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# Diamond Dybvig Model (1983) - PowerPoint PPT Presentation

Diamond Dybvig Model (1983). Captures elements of what a bank does. Shows that there is a basic problem of bank runs. The model consists of two parties. Depositors Banks The model has three time periods: yesterday, today and tomorrow. Depositors.

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Diamond Dybvig Model (1983)

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### Diamond Dybvig Model (1983)

• Captures elements of what a bank does.

• Shows that there is a basic problem of bank runs.

• The model consists of two parties.

• Depositors

• Banks

• The model has three time periods: yesterday, today and tomorrow.

### Depositors

• Depositors placed money (say £1000) in a bank (yesterday) before learning when they need the money.

• Depositors either need their money today (impatient) or tomorrow (patient). There is a 50% chance of being either type.

• The ones that need their money tomorrow can always take the money today and hold onto it.

• The ones that need money today get relatively very little utility for the money tomorrow.

### Banks

• Banks have both a short term and a long term investment opportunity for the money.

• The short term investment (reserves) is locking the money in the vault. This investment returns the exact amount invested.

• The long term investment returns an amount R tomorrow. It is illiquid and returns only L<1 today.

### Deposit Contract

• The depositors invested £1000 yesterday have a contract with the bank.

• The depositors can withdraw their money today and receive £1000 or wait until tomorrow and receive R*£1000.

### Bank’s decision

• How can the bank meet this contract?

• The bank can divide into two parts.

• Take half and keep it as reserves.

• Take the other half and put it in the long term investment.

• Say there are 10 depositors: 5 patient and 5 impatient. The bank puts £5000 in the vault and invests £5000.

• Demands today are 5*1000, and 5*R*1000. The bank has 5000 and R*5000 tomorrow.

• Thus, a bank makes zero profit.

### Multiple equilibria

• This leads to multiple (Nash) equilibria.

• It is inherent in banking.

• Here is an example with 2 patient depositors (and 2 impatient depositors).

• This forms a 2x2 game between the patient depositors.

• R=1.5 and L=.5

Depositor 1

Tomorrow

Today

0

3/4

Today

3/4

1

Depositor 2

1

3/2

Tomorrow

0

3/2

R=1.5, L=.5

### Our experiment

credit crunch

Normal conditions

### What is not captured in the model

• Uncertainty in depositor’s preferences.

• Too many actually need the money today.

• Riskiness in technology.

• Perhaps there really isn’t enough to meet demand tomorrow.

• Implication: some bank runs will be rational.

### Early Solutions to Bank Runs

• Put money in the windows

• Slow up payments.

### Solutions.

• Make sure R is not risky.

• Pay early withdrawers less than 1 or pay late withdrawers less than R (and keep more reserves)

• Problems: not best contract.

• Suspend payments/ Partial Suspension.

• Problem when number needing money today is uncertain.

• Creditor Coordination.

• Long Term Capital Management ran into trouble in 1998.

• The NY FED organized a bailout with creditors.

• Lender of last resorts.

• Central bank will stop in and loan the bank money to replace deposits.

• This should work with depositors in the case of a problem with liqudity

• In 1975,

• April 14th, Credit Suisse announced lost some money in one of its branches. It didn’t mention details.

• April 25th, The Swiss Central Bank announced it was willing to lend money.

• This had the opposite result cauing share price to tumble 20%.

• Deposit Insurance.

• This works well. Risk-Sharing between banks.

### Better Contract

• Why should the bank pay the depositors withdrawing early only 1?

• The bank can pay them more.

• This would “insure” a depositor against needing the money early.

• For R=1.5, what would the full insurance contract look like. In other words, the payment is the same in either period.

• The amount would solve (2000-X)*R=X

• This amounts to a gamble of having either 1000 or 1500 or 1200 for sure. Risk-averse enough people would prefer 1200.

• Note the best contract (and perhaps fairest) will pay depositors withdrawing today somewhere 1000 and 1200.

### Hidden assumption

• Depositors withdraw sequentially: a bank cannot count the number of people wanting to withdraw today and then decide how much to pay them.

• Otherwise, they can just pay them 5000/N where N is the number withdrawing early (for the 10 depositor case).

### Insurance Problem: Moral hazard

• Todd buys theft insurance for his laptop.

• Because he buys the insurance, he is more likely to leave the laptop in his car.

• Ideally, he would like to commit to not leaving the computer in his car.

• Sometimes, we can contract on it.

• Other times, we can’t.

• Do we have a moral hazard problem with deposit insurance?

• Marc is the manager of a Springfield S&L.

• Marc pays higher interest than a bigger and safer bank claiming his small size helps him cut costs.

• Springfield has deposit insurance (100%).

• Todd puts money in Springfield.

• Springfield lends money to a dodgy lecturer at Springfield State University at a higher rate.

• When there is no default, everyone wins.

• When there is a default, Todd still gets paid.

• Without insurance, Todd wouldn’t invest if he sees Springfield’s risky behavior.

### Model of Moral Hazard.

• The bank can choose any investment x, where 3>x=>1.

• Any investment costs £.95 and is either successful and pays of x or unsuccessful and pays £0.

• The probability of the investment being successful is

P(X)=(3-x)/2.

• Choosing x=1 is safe, choosing x close to 3 is unsafe.

• Todd is close to risk neutral and wants to earn at least as much as £1 (in expectation) which the other banks are offering as a risk free investment. He wants R where R*P(x)=1.

• Without insurance, the bank maximizes

• P(X)*(X-R) where R=1/P(x)

• With insurance, Todd only needs R=1. So the bank maximizes

• P(X)*(X-R) where R=1

### Savings and Loans scandal

• In the 1980s about 1000 S&L’s went bankrupt.

• They originally lent money out at fixed rates of 6% and paid deposits 3%.

• With inflation, they lost money.

• Took gambles to catch up, went to Vegas.

• They were able to take high risk due to the deposit insurance.

• This cost US taxpayers \$120 billion.

### Solution to Moral Hazard

• One solution is for insurance to not be 100% (co-pay as in the UK).

• However, this requires the depositors to be savvy and this still keeps the multiple equilibrium problem.

• In the US, in 2006 Bush signed a law allowing the FDIC to charge premiums based upon risk.

### Lender of Last Resorts: commitment

• Gambling Jim has a rich uncle.

• Jim’s uncle loves him very much.

• Jim blows his money in a poker game. His rich uncle bails him out.

• His uncle says that is the last time.

• Jim gambles again and loses. His rich uncle can’t bear to see Jim’s legs broken.

• The problem is that Jim knows his uncle will always be there for him.

• The uncle can either find some way to commit not to help Jim afterwards, or sacrifice Jim to stop his other nephews from gambling.

### Northern Rock and the Subprime crisis.

• Jim Cramer on subprime.

• Bill Poole says that it is risky lenders that got what they deserved.

• Jim Cramer more or less says everyone is in trouble.

• Bernanke is thinking about whether to cut rates.

### Subprime mortgages

• Miriam, a divorced mother, was offered a mortgage on a 2-28 deal: 2 years of a teaser rate of a mortgage and the rest floating.

• Finally, the dream of owning a home is a reality.

• Miriam did not have to verify income or assets. She got a piggyback loan to cover the down payment.

• She was told that she can refinance after two years and with the prices the way they are going get some money out as well.

• She took some extra credit cards and agreed.

• The bank took her mortgage packaged it up with others. The rating agency (paid by the bank) rates the package high. It is sold to a hedge fund.

• The bank now only collects the money.

### Sub-prime continued

• Unfortunately, rates went up and she couldn’t make the payments.

• Housing prices didn’t go up and she has no equity.

• The local bank doesn’t want to work out a deal so forecloses (actually collects extra fees doing so).

• These packages drop faster than one would have thought.

### Hedge fund

• Hedge funds are highly leveraged. The price of these securities drop more than they should given the state of the economy, interest rate, etc.

• People loaning the hedge funds want to take their money out. Forcing the hedge funds to sell more. This further suppresses the price.

• Hedge funds start to go broke.

• Banks also have these mortgages on their books. It isn’t clear who owns what.

• Banks don’t want to lend to each other for two reasons:

• Afraid of the financial state of other banks.

• Want to keep extra reserves in case they can’t borrow.

### Northern Rock

• Had mortgages not necessarily subprime.

• To provide funding for these mortgages, it had deposits and borrowed from other banks.

• The other banks were in essence another depositor.

• When the credit dried up, the other banks needed the money (became impatient).

• Northern Rock couldn’t continue to borrow.

• They had to borrow from the lender of last resorts.

### Analogy

• There are a limited number of homes around a lake.

• The owners of the homes only sometimes go there for a few days. They only know if they can go last minute.

• When they don’t go, they rent them out. Say that they stay there only 1/5 the time and there is 3 non-owners for every owner during a summer.

• The owners are indifferent to staying at their home or someone else’s home.

• Thus, the owners are not too worried about renting out their home since they can always rent someone else’s home.

### Lake Analogy

• Suddenly, a rumour spreads that it will be hard to rent.

• The owners want to stay very much when they are free and take their homes off the market.

• Since all the owners do so, there is no rental market and it is self-fulfilling.

### Northern Rock Bank Run

• Depositors now started to run.

• Was it rational for shareholders to run as well?

• Not enough deposit insurance.

• It also wasn’t clear how much lending was to Northern Rock.

• It isn’t clear how good their loans are.

What should the Bank of England do?

### Should Mervyn King act?

Questions:

• Was there a risk of contagion.

• Was it a solvency problem or a liquidity problem?

• Would it cost tax-payers?

• Does this set a bad example?

Actions: Cut Rate, Lend, Organize a bailout (LTCM)

Future:

• Rewrite the deposit insurance?

### Other applications

• Farepak.

• People saved during the year and got coupons at end worth their savings.

• Company used next year’s money to pay for this year’s coupons.

• Defined-Benefit Pension schemes/Social Security.

• Young pay for old.

• Usually mandatory to stop runs.

### What we learned

• Theoretical Model of Bank Runs.

• That these may actually happen (experiment).

• Possible solutions to the problem.

• The Moral Hazard problem.

• A bit about the current crisis.

### Homework.

• Take the DD model with L=.5 and R=2. Let us say that deposits are insured up to fraction f. For what values of f is there only one equilibrium and what values are there two equilibria? (Early withdrawers are guaranteed to get 1*f and late get 2*f.)

• How would you modify our classroom experiment to test different deposit insurance schemes? Under what parameters do you think we will get a bank run.