Question

A loan of $100,000 is made today. The borrower will make equal repayments of $3613.5 per month with the first payment being exactly one month from today. The interest being charged on this loan is constant (but unknown). For the following two scenarios, calculate the interest rate being charged on this loan, expressed as a nominal annual rate compounded monthly in percentage:

(a) The loan is fully repaid exactly after 31 monthly repayments, i.e., the loan outstanding immediately after 31 repayments is exactly 0.

(b) The term of the loan is unknown but it is known that the loan outstanding 2 years later equals to $28733.72.

Answer 1

(a)

The nominal monthly rate is calculated using RATE function in Excel :

nper = 31 (31 monthly repayments to fully repay the loan)

pmt = -3613.5 (Monthly payment. This is entered with a negative sign because it is a payment)

pv = 100000 (Loan amount. This is entered with a positive sign because it is the amount borrowed, hence it it is like a cash inflow)

fv = 0 (Loan outstanding after 31 months is zero)

RATE is calculated to be 0.72%. This is the nominal monthly rate. To get nominal annual rate, we multiply by 12.

Nominal annual rate is 8.70%

(b)

The nominal monthly rate is calculated using RATE function in Excel :

nper = 24 (2 year loan with 12 monthly repayments each year. Total number of payments = 12 * 2 = 24)

pmt = -3613.5 (Monthly payment. This is entered with a negative sign because it is a payment)

pv = 100000 (Loan amount. This is entered with a positive sign because it is the amount borrowed, hence it it is like a cash inflow)

fv = 28733.72 (Loan outstanding after 2 years. This is entered with a positive sign because it is the loan balance outstanding, and hence is like a cash inflow)

RATE is calculated to be 0.96%. This is the nominal monthly rate. To get nominal annual rate, we multiply by 12.

Nominal annual rate is 11.50%