Combining Signals

A desk running five signals faces a question that looks trivial and isn't: how much weight does each one get in the combined score used to rank stocks? Weight them equally and you've thrown away information about which signals are actually better. Weight them by an optimizer's best estimate of their historical Sharpe ratios and covariances and you've likely overfit to noise, because five years of monthly signal data is a very short sample to estimate a five-by-five covariance matrix precisely. Every real combination scheme lives somewhere between these two extremes.

Three approaches, in order of how much they trust the data

Equal weighting. Add up the z-scored signals and divide by the count. This throws away any information about which signal is more reliable, but it also can't overfit — there are no parameters to estimate, so there's nothing for estimation error to corrupt. It is a surprisingly strong baseline, and in several published comparisons of combination methods, equal weighting beats "optimal" weighting out-of-sample often enough that any fancier method has to actively prove it's better, not just assume it is.

IC-weighted combination. Weight each signal by its historical information coefficient, so a signal with IC of 0.06 gets roughly twice the weight of one with IC 0.03. This uses more information than equal weighting but needs only one number per signal (not a full covariance matrix), so it's less exposed to estimation error than a full optimization while still differentiating between a strong and a weak signal.

Regression / mean-variance combination. Formally optimize the weights to maximize the combined signal's information ratio, which requires estimating not just each signal's own IC but the full covariance matrix of ICs across signals — how correlated their predictive errors are with each other. This is the theoretically "correct" answer given the true parameters, and the worst-performing answer in practice when the parameters are noisy estimates from a short history, because the optimizer aggressively overweights whichever signal happened to look best (or least correlated with the others) in-sample, which is often just noise.

Worked example. A desk has two signals, both standardized to mean zero, standard deviation one across the stock universe. Signal A has historical IC of 0.06 and signal B has IC of 0.03; their forecast errors have a correlation of 0.4. Equal weighting gives the combined score $s = 0.5s_A + 0.5s_B$. IC weighting gives $w_A = 0.06/(0.06+0.03) = 0.67$ and $w_B = 0.33$, so $s = 0.67s_A + 0.33s_B$ — tilted toward the stronger signal but still using both. A full mean-variance optimization, accounting for the 0.4 error correlation, might push the weight on A to 0.85 or higher, because the optimizer sees B as mostly redundant once A's 0.4-correlated component is accounted for — but that 0.85 is only as trustworthy as the IC and correlation estimates feeding it, and both were estimated on the same handful of years of data.

Think of this fitted line as the mean-variance optimizer's job: find the combination of inputs that best explains the outcome. The residual scatter around the line is the part no combination of signals explains — and the tighter the historical sample, the more that residual scatter gets mistaken for a genuine, exploitable pattern rather than noise the optimizer happened to fit.

Why the fancier method often loses

The core problem is degrees of freedom. Estimating five signal weights well requires estimating not five numbers but the entire covariance structure between them — with $n$ signals that's roughly $n(n-1)/2$ correlation terms plus $n$ variances, and each one is estimated on the same short, overlapping history the signals themselves were built on. Small errors in those estimates get amplified by the optimization, which actively searches for whatever combination fit best in-sample — including combinations that fit best purely by chance. This is the same overfitting mechanism that makes naive mean-variance portfolio optimization notoriously unstable (see Factor Risk Models): more free parameters than the data can reliably pin down.

What this means in practice

Most production desks land on something between IC-weighting and lightly shrunk mean-variance combination, re-estimated infrequently (quarterly or annually, not daily) specifically to avoid chasing noise in the weights themselves. A weighting scheme that changes dramatically month to month is a sign the estimation is unstable, not that the signals' true quality is genuinely shifting that fast.

In interviews

Present the three methods in order of how much estimation risk each one carries, and be explicit that "more sophisticated" doesn't mean "better out-of-sample" — this is the point interviewers are testing for. If asked to combine two signals given their ICs and a correlation, compute the IC-weighted combination by hand, then note that a full mean-variance answer would need the correlation of forecast errors, not just of the raw signals, and that estimating it reliably takes more data than most desks have.