Question

suppose that you are given the option to borrow a fixed rate US mortgage of $80,000 at 12% for 25 years with monthly payments. Alternatively, you may borrow another fixed rate US mortgage of $90,000 for 25 years with monthly payments at a contract interest rate to be determined. The lender would like to have an effective annual yield of 25% on the incremental cost of borrowing (i.e., on the $10,000), reflecting the borrower’s increased default risk. Formulate how you would compute the contract interest rate on the entire $90,000 loan.

Answer 1

Problem 1:

Principal: USD 80,000; Rate: 12% p.a., payable monthly; Tenure: 25 years

Generally, the montly payments are calculated using the present value formula. The below formula provides the methodology to calculate the payments:

PV = USD 80,000; r = 12% p.a., which 1% p.m. n = 25 years, which is 25x12=300 months

Therefore the monthly payment is P = (1% x 80,000) / [ 1 - (1 +1%)^(-300)] = USD 842.58

For increased default risk on the incremental cost of borrowing (i.e., on the USD 10,000), an effective annual yield of 25% is required.

Hence, the monthly payment for the incremental borrowing can be determined with the above formula as under:

PV = USD 10,000; r = 25% p.a., which is 25%/12 = 2.08% p.m.; n = 25 years, which is 25x12=300 months

Therefore the monthly payment is P = (2.08% x 10,000) / [ 1 - (1 +2.08%)^(-300)] = USD 208.76

Therefore, the contract interest rate on the entire loan of USD 90,000 can be determined using the available information:

PV = USD 90,000; n = 25 years, which is 25x12=300 months; Monthly payment, P = 842.58+208.76 = USD 1051.34

The manual calculation for the rate per period is tedious and the formula for the same is as under:

The same can be calculated through Microsoft Excel using "RATE" function or through Scientific Calculator.

Accordingly, the contract interest rate per month is calculated as 1.13% p.m., which is 13.53% p.a.