on the branch — the signal decays to noise. There’s a sweet spot in the middle where the fruit is ripe and you’ve amortized the trip.

The mechanism. Hold for $h$ days and you capture more of the cumulative signal — gross alpha grows as $A_0(1 - e^{-h/\tau})$ — but you pay one round-trip cost spread over those $h$ days. Net alpha per day is:

$\text{net/day}(h) = \frac{A_0\big(1 - e^{-h/\tau}\big) - \text{cost}}{h}.$

Tiny $h$: the cost term, divided by a tiny $h$, dominates and net/day craters. Huge $h$: you’ve captured nearly all of $A_0$ but you’re dividing by a big $h$, so net/day fades. In between sits the peak $h^*$.

Two rules of thumb fall straight out of the formula:

• Faster decay (smaller τ) → rebalance faster. A perishable signal can’t wait; act before it’s gone.
• Higher cost → rebalance slower (hold longer). When each round trip is expensive, you must hold long enough to earn back the spread.

Worked example — find the optimal holding horizon. A signal’s full capturable gross alpha is $A_0 = 30$ bps, with decay constant $\tau = 8$ days, and the round-trip cost is $10$ bps. Compute net alpha per day, $\frac{30(1 - e^{-h/8}) - 10}{h}$, across candidate horizons:

Trade daily (h = 1) and you lose money — the 10-bp round trip dwarfs the 3.5 bps of gross alpha you’ve captured in one day. Hold 20 days and you’ve banked most of the alpha but spread it thin, so net/day sags back to 0.88. The peak sits near h ≈ 10 days, where net alpha per day is highest at ~1.14 bps/day. That horizon — not “as fast as possible” and not “as slow as possible” — is the cost-aware optimum, and it directly sets your rebalance frequency and your turnover.

Putting it together

The full pipeline reads top to bottom: take many raw signals, each with a humble IC of 0.02–0.05; orthogonalize and combine them into a composite alpha (equal-weight as the robust default — the √n diversification bonus only fires on independent signals); measure the composite’s decay curve to get its half-life; then set the rebalance frequency at the cost-aware optimum, where net alpha per day peaks. That chosen horizon fixes your turnover — and turnover, scaled up by capital, is what determines your capacity, the bridge to the next lesson.

You now know how to turn a pile of weak, perishable signals into one tradable composite and how often to trade it. But every answer here quietly assumed your bets stay independent and your trades stay cheap — and both assumptions break down as you scale up or as other funds pile into the same trade. What happens when everyone discovers the same signal at once, when crowding collapses your breadth and turns a placid strategy into a stampede? That violent question — capacity, crowding, and the Quant Quake — is where we head next.