7.9.2 From Forecasts to Positions

Basic Prerequisites: Statistics & Inference (covariance, OLS, correlation). Builds directly on 01 · From Zero.

1. Intuition & Practical Objective

You almost never trade one model. A real desk runs several signals on the same asset - a momentum model, a value model, a fundamentals model, an alt-data model - and must turn all of them into one position. This page answers two questions:

How do I combine several forecasts into a single best forecast? (forecast combination)
Given a forecast, how big should the position be? (bet sizing)

The intuition for (1): do not trust each model equally. Weight each forecast by how reliable it is - by the inverse of its error variance. A model whose past errors were small gets more weight; a noisy model gets less. This is the forecast-combination analog of the inverse-variance portfolio rule you already know from Risk Parity.

The intuition for (2): a forecast of “up 3%” is not an instruction to bet everything. Bet size should grow with conviction (how far the predicted probability is from a coin flip) but stay bounded - López de Prado’s meta-labeling machinery (AFML Ch 10) maps a model’s predicted probability $p$ into a position $m \in [- 1, 1]$ via a normal test statistic.

2. Mathematical Ground Truth & Derivations

2.1 Combining forecasts

Let $f_{1}, \dots, f_{K}$ be forecasts of the same quantity from $K$ models, and let $Σ_{e}$ be the covariance of their errors. The combined forecast is a weighted sum

\hat{f} = k = 1 \sum K w_{k} f_{k} = w^{⊤} f, k \sum w_{k} = 1.

Inverse-variance (optimal, uncorrelated errors). If errors are mutually uncorrelated, the variance of the combination is minimized by weighting each forecast by $1/ σ_{k}^{2}$ :

w_{k} = \frac{1/ σ _{k}^{2}}{\sum _{j} 1/ σ _{j}^{2}}, Var (\hat{f}) = \frac{1}{\sum _{j} 1/ σ _{j}^{2}} .

This is the same form as the inverse-variance portfolio: reliability (precision) weights, precision = inverse variance.

Granger–Ramanathan (OLS) combination. The most general linear combination regresses realized returns on the $K$ forecasts (Granger & Ramanathan, 1984):

β = ar g β min r - Fβ^{2},

where $F$ is the design matrix of forecasts. An unconstrained OLS fit can produce large and even negative weights - it overfits the in-sample covariance of forecast errors, which is exactly why, on the verified example in §3, it scores worse out-of-sample than inverse-variance.

The key contrast. Inverse-variance weights are closed-form and robust - they only need each model’s own error variance. Granger–Ramanathan is data-hungry and fragile - it estimates the full error covariance and re-weights everything, so it is more powerful when you have a long clean history and more dangerous when you do not.

2.2 Sizing a bet from a predicted probability (AFML Ch 10)

For a two-outcome classifier, let $p$ be the predicted probability that the label is $+ 1$ . Test the null that the model is no better than a coin, $H_{0} : p = \frac{1}{2}$ , with the z-statistic

z = \frac{p - \frac{1}{2}}{p ( 1 - p )} \sim N (0, 1),

then set the bet size (position) via the normal CDF $Φ$ :

m = 2Φ (z) - 1 \in [- 1, 1] .

$p = 0.50$ (coin flip) → $z = 0$ → $m = 0$ (no position).
$p = 0.90$ (high conviction) → $z = 1.33$ → $m = + 0.82$ (almost a full position).
$p = 0.55$ (weak conviction) → $z = 0.10$ → $m = + 0.08$ (tiny position).

The map is monotone, bounded, and self-scaling: it turns a classifier’s confidence into a position that respects your notional cap. In meta-labeling, $p$ is the output of a secondary classifier that predicts whether the primary model’s side call will succeed - so bet size is set by a model explicitly trained on the reliability of the primary signal, decoupling “which side” from “how confident the side call is”.

3. Computational Implementation - combining three models into a position

numpy. Three models forecast one asset; we estimate error variances in-sample, weight by inverse-variance, and compare against equal-weight, the best single model, and an (overfit) OLS combination. Numbers re-executed and verified.

What the numbers teach.

Combining helps. Even equal-weight beats the best single model (IC $0.225$ vs $0.164$ ) - diversification across models, exactly like across assets.
Weighting by reliability helps more. Inverse-variance (IC $0.232$ ) edges out equal-weight ( $0.225$ ), and it does so with fewer assumptions (only each model’s own variance).
OLS combination can backfire. Granger–Ramanathan (IC $0.214$ ) fits the full in-sample error covariance, overfits, and lands below inverse-variance out-of-sample. This is the canonical “overfitting the combination weights” failure - see 05 · Failure Modes.

Bet sizing from a predicted probability (AFML Ch 10, verified)

4. Failure Modes & First-Principles Breakdowns

Forecast-to-position mismatch. The single most common error. A correct forecast (good IC) is routinely turned into a wrong position by ignoring reliability: all-in on a weak signal, or a correct side call with the wrong size (López de Prado’s poker analogy, AFML Ch 10.1: two strategies with identical forecasts have opposite PnL purely from sizing).
Overfitting the combination weights. Granger–Ramanathan/stacking estimates the full error covariance from limited data; the estimated weights fit noise, so the in-sample combination looks great and the out-of-sample combination underperforms a robust equal- or inverse-variance blend. The verified GR vs inverse-var gap above is this failure.
Correlated errors break inverse-variance. The optimality of inverse-variance assumes uncorrelated forecast errors. If two models are actually the same model in disguise (same features, same leakage), you double-count - the combination gives them too much joint weight.
Using in-sample variance for sizing. Estimating $σ_{k}^{2}$ (and bet sizing) on the data you tune the model on guarantees optimistic sizes. Estimate reliability with purged, out-of-sample data (Purged CV & Backtest Hygiene).

5. References

Granger & Ramanathan, “Improved Methods of Combining Forecasts,” J. Forecasting 3(2):197–204, 1984
López de Prado, Advances in Financial Machine Learning (2018)
Bates & Granger, “The Combination of Forecasts,” Operational Research Quarterly 20(4):451–468, 1969
López de Prado, Machine Learning for Asset Managers (2020)

6. Connected Graph Bridges

Back: 01 · From Zero
Forward: 03 · Ensembling Models · 04 · ML for Covariance & Factors
Position sizing theory: Kelly Criterion & Bet Sizing · Financial-ML Pitfalls & Low SNR

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