What SHAP Values Actually Compute

Question

You have a gradient-boosted model predicting next-day returns from a few dozen signals, and a portfolio manager asks you to explain a single prediction: "why did the model go long this name today?"

Explain what a SHAP value computes. State the Shapley formula and the intuition behind it, the additivity property, how SHAP differs from global importances like MDI/MDA, why exact computation is expensive and how TreeSHAP helps, and the caveats with correlated features and non-stationary financial data.

Accepted Answer

SHAP (SHapley Additive exPlanations) answers a local question: for one specific prediction, how much did each feature push the output away from a baseline, and in which direction? It does so by importing the unique "fair payout" rule from cooperative game theory — the Shapley value — and treating the model's features as players in a game whose payout is the prediction.

The game-theory setupImagine the features form a coalition that together produces the model output. For any subset (coalition) $S$ of features, define a payout $v(S)$ = the model's expected prediction when only the features in $S$ are known (the others are marginalised out, typically integrated over the background data). Adding feature $i$ to coalition $S$ changes the payout by its marginal contribution $v(S \cup \{i\}) - v(S)$. The problem is that this marginal contribution depends on which other features are already present. The Shapley value resolves this by averaging the marginal contribution over every possible coalition, weighted so that every ordering of the features is counted equally: \begin{equation} \phi_i = \sum_{S \subseteq F \setminus \{i\}} \frac{|S|! \,(|F| - |S| - 1)!}{|F|!} \Big[ v(S \cup \{i\}) - v(S) \Big] \end{equation} where $F$ is the full set of features. The combinatorial weight is exactly the probability that, if you build up the coalition by adding features in a uniformly random order, feature $i$ arrives right after the set $S$ has already joined. Equivalently: line up all $|F|!$ orderings of the features, and for each ordering record how much the prediction jumped when feature $i$ was added; $\phi_i$ is the average of those jumps. That averaging over orderings is what makes the attribution fair and order-independent.

Additivity (efficiency)The Shapley values for one prediction sum to the gap between that prediction and the baseline: \begin{equation} \hat{f}(x) = \phi_0 + \sum_{i=1}^{|F|} \phi_i, \qquad \phi_0 = E[\hat{f}(X)] \end{equation} So if the model's average output across the background data is, say, $+0.1\%$ expected return and today it predicts $+0.8\%$, the $+0.7\%$ excess is split exactly among the features: perhaps momentum contributed $+0.5\%$, value $+0.3\%$, and a crowded-flow signal $-0.1\%$, and these add up. This exact decomposition is what lets you hand the PM a complete, auditable story for one trade.

How SHAP differs from MDI / MDAMDI (mean decrease in impurity) and MDA (mean decrease in accuracy / permutation importance) are global scores: one number per feature describing its average importance over the whole dataset. They cannot tell you why this prediction happened, and they have known biases (MDI inflates high-cardinality and continuous features; MDA reflects model accuracy loss, not direction). SHAP is local (per-prediction, signed, so it tells you direction not just magnitude) and consistent: if you change the model so that a feature's true marginal contribution never decreases, its SHAP importance is guaranteed not to decrease — a property MDI does not satisfy. You can still aggregate SHAP globally (mean absolute SHAP per feature) to get an importance ranking that inherits this consistency, giving a more trustworthy global view than MDI as a bonus.

Why exact SHAP is expensive, and TreeSHAPThe formula sums over all $2^{|F|}$ subsets of features, and each term needs an evaluation of $v(S)$. That is exponential in the number of features and hopeless for a few dozen signals evaluated over thousands of rows. The breakthrough is TreeSHAP: for tree ensembles (random forests, gradient boosting) it exploits the tree structure to compute exact Shapley values in polynomial time — roughly $O(T L D^2)$ for $T$ trees, $L$ leaves and depth $D$ — by pushing all coalitions through each tree simultaneously and tracking, at every node, the proportion of subsets that would flow down each branch. This turns an intractable combinatorial sum into a fast tree traversal, which is why SHAP is practical precisely for the boosted models used in return prediction.

Caveats for finance

 	Correlated featureswhen two signals are highly correlated, the marginal-contribution game must break a tie: the joint effect gets split between them, so neither looks individually dominant, and the standard ("interventional") SHAP can evaluate the model on unrealistic feature combinations that never occur in the market. Substitutes share credit, which can understate a signal's apparent importance.
 	Baseline choice mattersSHAP values are always relative to a background distribution. Pick the background from a calm regime and the same trade gets explained differently than against a stressed background.
 	Non-stationarityfinancial relationships drift, so a clean SHAP attribution today may reverse next quarter; SHAP explains the model, not an immutable economic law, and an explanation is only as stable as the regime it was computed in.
 	Explanation, not causationSHAP tells you how the model used a feature, not that the feature causes returns. Treat it as a debugging and communication tool, not a causal claim.

The summary line: a SHAP value is the game-theoretically fair share of a single prediction's deviation from baseline that is attributable to each feature, computed by averaging marginal contributions over all coalitions — local, signed, additive and consistent, and made tractable for tree models by TreeSHAP.

Title	Category	Subcategory	Difficulty	Status
MDI versus MDA	Machine Learning	Feature Importance	Easy	Unlock
The Substitution Effect	Machine Learning	Feature Importance	Hard	Unlock

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What SHAP Values Actually Compute

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