Turning a Score Into an Expected Return

A model ranks a stock in the top decile on some score. Portfolio construction, though, does not consume ranks — it consumes an expected return in percent, because that is the unit position sizing and risk budgeting need to trade off against volatility and correlation with everything else in the book. The step that turns "this ranks well" into "expect +0.6% over the next month" is a deliberate modelling choice, and different choices produce meaningfully different books even when the underlying ranking is identical.

Three ways to make the jump, and what each assumes

Linear scaling (the Grinold approach). Expected return is set proportional to the standardised score, scaled by the signal's information coefficient and the stock's volatility. This assumes the relationship between score and return is a straight line all the way through the distribution — a name at $z=3$ gets exactly three times the expected return of a name at $z=1$. Simple, and usually a reasonable approximation near the centre of the distribution.

Binning into deciles or quantiles. Rather than assuming linearity, group names into buckets (deciles are typical) and assign each bucket the historical average realised return of names that landed in that bucket in the past. This makes no linearity assumption at all — it lets the data say whether the relationship flattens out, curves, or even reverses in the tails.

A fitted nonlinear function. Regress historical realised returns on the score directly, allowing for curvature (a spline, or a simple quadratic term), and read the expected return off the fitted curve for any new score. This captures whatever shape actually exists but requires enough history to fit reliably, and risks fitting noise in the tails where data is sparse.

A worked example: checking the assumption

A researcher bins five years of a signal's history into deciles and computes the average forward monthly return realised by names in each decile.

Decile	Avg. score $z$	Realised avg. return
1 (lowest)	−1.8	−0.31%
3	−0.6	−0.09%
5 (middle)	0.0	+0.02%
7	+0.6	+0.14%
10 (highest)	+2.1	+0.24%

A straight line through the middle deciles fits well — returns rise roughly proportionally with score from decile 3 through 7. But the top decile breaks the pattern: a score of +2.1 (versus +0.6 in decile 7) would predict roughly 3.5× the return under a linear extrapolation, i.e. close to +0.49%, but the realised average is only +0.24% — barely more than decile 7 delivered. The relationship flattens at the top.

A linear (Grinold-style) mapping applied naively here would systematically overstate expected return, and therefore position size, in the very names the model is most excited about. The fix used in practice is to fit the binned relationship directly, or apply a linear mapping only up to some score threshold and cap the assigned expected return beyond it.

Drag the points to add outliers far from the fitted line: this is what the top-decile flattening looks like in miniature — the OLS line, fit mostly by the dense middle of the data, extrapolates poorly to the sparse extremes, exactly where a mis-specified score-to-return mapping does the most damage.

Where it bites

This choice matters most for strategies with concentrated, high-conviction positions, where a handful of extreme-scoring names carry disproportionate portfolio weight — an overstated mapping in the tails directly inflates the risk taken in exactly those names.