How is risk divided in a bank? Discover the magic of Shapley!

How is risk divided in a portfolio? In finance, risk does not simply add up. To see who is responsible, experts use risk allocation. One of the most famous techniques is Shapley allocation, inspired by a game‑theory concept. Imagine sharing a pizza among friends: Shapley tells how much each friend contributed to the pizza’s value.

The paper “Sharpening Shapley Allocation” compares several allocation strategies. The authors built two test settings: a simplified one to highlight basic behaviours, and a realistic one with real portfolios under Basel 2.5 and FRTB rules. They also solved two tough problems: handling negative risk values for a unit, and managing multi‑level hierarchies in a bank.

The results show that Shapley is the best compromise. It is easy to understand, has solid mathematical properties, and represents real risk well. Moreover, the computation is fast when an efficient Monte Carlo simulation is used, even with hundreds of business units.

Although the study focuses on market risk, the method can be applied to credit, liquidity or valuation risk. In practice, Shapley helps banks see who truly carries the risk, improving transparency and decision‑making.

Shapley allocation originates from cooperative game theory, where the value of a coalition of players is distributed fairly among its members. In finance, the "players" are business units or individual assets of a portfolio, and the coalition's value is the aggregated risk measure, typically Value‑at‑Risk (VaR) or Expected Shortfall (ES). The Shapley formula computes, for each unit, the average marginal contribution to risk over all possible entry permutations into the coalition. This guarantees desirable properties: efficiency (the sum of contributions reconstructs total risk), symmetry (identical units receive the same contribution) and nullity (a unit that does not affect risk gets zero).

The paper tackles two practical challenges. First, negative allocations, which appear when an asset reduces overall risk (e.g., a hedge). The authors propose a monotonic transformation of the Shapley value that preserves additivity while eliminating negative signs, keeping the interpretation of "how much risk is carried". Second, the multi‑level hierarchy typical of large institutions, where units are organized into departments, divisions, and business lines. A recursive algorithm applies Shapley at each level, ensuring consistency between lower‑level and higher‑level allocations.

From a computational standpoint, exact Shapley calculation is NP‑hard for large portfolios, as it requires evaluating n! permutations. The study shows that a Monte Carlo simulation with stratified sampling drastically reduces the required scenarios while keeping estimation errors below 1 % even with thousands of units. Scaling analysis reveals an almost linear growth of computation time with the number of units, making the method feasible for Basel 2.5 and FRTB reporting. Finally, the authors highlight Shapley's versatility: it can be extended to non‑linear risk measures, stress‑testing scenarios, and derivative valuation contexts, providing a unified framework for risk management across the organization.