Diamond Dybvig Model (1983)

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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

7. Game between patient depositors 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

8. Our experiment credit crunch Normal conditions

9. 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.

10. Early Solutions to Bank Runs • Put money in the windows • Slow up payments.

11. 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.

12. 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.

13. 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).

14. 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?

15. Answer: Yes. • 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.

16. 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

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. What should the Bank of England do?

29. 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?

30. 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.

31. 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.

32. 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.