Question

You want to buy a car, and a local bank will lend you $20,000. The loan would be fully amortized over 5 years (60 months), and the nominal interest rate would be 12%, with interest paid monthly.

a)What would be the monthly loan payment?

b)What would be the loan’s EAR?

c) If the nominal interest rate of the loan is quoted as 12% with semiannual compounding assumption, what is your monthly payment?

Answer 1

Answer a.

Amount borrowed = $20,000
Annual interest rate = 12%
Monthly interest rate = 1%
Time period = 60 months

$20,000 = Monthly payment * PVIFA(1%, 60)
$20,000 = Monthly payment * (1 - (1/1.01)^60) / 0.01
$20,000 = Monthly payment * 44.955038
Monthly payment = $444,89

Answer b.

EAR = [1 + (APR/n)]^n - 1, where n is number of compounding period per year
EAR = [1 + (0.12/12)]^12 - 1
EAR = 1.01^12 - 1
EAR = 1.1268 - 1
EAR = 0.1268 or 12.68%

Answer c.

APR = 12% compounding semiannually

EAR = [1 + (APR/n)]^n - 1
EAR = [1 + (0.12/2)]^2 - 1
EAR = 1.06^2 - 1
EAR = 1.1236 - 1
EAR = 0.1236 or 12.36%

Monthly interest rate = (1 + EAR)^(1/12) - 1
Monthly interest rate = 1.1236^(1/12) - 1
Monthly interest rate = 1.00976 - 1
Monthly interest rate = 0.00976 or 0.976%

$20,000 = Monthly payment * PVIFA(0.976%, 60)
$20,000 = Monthly payment * (1 - (1/1.00976)^60) / 0.00976
$20,000 = Monthly payment * 45.250536
Monthly payment = $441.98