One month, one correlation
Take this month's 500 scores and next month's residual returns. Rank both, then correlate the ranks. That's the IC.
This month it's about 0.03. The scatter looks like pure noise. For a good signal, it should.
The IC
One number per month
The information coefficient asks one question every month: did higher scores come before higher returns?
Repeat it every month
120 months gives 120 ICs. About one month in four is negative, even though this signal works.
A single month tells you almost nothing. The series is what you judge.
Summarize the series
Four numbers describe it: the mean IC, its volatility, their ratio (the ICIR), and a t-stat.
The t-stat is the ICIR times the square root of the number of months. Here it's well above 3.
Rule of thumbIn equities, a mean IC of 0.02 to 0.05 is a good signal. An IC of 0.20 usually means a bug, very often look-ahead.
Why a tiny IC is enough
The fundamental law says IR ≈ IC × √breadth. Skill per bet is small, so you make many bets.
Log scale, from 10 to 10,000.
Interview checkBreadth counts independent bets. 500 stocks rebalanced monthly is 6,000 bets on paper, but stocks move together, so effective breadth is far smaller. The law gives an upper bound.
One month · 500 stocks · score rank vs next-month residual return
IC this month
0.00
Stocks
500
Variance explained
0.0%
IC squared
Monthly IC · 120 months
Mean IC
0
IC volatility
0
ICIR
0
mean ÷ volatility
t-stat
0
ICIR × √120
0% of months have a negative IC.
The fundamental law of active management
0.95IR = 0.030 × √1,000
Read it both ways. At 10 bets a year you need a huge IC to matter. At thousands of bets, an IC that looks like noise in any one month adds up to a strong information ratio.
The sort
Sort into buckets
The IC is one number. A quantile sort shows where along the score the returns come from.
Ten buckets
Each month, sort the 500 stocks into ten deciles by score. Record each decile's average next-month residual return.
Average those over ten years and you get this shape.
The spread is the strategy
Go long the top decile and short the bottom. D10 minus D1 is the long-short return, here about 1% a month before costs.
That spread, not the IC, is what shows up in P&L.
Check the shape
Signal B has a similar spread, but it all comes from the bottom decile. The other nine are flat.
That's a short-only signal in disguise. Shorts are harder to borrow and often crowded, and 90% of B's scores carry no information.
Monotonic is the bar
Returns should rise more or less steadily from D1 to D10. A little wiggle in the middle deciles is noise. A kink at one end is a finding.
Say it this way"IC is 0.03 with a t of 6. The decile spread is roughly monotonic, with both tails contributing, so it isn't living on the short side."
Signal
Average next-month residual return by decile · % per month
D10 − D1
0
% per month
From the long side
0%
of the spread
Rank corr, decile vs return
0
1.0 = perfectly monotonic
The clock
How long does the signal last?
IC at one horizon isn't enough. The decay curve sets how long you hold, and how long you hold sets what you pay in costs.
Measure IC further out
Correlate today's score with returns 1, 2, up to 12 months ahead.
Residual momentum fades slowly. Its IC is still useful several months out, so you don't have to act on it immediately.
Compare a fast signal
Short-term reversal starts with a higher IC, but it's mostly gone after one month.
To capture it, you have to trade every month, and in practice more often than that.
Autocorrelation sets turnover
Residual momentum's score this month correlates about 0.9 with last month's, because 10 of its 11 months overlap. Most of the top quintile stays there.
Reversal's score is close to uncorrelated month to month, so its book turns over almost completely.
Match the holding period to the decay
Hold too briefly and you pay costs for alpha you'd have kept anyway. Hold too long and the signal is gone before you exit.
Rule of thumbA slow signal can be smoothed or traded less often at little cost. A signal that's gone before you can trade it isn't one you can use. Page 6 puts prices on this.
IC by horizon · months ahead
Residual momentum
Short-term reversal
Residual momentum
Score autocorrelation0.90
Top quintile kept0%
Short-term reversal
Score autocorrelation0.00
Top quintile kept0%
The novelty test
Is it new?
A signal can predict returns and still be worth nothing, if it's a factor you could buy or a signal you already run.
Run page 1's test
Take residual momentum's monthly long-short returns and regress them on the Fama-French three factors.
Alpha is about 7% a year, with a t-stat of about 3. None of the three factors explains it.
Add plain momentum
Momentum claims part of it, with a loading of about 0.3. Alpha falls to about 4% a year, and the t-stat stays above 2.
So residual momentum is related to momentum, but it isn't the same thing. That's what you hoped to see.
New alpha can still add little
Suppose your book already runs a signal with an IR of 0.6, and the new one also has an IR of 0.6. What it adds depends on how correlated the two are.
At 0.8 an equally good signal barely moves the combined IR.
The mathThe new information is the part of the new signal your old one can't explain: (IR₂ − ρ·IR₁) ÷ √(1 − ρ²). The combined IR is √(IR₁² + new²).
The verdict
Two questions decide whether a signal that predicts is also worth adding.
Say it this way"It keeps about 4% alpha after FF3 plus momentum, and it's 0.5 correlated with what we run, so it adds about 0.1 to the book's IR before costs."
Residual momentum long-short · 120 months · simulated
Factors in model
Alpha, per year
0
Alpha t-stat
0
R²
0
Same test as page 1. There it showed a strategy was mostly factors. Here most of the return survives, which is the result you want.
Adding a signal to a book · information ratios
Existing signalIR₁
0.60
New signal aloneIR₂
0.60
New informationafter removing IR₁'s part
0
Combined book√(IR₁² + new²)
0
Gain from adding it
+0.00
to the book's IR
A signal that predicts · is it worth adding?
Does alpha survive the known factors?
No
It's a factor. Its returns are worth what the factor costs to buy, which is close to nothing.
Yes
Is it highly correlated with what you already run?
Yes
Small marginal value. Orthogonalize it against your book and test what's left.
No
Real addition. Take it to page 4 and try to kill it.