Key Insights:

Central Argument and Key Findings

This study conducts a systematic investigation of risk allocation strategies, concluding that Shapley allocation offers the best compromise between simplicity, mathematical properties, risk representation, and computational cost, even within large, complex financial institutions. Contrary to common wisdom that Shapley is computationally prohibitive, the paper demonstrates its practical feasibility by advocating a switch from the analytical formula to an efficient Monte Carlo simulation for portfolios with a large number of business units.

The research provides three principal contributions of significant practical value for risk management:

1. Novel solutions for managing negative risk allocations that preserve the essential "full allocation" property.
2. A coherent multi‑level allocation framework designed for the deep hierarchical structures of financial institutions.
3. Empirical validation using realistic trading portfolios under both Basel 2.5 and FRTB regulatory frameworks, demonstrating the methodology's real‑world applicability.

The methodological framework presented is general and applicable beyond market risk to other financial contexts, including valuation risk, liquidity risk, credit risk, and counterparty credit risk.

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Overcoming the Computational Hurdle of Shapley Allocation

A primary objection to the Shapley method has been its computational cost, which scales exponentially with the number of risk units (n). The analytical formula requires a number of risk measure calculations on the order of n × 2^(n‑1), which becomes unmanageable for even a relatively small number of business units.

This study's key insight is that this barrier is surmountable. The paper proves that a switch to a Shapley Monte Carlo approach makes the strategy viable for a challenging number of business units (e.g., tens of desks).

• Linear Scaling: The computational cost for the Monte Carlo method scales linearly with the number of risk units (n × N_MC / 2), where N_MC is the number of simulation samples.
• Practical Crossover: The Monte Carlo approach becomes more efficient than the analytical method for n as low as 10‑14, depending on the number of simulation samples required for convergence.
• Efficient Algorithm: The paper utilizes a specific implementation (Algorithm 1) featuring antithetic sampling, which improves convergence and significantly reduces the number of required risk measure calculations. Crucially, this algorithm preserves the full allocation property for each individual Monte Carlo sample path, not just on average.

Scaling of Computational Costs: Analytical vs. Monte Carlo The following figure from the study illustrates the crossover point where the Monte Carlo method becomes preferable.  (Note: As per Markdown‑only formatting rules, this is a descriptive placeholder for Figure 2 in the source document.)

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Contribution 1: A Practical Framework for Negative Allocations

While theoretically meaningful, negative risk allocations‑which occur when a risk unit acts as a hedge within a coalition‑are often problematic for practical reporting and capital management. The study addresses this long‑standing challenge by proposing two novel, structured approaches that ensure non‑negative outcomes while preserving the full allocation property.

The set of risk units with negative Shapley allocations is denoted by 𝒩−.

These methods provide a robust and theoretically sound alternative to ad‑hoc adjustments or ignoring the issue, a common practice described by Denault (2001) as a "crossed finger" approach.

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Contribution 2: A Consistent Multi‑Level Allocation Strategy

Financial institutions are organized in deep hierarchies, and standard risk allocations performed independently at each level are often inconsistent. Specifically, the sum of risk allocated to child business units does not typically equal the risk allocated to their parent unit.

The paper develops a framework with three distinct strategies to ensure the Multi‑Level full allocation property (Σ K_child = K_parent) is maintained.

The choice between these alternatives can be guided by specific business objectives and internal managerial decisions.

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Contribution 3: Empirical Validation in Regulatory Contexts

The study validates its proposed frameworks through application to a realistic, large‑scale trading book under two key regulatory regimes.

Application 1: Market Risk Capital under Basel 2.5

• Risk Measure: VaR (1%) + sVaR (1%). The highly complex Incremental Risk Charge (IRC) component was excluded, but the paper proposes a hybrid allocation strategy for such cases: use Shapley for tractable components (VaR, sVaR) and a simpler Proportional allocation for intractable components (IRC).
• Portfolio Structure: A multi‑asset portfolio organized into three hierarchical levels of 4, 12, and 24 trading desks.
• Key Results:
  • The methodology was successfully applied across all levels, using exact Shapley for the 4- and 12‑desk levels and Shapley Monte Carlo for the 24‑desk level, proving its scalability.
  • Unlike Proportional allocation, Shapley correctly identified and quantified the hedging effects of portfolios with negative correlations (e.g., PTF_4 in the top level), leading to a significantly different and more accurate risk representation.
  • The study confirmed that the largest contributors to the final Shapley value are the "Standalone" and "Marginal" terms, which represent a risk unit's contribution on its own and its contribution to the grand coalition, respectively.

Application 2: Market Risk Capital under FRTB

• Risk Measure: Sensitivity‑Based Approach (SBA) for Delta Equity risk under the Standard Approach (SA).
• Portfolio Structure: The equity positions from the same trading book, organized into two hierarchical levels (4 and 9 desks).
• Key Results:
  • The Shapley method was successfully applied to the FRTB SBA framework, demonstrating its relevance for the new capital regime.
  • Results showed that Shapley allocation reduces the risk attributed to dominant portfolios (e.g., PTF_3, which had the largest absolute delta exposure) and redistributes it to other desks, providing a more nuanced view of capital consumption that accounts for portfolio interactions.

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Comparative Analysis of Allocation Strategies

The paper provides a detailed comparison of major allocation strategies, the key findings of which are summarized below.