Exact Trade-Level Attribution of FRTB-IMA Capital

Yuhe Sui

Papers from arXiv.org

Abstract: The internal-models approach of the Fundamental Review of the Trading Book (FRTB-IMA) determines market-risk capital at the aggregate level of a bank's approved trading desks, but it does not say how that charge should be attributed to individual trades. Risk managers, capital planners and validators often need one, and it is hard to obtain: capital passes through expected-shortfall, stress-scaling, non-modellable, default-risk, history and standardised-approach layers that are nonsmooth and depend on past positions. We present a formula-complete attribution of FRTB-IMA capital to (observation date, trade) positions that reconciles to computed capital. The calculation is written as a computational graph; after one forward evaluation, a single reverse pass propagates allocation vectors through every node by local Euler rules (expected-shortfall dual weights, active-branch weights at floors and maxima, the default-quantile scenario), avoiding one capital re-evaluation per trade. Under stated homogeneity assumptions the ledger provably sums to capital and to any desk or product grouping, and equals the Euler (gradient) allocation at smooth points; it is exact relative to the implemented calculation and this rule, not a regulatorily prescribed attribution. Its past-date entries capture capital allocated to earlier observation dates, which today's-book sensitivities miss by an exact identity at smooth points. At regulatory ties, where the marginal is not unique, we report the realisable marginal vertices, identified by exact linear-programming tests, instead of one selection. On 14 smooth 96-trade synthetic benchmark books, the median share of capital allocated to past observation dates is 61% (range 21%-99%), and ledgers reconcile to within 5.6e-16 relative error.

Date: 2026-09
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