The Encyclopedia of Economic Principals

Chapter 49

Liquidity, Fire Sales, and Financial Fragility

When selling to raise cash lowers the value of what is left.

Four of the chapter's worked examples, made interactive: a fire sale that marks down collateral, a margin spiral, delegated monitoring with a run, and a credit crunch with and without a central bank facility.

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

A fire sale marks down everyone's collateral

Can a fund cure a covenant breach by selling into the same thin market that caused it?

Each seller treats the price as given, but with limited specialist cash every extra machine lowers the mark on all remaining collateral. B's sale makes its own covenant worse, an externality that more buyer cash removes.

Equation, written in LaTeX: p=\frac{8{,}000}{100}=80\text{ dollars per machine}.

Equation, written in LaTeX: 0.80(80)(100)=6{,}400\text{ dollars},

Equation, written in LaTeX: 7{,}500-80x=0.80(80)(100-x).

Equation, written in LaTeX: p=\frac{8{,}000}{169}\approx47.34\text{ dollars per machine}.

Equation, written in LaTeX: 0.80(31)(47.3373)\approx1{,}173.96\text{ dollars}.

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p is the cash-in-the-market price: specialist cash divided by machines offered, capped at the 100 dollar use value. Fund A sells 100 machines. Fund B owns 100, owes 7,500 and must keep debt at or below 80 percent of marked collateral. x is the number of machines B sells.

Predict first. With 8,000 dollars of specialist cash, does B's planned sale of 69 machines cure the breach?

Your prediction

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Figure: A fire sale marks down everyone's collateral. Left: price per machine 80.00 when A sells alone and 47.34 when B also sells. Right: B's debt against 80 percent of marked collateral before and after; remaining shortfall 3,059.76.
Specialist buyers' cash (dollars): 8,000
Constructed example: the chapter's hypothetical funds (8,000 and 12,000 of specialist cash, debt 7,500, 80 percent covenant); cash of 9,000 and 10,000 is added for comparison.

Calculated values

First mark
$80.00
Covenant capacity at first mark
$6,400.00
Breach
$1,100.00
Machines B plans to sell
68.75, so 69
Clearing price with both sellers
$47.34
Remaining shortfall
$3,059.76

With $8,000 of specialist cash, A alone clears at 8,000 / 100 = 80.00. B's capacity is 0.80 x 80.00 x 100 = 6,400.00, a breach of 7,500 - 6,400.00 = 1,100.00. Solving 7,500 - 80.00x = 0.80 x 80.00 x (100 - x) gives x = 68.75, so B plans 69 sales. With 169 machines for sale the price falls to 8,000 / 169 = 47.3373. Debt becomes 4,233.73 against capacity 1,173.96, so a shortfall of 3,059.76 remains: the cure does not work once B's own sale moves the price.

Worked steps

  1. First mark = 8,000 / 100 = 80.00
  2. Capacity = 0.80 x 80.00 x 100 = 6,400.00
  3. Breach = 7,500 - 6,400.00 = 1,100.00
  4. x = 1,100.00 / (80.00 x 0.20) = 68.75, round up to 69
  5. Clearing price = 8,000 / 169 = 47.3373
  6. Debt = 7,500 - 69 x 47.3373 = 7,500 - 3,266.27 = 4,233.73
  7. Capacity = 0.80 x 31 x 47.3373 = 1,173.96
  8. Shortfall = 4,233.73 - 1,173.96 = 3,059.76

Use the idea

Before relying on asset sales to meet a covenant or margin call, ask who else will sell to the same buyers and how much cash those buyers have.

Where the conclusion applies

Cash-in-the-market pricing with a fixed specialist budget, settlement at one price, whole machines, all proceeds repay debt.

Check your understanding: With 9,000 dollars of specialist cash, how many machines must B plan to sell and what shortfall remains?
p = 90; capacity 7,200; breach 300; x = 300 / 18 = 16.67, so 17; new price 9,000 / 117 = 76.92; debt 6,192.31 against capacity 5,107.69; shortfall 1,084.62.

Chapter 49 source: section "Fire-sale externality".

Demonstration 2 of 4

Margin spirals force extra selling

How much of a dealer's forced sale comes from the lender raising the margin?

A price fall shrinks equity, and a higher margin shrinks what each dollar of equity can finance. Together they force sales that can lower prices again: market liquidity and funding liquidity feed each other.

Equation, written in LaTeX: 0.10(10\text{ million dollars})=1\text{ million dollars},

Equation, written in LaTeX: 9.5-\frac{0.5}{0.10}=4.5\text{ million dollars}.

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The dealer borrows 9 million dollars against a bond position marked at 9.5 million (book). Equity is the mark minus debt, and a margin m limits the position to equity / m.

Predict first. At the book's 9.5 mark, how much of the 7 million sale at a 20 percent margin is caused by the margin increase alone?

Your prediction

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Figure: Margin spirals force extra selling. Bars: marked position 9.5, financeable position 2.50 and required sale 7.00 million dollars at a 20 percent margin.
New margin: 20%, Marked position (million dollars): 9.5
Constructed example: the chapter's hypothetical dealer (mark 9.5, debt 9, margins 10 and 20 percent); margins of 15 and 25 percent and marks of 9.7 and 9.8 are added for comparison.

Calculated values

Equity
0.50
Largest financeable position
2.50
Required sale
7.00
Sale at a 10 percent margin
4.50
Extra sale from the margin change
2.50

Equity is 9.5 - 9 = 0.50 million. At a 20 percent margin it finances 0.50 / 0.20 = 2.50 million, so the dealer sells 9.5 - 2.50 = 7.00 million. At 10 percent the sale would be 9.5 - 0.50 / 0.10 = 4.50, so the margin change adds 2.50 million of selling.

Worked steps

  1. Equity = 9.5 - 9 = 0.50
  2. Financeable = 0.50 / 0.20 = 2.50
  3. Sale = 9.5 - 2.50 = 7.00
  4. Sale at 10 percent = 9.5 - 0.50 / 0.10 = 4.50
  5. Extra = 7.00 - 4.50 = 2.50

Use the idea

Stress a leveraged position for a price fall and a margin rise at the same time, not one at a time.

Where the conclusion applies

Debt stays at 9 million; the snapshot ignores the further price impact of the sale itself.

Check your understanding: With the mark at 9.8 and margin 15 percent, what must the dealer sell?
Equity 0.8; financeable 0.8 / 0.15 = 5.33; sale 9.8 - 5.33 = 4.47 million dollars.

Chapter 49 source: section "Market-liquidity and funding-liquidity spiral".

Demonstration 3 of 4

Delegated monitoring and maturity mismatch

Why does a bank save monitoring costs, and how much does a run cost it in forced loan sales?

One specialist reviewing each borrower replaces 100 repeated reviews, which is why the bank assembles illiquid loans. Funding them with demandable deposits means a large withdrawal forces sales below face value.

Equation, written in LaTeX: 100(20)(50)=100{,}000\text{ dollars}.

Equation, written in LaTeX: 20(400)+2{,}000=10{,}000\text{ dollars}.

Equation, written in LaTeX: 60{,}000-20{,}000=40{,}000\text{ dollars}.

Equation, written in LaTeX: \frac{40{,}000}{0.75}=53{,}333.33\text{ dollars}.

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100 savers deposit 1,000 each; the bank makes 20 loans of 5,000. A saver's review costs 50 per loan, a specialist's 400, and governance 2,000. The bank holds 20,000 of reserves and 80,000 of loans that sell immediately at a price per dollar of face.

Predict first. Do 15,000 dollars of withdrawals force any loan sale?

Your prediction

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Figure: Delegated monitoring and maturity mismatch. Left: monitoring cost 100,000 direct versus 10,000 delegated. Right: 20,000 of reserves and 80,000 of loans, of which 53,333.33 of face is sold at price 0.75.
Early withdrawals (dollars): 60,000, Immediate sale price per dollar of face: 0.75
Constructed example: the chapter's hypothetical bank (withdrawals 15,000 and 60,000, price 0.75); withdrawals of 40,000 and prices 0.85 and 0.95 are added for comparison.

Calculated values

Direct monitoring cost
$100,000
Delegated cost
$10,000
Saving
$90,000
Cash gap
$40,000.00
Loan face sold
$53,333.33
Loss from the sale
$13,333.33

Direct review costs 100 x 20 x 50 = 100,000 and delegation 20 x 400 + 2,000 = 10,000, a saving of 90,000. Withdrawals of 60,000 exceed the 20,000 of reserves by 40,000, so the bank sells 40,000 / 0.75 = 53,333.33 of loan face, losing 53,333.33 - 40,000 = 13,333.33.

Worked steps

  1. Direct = 100 x 20 x 50 = 100,000
  2. Delegated = 20 x 400 + 2,000 = 10,000
  3. Gap = 60,000 - 20,000 = 40,000
  4. Face sold = 40,000 / 0.75 = 53,333.33
  5. Loss = 53,333.33 - 40,000 = 13,333.33

Use the idea

Compare a lender's liquid reserves with plausible withdrawals, and price the loss from selling loans quickly at a discount.

Where the conclusion applies

Loans sell at one price per dollar of face; the governance cost of 2,000 is fixed.

Check your understanding: With 40,000 of withdrawals and a 0.85 price, how much loan face is sold?
Gap 40,000 - 20,000 = 20,000; 20,000 / 0.85 = 23,529.41 of face.

Chapter 49 source: section "Financial Intermediation and Banking".

Demonstration 4 of 4

Liquidation losses become a credit crunch

How does a forced loan sale shrink a bank's lending, and what does a central bank facility change?

Selling loans below face turns a liquidity need into a capital loss, and with a capital requirement each lost dollar of equity removes ten dollars of lending. A lender of last resort that lends against the loans avoids the sale and the loss.

Equation, written in LaTeX: 20+100-110=10\text{ million dollars}.

Equation, written in LaTeX: \frac{40}{0.90}=44.44\text{ million dollars}

Equation, written in LaTeX: 44.44-40=4.44\text{ million dollars}.

Equation, written in LaTeX: 55.56-50=5.56\text{ million dollars}.

Equation, written in LaTeX: 100-50-40=10\text{ million dollars}.

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The bank holds 20 of cash and 100 of loans against 110 of claims (million dollars); 60 is due today. Loans sell at a price per dollar of face, and a 10 percent capital requirement caps lending at equity / 0.10.

Predict first. At which sale price does the forced sale wipe out all equity?

Your prediction

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Figure: Liquidation losses become a credit crunch. Equity and lending capacity before (10 and 100) and after the 60 million payment (5.56 and 55.56), facility off, sale price 0.90.
Sale price per dollar of loans: 0.9, Central bank facility: Off
Constructed example: the chapter's hypothetical bank (price 0.90, facility off and on); prices 0.80 and 1.00 are added for comparison.

Calculated values

Funding gap
40.00
Loan face sold
44.44
Liquidation loss
4.44
Equity after
5.56
Lending capacity
55.56

Cash covers 20 of the 60 due, leaving a gap of 40. At 0.90 per dollar the bank sells 40 / 0.90 = 44.44 of loans, a loss of 44.44 - 40 = 4.44. Loans left are 100 - 44.44 = 55.56, so equity of 55.56 - 50 = 5.56 supports 5.5556 / 0.10 = 55.56 million of lending. The book rounds equity to 5.56 first and prints 55.6.

Worked steps

  1. Face sold = 40 / 0.90 = 44.44
  2. Loss = 44.44 - 40 = 4.44
  3. Loans left = 100 - 44.44 = 55.56
  4. Equity = 55.56 - 50 = 5.56
  5. Capacity = 5.5556 / 0.10 = 55.56

Use the idea

Translate a funding gap into face sold, loss, equity and lending capacity before judging whether a bank needs liquidity or capital.

Where the conclusion applies

One sale price, a binding 10 percent requirement, and loans worth face at maturity. If loans are worth less, the facility leaves an insolvency problem.

Check your understanding: With price 0.80 and no facility, what is equity and lending capacity?
Face sold 40 / 0.80 = 50; loans left 50; equity 50 - 50 = 0; capacity 0.

Chapter 49 source: section "Liquidity Crisis / Credit Crunch".