Question

Suppose your request for a $ 5,000 loan has been approved by the bank. The terms of the loan specify an annual effective interest rate of% 12,683 (based on monthly compound periods) and 48 monthly payments
a) Express interest rate as a nominal interest rate
b) Compute the corresponding monthly payment
c) Estimate the total amount to be paid in interest
d) Determine at the end of the fourth month, what amount of the monthly payment corresponds to the principal payment and how much corresponds to the payment of interest

Answer 1

(a)

Effective interest rate = [1 + (r/N)]^N - 1, where N: Number of compounding periods in a year and r: Nominal interest rate

When EAR = 12.683% = 0.12683 and N = 12,

0.12683 = [1 + (r/12]¹² - 1

1.12683 = [1 + (r/12]¹²

Taking 12th root,

1.0100 = 1 + (r/12)

r/12 = 0.0100

r = 0.12

r = 12%

(b)

Monthly (nominal) interest rate = 12%/12 = 1%

Monthly loan payment ($) = Amount of loan / PVIFA(1%, 48) = 5,000 / 37.974** = 131.67

(c)

Total (principle + interest) payable after 48 months = $131.67 x 48 = $6,320.16

Total interest payable = Total (principle + interest) payable after 48 months - Amount of Loan = $6,320.16 - $5,000

Total Interest payable = $1,320.16

(d)

Interest in month N = Beginning balance in month N x 1%

Principal in month N = Monthly payment - Interest in month N

Interest in month 1 ($) = 5,000 x 1% = 50

Principal in month 1 ($) = 131.67 - 50 = 81.64

End-of-month balance in month 1 = Beginning-of month balance in month 2 = $(5,000 - 131.67) = $4,868.33

Interest in month 2 ($) = 4,868.33 x 1% = 48.68

Principal in month 2 ($) = 131.67 - 48.68 = 82.99

End-of-month balance in month 2 = Beginning-of month balance in month 3 = $(4,868.33 - 131.67) = $4,736.66

Interest in month 3 ($) = 4,736.66 x 1% = 47.37

Principal in month 3 ($) = 131.67 - 47.37 = 84.3

End-of-month balance in month 3 = Beginning-of month balance in month 4 = $(4,736.66 - 131.67) = $4,604.99

Interest in month 4 ($) = 4,604.99 x 1% = 46.05

Principal in month 4 ($) = 131.67 - 46.05 = 85.62

**From PVIFA Factor table