How Eisenberg–Noe Clearing Algorithms Propagate Interbank Defaults: Liability Matrices, Fixed Points, Fictitious Defaults, Uniqueness and Network Blind Spots

Reader question: If Bank A cannot pay Bank B in full, Bank B may then be unable to pay Bank C. How can an algorithm calculate all of those interdependent payments without deciding them one bank at a time in the wrong order?

The Eisenberg–Noe model turns an interbank system into a fixed-point problem. Every bank’s payment depends on the payments it receives from others, while those other banks’ payments depend on what they receive in return. The correct clearing vector must therefore be solved simultaneously.

This article owns the static proportional interbank-clearing problem: nominal bilateral liabilities + external assets → a clearing payment vector, a default set, creditor recoveries and diagnostics about how a shock travels through contractual obligations.

It does not own fire-sale contagion, intraday payment queues, bank-resolution waterfalls, liquidity regulation or behavioural bank runs. Those are different mechanisms. The purpose here is to understand one of the cleanest mathematical foundations of financial-network contagion.

This is public mathematical and computational education. It is not financial advice, a forecast of any real bank’s solvency or a substitute for supervisory data.

1. Why sequential reasoning fails

Suppose A owes B, B owes C and C owes A. If A is short of cash, we cannot determine A’s final payment until we know what A receives from C. But C’s payment depends on what C receives from B, and B’s payment depends on A.

The obligations form a cycle.

A naive algorithm such as “resolve A, then B, then C” can make the answer depend on an arbitrary processing order. A clearing model instead searches for a vector in which every bank’s payment is simultaneously consistent with its available resources.

2. The liability matrix

Let:

L_ij ≥ 0

be the nominal amount bank i owes bank j.

The matrix L is directed. In general:

L_ij ≠ L_ji.

For each bank i, total nominal interbank liabilities are:

p̄_i = Σ_j L_ij.

The vector p̄ tells us what every bank would pay if all obligations were honoured in full.

3. Relative liabilities turn payments into creditor shares

When p̄_i > 0, define:

Π_ij = L_ij / p̄_i.

Each row of Π gives the proportions in which bank i‘s total payment is distributed across its creditors.

If A owes 70 to B and 30 to C, then A’s relative liability weights are 0.7 and 0.3. If A can pay only 50 in total under the proportional-repayment assumption, B receives 35 and C receives 15.

This proportionality is an assumption of the basic model. Real contracts can have collateral, seniority and close-out rules that violate it.

4. External assets provide resources from outside the network

Let:

x_i ≥ 0

be bank i‘s external assets available for meeting obligations in the model.

If the clearing payment vector is p, the interbank inflow received by bank i is:

Σ_j Π_ji p_j.

So the resources available to bank i are:

x_i + Σ_j Π_ji p_j.

5. The clearing fixed-point equation

A bank cannot pay more than it owes, and under limited liability it cannot pay more than its available resources. Therefore:

p_i = min[p̄_i, x_i + Σ_j Π_jip_j].

In vector notation:

p = min[p̄, x + Π^Tp],

where the minimum is applied component by component.

This is the heart of the algorithm. The unknown payment vector p appears on both sides because one bank’s ability to pay is another bank’s incoming asset.

6. The economic assumptions encoded by the equation

The basic Eisenberg–Noe clearing mechanism is built around three important principles.

Limited liability: a bank does not pay more resources than it has available in the model.

Absolute priority in aggregate: if a bank has enough resources, it pays its nominal liabilities in full; if not, it devotes the available amount to creditors rather than withholding value.

Proportional repayment: creditors of the same debtor receive the debtor’s reduced payment in the proportions specified by the liability matrix.

These principles make the clearing problem tractable. They are also exactly where real-world institutional details can break the model.

7. A three-bank worked example

Consider three banks:

• A owes 100 to B;
• B owes 80 to C;
• C owes 60 to A.

External assets are:

x = (20, 10, 0).

Because each bank has only one interbank creditor, every relevant relative-liability weight is 1.

The clearing equations are:

p_A = min(100, 20 + p_C)

p_B = min(80, 10 + p_A)

p_C = min(60, p_B).

The vector:

p = (80, 80, 60)

satisfies all three equations:

• A receives 60 from C plus 20 externally and can pay 80 of the 100 it owes;
• B receives 80 from A plus 10 externally, enough to pay its full 80;
• C receives 80 from B, enough to pay its full 60.

Only A defaults in the clearing sense because p_A < p̄_A.

8. Shock one bank and the default set changes

Now reduce A’s external assets from 20 to zero:

x = (0, 10, 0).

A clearing vector is now:

p = (60, 70, 60).

A receives 60 from C and pays 60 rather than 100. B then receives only 60 from A plus its 10 external assets, so B can pay only 70 rather than 80. C still receives enough to pay its full 60.

A shock that initially removed 20 from A’s external resources therefore caused an additional 10-payment shortfall at B.

This is contractual contagion: losses propagate because one institution’s liability is another institution’s asset.

9. Why the answer is a fixed point

Define the clearing map:

Φ(p) = min[p̄, x + Π^Tp].

A clearing vector satisfies:

p = Φ(p).

That is a fixed point.

The mapping is monotone: if all incoming payments increase, no bank’s feasible outgoing payment falls. This order structure is one reason fixed-point methods work well.

10. Simple monotone iteration

For teaching purposes, one can begin at full nominal payments:

p⁽⁰⁾ = p̄

and repeatedly apply:

p^(k+1) = Φ(p^(k)).

The sequence can move downward toward a clearing solution. Starting from zero can instead move upward toward a clearing solution under appropriate conditions.

This iteration is intuitive and useful for diagnostics, but the original Eisenberg–Noe work also gives a more structured fictitious-default algorithm that identifies defaulting institutions and solves the corresponding linear system.

11. The fictitious-default idea

Start by assuming everyone pays in full. Calculate each bank’s resources under that assumption.

Any bank whose resources are insufficient is placed in a provisional default set.

Given that default set, solvent banks pay in full while defaulting banks pay only what their resources permit. This converts the nonlinear minimum equation into a linear system conditional on the chosen default set.

Recompute payments. If additional banks become insolvent because their expected inflows have fallen, expand the default set and solve again.

The algorithm terminates when the default set no longer changes.

The conceptual advantage is important: defaults are discovered endogenously from the network rather than imposed one at a time.

12. Existence is not the same as uniqueness

The original Eisenberg–Noe framework establishes existence of clearing payment vectors and gives uniqueness under regularity conditions. It is tempting to compress that into “the clearing vector is always unique.” That statement is too strong.

Later mathematical work has studied when clearing solutions can be multiple in more general financial systems and how strongly connected components and outside obligations affect uniqueness.

A numerical implementation should therefore distinguish:

• whether at least one clearing solution exists;
• whether the model assumptions guarantee uniqueness;
• whether the algorithm is returning a greatest or least clearing solution when multiplicity is possible.

This is a useful mathematical lesson: solver convergence to a fixed point is not automatically proof that the economic model has only one fixed point.

13. A graph interpretation

The network can be represented as a directed weighted graph.

• Nodes are banks.
• An edge i → j carries nominal liability L_ij.
• External assets enter nodes from outside the graph.
• Payment shortfalls travel in the same direction as obligations, while asset losses are experienced by creditors at the receiving end.

Cycles matter because they make payments mutually dependent. Strongly connected components identify groups of banks that can reach one another through directed liability paths.

But network centrality alone is not a clearing result. A highly connected bank may still be harmless if its external resources are strong and its obligations are small relative to assets. The algorithm needs balance-sheet magnitudes, not just graph topology.

14. Inputs and outputs

Inputs can include:

• institution identifiers;
• bilateral nominal liabilities;
• external assets available to the model;
• the treatment of outside creditors;
• netting assumptions;
• seniority and collateral simplifications;
• shock scenarios;
• numerical tolerance;
• solution-selection rule if uniqueness is not guaranteed.

Outputs can include:

• clearing payment vector;
• defaulting institutions;
• payment shortfalls;
• creditor recoveries;
• network loss amplification;
• default cascade order under an iterative representation;
• fixed-point residuals;
• sensitivity to exposure reconstruction;
• alternative clearing solutions where relevant.

15. Verification invariant: every payment must satisfy the clearing equation

After the solver returns p*, recompute:

r = p* − min[p̄, x + Π^Tp*].

The residual r should be zero up to numerical tolerance.

Also verify:

0 ≤ p* ≤ p̄.

These are simple but powerful invariants. A solver that reports a negative payment or a payment greater than the nominal obligation has failed mechanically before any economic interpretation begins.

16. Evidence polarity

Evidence for confidence includes small fixed-point residuals, agreement between independent solution methods, stable results under tighter numerical tolerances, correct recovery of hand-calculated toy networks, exposure data that reconcile to balance-sheet totals, and conclusions that survive reasonable alternative exposure reconstructions.

Evidence against confidence includes missing or inconsistent bilateral exposures, large differences between plausible network reconstructions, solver dependence on arbitrary node order, payment vectors that violate bounds, unexplained multiple solutions, default cascades that vanish when one small exposure is corrected, or claims about systemic contagion that actually come from channels absent from the model.

17. Counterexample: missing bilateral data

Public balance sheets may reveal that Bank A has 100 of total interbank assets and Bank B has 80 of total interbank liabilities without revealing who owes whom.

A reconstruction algorithm such as maximum entropy can fill in a liability matrix consistent with row and column totals. But many different matrices can match the same totals and produce different contagion paths.

BIS research on reconstructing interbank networks shows why this matters: assumptions about unknown bilateral exposures can materially affect estimated contagion.

Falsifier: solve the clearing model across multiple network reconstructions. If default amplification changes sharply, the result is exposure-assumption-sensitive rather than a robust fact about the system.

18. Counterexample: proportional repayment ignores seniority

Suppose one creditor is secured and another unsecured. In a real insolvency, they may not receive the same proportional recovery.

The basic Eisenberg–Noe rule would allocate reduced payments proportionally to nominal claims, which can misrepresent contractual priority.

Falsifier: compare the model’s creditor ranking with the actual legal waterfall. If legal seniority materially changes recoveries, a richer clearing model is needed.

19. Counterexample: no bankruptcy costs

In the basic model, a defaulting bank’s available assets are distributed to creditors without an explicit deadweight bankruptcy haircut. Real defaults can destroy value through legal costs, delays, forced liquidation and operational disruption.

Rogers–Veraart-type extensions add default costs and can generate greater loss amplification.

Falsifier: stress the recovery fraction available after default. If conclusions reverse under plausible bankruptcy costs, the zero-cost baseline is not sufficient.

20. Counterexample: fire sales are a separate feedback loop

A bank can meet interbank obligations by selling securities. If many banks sell the same assets, prices can fall and damage balance sheets that were not initially in default.

That mechanism is absent from the basic static clearing equation, which treats external assets as fixed numbers.

Falsifier: if the scenario requires asset liquidation large enough to move market prices, add a price-mediated contagion model rather than attributing the resulting losses to pure Eisenberg–Noe clearing.

21. Counterexample: liquidity timing matters

A bank may be solvent in end-of-day net terms but unable to make a payment at 10:00 a.m. because an incoming payment arrives at 3:00 p.m. Static clearing compresses timing into one simultaneous settlement problem.

Falsifier: if intraday sequence, payment queues or collateral timing determine default, use a dynamic liquidity model.

22. Counterexample: obligations can change after default

Close-out netting, derivatives termination, collateral calls and cross-default clauses can change liabilities when an institution defaults. In the basic model, the nominal liability matrix is fixed.

Falsifier: recompute contractual obligations under the relevant default state. If the liability matrix itself changes materially, the static fixed-liability assumption is violated.

23. Stress testing the network

A clearing model becomes more informative when it is used comparatively.

1. Calculate the baseline clearing vector.
2. Shock one or more external asset positions.
3. Recalculate clearing.
4. Measure first-round external losses.
5. Measure additional interbank payment shortfalls.
6. Identify newly defaulting nodes.
7. Repeat across shock sizes, sectors and network reconstructions.

A useful output is an amplification ratio:

network-amplified loss / initial external shock.

The exact definition must be explicit because losses can be measured as payment shortfalls, equity losses, outside-creditor losses or another quantity.

24. Alternatives and extensions

Rogers–Veraart clearing introduces bankruptcy costs.

Senior/subordinated clearing models represent creditor priority.

Fire-sale network models add price-mediated contagion through overlapping portfolios.

Dynamic clearing models introduce time and changing obligations.

Payment-system queue models focus on intraday liquidity and settlement order rather than end-state solvency.

Agent-based models can add strategic behaviour, liquidity hoarding and endogenous balance-sheet responses, at the cost of many additional assumptions.

25. Connections to the surrounding Bukit Timah Tutor knowledge estate

The broader interbank-network article surveys direct exposures, overlapping portfolios and fire-sale feedback. This page isolates the fixed-point clearing mechanism for nominal obligations.

Settlement and liquidity are separate from solvency clearing. Multilateral payment-netting algorithms explain obligation matrices, queues and gridlock in payment systems.

Large-exposure controls provide another view of network concentration: Basel large-exposure algorithms constrain concentrations before a default network is solved.

The full lane is indexed at Finance & Banking Algorithms | Applied Mathematics in Real Financial Systems.

26. What would falsify confidence?

Confidence should be withdrawn if bilateral exposures do not reconcile; if the clearing vector fails its own fixed-point equation; if the answer depends on arbitrary processing order; if regularity or uniqueness assumptions are absent but a single solution is presented as inevitable; if seniority, collateral, bankruptcy costs, changing obligations or fire-sale feedback dominate the scenario; or if plausible exposure reconstructions produce materially different contagion outcomes.

27. Verification and update triggers

Preserve the liability matrix, relative-liability matrix, external asset vector, outside-creditor treatment, shock vector, solver tolerance and fixed-point residuals for each run. Unit-test the engine on networks with known analytic solutions, chains, cycles, disconnected components and zero-liability nodes.

Trigger model review when bilateral exposure data change materially, netting or collateral rules change, a new legal seniority becomes relevant, observed defaults reveal material bankruptcy costs, network topology shifts, outside liabilities are reclassified, or the intended use expands from static solvency to liquidity timing or price-mediated contagion.

28. Primary and high-quality references

• Larry Eisenberg and Thomas H. Noe, Systemic Risk in Financial Systems, Management Science, 2001. Original clearing-payment framework.
• Recent mathematical work on uniqueness and general clearing systems: Uniqueness of Clearing Payment Matrices in Financial Networks, Mathematics of Operations Research.
• Bank for International Settlements, Filling in the Blanks: Network Structure and Interbank Contagion, on reconstructing missing bilateral exposures and the consequences for contagion estimates.
• Bank for International Settlements, Contagion Accounting, distinguishing and measuring channels of financial contagion.
• Bank of England, Financial Interlinkages in the UK’s Interbank Market and the Risk of Contagion, empirical network-stress research.

Educational boundary: A clearing network is a model of contractual payment consistency under explicit assumptions. It should not be used to infer the condition of a real institution without authoritative exposure, legal, collateral, liquidity and valuation information.