## Sunday, March 30, 2008

### Adverse Selection, Loan Defaults, and Credit Rationing

What is the source of foreclosures, interest rate resets or falling prices? In this post, I said:

It's easy to explain how interest rate resets could increase foreclosure rates since the monthly housing payment will change as a result of the reset, but why do falling prices cause increased foreclosures? Falling prices don't change monthly payments, so why do more people default?

The post then explains how falling prices can increase defaults and Richard Green provides academic work supporting the mechanism.

But falling prices and interest rate resets are not mutually exclusive explanations for rising foreclosure rates, both could be at work, and Brad DeLong presents a model that explains how, through adverse selection, rising interest rates can cause increases in defaults.

The purpose of this post is to further illuminate Brad's discussion and to explain how the adverse selection mechanism operates.

To do so, rather than try to impress all of you by building my own model, I'll avoid reinventing the wheel and will instead base this discussion on a version of the Stiglitz and Weiss (1981) model presented in Carl Walsh's Monetary Theory and Policy text (so full credit should be given to those authors). The presentation is mathematical, but along the way I will try to provide the intuitive underpinnings, so hopefully the points will be clear even if the mathematics is not.

The basic point of the model is to illustrate two things. First, how an increase in the interest rate can increase defaults. This is the main point. Second, how equilibrium credit rationing can occur, i.e. how financial markets can settle on an equilibrium where there are buyers willing to take out loans at the going interest rate, but nobody willing to lend them money at that rate, and the excess demand for loans is not resolved through rising interest rates.

I should add that there are other mechanisms that can be used to explain these results, moral hazard and monitoring cost models for example (e.g., as interest rates increase, borrowers are induced to take on more risk in moral hazard models, and the increased risk taken on by borrowers increases defaults) so this should not be considered exhaustive.

The Model:

1. The model contains a single type of lender, and different types of borrowers. The lender's expected return on loans is a function of the interest rate and the probability of repayment. The probability of repayment will vary across individuals.

2. Borrowers come in two types:

Type g: this type repays loans with probability qg
Type b: this type repays loans with probability qb

It is assumed that  qg > qb, so that the good borrowers (g types) are more likely to repay than the bad borrowers (b types).

3. If lenders can observe the type of borrower they are lending to, they will charge each a different interest rate to reflect differences in risk, and the market will clear without credit rationing. It will be fully efficient.

4. For example, we'll assume the supply of credit is perfectly elastic (the supply curve for credit is horizontal): Assuming risk neutral lenders, and that they lend to a large number of borrowers (so the law of large numbers applies), the lender will charge r/qg to the good borrowers, and  r/qb to the bad borrowers and will realize an expected return of r for each group (i.e., the lender receives r/q with probability q so the expected return is q(r/q)=r). There is no credit rationing, the lender simply charges risky borrowers more than good borrowers to compensate for the extra risk.

5. Now assume the lender cannot observe the different types of borrowers. What we will now show is that as the interest rate, r, increases, the fraction of bad borrowers in the loan applicant pool also increases, so the probability of default goes up.

That is the main point - when interest rates rises, the good borrowers drop out (this is the adverse selection mechanism, the good borrowers self-select out of the pool leaving a greater fraction of risky borrowers in the market).

But we can also get equilibrium credit rationing with this as well. Here's how. As r increases, the return to the lender goes up, so an increase in r increases profit. But, as we will show, the increase in r also causes the fraction of bad borrowers to go up and this increases the default rate and lowers profit. So, the net effect of an increase in r on profit depends upon which of these two effects is stronger, the increase in profit from charging a higher interest rate, or the decrease in profit from higher defaults.

To get rationing, what we have to show is that as r increases, initially profits go up since the higher price effect dominates the increase in defaults. But there comes a point in the lender's profit function where any further increase in r is not profitable, the loss from defaults increasing is larger than the profit increase from the higher interest rate, so the lender won't raise the interest rate even if there is an excess demand for loans at the current r being charged in the marketplace.

This is the credit rationing. Even if there is excess demand for loans, the lender will not raise r above the critical value of r, call it r*, where profit begins falling.

Let's show this mathematically.

6. Let g be the fraction of good borrowers among all borrowers. In order to earn an expected return of r, the lender charges borrowers (which cannot be distinguished and hence face the identical loan rate) r1 such that:

gqgr1 + (1-g)qbr1 = r = expected return if lender charges  r1 to all types.

The lender should charge:

r1 = r/[gqg + (1-g)qb]

7. Using this strategy, the lender will thus earn r if borrowers are chosen (walk through the doors of the bank) randomly. But they don't show up randomly, so this is not the end of the story.

Notice that r/qg < r1< r/qb. Good borrowers are paying too much, and bad borrowers are paying too little. Thus, good borrowers are more likely to drop out of the market, and the fraction of good borrowers will diminish over time increasing average default rates  (perhaps because they are good borrowers they can find other, cheaper ways to finance investment) .

This is a classic lemons problem - good borrowers leave the market increasing the average risk and default rates of borrowers still in the market, interest rates go up to compensate for the higher risk, more borrowers leave the market, average risk goes up, more borrowers leave, etc. - and it will lead to market failure.

8. Now let's change the model slightly to illustrate equilibrium credit rationing. Loans are characterized by more than just the interest rate, and here we will characterize loans by three parameters, the interest rate lenders charge on loans,  r1, the size of the loan L, and the required collateral on the loan, C.

9. The probability that a loan is repaid depends upon the return yielded by the borrower's risky project. Let a particular project yield a return of R. Then the lender will be repaid if

L(1 + r1) < R + C

That is, the lender is repaid if the value of the loan is less that what the borrower has to give up in default (the lender gets to claim any return, R, that the borrower made on the project plus the value of the collateral). This just says that the borrower repays when losses are smaller from doing so.

10. Now suppose that the return, R, is risky:

Return R = R1+x with probability 1/2
Return R = R1-x with probability 1/2

Then the expected return is R1, and the variance of returns is x2. As x increases, there is a mean-preserving spread in the distribution, i.e. risk goes up, but the expected return is not changed.

11. Next, to limit the outcomes to the ones we are interested in, assume that

R1-x < (1+r1)L - C

This means that the borrower will always choose to default when a bad outcome is drawn (-x), and will always repay when there is a good outcome (+x).

12. Thus, under a good outcome the borrower earns

R1+x - (1+r1)L

(this is the return on project minus the cost of the loan) and under a bad outcome, the borrower loses -C, i.e. loses the collateral on the loan.

13. Then the borrower's expected profit is

EπB = (1/2)[R1+x - (1+r1)L] + (1/2)[-C]

[The superscript means borrower]. That is, the borrower gets the good outcome shown in the first set of brackets 1/2 the time, and the bad outcome of -C shown in the second set of brackets the other half of the time.

14. Define x*(r,L,C)(1+r1)L - C - R1. That is, x* is the value of x such that EπB > 0 whenever x > x*, and EπB < 0 whenever x < x*. It's the point where profit turns negative.

Another way to say the same thing is that, with x* defined in this way, EπB = (1/2)(x-x*). x* gives the level of risk (how big the bad outcome must be, i.e. the size of x) where it becomes worthwhile for the borrower to walk away from the loan and default.

15. Notice that x* is increasing in r1. This means that as r1 increases, those with smaller x values drop out (i.e. those facing less risk), but the riskier borrowers (those with larger x values) remain in the pool. The mix of borrowers changes toward riskier borrowers and defaults will increase.

16. What about the lender? The lender's expected profit is

EπL = (1/2)[(1+r1)L] + (1/2)[C+R1-x] - (1+r)L

The first term is the return in the good state, the second is the return in the bad state (both happen with probability 1/2), and the third term is the opportunity cost of the funds it lends out (so the return is r, not r1, since the opportunity cost is the market return, r).

That is, the lender receives a fixed amount in the good state, (1+r1)L, but as x increases, the lender does increasingly worse in the bad state where it receives C+R1-x (i.e. as x increases, profit falls). This means that EπL is decreasing is the level of risk, x.

17. Now, let there be two groups of borrowers . Good borrowers are low risk (have small x values), bad borrowers are high risk (have large x values). Designate the x-values for each group as xg and xb, where xg < xb.

From the condition that EπB = (1/2)(x-x*), if r1 is low enough,

xg < xb < x*(r, L, C)

In this case, all loans are repaid, and all loans are profitable. If each type of borrower is equally likely to be in the market, then expected profit for the lender is

EπL = (1/2)[(1+r1)L+C+R1] - (1/4)[xg + xb] - (1+r)L

This is increasing in r1.

18. But, as r1 increases, we will eventually reach the point where xg = x*(r, L, C) and the good types drop out of the market and stop borrowing (this is adverse selection at work). In this case, expected profit falls to

EπL = (1/2)[(1+r1)L+C+R1] - (1/2)[xb] - (1+r)L

Thus, EπL falls discretely when xg = x*, i.e. profit falls discretely when r1 increases and reaches

r1 = (1/L)[xg - C + R1] - 1

since this is the point where low risk types exit the market (the discrete jump comes from having two groups - with a continuum of risky borrowers, the discrete jump would be replaced by a maximum profit point, i.e. a single-peaked profit function).

19. We can show this graphically:

For loan rates between 0 and r1, no loans are profitable and none will be made. For loan rates between r1 and r*, both types of borrowers are in the market, and all loans are profitable (and profit is increasing in r).

For loan rates between r* and r2, loans are unprofitable, so no loans would be made. For loan rates above r2, loans are profitable, but only the risky group will be in the market.

Thus, credit rationing is possible at equilibrium. If loan demand is robust, lenders will increase r until it hits r*. At r*, there can be excess demand, but lenders will not raise the loan rate unless demand is so strong that rates can be profitably increased all the way to r2. Thus, if demand is strong enough to produce excess demand at r1, but not strong enough to push rates all the way to r2 or above, there will be credit rationing at equilibrium.

Conclude briefly:

We have shown two things. First, when the interest rate increases, adverse selection mechanisms can cause good borrowers to drop out of the loan pool increasing the riskiness of the average borrower. This increases default. Thus, this shows how an increase in the interest rate can increase default rates.

[Note: I wouldn't apply this model directly to mortgage markets as is, interest rate resets would be modeled a little bit differently, but the basic mechanism would be the same and is well illustrated by this and Brad DeLong's example.]

Second, the adverse selection mechanism can explain the presence of equilibrium credit rationing. There are other explanations too, e.g. moral hazard and monitoring cost models can explain equilibrium credit rationing, so there are other ways to get this result.

Posted by on Sunday, March 30, 2008 at 04:32 PM in Economics, Financial System, Market Failure | Permalink  TrackBack (0)  Comments (17) You can follow this conversation by subscribing to the comment feed for this post.