Shapley Value Risk Allocation

Splitting a portfolio's total risk across its positions or strategies isn't as simple as it sounds, because risk isn't additive — two positions together can be riskier or safer than the sum of their standalone risks, depending on correlation. A common shortcut, marginal contribution to risk, measures each position's contribution by adding it last to all the others — but that answer depends on the arbitrary choice of what order you imagine positions being added in. The Shapley value, borrowed from cooperative game theory, fixes this by averaging a position's marginal contribution over every possible order in which it could be added:

$$\phi_i = \frac{1}{n!} \sum_{\text{orderings}} \big[ \text{Risk}(S \cup \{i\}) - \text{Risk}(S) \big] ,$$

where $S$ is whatever set of other positions comes before $i$ in a given ordering. In plain English: instead of picking one arbitrary sequence for "who joined the portfolio last," average the marginal risk added across all sequences, so the answer doesn't depend on which order you happened to imagine.

Worked example

Two correlated strategies, A and B, each have standalone risk of 10 (in some risk units), but combined portfolio risk is 16 rather than 20, because of diversification. Added first, A contributes 10 and B then contributes 6 (16 − 10); added the other order, B contributes 10 and A then contributes 6. The Shapley value averages these: each of A and B gets credited with $\frac{10+6}{2} = 8$ — splitting the diversification benefit evenly between them, unlike a marginal-contribution number computed under only one arbitrary order, which would have handed one of them all 10 and the other only 6.

Note the two allocations sum exactly to the portfolio total (8 + 8 = 16), a property ("full allocation") that makes Shapley values useful for capital budgeting across desks.