Liquidity, solvency, fire-sale and fair prices, and TARP: a simple model

Nick Rowe follows up his post on the origins of the crisis with some thoughts about how to deal with it:

How is the Troubled Asset Relief Program supposed to resolve the problem of bank failures? I haven’t seen any clear answer, so thought I would sketch out the skeleton of a simple model where TARP would rescue banks, at no cost to the taxpayer.

There are two types of savers: long-term savers, who know they will not need their savings for many years; and short-term savers, who face a stochastic need to withdraw their savings in one year. Those chances are not perfectly correlated, so a relatively predictable percentage of short-term savers will need to withdraw their savings every year.

There are two types of investment projects: long-term projects, which mature in many years and earn a high rate of interest; and short-term projects, which last one year and earn a low rate of interest.

So far the model is very similar to the Diamond-Dybvig model of bank runs. The long-term savers set up a bank, and accept deposits from short-term savers. The deposits can be withdrawn on demand after one year. Most of the assets of the bank are invested in the long-term projects; only a small fraction, sufficient to cover predictable withdrawals from short-term savers, is invested in the short-term projects. In a “good” equilibrium, only those short term savers who truly need to withdraw funds will do so. But there is also a “bad” equilibrium, with a bank run, where every short-term saver rushes to be first in line to withdraw funds because he expects every short-term saver to run to the bank too, so the belief is self-fulfilling.

So far, the bank can become illiquid, but cannot become insolvent. Eventually the long-term investments will earn sufficient to pay off the depositors. The worst that can happen is that the bank needs to sell its long term investments, and use the proceeds to pay its depositors. Those long-term investments could be bought by other long-term savers. They could also be bought by short-term savers, who know that they could re-sell those assets to other short-term savers the following year, if they needed funds. In fact, it is hard to see why banks are necessary at all; the short-term savers could simply invest directly in long-term investments, and sell them to any time they needed funds.

Now introduce risk. Suppose that some of the long-term investment projects go bad, and become worthless. If a small number of investments go bad, the long-term savers bear all the risk, and the deposits are safe. But if enough investments go bad, the losses are greater than the capital invested in the bank by the long-term savers, and the bank would be insolvent, and unable to pay its depositors even if it sold its remaining good assets. With risk, it is even harder to see why banks are necessary. Banks merely cause a rather peculiar distribution of risks, where long-term savers bear all the risk until they are bankrupt, and then the short-term savers bear all the remaining risk. It would make more sense for all savers to hold a mix of short-term investments for safety, and mutual funds in the long-term investments for return, since they can sell the mutual funds if they need cash.

Now introduce a fixed cost of learning whether a long-term investment is good or bad. Short-term savers will now avoid buying long-term investments, because if they need to re-sell those investments the new buyer would need to pay the cost of learning whether they are good or bad. Every time the long-term investment changes hands, there is a transactions cost, because the buyer cannot trust the seller. It would now be efficient for the long-term savers to hold long-term investments, because they only need to pay that transactions cost once. Now there is a genuine role for banks. The long-term savers start a bank, which pays the transactions cost on long-term investments once only, while accepting deposits from short-term savers.

So in 2007 the bank is running smoothly in the “good” equilibrium (government deposit insurance, and the assurance that the central bank will lend freely to a solvent bank, prevents the “bad” equilibrium of a bank run). Then in 2008 a shock hits. Some of the long-term investments, which the bank thought were good, turn out to be bad. This shock is public knowledge. The bank knows which of its long-term investments are bad, but the public does not.

Is the bank still solvent? If it sold all its investments, could it pay its depositors? If the bank is small, and the shock hit only that one bank, solvency can easily be determined. The bank sells its long-term investments to long-term savers, who do the research to find out how many of those investments are good. The price will be the expected value of the investments, if held to maturity, minus a one-time transactions cost. A solvent bank can be wound up, or bought by another bank, or have its assets valued at market prices and shown to be solvent.

Now suppose that the same shock hits all banks. There may not be enough long-term savers able and willing to buy the long-term assets of all the banks at anything like their expected value. This will be especially true if all long-term savers had invested all their savings in banks, because then all long-term savers would be in the same position, and would have suffered a loss of their wealth. The only people left to buy the long-term investments would be short-term savers.

Assume the short-term savers have a 50% need of funds in any one year, and so on average they withdraw their savings after two years. Assume long-term investments mature in 30 years. A long-term investment held by a sequence of short-term savers would change hands 15 times, and each time there would be the transactions cost of learning whether the investment was good or bad. So short-term savers would be willing to pay the expected value minus 15 times the transactions costs. That is the fire-sale price of a long-term investment. It could easily be zero. And the market for those long-term assets could be very thin. The demand curve from short-term savers would be elastic, but at a very low demand price. The supply curve would be very elastic at the expected value to maturity, but would be very inelastic at the fire-sale price, for only the most desperate banks would be willing to sell at that price, and then only sell the minimal amount.

So a bank could be quite solvent, in the sense that if it held its long-term investments to maturity it could repay its depositors, but at the same time appear insolvent under mark-to-market accounting, at fire-sale prices. This problem only arises when there is a system-wide banking problem; it does not arise for a single (small) bank.

The solution is to find another source of long-term savers, willing to hold the long-term investments to maturity. That’s where TARP comes in. If there exists some price of the bank’s assets, between the fire-sale price and the value if held to maturity, at which the bank is solvent, then if TARP buys assets just slightly above that price, an otherwise insolvent bank (at market prices) is made solvent, and the taxpayer makes a profit. Once the bank is shown to be solvent, its liquidity problems can be solved easily, either by the restored faith of its depositors, or by loans from the central bank.

The fundamental problem is that it makes sense for long-term savers to do the research, and for those who do the research to hold the risk. But if a lot of risks go bad, we run out of wealthy long-term savers.

Comments

[westslope]

Thanks Nick, much appreciated. Very much appreciated.

Does this model suggest why buying the 'toxic assets' themselves as opposed to simply recapitalizing banks is a good or better idea?

Or do the presence of significant transaction costs and fears of political rent-seeking suggest why the state might wish to delegate investment research costs to private, presumably better qualified, banks?

[Nick Rowe]

Thanks westslope. When building the model, I hadn't thought about comparing buying troubled assets vs buying equity in the bank. I haven't worked it out, but my guess is that if the government bought equity in the bank, it would get more "bang for the buck invested", but at higher transactions costs. Good question. I'm not sure of the answer.

[booster]

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