Question

A bank faces two types of borrowers, type A and B. Both want a $225 loan. Type A repays the loan 90% of the time and type B only repays with probability 0.76. The bank doesn't observe type, but believes fraction x is type A. What does x need to be so that the bank can afford a pooling interest rate of 27.1%? Please explain, someone else answered incorrectly .29, but the correct answer should be .19 according to my homework.

Answer 1

Solution:

Note that a bank would always want to at least recover it's loan amount, that is, even if the interest on loan is forgone, bank would always want to earn back the principal amount, in case of defaulters chance.

Let x fraction of people be of A type, then remaining (1-x) fraction would be of B type.

Further the expected amount to be received from all people (including both types) should equal the principal amount (as already stated, bank would want to cover that much amount at least; here we take it with equality to show indifference.

So, expected amount to be received = 225, at indifference ... (*)

Now, this expected amount to be received in total = expected amount to get back from type A + expected amount to get back from type B

From a type A person, expected amount recovered = 0.90*225*(1 + 0.271) + (1 - 0.90)*0

(this is obtained as with 0.90 probability, bank gets back the principal (along with interest, since repayment) and with (1-0.90) probability, bank gets back 0)

Expected amount recovered = 257.3775

Similarly, from a type B person, expected amount recovered = 0.76*225*(1 + 0.271) + (1 - 0.76)*0

(this is obtained using same reasoning as mentioned for type A)

Expected amount recovered = 217.341

Note that the interest rate applied is 27.1%, is same for both types, because we are talking about pooling equilibrium here, and not a separate equilibrium.

Now, total expected amount recovered (from all people) = fraction of A type*amount recovered from A type + fraction of B type*amount recovered from B type

Expected total recovery = x*257.3775 + (1 - x)*217.341 = 40.0365*x + 217.341

Then, using (*), we have 40.0365*x + 217.341 = 225

Solving this, we get, x = (225 - 217.341)/40.0365 = 0.1913

So, x = 0.19 (approximately), which matches the correct answer provided in the book.