Quick answer: bank stress can travel through a financial system by more than one route. One bank may owe another money directly. Several banks may hold the same assets and sell them simultaneously. Banks may hoard liquidity, withdraw interbank funding or tighten collateral terms. A falling asset price can then create losses for institutions that never lent to the original failing bank. Network models represent banks as nodes, financial relationships as edges, common asset holdings as a second layer of connections, and stress propagation as a sequence of losses, funding withdrawals and price feedback.

A bank can be connected to another bank without lending it a single dollar—if both must sell the same asset at the same time.

This article deliberately does not own “How a bank run works”

The wider eduKate estate already has a canonical public explanation of the depositor coordination problem in Understanding Bank Runs: Panic, Liquidity, and Repairs. That article explains why withdrawal demand can accelerate faster than repair capacity.

This Bukit Timah Tutor article begins one layer later and asks a different mathematical question:

Once one institution is stressed, through which network channels can losses or defensive actions reach other institutions?

Why this belongs in mathematics

Financial contagion uses graph theory, matrices, fixed points, balance-sheet identities, optimisation, nonlinear price impact and threshold dynamics. It also provides a useful warning against naturalised thinking. “Connected” does not automatically mean “fragile.” Connections can distribute liquidity and risk in normal times. The same connections can transmit stress under different conditions.

1. Represent the banking system as a graph

Let each bank be a node. Draw a directed edge from Bank i to Bank j if i owes j an interbank payment or has borrowed from j. Assign a weight w_ij equal to the amount owed.

The system can then be stored in an exposure matrix W. Row i describes obligations leaving Bank i; column j describes claims held by Bank j.

A bank with many links is not necessarily the most systemically important. Edge size, direction, loss-absorbing capital, liquidity and network position matter. A moderately connected bank sitting between otherwise separate clusters can sometimes matter more than a densely connected peripheral bank.

2. Direct-credit contagion

Consider three banks:

Bank A owes Bank B S$20 million.
Bank B owes Bank C S$15 million.
Bank C has no direct claim on Bank A.

If Bank A fails and only 40% of its obligation is recovered, Bank B suffers a S$12 million credit loss. If B has only S$10 million of capital available to absorb that shock, B may itself become distressed. Bank C can then be affected through its claim on B.

This is the intuitive domino channel. It exists, but it is not always the dominant contagion mechanism.

3. Evidence polarity: direct interbank contagion is not always large

A good model needs evidence that can disagree with its favourite story. Early BIS work using bilateral US federal-funds exposures found direct contagion from simulated bank failures to be economically small in the sample studied. Later research likewise shows that the importance of direct interbank links depends strongly on network structure, exposure size, capital and the assumptions used when bilateral exposures are incomplete.

See Interbank exposures: quantifying the risk of contagion and Filling in the blanks: network structure and interbank contagion.

The lesson is important: do not build a contagion model that assumes domino defaults must dominate merely because dominoes are easy to draw.

4. Liquidity hoarding creates another network

Suppose Bank B is uncertain about its own future cash needs after Bank A becomes distressed. Even before A defaults, B may stop renewing short-term loans to Bank D, increase haircuts or keep cash that it would normally lend.

Bank D can therefore lose funding without holding any claim on A. Stress has travelled through defensive behaviour, not realised credit loss.

BIS network research identifies liquidity hoarding alongside interbank linkages and fire-sale externalities as important contagion channels. See Bank networks: contagion, systemic risk and prudential policy.

5. Overlapping portfolios create hidden edges

Now imagine Banks X and Y have no direct interbank loan. Both, however, own large amounts of the same bond.

If X suffers a separate shock and must sell that bond quickly, its sale can push the market price down. Y marks its own bond holdings to the lower price and suffers a loss. If Y is highly leveraged, it may need to sell too. That pushes the price lower again.

The hidden edge is therefore:

Bank X → common asset price → Bank Y.

No contractual claim is required.

6. Fire-sale feedback is a nonlinear loop

A simplified fire-sale loop is:

Initial loss → leverage rises → forced sale → market price falls → peer losses → peer leverage rises → more sales.

If price impact were perfectly linear and small, the loop might damp quickly. In stressed markets, market depth can disappear and price impact can become nonlinear. Selling twice as much may move the price by more than twice as much.

This produces a threshold effect. A modest initial shock may remain local; a slightly larger shock can trigger enough forced selling to cross into self-reinforcing system loss.

7. A simple price-impact model

For a teaching model, let q be the quantity of an asset sold and λ measure market-price sensitivity. A simple linear approximation is:

ΔP = −λq.

If Bank A sells 100 units and λ = 0.002 price units per unit sold, the estimated price move is −0.2. But λ is not a constant of nature. In stress, liquidity can thin and λ can increase sharply.

That is why fire-sale calibration is one of the hardest parts of systemic-risk modelling. The model needs to know how market prices respond when many institutions try to exit together.

8. Common holdings can dominate direct network exposure

A 2020 BIS study built an accounting stress-testing framework that combined direct interbank exposures with common asset holdings and fire-sale dynamics. In its euro-area application, direct interbank contagion contributed little compared with losses transmitted through common portfolios and fire-sale price effects.

See Contagion Accounting.

This is a valuable falsifier against an overly naturalised “bank A fails, therefore bank B fails because A owed B money” narrative. Sometimes the more important route is that A and B owned the same thing.

9. Network topology changes the propagation path

Several network properties matter:

Degree. How many counterparties connect to each bank?
Weighted degree. How large are those exposures?
Centrality. Does a bank sit on many important paths?
Core–periphery structure. Are a few highly connected core banks surrounded by less connected institutions?
Concentration. Is most funding dependent on a small set of counterparties?
Assortativity. Do highly connected banks connect mainly to other highly connected banks or to smaller institutions?
Clustering. Are exposures concentrated inside tightly linked groups?

A dense network can sometimes distribute shocks because losses are shared across many counterparties. The same density can also create more routes for stress. Connectivity has no universal sign; capital, liquidity and exposure concentration determine whether it stabilises or amplifies.

10. Missing bilateral data can reverse the conclusion

Often regulators or researchers know each bank’s total interbank assets and liabilities but do not observe every bilateral exposure. They must reconstruct a plausible network.

This is dangerous because different reconstructed networks can produce different contagion results. BIS research comparing maximum-entropy and minimum-density reconstructions found that one can underestimate contagion while the other overestimates it, creating a useful range rather than a false single answer.

The missing-data problem is itself part of systemic risk: if the network is not observed, precision in the contagion simulation can be cosmetic.

11. A multi-layer contagion algorithm

Build bank balance sheets. Assets, liabilities, capital and liquidity must reconcile.
Map direct exposures. Interbank loans, derivatives, secured funding and payment obligations form contractual edges.
Map common assets. Create bank-to-asset holdings matrices.
Apply the initial shock. Default, asset-price loss, deposit outflow or funding withdrawal begins the scenario.
Calculate direct losses. Apply recoveries and contractual exposures.
Update capital and liquidity. Determine which banks breach constraints or internal targets.
Model defensive behaviour. Banks may hoard liquidity, reduce lending or sell assets.
Apply price impact. Forced sales move common asset prices.
Revalue peers. Common holdings transmit the price change.
Repeat until stable. The model iterates until no new failures, sales or funding withdrawals occur—or the system enters a collapse state.
Separate channels. Report how much loss came from direct exposures, funding, common holdings and fire sales.
Challenge the network. Re-run under alternative unseen-edge assumptions and market-depth calibrations.

12. Historical networks were useful before they were dangerous

Interbank relationships are not an accidental defect. Historical correspondent networks helped banks process payments, access money markets and meet liquidity needs. BIS research on the US National Banking Era shows that these networks performed useful functions while also affecting the way liquidity risk propagated during panics such as 1893.

See Interbank networks in the national banking era.

This is another anti-normalisation lesson: the same system feature can be productive in one state and fragile in another.

13. Creative-work lens: the bank-run scene is only the visible surface

It’s a Wonderful Life and Mary Poppins both contain memorable bank-run scenes. They are useful for visualising depositor coordination: once people fear that others will withdraw, waiting becomes dangerous. But those scenes stop mostly at the bank’s front door.

The network problem begins behind that door. Which banks funded the institution? Who holds the same securities? Which counterparties cut limits? Which payments still need to settle? Which asset sale forces another bank to mark down its portfolio?

The creative work makes the visible panic memorable. Network mathematics reconstructs the invisible propagation.

14. Failure modes

Single-channel thinking. The model includes direct defaults but ignores common holdings and liquidity hoarding.
Static-network assumption. Banks keep lending normally while the scenario says confidence is collapsing.
Linear price impact. Market depth is assumed constant during mass selling.
Missing-edge certainty. Reconstructed bilateral exposures are treated as observed facts.
Connectivity moral. Dense networks are assumed inherently good or inherently bad.
Capital-only modelling. Solvent banks are assumed able to meet every short-term cash need.
Consolidated blindness. Foreign subsidiaries or off-balance-sheet exposures disappear from the graph.
Behavioural sameness. Every bank is assumed to respond to stress with the same sale or funding rule.

15. Diagnostics and falsifiers

Which bank produces the largest loss if removed from the network?
Does the answer change when indirect fire-sale channels are added?
How much system loss comes from direct claims versus common assets?
Which assets are owned by the largest number of leveraged banks?
Which banks depend on the same short-term funding providers?
How sensitive are results to the assumed price-impact coefficient?
Do alternative plausible network reconstructions reverse the ranking of systemic nodes?
What happens when banks hoard liquidity before any actual default occurs?

Suppose someone claims, “Bank Z has almost no direct exposure to the failing bank, so it is insulated.” A falsifier is a scenario in which Z holds the same illiquid asset and is forced below a capital or liquidity threshold by the fire-sale price decline. Direct distance in one network layer does not imply system distance in every layer.

16. Verification and update triggers

reconcile bilateral exposures with regulatory and payment data where available;
compare several network-reconstruction methods when links are missing;
update common-asset holdings frequently enough to capture concentration;
calibrate fire-sale price effects against stressed market evidence;
test alternative bank behavioural responses;
include liquidity and collateral calls alongside solvency losses;
re-run after major mergers or funding-market shifts change network topology;
separate observed facts from reconstructed edges in every result.

Connections across the finance-and-banking algorithms lane

Understanding Bank Runs: Panic, Liquidity, and Repairs — canonical upstream explanation of depositor-run mechanics.
Bank liquidity stress testing — how individual institutions model cash survival.
Counterparty credit risk — the bilateral exposure layer inside the wider graph.
Market-risk stress scenarios — the price shocks that can initiate common-asset losses.
Payment systems as graphs — another network where timing and routing determine system behaviour.

Research anchors

The deeper lesson

Systemic risk is not a property stored inside one bank. It can emerge from relationships: who owes whom, who owns what, who depends on which funding source, and who must sell when a constraint is breached. Graph theory makes the visible links explicit. Common-asset modelling reveals hidden links. Fire-sale dynamics show that the network itself changes as participants react. The strongest contagion model therefore asks not only, “Who is connected?” but “Connected through what mechanism, under which state of the world, and with what evidence?”

Educational note: This article explains banking mathematics and systemic-risk research. It is not financial advice, investment advice, a prediction of bank failure or institution-specific regulatory guidance.

Small Group Tutorials

How Interbank Networks Transmit Bank Stress: Exposure Graphs, Overlapping Portfolios, Fire-Sale Feedback and Contagion