Question

Problem 2: The quarterly payment on a 15‐year construction loan is $1867.50. The loan’s interest rate is a 6.40%

annual percentage rate (APR) and payments are end‐of‐quarter. (a) What is the builder’s loan amount? (b)

What is the loan’s effective annual rate (EAR)? (c) What will the loan balance be immediately after the 39th

payment is made?

Answer 1

Answer (a):

Quarterly payment at the end of the quarter = $1867.50

Loan duration = 15 years = 15 * 4 = 60 quarters

Quarterly interest rate = 6.40% / 4 = 1.60%

Loan amount will be the present value of the all quarterly payments.

PV of ordinary annuity = Periodic payment * (1 - 1 /(1 + Periodic interest rate) ^{Number of periods}) / Periodic interest rate

= 1867.50 * (1 - 1 /(1 + 1.60%) ⁶⁰) / 1.60%

= $71,687.07

Builder’s loan amount = $71,687.07

Answer (b):

Effective annual rate = (1 + APR / Number of compounding periods per year) ^{Number of compounding periods in a year} - 1

= (1 + 6.40% / 4) ⁴ - 1

= 6.5552%

Loan’s effective annual rate (EAR) = 6.56%

Answer (c):

Loan balance = Present value of remaining quarterly payments to be made

Remaining quarterly payments to be made immediately after the 39th payment = 60 - 39 = 21

Loan balance immediately after the 39th payment = 1867.50 * (1 - 1 / (1 + 1.60%) ²¹) / 1.60% = $33,086.70

Loan balance immediately after the 39th payment = $33,086.70