The Encyclopedia of Economic Principals

Chapter 52

Bank Runs, Collateral Cycles, Crisis Backstops, and Sovereign Links

Trace how falling collateral, rising haircuts and self-fulfilling withdrawals turn stress into crisis.

Four of the chapter's worked examples, made interactive: a collateral cycle, a run on repo, a Diamond-Dybvig bank run and a lender of last resort facing a fire sale. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Collateral prices amplify a cash loss

How much financing does a firm lose when a cash shock also lowers the price of its collateral?

Debt capacity is tied to collateral value. When many leveraged owners cut their bids at once, the price of the collateral falls, and every owner's borrowing limit shrinks with it, adding a second loss to the first.

Equation, written in LaTeX: 12(80{,}000)=960{,}000\text{ dollars}.

Equation, written in LaTeX: 0.65(960{,}000)=624{,}000\text{ dollars}.

Equation, written in LaTeX: 0.65(816{,}000)=530{,}400\text{ dollars}.

Scroll sideways for the whole equation

The firm owns 12 parcels used as collateral and can borrow the pledgeable share (loan to value) of their market value. Financing capacity is internal funds plus that debt. A demand shock removes 90,000 dollars of cash; the parcel price may also fall.

Predict first. At the book's 68,000 price and 65 percent pledgeability, is the repricing loss bigger or smaller than the 90,000 cash loss?

Your prediction

Choose an example

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Figure: Collateral prices amplify a cash loss. Stacked bars of internal funds and collateral debt. Capacity falls from 984,000 to 800,400: 90,000 from the cash shock and 93,600 from repricing.
Parcel price after the shock (dollars): $68,000, Pledgeable share: 65%
Constructed example: the chapter's hypothetical logistics firm (12 parcels at 80,000 then 68,000, 65 percent pledgeable, 360,000 of internal funds, a 90,000 cash shock); prices of 62,000 and 74,000 and pledgeable shares of 50 and 80 percent are added for comparison.

Calculated values

Capacity before
$984,000
Capacity after
$800,400
Cash-loss part
$90,000
Repricing part
$93,600
Total decline
$183,600

Before: 360,000 + 0.65 x 12 x 80,000 = 360,000 + 624,000 = 984,000. After: 270,000 + 0.65 x 12 x 68,000 = 270,000 + 530,400 = 800,400. Capacity falls 183,600: 90,000 from the cash loss and 93,600 from repricing, so the repricing loss is larger than the cash loss.

Worked steps

  1. Debt before = 0.65 x 12 x 80,000 = 624,000
  2. Capacity before = 360,000 + 624,000 = 984,000
  3. Debt after = 0.65 x 12 x 68,000 = 530,400
  4. Capacity after = 270,000 + 530,400 = 800,400
  5. Repricing part = 624,000 - 530,400 = 93,600

Use the idea

When stress-testing a borrower, mark its collateral at the price that would clear if similar owners sell together, not at the calm-market price.

Where the conclusion applies

A fixed loan-to-value ratio and an assumed market-clearing price; the size of the feedback depends on pledgeability, holdings and the price, and is not a universal multiplier.

Check your understanding: If buyers held the price at 80,000, what is capacity after the shock?
270,000 + 0.65 x 960,000 = 270,000 + 624,000 = 894,000 dollars.

Chapter 52 source: section "Kiyotaki-Moore collateral cycle".

Demonstration 2 of 4

Haircuts drain a dealer

How much cash must a dealer find when lenders raise the haircut and the collateral is marked down?

Repo is rolled over every night, so its terms can change overnight. A higher haircut cuts what lenders will advance against the same collateral, and price falls cut it further, turning a modest markdown into a large cash demand.

Equation, written in LaTeX: (1-0.04)(240)=230.4\text{ million dollars}.

Equation, written in LaTeX: \frac{240}{9.6}=25.

Equation, written in LaTeX: 0.88(238)=209.44\text{ million dollars}.

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A dealer funds collateral worth 240 million dollars with overnight repo. Lenders lend the value times one minus the haircut. At rollover the haircut may rise and the collateral may be marked down; the dealer must replace the difference in cash. Market equity is collateral value minus the old repo.

Predict first. When the dealer fails to roll over at the book's 12 percent haircut and 238 million value, is it insolvent?

Your prediction

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Figure: Haircuts drain a dealer. Bars of the old repo 230.40 million, new funding 209.44 million and market equity 7.60 million; the gap of 20.96 splits into haircut 19.20 and markdown 1.76.
New haircut: 12%, Collateral value at rollover (million dollars): 238
Constructed example: the chapter's hypothetical dealer (240 million of collateral, haircuts of 4 and 12 percent, value 238 million at rollover); haircuts of 8 and 20 percent and a value of 230 million are added for comparison.

Calculated values

Old repo
230.40
New funding
209.44
Cash to replace
20.96
Haircut part
19.20
Markdown part
1.76
Market equity
7.60
Solvent at market value
yes

New funding is (1 - 0.12) x 238 = 209.44 against the old (1 - 0.04) x 240 = 230.40, a gap of 230.40 - 209.44 = 20.96. The haircut accounts for 19.20 and the markdown for 0.88 x 2 = 1.76. The dealer must replace 20.96 million in cash while its market equity is still 7.60 million: a payment problem before insolvency.

Worked steps

  1. Old repo = 0.96 x 240 = 230.40
  2. New funding = 0.88 x 238 = 209.44
  3. Gap = 230.40 - 209.44 = 20.96
  4. Haircut part = 230.40 - 0.88 x 240 = 230.40 - 211.20 = 19.20
  5. Markdown part = 0.88 x 2 = 1.76
  6. Market equity = 238 - 230.40 = 7.60

Use the idea

Measure a funding run by the cash a borrower must replace at new haircuts, not only by the fall in its securities' prices.

Where the conclusion applies

No spare cash or unsecured line, a single collateral pool and one rollover date.

Check your understanding: At a 12 percent haircut and 238 million value, how much of the gap comes from the price fall?
0.88 x 2 = 1.76 million; the total gap is 230.4 - 209.44 = 20.96 million.

Chapter 52 source: section "Run on repo".

Demonstration 3 of 4

A run on a sound bank

Why can a bank with good assets still fail when depositors expect a run?

Deposit contracts promise more at date 1 than an interrupted asset can pay to everyone. If all depositors run, the bank liquidates everything, pays a first group in full and leaves the rest short, so running is individually rational once a run is expected.

Equation, written in LaTeX: c_2=\frac{87.36}{64}=1.365.

Equation, written in LaTeX: 0.20(1.10)+\frac{0.80(1.365)}{1.40}=0.22+0.78=1.

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80 depositors place one unit each. The bank's asset returns R = 1.40 at date 2 or one unit if interrupted at date 1. A share lambda of depositors is impatient and withdraws c1 at date 1; the patient rest share the date-2 output as c2.

Predict first. At the book's values, is c2 above c1, so that patient depositors prefer to wait if others wait?

Your prediction

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Figure: A run on a sound bank. Payments per depositor. Without a run, early claimants get 1.10 and patient ones 1.365. In a run the first 72 get 1.10 and the rest share 0.80.
Promised early payment: 1.10, Share truly impatient: 0.20
Constructed example: the chapter's hypothetical bank (80 depositors, R 1.40, lambda 0.20, c1 1.10); early payments of 1.00 and 1.20 and impatient shares of 0.10 and 0.30 are added for comparison.

Calculated values

Interrupted for early claims
17.60
c2
1.365
Total consumption, no run
104.96
Total in a run
80
Loss from a run
24.96
Full payments in a run
72
Left after full payments
0.80

16 impatient depositors take 16 x 1.10 = 17.60, leaving 62.40 invested, which pays 62.40 x 1.40 = 87.36. Shared among 64 patient depositors, c2 = 87.36 / 64 = 1.365, above c1, so a patient depositor waits if others wait. In a run the bank can pay only 72 claims of 1.10 in full (79.20), and total consumption falls from 104.96 to 80, a loss of 24.96.

Worked steps

  1. Interrupted = 16 x 1.10 = 17.60
  2. Date-2 output = (80 - 17.60) x 1.40 = 87.36
  3. c2 = 87.36 / 64 = 1.365
  4. No-run total = 17.60 + 87.36 = 104.96
  5. Run: 72 x 1.10 = 79.20, leaving 0.80
  6. Loss = 104.96 - 80 = 24.96

Use the idea

Judge a bank's run risk by comparing its promised liquid payouts with what its assets fetch if liquidated at once, not only by its solvency at maturity.

Where the conclusion applies

Sequential service, a liquidation value of one per unit, and no deposit insurance or lender of last resort.

Check your understanding: In a run with c1 = 1.10, how many full payments can the bank make?
80 / 1.10 = 72.7, so 72 payments of 1.10 = 79.2, leaving 0.8.

Chapter 52 source: section "Diamond-Dybvig bank-run mechanism".

Demonstration 4 of 4

Fire sale or central bank window

How much equity does a solvent bank lose if it must sell loans to meet a run, and what does a lender of last resort save?

A forced sale turns a liquidity shortage into a solvency loss, because loans worth a dollar fetch less in a panic. A facility that lends against the same loans at a haircut keeps them on the balance sheet and preserves equity, apart from interest.

Equation, written in LaTeX: \frac{20}{1-0.20}=25\text{ million dollars}.

Scroll sideways for the whole equation

The bank holds 18 million of cash and 90 million of loans against 80 million of deposits. A run withdraws part of the deposits. It can cover the cash gap by selling loans at a fire-sale price per dollar of normal value, or by borrowing from a facility that lends against loans with a 20 percent haircut at 8 percent a year for 60 days.

Predict first. How much equity does a fire sale at 65 cents destroy in the book's 38 million run?

Your prediction

Choose an example

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Figure: Fire sale or central bank window. Bars of bank equity: 28 before the run, 17.231 after a fire sale at 0.65, 28 with a central bank advance backed by 25.000 of collateral.
Fire-sale price per dollar: 0.65, Withdrawal size (million dollars): 38
Constructed example: the chapter's hypothetical bank (cash 18, loans 90, deposits 80, a run of 38, a fire-sale price of 0.65, a 20 percent haircut at 8 percent for 60 days); fire-sale prices of 0.50 and 0.80 and runs of 28 and 48 are added for comparison.

Calculated values

Cash gap
20
Loans sold
30.769
Equity after fire sale
17.231
Value destroyed
10.769
Solvent after fire sale
yes
Collateral pledged
25.000
Equity with the facility
28.000
Interest, 60 days
0.263

The run of 38 takes the 18 of cash and leaves a gap of 20. Selling loans at 0.65 needs 20 / 0.65 = 30.769 of loans, leaving 59.231 against 42 of deposits: equity 59.231 - 42 = 17.231, so 10.769 is destroyed. With the facility the bank pledges 20 / 0.80 = 25.000 and keeps equity 90 - 42 - 20 = 28 before interest of 20 x 0.08 x 60 / 365 = 0.263.

Worked steps

  1. Gap = 38 - 18 = 20
  2. Loans sold = 20 / 0.65 = 30.769
  3. Fire-sale equity = (90 - 30.769) - 42 = 17.231
  4. Destroyed = 28 - 17.231 = 10.769
  5. Collateral = 20 / (1 - 0.20) = 25.000
  6. Facility equity = 90 - 42 - 20 = 28
  7. Interest = 20 x 0.08 x 60 / 365 = 0.263

Use the idea

Lend freely to solvent institutions against good collateral at a penalty rate, and check solvency first: lending cannot repair a bank whose assets are worth less than its deposits.

Where the conclusion applies

Loans worth their normal value if held, a fixed fire-sale price and simple interest. If the loans were truly worth less, the facility would only delay recognising the loss.

Check your understanding: With a 20 percent haircut, how much collateral backs a 20 million advance?
20 / (1 - 0.20) = 25 million.

Chapter 52 source: section "Lender of Last Resort".