A desk cannot cut capital that the history branch still assigns to yesterday's positions. On Sui's 14 smooth synthetic books, the median share of FRTB-IMA capital charged to earlier observation dates is 61%. Today's-book sensitivities cannot reach that share while the averaged-history branch stays active. Its size, though, depends on how the books were generated. The paper also presents a trade-level ledger that reconciles within 5.6e-16. Sui's counterexample makes clear how little reconciliation alone tells a desk about which trade to cut.
Following capital back to each trade
Sui puts the full IMA calculation into a graph. Its starting point is 90 expected-shortfall blocks: six risk-class groupings, three factor-set and period combinations, and five liquidity horizons. Horizon weights are (1,1,2,2,6). The graph then applies a stress ratio floored at one, combines classes into the internal-models charge, and adds the non-modellable charge and a 0.999 default quantile. For history, it takes maxima involving the latest value and a 60-day average (multiplier 1.5), plus a 12-week default average. The amber surcharge and a minimum against standardised capital come last.
A forward pass records each value and active branch; a reverse pass assigns capital back to the leaves. Tail scenario weights govern ES blocks. Floors and maxima follow the active branch, while the default charge follows the scenario at the quantile. Each local rule is an Euler identity under the stated positive-homogeneity assumptions, so the allocations still sum to capital when they reach the leaves. Stored books make those leaves (date, trade) positions. At smooth points the result also equals a gradient allocation, provided the standardised inputs arrive with gradient allocations. Sui calls the ledger "exact relative to the implemented calculation and this rule" and disclaims any regulatory standing for the attribution.
Sui separates five questions: local marginal, removal effect, accounting ledger, historical origin and hedge choice. Consider capital max(w1, w2) at (1,1), where K = 1. Remove either trade and capital remains 1. Both removal effects are zero, summing to 0, although the marginal lies on a segment between (1,0) and (0,1). A 50/50 ledger reconciles by convention.
Why does yesterday's book hold today's capital?
The history toy gives the clearest answer. Set C = max(I1, 1.5 x (I0 + I1)/2), with yesterday's charge I0 = 3 and today's I1 = 1. The average branch wins: C = 3. The origin ledger assigns 2.25 to yesterday and 0.75 to today. Today's marginal and removal effect are each 0.75. The other 2.25, or 75% of capital, stays frozen while that branch is active; a cut to today's book cannot take C below it.
Sui proves the relationship beyond the toy. At a smooth point, the allocation on past dates exactly equals capital minus the sum of today's-book sensitivities. Across the 14 smooth 96-trade books, the gap ranges from 21% to 99%, with a median of 61%. The three history-heavy books reach 94.0%, 95.3% and 98.6%. Analytical Euler is exact on all 14 smooth books at about 2.9e-18, yet it cannot fill a full-capital ledger from today's book alone. Li and Xing's method applies to only one main book. On that book its marginal error is 0.17, and its today's-book split falls short of capital by 0.79K. Sui traces that limited coverage to a stress-ratio model without the floor.
The benchmark's past dates come from seeded perturbations of today's notionals. "history-heavy" is one of six named synthetic portfolio types. Those choices shape the 61%. The identity determines the gap once the books are given; it does not determine how large the gap will be, and the benchmark includes a low of 21%. Whether a real desk often occupies that branch remains open. Its own 60 days of stored charges would answer the question.
Reconciliation has a narrower job
The accounting ledger's maximum relative reconciliation gap is 4.3e-16 on the 18 main books and 5.6e-16 on the 42 tie books. Sui is explicit that the comparison uses the same engine's capital. A correct sum is necessary for the ledger, yet it cannot certify a marginal; the paper labels allocations as either true marginals or accounting splits. For example, the amber surcharge allocation includes (B minus J) times the amber coefficient's allocation, a term whose total is zero. Removing it leaves every reconciliation check intact while shifting desk subtotals by plus or minus (B minus J)/8 in Sui's two-desk example. In a two-parent toy graph, independent branch choices can yield (2, -1). That allocation sums to capital and is no marginal. Sui says the FRTB graph contains this structure because J enters its final layer with both signs.
A separate test bears on the marginals. For 36 tie books placed exactly on a kink, the engine's marginal is within 1.55e-16 of a realisable set constructed without the engine. The exact phase-one linear program used for realisability can miss some valid marginal vertices. Four of the 18 main books also lack a marginal reference answer. And on each main book, the engine declines 35 to 58 single-trade capitals when a default-quantile tie exceeds its enumeration bound.
Shapley allocation over trades is undefined on every main book. Of the 96 single trades, 14 to 47 have no reduced-set exposure, leaving a zero denominator in the stress ratio. Schulze's marginal measures produce answers for all 14 smooth books, with median error 0.083 and maximum 0.50. The paper attributes those figures to the target: the constructions cover only the source-supported portion of the graph. Across this benchmark, each established method addresses at most two of the five measures. None computes the removal effect, origin ledger or hedge choice; only incremental allocation supplies a full-capital ledger.
Certified repricing, without a speed result
Sui's certified repricing scheme uses a few exact anchors to bracket convex-in-spot option prices, then reprices only tail rows. It certified all six eligible American-option books, recording zero guarantee violations across 90 block checks each. Against the exact engine, the maximum allocation difference was 1.4e-17 to 2.2e-16 on five books and exactly 0 on the sixth. The six basket and barrier books are outside the scheme's class. These option books used calibration settings rather than a final pre-registered run. The certifier has yet to receive adversarial review and includes no explicit pricer-error term.
Sui makes no speed claim. An earlier pilot used a different pilot benchmark and figures the paper calls non-authoritative. It found an advantage only for one-off queries: the first-query time ratio was 0.395 on one family. Across 20 repeated queries, the certified route took 1.46 to 1.54 times as long as memoised full revaluation. A spline surrogate came within about 3e-5 to 4e-4 on the internal-models charge using 5% of the pricings, without a guarantee.
For hedge choice, the paper ranks capital across 40 libraries of 8 candidate actions per main book. All 18 have a unique winner. In the first of its 40 libraries on ordinary book 1, the best of eight actions removes a random subset of trades and takes capital from 105.39 to 92.48. We found no P&L, hedge-cost or hedge-effectiveness measurement for those rankings.
We could not run any of this ourselves.
A test on a real desk would require dated, trade-level bank books, full scenario losses, non-modellable and default scenarios, and standardised-approach inputs. We hold none of them.
Sui has an accounting ledger that reconciles to the implemented formula and an identity with a direct consequence for desk heads. The 61% figure still has no demonstrated counterpart on a production book. Sui makes that boundary clear: the mathematics is established on synthetic books, behaviour on a production book is untested, and validation on real desk groupings remains an open problem.