Question

An entrepreneur is quoted a loan rate of 12% at the local bank, while the bank pays depositors 6% per annum.

a. If in bankruptcy the entrepreneur will not pay back anything, but otherwise everything will be repaid, then what does the bank believe the probability of failure to be?

b. What is the quoted default premium?

c. Compute the expected default premium. (Note that when you lose all your money plus the default premium, your rate of return can be below –100%. This is not only reasonable but necessary to get an average default premium that is what it should be.)

Answer 1

Part (a)

Let's assume the probability of failure (default) to be p.

Hence, risk free return on deposits = 6% = p x 0% + (1 - p) x loan rate = p x 0% + (1 - p) x 12%

Hence, p = 6% / 12% = 0.50.

Hence, the bank believe the probability of failure to be 0.50 or 50%

Part (b)

Quoted default premium = Loan rate in excess of risk free deposit rate = 12% - 6% = 6%

Part (c)

In case of default, loss = -100% as nothing is paid

IN case of no default, money received = Principal + Return = 100% + 12% = 112%

Hence, expected default premium = p x (-100%) + (1 - p) x 112% = 0.5 x (-100%) + (1 - 0.5) x 112% = 6%