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# Thoughts on Banking: An Example

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My earlier post left people confused. Let me give an example to clarify my model of financial intermediation with no information asymmetries.

We plant fruit trees this year. The total cost to plant them is \$5 million. Next year, there is a 60 percent chance the trees will be worth \$10 million and a 40 percent chance that they will be worth \$1.1 million. The one-year, risk-free interest rate is, say, 10 percent. I'll wave my hands about where that interest rate comes from.

The trees are financed with \$4 million in equity, held in a mutual fund, and \$1 million in debt. There cannot be more than \$1 million in debt, because there are no information asymmetries, so that everybody knows that more than \$1 million in debt would be a false promise.

A mutual fund handles all of the equity. People who are willing to take risk take lots of equity. Others prefer debt.

At the initial probabilities, the expected value of the trees in one year is 0.6 times \$10 plus 0.4 times 1.1, or \$6.44 million. Subtracting the value of the debt, the expected future value of the equity is \$5.44 million. Its present value is \$4.0 million, because of the combination of the discount rate of 10 percent and a risk premium of about 20 percent. I'll wave my hands about where the risk premium comes from.

Every day brings new information about the probability of success. If the probability of a good outcome goes up, the value of the equity goes up. If the probability goes down, the value of the equity goes down. For example, if the first day the probability of success gets revised down from 0.6 to 0.3, the future value of the equity becomes \$3.77 million. The present value, risk-adjusted, might be about \$2.8 million.

A bank handles everyone's accounts. Fruit is the medium of exchange and a store of value. The dollar is the unit of account. People can have either positive or negative balances at the bank. Those with positive balances earn instantaneous interest that works out to 10 percent at an annual rate. Those with negative balances pay 10 percent interest. Every instant, balances change, and new interest is calculated. The bank collects fees to cover its costs. All of this is risk-free. The bank makes no economic profit and need not fear a loss.

After the negative balances of some people have been netted against the positive balances of other people, the total positive balance in the bank will be \$1 million, which is the amount of risk-free debt that can exist in this economy. After one year, nobody will have a negative balance. All loans will be paid off, either before or at that time.

The entrepreneurs who own fruit trees are the ones who owe the net \$1 million. (If the bank includes their debt in its accounts, then the bank shows a net balance of zero.) The entrepreneurs are sure to be able to repay this loan. Consumers who have equity in fruit trees can borrow against that equity from other consumers who have more fruit than they wish to consume. However, they can only borrow for an instant, and they can borrow no more than the minimum possible value for their equity in the next instant.

For example, suppose that I have \$100 in equity, but it could lose 15 percent of its value in an instant. In that case, I can only borrow \$85 for an instant. Of course, if my equity retains its value or appreciates for a while, then I can roll over my loan, or even take out a bigger loan.

Perhaps the right way to think about this bank is to treat it as having no assets and no liabilities. It is simply a broker/accountant for the principals in the economy. If there were no bank, people would lend to one another. If there were no bank, the debt of the fruit tree entrepreneurs to lenders would be recorded in some other accounting book. Maybe calling this institution a bank is what is confusing--you tend to want to attribute to it properties of real-world banks that are not appropriate for this bank.

Truly,all the action in the economy is in the mutual fund. Each instant between now and when the fruit trees mature, new information arrives about probabilities of success. This new information changes the value of the mutual fund. That in turn changes people's consumption paths. If the news is good, people are more willing to consume during the year and risk having less to eat when the trees mature. If the news is bad, people worry more about not having enough to eat when the trees mature, so they cut back on consumption today.

The point of all this is that in an economy with no information asymmetries, financial institutions are pretty trivial. You cannot have a financial institution that introduces new risk into the economy, unless there are people who enjoy gambling--in which case you would get a casino, not a bank or mutual fund. You can only have financial institutions that can fail or debt contracts that might not be honored if there is asymmetric information. In the earlier post, I tried to explain how default-able debt or banks that might fail can arise when there is asymmetric information.

The information asymmetry might be real--Sue knows more than Fred about something real in the economy. Or it might be due to bias or irrationality--Sue thinks she knows more than Fred, but she really doesn't, and Fred takes advantage of her. I actually think that a lot of real-world financial institutions exploit the latter type of information problem.

### Comments and Sharing

COMMENTS (11 to date)
writes:

By postulating that the liquidation value of the fruit trees (presumably as firewood) exceeds the value of the debt tranche of their financing, you've taken most the fun out of the problem. This is certainly not true in the case of housing, for example!

Fixing the interest rate has removed a great deal of the remaining fun. And, perhaps most importantly, I don't see a consideration of what happens if all the depositors decide to withdraw all their dollars/fruit from the bank. Remember, this is basically our original problem.

Are the fruit farms liquidated? If not, who comes up with the money/fruit? Can the bank print it, or at least monetize its own liabilities? There seems to be an invisible Fed lurking at various points in this model.

As for information asymmetry, universal perfect knowledge of the future is also required in order to prevent loss. It is easy to imagine events whose knowledge is perfectly symmetrical, but which cause securities to "gap down." If Apple announced today that Steve Jobs was dying of cancer, the stock went from 100 to 50 on the news, and I had a stop-loss at 80, I would expect to see that order executed at 50, not 80. Even if Apple's secrecy until the announcement violates the asymmetricity postulate, there is always an instant when the doctor notices the mass on the slide. The shareholders, looking over his shoulder, gap down. My order at 80 will not find a buyer until 50.

And if I was using this equity as collateral for a loan of 75, I will, or at least may, default.

The assumption of continuity is just not valid. And financial engineering mechanisms designed to create the illusion of continuity tend instead to create extreme hairtrigger instability. See, again: portfolio insurance. If everyone decides to exit the same trade at the same time, they are not all going to get out at the last quoted price.

This is why I prefer to build the problem up from the ground, describing the individual transactions, instead of starting with these portfolio-theory constructs which include so many simplifications and assumptions.

ed writes:

Why are you assuming that the debt level must be low enough that there is no chance of default?

For example, what if someone notices that interest payments are deductible, but dividend payments are not. Now there is an incentive to finance with risky debt rather than equity. You don't need information asymmetries to explain that.

winterspeak writes:

Arnold: I see in this example you've focused more closely on a model of a bank as a "tally" mechanism -- matching credits with debts and nothing else.

And you are right, in a perfect tally, the bank takes no risk. It's almost as if there is no bank, lenders and borrowers are transacting directly by creating (and extinguishing) debits and credits. The tally bank is just a ledger keeping track of all of this.

(I actually think that this is a pretty good model of banks in a fiat currency world -- certainly much more realistic than the gold bars in vaults model we instinctively carry in our heads.)

But to Mencius' point, this does not absolve us of the bank run risk of maturities are transformed in this tally. The tally is not perfect (riskless) unless the credits and debits truly match, so both the size has to match, and when they are due/can be called in has to match.

My fruit trees take one year to grow. It matters whether the \$1M in debt will be called in during the growth period or not. Suppose that debt is also on the tally sheet (our bank ledger) -- doesn't it matter if it's called in? What if the ledger's creditors demanded it immediately? Would the equity price smoothly glide from a higher number to a lower number, or would it jump?

Our ledger is not a central bank and cannot produce money ex nihilo. If there is a run, it will be caught short.

Perhaps you mean to say that, because information is perfect, there will never be a run. Is that your argument?

-winterspeak

Arnold Kling writes:

Indeed, there should never be a run. With symmetric information and continuous trading, you never make a loan that cannot be repaid. To do so would be taking unnecessary risk. And in Black's world, you never take unnecessary risk.

In fact, that is one way to understand Black's world. Only necessary risks are taken. And in this model, the only necessary risk is the risk of a bad outcome for the fruit trees. As the probability of that bad outcome fluctuates during the year, people may trade consumption levels (and hence exposure to risk) with one another, based on differences in their tolerance for risk. Otherwise, nothing interesting happens. Certainly no loan defaults or bank runs.

With symmetric information, there is no maturity mismatching. Loans are instantaneous. The intertemporal risks are all loaded into the equity in the mutual fund.

Of course, in the real world we observe maturity mismatching and bank runs. But my point is that real-world banking requires some information asymmetry. It could be that the real-world financial system is an efficient allocation of information costs. I suspect, however, that it is instead a response to a demand for Ponzi schemes. When a risky system that appears to offer high returns with low risk competes with a robust system, the latter loses.

winterspeak writes:

Arnold: Thank you for clarifying.

I'm still not sure that "never taking an unnecessary risk" means you MT. I'm not sure it does not mean you simply match maturity so you take out bank-run risk as well.

In your example, all intertemporal risk is in equity, which can go to zero, and the debt still gets paid. As Mencius says, this takes a lot of the fun out. This crises came after a credit bubble, not an asset bubble, so how would your example work if the creditors would need to take a haircut in the event of a default? I think it may make a rush for the exits a viable strategy in the model.

In this case, a straight ledger + perfect information is not enough to make things riskless. You also need to protect against a rush for the exits.

Is that right?

Arnold Kling writes:

I am not sure what a "rush for the exits" might mean in this economy. Does it mean that, in spite of the fact that the debt is sure to be repaid when the trees mature (even in the worst possible disease outcome), people don't want to hold the debt? If so, then why did they agree to hold it in the first place? What news could possibly make them change their minds?

This model is very limited in terms of news events. I don't think there is anything that people can learn that would make them want to "rush for the exits." All they can learn is that the probability of healthy trees has instantaneously gone up or down.

The point I am trying to make about the zero-information-asymmetries world is that there is perfect transparency. Your bank is either solvent or it isn't, and you can see right through to know the difference. Its assets are perfectly liquid, because everyone knows what they represent. So if it's solvent, it's also immune to a run--it just sells its assets to meet the needs of anyone trying to withdraw money.

We're used to thinking of banks as having lots of illiquid assets. But that's because we're used to a world of imperfect transparency. Which is quite realistic, I should add. I just want to start with the unrealistic case in order to have a baseline.

winterspeak writes:

Arnold: Sorry for not making myself clear.

In your model, you have debt that will 100% be repaid, plus equity that can go to zero.

My question is, suppose the debt might not be 100% repaid (equity can still go to zero). It would be partially repaid, but not 100% repaid.

So instead of the 40% chance it's worth \$1.1M, suppose there is a 40% chance it is worth \$0.9M.

The venture is financed in part by \$1M in debt.

My question is, as we approach failure mode, what happens to that debt? Is it tradable, like the equity is? Or is it locked in for the duration of the venture?

I want to understand the debt side of this equation because debt has obligations associated with it that equity does not, and we're looking at what happens when debt obligations are reneged on.

-winterspeak

fundamentalist writes:

Arnold: "The point I am trying to make about the zero-information-asymmetries world is that there is perfect transparency."

With symmetric information there are no risks, but as Hayek points out there are no profits either.

winterspeak writes:

Fundamentalist: The question is whether or not you need to maturity match to take unnecessary risk out of debt instruments.

In my interpretation of Arnold's model, he gets around this problem by making debt riskless, and converting all inter-temporal risk to equity. In Arnold's interpretation, this inter-temporal equity market functions smoothly as it has perfect information (I think).

Arnold argues that he's making a point about information asymmetry. To me, it looks like he's using capital structure to eliminate the key problem we are discussing. I'm trying to square this circle.

writes:

It could be that the real-world financial system is an efficient allocation of information costs. I suspect, however, that it is instead a response to a demand for Ponzi schemes. When a risky system that appears to offer high returns with low risk competes with a robust system, the latter loses.

I think we can agree on that!

I'm not sure I'd describe a financial system of systematic maturity mismatching as a "Ponzi scheme." A true Ponzi scheme is even more unstable, because it depends on continuous expansion. But the word "scheme" definitely strikes me as none too strong. "Bagehot scheme," maybe?

writes:

Let me see if I have it right.

The \$1-million loan the entrepreneur took out can be called at any time.

In fact, all loans are like that in Black's model.

If the loan is called before the trees bears fruit, then obviously the loan cannot be paid in fruit. It will have to be paid in firewood or in equity.

If it is still possible that the trees will bear fruit, the originator of the loan will tend to prefer equity over firewood.

There is a mechanism by which a sufficiently large fraction of the equity holders can compel the entrepreneur to convert equity into firewood. If the cost of employing the mechanism is zero, then equity is always preferable to firewood (because an option never has negative value).

Is that right?

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