The Verification Venue · where n squared meets n to the three halves
The Square Root of the Payment System
Between n banks paying each other at random, total gross obligations grow like n squared, but the money that must actually move under multilateral netting grows like n to the three halves. The gap between those two curves is the square root of n, and closing that gap is the entire business of a clearing house. Drag the controls and watch a seeded Monte Carlo sweep fit both exponents live, interval and all; every number on this page is recomputed in front of you.
Two banks that owe each other money can cancel most of it on a napkin. Three hundred banks owing each other at random cannot, not by hand, and the question of how much cancellation buys as a system grows has a sharp answer in a simple model. This page runs that model while you watch: a fixed seed generates every payment, exact integer arithmetic totals the obligations, and a least squares fit reads the exponents off a log-log sweep. Start by dragging the sweep width and watching the fitted net exponent settle. Then go further down, break a bank on purpose, and run the same default through both settlement regimes.
Five economies, one model: n 120 · k 8
The presets set the sweep's width and the payment count per pair; nothing else about the model changes between them. ↓
Net exponent (fitted)
···
a fit near 1.5 measures this model, it does not prove a theorem
Reduction factor at n max
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gross settled divided by net settled
Gross settled at n max
···
u = 100 cents is the nominal model constant; ceiling to whole cents puts the realized mean near 100.5, and the identity below is evaluated at the realized x̄, never at the constant
Net settled (multilateral)
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sum of positive net positions, exact integer cents
Drag right and the sweep reaches bigger economies; the fitted net exponent tightens as the lever arm in log n grows.
More payments per pair lifts both curves but moves neither exponent; the reduction factor grows like the square root of k inside the n scaling.
The gross side is an identity, not a statistic. Fix k payments between each ordered pair of banks and a mean payment x̄, and the gross total is exactly k·n·(n-1)·x̄, because that is simply how many payments there are. The net side is where the square root enters. Bank i's multilateral net position is the sum of about 2k(n-1) independent signed payments, so its typical size grows like the square root of n. A bank settles only the positive part of its net; the positive part of a symmetric fluctuation is proportional to its standard deviation; and n banks each of size root-n sum to n times root-n, which is n to the three halves.
So netting settles roughly √n times less money than gross settlement would for the same underlying payments, and that ratio is the liquidity a clearing system releases. Three precisions keep this honest. First, the 100-cent figure is the nominal model constant; drawing whole cents by ceiling lifts the realized population mean to about 100.5, and every identity on this page is evaluated at the realized mean x̄, never at the constant. Second, the same central-limit argument that produces three halves also fixes the constant, approximately: net settlement sits near x̄·n·√(k(n-1)/π) and the reduction factor near √(π·k·(n-1)), and the reduction readout prints that level beside the measured ratio. Third, because a finite sweep never lands exactly on an asymptotic law, the committed verifier fits the reduction exponent on its own seed, over n = 20 to 120, and measures 0.457; the live readout beside the plots prints this page's own fit and interval for the same quantity. Notice what the readouts do and do not claim: the gross exponent is pinned at 2 by the identity and has almost no room to move, while the net exponent is a measurement on this model, printed with its confidence interval and the range of n it covers. Real payment networks are nowhere near this independent; payments correlate through trade-credit chains and intraday liquidity management, so treat the square-root law as a baseline for the random model, never as a forecast about any real system.
What netting does with a default
Netting buys its efficiency with concentration, and this is the trade made visible rather than asserted. Under gross settlement, a bank that cannot pay spreads its shortfall across every counterparty it owes, a little to each. Under multilateral netting the same bank owes only its net position, and whatever it cannot pay comes out of the net creditors, pro rata. Same economy, same shortfall, very different wound. Pick a victim bank and a shortfall below, and read the two loss profiles side by side. Then hunt with the victim slider for a bank whose net position is positive: under netting such a bank owes nothing, so its default wounds the net system not at all, while gross settlement bleeds exactly as before.
Banks losing money
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gross vs net, same economy, same shortfall
Total first-round loss
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gross vs net, in u
Worst-hit bank's share
···
the concentration netting buys its efficiency with
Victim's net position
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a net receiver owes nothing under netting
···
Slide through the banks. Somewhere in this economy is a net receiver: find it and watch the net-regime loss collapse to zero while gross barely notices the difference.
The cash the victim cannot find, measured in multiples of this economy's realized mean payment.
The experiment is deliberately spare, and the rules are stated so you can argue with them. The victim meets all but S of its gross outflow, cut proportionally across its outgoing payments (gross regime), or all but S of its net obligation, shared pro rata across net creditors in proportion to their claims (net regime). Losses are first round only: nobody cascades, nobody holds capital buffers, and the clearing house guarantees nothing. Concentration is reported as a Herfindahl index of the loss shares. These are modelling choices, not facts about any real system, and they are listed in the disclosure below the check.
The check · every number recomputed in front of you
The table regenerates from the same seeded sweep the plots use. The gross columns are the anchor: k·n·(n-1)·x̄ is an identity, so the simulated column must land on it, and the residual shown is Monte Carlo scatter in the realized mean payment, matched against four standard deviations of that scatter. The net columns have no exact law in this model; the reference there is this page's own fitted line, printed so the eye can judge the fit rather than trust it.
| n | gross, simulated (u) | gross, exact law (u) | Δ | net, simulated (u) | net, fitted line (u) |
|---|
Live anchor at your current largest economy:
Every figure here is computed in your browser from the seeded model in this page's source, and the same computation is re-run, independently, by the committed verifier. Assumptions named where they bite: payments are independent draws, exponential in shape, quantised upward to whole cents; the sweep averages ··· runs per point; the confidence interval is a normal approximation to the least squares slope. Run it yourself: node research/the-square-root-of-the-payment-system/verify-the-square-root-of-the-payment-system.mjs.
What's exact here, what's idealised, and what's representative
Exactly true. The gross law k·n·(n-1)·x̄ is an identity for any fixed payment count per pair, not a fit. Multilateral net positions sum to zero exactly, and the total net settlement equals the magnitude of the sum of the negative positions, exactly. All money arithmetic on this page is whole cents carried in doubles, so every total is an exact integer sum with no rounding. The fitted exponents come from ordinary least squares on the logs, and the interval is the exact standard error of that slope times the 97.5 percent normal quantile; the approximation is in treating the slope as normal, not in the algebra.
Idealised. Payments are independent across pairs, directions and time, which real networks are not. Payment sizes are exponential, quantised up to whole cents; the ceiling moves the realized mean from the nominal 100 cents to about 100.5, which is why every identity here is stated at the realized mean. There is no intraday timing, no collateral, no reserves, no central guarantor. Defaults are first round only, with proportional rationing under gross and pro-rata sharing on net claims under net; no bank cascades into a second round. The confidence interval treats the sweep points as independent, which the shared seed makes almost but not exactly true.
Representative, not universal. The three-halves exponent rests on the variance of a single payment being finite; within any such finite-variance model the exponent is 1.5 and only the constant moves. Real payment size distributions are strongly heavy-tailed, and with sufficiently heavy tails the exponent itself can change; this page does not model that case. The constant out front depends on the size distribution and on k. No number from any real payment system appears on this page, on purpose: the model is the claim.