One number, many causes
KSTL rose 7.4% in March. That number mixes the market, the type of stock KSTL is, and news about the company itself.
Pulling those apart is the first move in any signal research.
The decomposition
Every return has two parts
Part of a stock's move belongs to the market and to the kind of stock it is. The rest belongs to the stock.
The market's share
KSTL has a market beta of 1.4. The market rose 2.0% in March, so 2.8 points of the move came from the market alone.
Anyone holding an index fund got that part.
Styles explain more
Add size, value and momentum. Each contribution is the stock's exposure times that factor's return for the month.
Together the four factors explain 4.9 of the 7.4 points.
The model
Exposures times factor returns, plus a residual.
r = β₁f₁ + β₂f₂ + … + βₖfₖ + εExposures times factor returns, plus a residual.
What's left is the residual
The last 2.5 points are KSTL's own: an earnings beat, a product launch, or plain noise. Alpha research lives entirely in that green row.
The residual is only as good as your betas. Drag the beta and watch the residual absorb the error.
The true beta is 1.40. Any other value pushes market return into the residual.
Interview checkIf your betas are wrong, your residual still carries factor exposure, and the "alpha" you find there is a factor in disguise.
KSTL · March · return contributions, % points
02468
Explained
0.0
Residual
7.4
Share explained
0%
The three builds
Three ways to build a factor model
They share one equation. They differ in which pieces you know up front and which you estimate.
One equation for every stock
Stack every stock's returns into a matrix R, with stocks down the side and months across.
A factor model writes it as R = B · F + E: exposures times factor returns, plus residuals. You always observe R. The question is what else you know.
Time-series: you know the factor returns
Fama-French factors are built from sorted portfolios, such as small minus big or cheap minus expensive. Their returns are F, and you know them.
For each stock, regress its returns on F through time to estimate its betas. That is one regression per stock, run along a row.
Cross-sectional: you know the exposures
Barra-style models start from characteristics you can see today: industry, size, book-to-price, past returns. Those are B, and you know them.
Each month, regress every stock's return on its exposures to estimate that month's factor returns. That is one regression per month, run down a column.
Statistical: you know neither
PCA pulls B and F out of the return matrix together. The first component explains as much variance as possible, the second explains the most of what's left, and so on.
The factors fit well, but they come without names. "Component 3 loaded" is hard to explain to a PM.
Pick by job
No build wins everywhere. Each one fits a different job on the desk.
Say it this way"For attribution I'd use Fama-French plus momentum. For the risk model, a fundamental model. For residuals in a fast stat-arb book, PCA."
Return matrix R · 8 stocks × 12 months
RReturnsKnown
BExposures?
FFactor returns?
EResidualsLeft over
Three builds · which job each one fits
| Build | Known | Estimated | Factors are | Best for |
|---|---|---|---|---|
| Time-series | F, from sorted portfolios | B, one regression per stock | Few, named | Explaining a strategy's returns |
| Cross-sectional | B, from characteristics | F, one regression per month | Many, named | Risk models and neutralizing |
| Statistical | Nothing but R | B and F together | Few, unnamed | Fast stat-arb residuals |
Trade-off. Time-series betas lag, because they need years of history. Cross-sectional exposures update the day a company changes. Statistical factors fit best in sample and drift the most out of it.
The attribution
Explaining a return stream
Regress any strategy on the factors. The intercept is its alpha. Everything else could have been bought.
A strategy that looks great
Here are ten years of monthly returns averaging about 11.6% a year.
With no factors in the model, the whole average counts as alpha. The grey dashed line is the strategy and the green line is what you'd credit to skill. Right now they're the same line.
Add the market
Regress the returns on the market. A beta of about 0.4 claims part of the return, and alpha drops to about 8.6% a year.
The t-stat is still above 2, so it still looks like skill.
Add size, value and momentum
With all four factors in, the picture changes. The strategy was mostly a value and momentum tilt.
Alpha falls to about 1% a year, with a t-stat under 1. The green line is almost flat.
Say it this way"It earns 11.6%, but about 10 points of that are market, value and momentum exposure. What's left is roughly 1% and not distinguishable from zero."
Alpha depends on the model
Toggle the factors on the panel yourself. Every combination gives a different alpha for the same returns.
So an alpha number means nothing until you name the model behind it.
Rule of thumbNever quote alpha without its model. "4% alpha against CAPM" and "4% alpha against FF5 plus momentum" are very different claims.
Simulated strategy · 120 months · OLS on monthly returns
Factors in model
Alpha, per year
0.0%
Alpha t-stat
0.0
R²
0%
Strategy, cumulative
Explained by factors
Alpha plus residual
The target
Alpha is the residual you can predict
Not the whole residual. Only the small part of it you can see coming.
Most of the residual is news
KSTL's residual swings about 6% a month. Almost all of that is news nobody could forecast.
Alpha is the small predictable part, the green line. It looks tiny, but across hundreds of stocks a sliver this size is a business. Page 3 shows why.
Factor exposure is cheap
Market exposure costs a few basis points a year through an index fund. Value and momentum tilts cost a little more through smart-beta ETFs.
A strategy that is secretly a factor tilt is worth what the tilt costs to buy, which is close to nothing. Funds pay for the residual.
Meet the running signal
Residual momentum ranks stocks on their own recent news instead of their raw returns, following Blitz, Huij and Martens.
It rewards stocks whose residuals have been strong, not stocks that rode a factor. Plain momentum can't tell those apart.
Why skip the last month?One-month returns tend to reverse. Including month t−1 would fight the signal. That reversal effect comes back on page 6 as a costs lesson.
Where this page comes back
Every later page reuses the factor model you just built. Neutralizing, measuring, simulating and sizing all run through B, F and E.
KSTL residual · 48 months · illustrative
Residual volatility
≈ 6% / mo
Predictable part
≈ 0.4% / mo
Correlation
≈ 0.05
between forecast and outcome
Residual return
Predictable part
What each part of a return costs to own
The test. Before you pitch a signal, ask what it would cost to get the same returns from factors alone.
Residual momentum · one stock, one month
t−36t−12t−1
Estimation window
Residuals summed
Skipped
Regress the past 36 months of returns on the Fama-French three factors.B from a time-series regression
Keep the residuals, the part the factors didn't explain.E
Sum the residuals from month t−12 to t−2.skip t−1
Divide by the residual volatility over the same window.score = Σε ÷ σ(ε)
Page 1's model · where it reappears
R = B · F + E
Page 2 · Idea to signalNeutralize the score against
B so it bets only on your idea.Page 3 · Does it predict?Measure IC against residual returns
E, then test whether momentum explains residual momentum.Page 5 · Luck or skillSimulate fake histories from
B · F + E to see what luck looks like.Page 6 · Risk and portfolioBuild the covariance matrix as
B ΣF Bᵀ + D.