question about bid-ask spread, what do you think?

Question

Bank Z receives U$1,000 in deposits from Tom and promises to pay him U$1,100 after one year. With this money, the bank will lend U$10 to another 100 people at a rate j_e, such that each of these 100 people will have to pay (1 + j_e) * 10 USD back to the bank after one year. However, not all borrowers repay their loans. Assume the default rate (the percentage of borrowers who do not pay the bank after one year) is 8%.

Knowing that the bank wants to honor its promise to repay Tom and still make a profit of 6% of the value of deposits, 1) what should the bank spread be? and 2) if the default rate rises by 10 percentage points, will the new spread increase by more or less than 10 percentage points?

a) Spread: 35.8%. It will increase by more than 10 percentage points.

b) Spread: 22.3%. It will increase by more than 10 percentage points.

c) Spread: 16.1%. It will increase by more than 10 percentage points.

d) Spread: 35.8%. It will increase by less than 10 percentage points.

e) Spread: 18.4%. It will increase by less than 10 percentage points.

Answer 1

The correct answer is (C).

First we calculate the rate lending rate j_e = the rate at which the bank lent to borrowers inorder to achieve 6% of profit with 8% of default rate.

1. The total amount the bank lent to the borrowers = the total amount Tom deposited = U$1,000.
2. The amount the bank lent to each borrower = U$1,000 ÷ 100 = U$10.
3. The number of borrowers expected to pay back with 8% of default rate = number of borrowers x (100% − 8%) = 100 x 92% = 92.
4. The amount each borrower expected to pay back = U$10 x (1+ j_e)
5. The total amount of pay back to the bank = 92 x 10(1+ j_e) = 920(1+ j_e).
6. The profit of the bank = 920(1+ j_e) − 1100 should each to the total amount the bank lent to the borrowers x the rate of projected profit 6% = U$1000 x 6% = U$60.

Thus we should have the equation

920(1+ j_e) − 1100 = 60. This is equivalent to 920 j_e = 240. Thus j_e = 240 ÷ 920 ≈ 26.087

The interest rate the bank pays Tom = (1100-1000) ÷ 1000 = 10%.

The bank spread = the lending rate j_e - the interest rate the bank pays Tom = 26.087 − 10% = 16.087%. This answers question (1).

If the default rate increases by 10%, the number of borrowers expected to pay back is now

100(100%-18%) = 100 x 82% = 82. Let j_e' be the new lending rate.

The total profit is changed to 82 x 10(1+ j_e') − 1100 = 60. This can be simplified to 820 j_e' = 340. Thus j_e' ≈ 41.463%. So the new bank spread = new lending rate − 10% = 31.463%. This is more than 10% increase from the original bank spread.

Therefore the correct answer is (c).