Math/ Economic brain teaser - due to the credit crisis

[SavvyAA3]

Can anyone sense a way to solve this. It would be great help to see your reasoning behind your assumptions. As a result of the credit crisis there are many asymetries in the loan market as a reult this set of question have arisen:

Information - take this as true
There are two types of investment projects. Safe investment projects yield an
output q in all states of the world. Risky investment projects yield an output of 2q
in the “good” state of the world (probability = ½), and 0 in the “bad” state of the
world (probability = ½). There are 2000 potential investors. 1000 investors are
aware of safe projects, with quality q distributed uniformly on [1,2] (each investor
only knows of one project). 1000 investors are aware of risky projects, with
quality q distributed uniformly on [1,2] (each investor only knows of one project).
Each project requires an investment of 1. There is no moral hazard, as each
investor only knows of one project of one type.
In lending money, lenders think about the average (across all states of the
world) that they receive from borrowers minus the amount they lend (i.e. 1). If
ibar is this net return, then S = 2000*ibar. For example, if investors receive an
average payment of 1.25, then ibar = .25 & S = 500.
All potential investors have equity equal to E which will be used to guarantee
loans, i.e. they will use E to pay their debt if the payoff of their investment project
falls short of the amount necessary to repay their loans. The following holds:
1 > E > 0.


1. If lenders were to offer loans at an interest rate i equal
to 0 (i.e. a loan of 1 is repaid with 1), meeting the demand for such loans
(whatever it might be) the average return on such loans, ibar, would be
given by:
a) 0
b) -E
c) -1
d) E/2
e) -E/2
f) (E-1)/2
g) None of the above.

2. If lenders were to offer loans at an interest rate i equal
to 1 (i.e. a loan of 1 is repaid with 2), meeting the demand for such loans
(whatever it might be) the average return on such loans, ibar, would be
given by:
a) 0
b) - E
c) (1-E)/2
d) E/2
e) -E/2
f) (E-1)/2
g) 1
h) There would be no loans, as the demand for loans at that price would
equal 0.
i) None of the above.

3. If lenders were to offer loans at an interest rate i (i.e. a
loan of 1 is repaid with 1+i), where i is less than or equal to 1 & greater
than 0, demand for such loans from investors would be given by:
a) 1000*(1-i) + 1000*(1-(i+E-1)/2)
b) 2000*(1-i)
c) 1000*(1-i) + 1000*(1-(i+E)/2)
d) 1000*i + 1000*(1+E)
e) 1000*(1-i) + 1000*(1-E/2)
f) None of the above

4. If E = 2/3, does credit rationing occur (i.e. there is no
interest rate at which demand equals supply):
a) Yes, credit rationing occurs.
b) No, credit rationing does not occur.
c) There is not enough information given to be able to tell.

 

[davee123]

Can anyone sense a way to solve this. It would be great help to see your reasoning behind your assumptions.

I'm not sure I really understand the problem...

There are two types of investment projects. Safe investment projects yield an
output q in all states of the world. Risky investment projects yield an output of 2q
in the “good” state of the world (probability = ½), and 0 in the “bad” state of the
world (probability = ½).

Ok. First of all, when you say "output" what do you mean? For investors or for lenders? How are you differentiating lenders versus investors versus borrowers? And, an "investment" for a lender may be very different from the "investment" made by an investor.

I'm assuming here that when you say "investment project", you're talking about a loan wherein a lender (L) lends N dollars to a borrower (B) at an interest rate of I for a duration of time T. Then, multiple investors (V) are given the option by L to invest amounts (A) into said loan, effectively sharing both the risk and profits with L. Hence, L's profits will be reduced, but their risk is similarly reduced. A loan that gets paid down immediately earns very little interest, and is low yield (but yields SOME money), and loans that get paid off "on time" make an expected profit. And of course loans that default may lose money rather than make it. So, when you say "0" vs "q" vs "2q", what do you mean, exactly? Does "0" imply a net loss, or simply imply no net gain? If a loss, would "q" imply no net gain? Is that gain for the loan overall, the lender, or the investor? How is that "output" divided between the investors and the lender?

All potential investors have equity equal to E which will be used to guarantee loans, i.e. they will use E to pay their debt if the payoff of their investment project falls short of the amount necessary to repay their loans. The following holds:
1 > E > 0.

I'm fuzzy here-- usually at least some percentage of the loan is paid off before a loan defaults, at which time "E" would be handed over to the lender to compensate them. Hence (disregarding legal fees, etc of course), they may make a net profit if the loan defaults, if (say) 95% of the loan is paid off (which represents a higher value than the loan was for initially, mind you, thanks to interest), and E is worth (say) 20% of the expected return, the lender yields a 115% return rather than their expected 100%. How are you working this in?

1. If lenders were to offer loans at an interest rate i equal
to 0 (i.e. a loan of 1 is repaid with 1), meeting the demand for such loans
(whatever it might be) the average return on such loans, ibar, would be
given by:
a) 0
b) -E
c) -1
d) E/2
e) -E/2
f) (E-1)/2
g) None of the above.

Well... you haven't clarified how this ties in with "q" in the beginning-- effectively I'm assuming that you mean that q=0. Hence, 1/2 are "safe" investments, and 1/2 are "risky" investments, meaning that 1/4 of the loans default. Since the remaining 3/4 make 0 profit (wherein ibar = 0), and the remaining 1/4 would lose money (E-1), you'd get the average of those-- IE (0+0+0+E-1)/4 = (E-1)/4.

2. If lenders were to offer loans at an interest rate i equal
to 1 (i.e. a loan of 1 is repaid with 2), meeting the demand for such loans
(whatever it might be) the average return on such loans, ibar, would be
given by:
a) 0
b) - E
c) (1-E)/2
d) E/2
e) -E/2
f) (E-1)/2
g) 1
h) There would be no loans, as the demand for loans at that price would
equal 0.
i) None of the above.

Alright, you haven't addressed "demand for loans" at all, so I've got to totally ignore that part of the question as it hasn't been defined. Again, let's assume that 1/2 of these are "safe" loans as you stated, which mean they have an expected ibar of 1. And the remaining 1/2 are risky loans, half of which default, and half of which yield double-- so ibars of 2 and (E-1) respectively. Of course that's assuming the defaulting loans were paid with 0 money down and defaulted immediately, since there's apparently no correlation between E and the time it takes to default. So the average by that would be (1+1+2+(E-1))/4 = (3+E)/4.

3. If lenders were to offer loans at an interest rate i (i.e. a
loan of 1 is repaid with 1+i), where i is less than or equal to 1 & greater
than 0, demand for such loans from investors would be given by:
a) 1000*(1-i) + 1000*(1-(i+E-1)/2)
b) 2000*(1-i)
c) 1000*(1-i) + 1000*(1-(i+E)/2)
d) 1000*i + 1000*(1+E)
e) 1000*(1-i) + 1000*(1-E/2)
f) None of the above

Ok, as stated above, you've still never given any sort of correlation for "demand", so I'll have to skip this one.

4. If E = 2/3, does credit rationing occur (i.e. there is no
interest rate at which demand equals supply):
a) Yes, credit rationing occurs.
b) No, credit rationing does not occur.
c) There is not enough information given to be able to tell.

Again you're talking about something that you haven't even mentioned in your assumptions, so... I can't really say anything. You'll have to clarify what "demand" and "supply" are, and why the interest rate would affect it.

Also, what's with the multiple choice stuff? Are you making this up, or did you get this from somewhere else?

DaveE

 

[SavvyAA3]

Supply: S=2000*ibar

Demand: Those who enter the loan market

q: the quality of the investment project - this lies on a uniform distribution on scale [1,2]
this is what you can use to determine who enters the market, as for example if i = 0, you know all 2000 people will enter the market because the quality of their investment project should yield a return greater than 1. Those in the risky world with bad projects do not know it is a bad project when they are taking the loan because q has not yet been realized.

I hope this helps you to attempt the last part of the question.

 

[davee123]

Supply: S=2000*ibar

So... you mean supply as in the net amount of profit available to investors?

Demand: Those who enter the loan market

I ... think ... you mean that "Demand" in this case is fixed at 2000, then? IE there are 2000 potential investors?

q: the quality of the investment project - this lies on a uniform distribution on scale [1,2]
this is what you can use to determine who enters the market, as for example if i = 0, you know all 2000 people will enter the market because the quality of their investment project should yield a return greater than 1. Those in the risky world with bad projects do not know it is a bad project when they are taking the loan because q has not yet been realized.

? You're saying that if there's an interest rate of 0 that people will want to invest in loans? NOBODY will invest in loans if the interest rate is 0 because they're all guaranteed at BEST to earn NO profit, and worst case LOSE money. Are you suggesting these 2000 people are the ones who are the borrowers? In that case, you've TOTALLY lost me.

Also, you still haven't related how q relates to any of this. With an interest rate of "i" such that X+i pays back a loan of X in full, how does this have potential to yield an "output" of q in a safe market versus 2q in a risky one? You've ... sort of ... implied ... VERY loosely ... that an "output" of 0 means that a loan has defaulted, but you haven't explained how, with a fixed rate of "i", you can get a different output for a risky investment versus a safe one. I basically assumed (because you didn't explicitly state it) that "risky" loans are effectively paid back with X+2i rather than X+i. But that's a horrifically bad assumption for me to make. I just don't know what else I should be assuming.

DaveE

 

[SavvyAA3]

Ok, this question is a bit tricky, but it is wise to first make certain distinctions.

There are loans available and there are investment projects. People borrow (i.e. take out a loan) to finance their spending now. They invest in a project (with the quality of that project varying over a range of [1,2]) to hopefully get a better return in the future.

At an interest rate of 0 all 2000 people in the world will take out (demand) a loan (not invest in the loan - but demand the loan). The reason why they do this is because all of them will expect to use their loan for consumption now and then repay it with the return from their investment project (which should be greater than 1 as the quality is distributed uniformly over the interval [1,2]). As stated earlier as there is a likelyhood for those in the bad state of the world that the investment project yeilds zero, then these people will have to use their equity to pay back the loan.

Demand is indeed fixed at 2000 but not all 2000 people will enter the loan market all the time. The interest rate determines whether or not they will enter the loan market. As stated above if this is zero, everyone enters the market/ demands for the loans because they believe from the nature of their earlier investment that when the loan falls due for repayment their investment will have yeilded a greater return and so they will certainly be able to repay the loan.

I am not making this question up myslef. I am a student at Wharton and this is a difficult brain teaser. The lecturer has not yet supplied the answer to this.

(Your answer to question one is what I too believe it to be).

 

[davee123]

There are loans available and there are investment projects. People borrow (i.e. take out a loan) to finance their spending now. They invest in a project (with the quality of that project varying over a range of [1,2]) to hopefully get a better return in the future.

Ok. So the investments in your example have nothing whatsoever to do with the loans that are being taken out? Borrowers are (say) investing in common stock? Like, buying shares of Apple or something? The way you had phrased things, I was assuming that these investors were investing in the loans themselves-- IE that these people were buying mortgage backed securities, assumably with a single loan in the pool to make things simpler.

At an interest rate of 0 all 2000 people in the world will take out (demand) a loan (not invest in the loan - but demand the loan). The reason why they do this is because all of them will expect to use their loan for consumption now and then repay it with the return from their investment project (which should be greater than 1 as the quality is distributed uniformly over the interval [1,2]). As stated earlier as there is a likelyhood for those in the bad state of the world that the investment project yeilds zero, then these people will have to use their equity to pay back the loan.

Ok, so each borrower is assumed to necessarily be investing their borrowed money in some sort of separate investment. Hence, borrower = investor in your example.

Demand is indeed fixed at 2000 but not all 2000 people will enter the loan market all the time. The interest rate determines whether or not they will enter the loan market. As stated above if this is zero, everyone enters the market/ demands for the loans because they believe from the nature of their earlier investment that when the loan falls due for repayment their investment will have yeilded a greater return and so they will certainly be able to repay the loan.

Yes, but by the same token, demand is an irrelevant question given the definitions. In this hypothetical example, everyone equally assesses their own situation. They all believe they can pay back their loans equally, and they're not aware of whether something's a risky investment or not. Hence, if anyone of these people thinks he's likely to gain money, they ALL think they're likely to gain money, and so will ALL participate. Effectively, demand is necessarily going to be 0 or 2000, unless there's something else that these people know in advance.

If they were aware of the best-case/worst-case scenarios for their investments, or the likelihood of their success, then it might affect their decisions. If (for example) they know they have the potential to earn 2q, that may affect their decision to invest as opposed to if they knew they ONLY had a potential to earn q.

But there's another issue here, and that's how q relates to i. People would (in theory) invest if they were guaranteed to get q *profit* and lose *i* in interest, but only if q > i. If q < i, then they've got a problem. You also haven't explained how q relates to the investment that they're making. If they invest worth 1 (your assumption of loan amounts), and they get 0 output, does that mean they still have a value of 1? Or do they instead now have a value of 0? Is q profit or is it total value?

[edit]
Ok. So, to clarify, here's the charts as I see them, to make things a little easier to understand.

"Pre" = Before the loan is issued
"Issued" = After the loan is issued, before investment is made
"Invest" = After the investment is made, before return on investment is known
"Return" = After the investment has matured, before loan is paid off
"Paid" = After terms of the loan have been settled.

Terms:
Amount Borrowed = X
Amount Invested = X
Total Interest = i

There's also the default payment and the profit that get made from the investment, but those are unclear by your definition. You might mean:

1A) Default Payment = E (IE borrower returns E to lender upon default of the loan)
- OR -
1B) Default Payment = X+E (IE borrower returns the loan amount plus E upon default of the loan)

Also, you might mean:

2A) Return from investment R = q (IE borrower gets q from a safe investment, where q > X)
- OR -
2B) Return from investment R = X+q (IE q is the amount of profit they get)

Assuming 1B and 2B, you'd draw a little chart like this:

       | Lender |    "winner"  |   | Lender |    "loser"   |
       |  has   |  has    owes |   |  has   |  has    owes |
-------+--------+-------+------+   +--------+-------+------+
Pre    | X      | E     | 0    |   | X      | E     | 0    |
Issued | 0      | E+X   | X+i  |   | 0      | E+X   | X+i  |
Invest | 0      | E     | X+i  |   | 0      | E     | X+i  |
Return | 0      | E+X+q | X+i  |   | 0      | E+X   | X+i  |
Paid   | X+i    | E+q-i | 0    |   | X+E    | 0     | 0    |

And there's obviously 4 different ways of interpreting the above-- 1A & 2A, 1A & 2B, 1B & 2A, and 1B & 2B.
[/edit]

DaveE