Question

You are looking to purchase a new car with the assistance of a $22,000, five-year loan at an APR of 5.4%, compounded monthly.

Required:

a. What is the monthly payment on this loan?

b. What is the amount of interest and principal on the loan’s first monthly payment?

c. What is the amount of interest and principal on the loan’s sixteenth monthly payment?

d. Show a second method to determine the amount of interest and principal on the loan’s sixteenth monthly payment.

e. How many months would it take to pay this loan down to $8,000? f. What is the total amount of interest and principal that you would pay over five years?

Answer 1

a. Monthly payment = (P*R*(1+R)^T)/((1+R)^T-1) where P = loan amount = $22,000; R = monthly interest = 5.4%/12 = 0.45%; T = Loan period = 5 years or 60 months (5 years *12 months a year)

= (22000*0.45%*(1+0.45%)^60)/((1+0.45%)^60-1) = $419.21

Using EXCEL =PMT(rate,nper,-pv)

where rate = 5.4%/12 (as it is monthly compounded); nper = 60 months (5 years * 12 months a year); PV = Loan value = $22,000

=PMT(5.4%/12,60,-22000) = $419.21

Monthly payment = $419.21

b. Interest & Principal on the loan's first monthly payment

Monthly payment = $419.21

Interest = Loan amount * monthly interest rate = $22,000*5.4%/12 = $99

Principal = Monthly payment - Interest =$419.21-$99 = $320.21

Thus, Interest on the loan's first monthly payment is $99 & Principal on the loan's first monthly payment is $320.21

c. Interest & Principal on the loan's sixteenth monthly payment:

This can be found by building the amortisation schedule of the loan as follows:

Month	Opening Principal	Monthly payment	Interest payment	Principal payment	Closing Principal
	(A)	(B)	(C) = (B)*5.4% / 12	(D)=(B)-(C)	(E) = (A)-(D)
1	22,000.00	419.21	99.00	320.21	21,679.79
2	21,679.79	419.21	97.56	321.65	21,358.14
3	21,358.14	419.21	96.11	323.10	21,035.04
4	21,035.04	419.21	94.66	324.55	20,710.48
5	20,710.48	419.21	93.20	326.01	20,384.47
6	20,384.47	419.21	91.73	327.48	20,056.99
7	20,056.99	419.21	90.26	328.95	19,728.04
8	19,728.04	419.21	88.78	330.43	19,397.60
9	19,397.60	419.21	87.29	331.92	19,065.68
10	19,065.68	419.21	85.80	333.42	18,732.26
11	18,732.26	419.21	84.30	334.92	18,397.35
12	18,397.35	419.21	82.79	336.42	18,060.93
13	18,060.93	419.21	81.27	337.94	17,722.99
14	17,722.99	419.21	79.75	339.46	17,383.53
15	17,383.53	419.21	78.23	340.98	17,042.55
16	17,042.55	419.21	76.69	342.52	16,700.03

(Opening Principal of Month 1 is the loan amount of $22,000 and of the subsequent months is same as closing principal of previous month)

Thus, Interest on the loan's 16th monthly payment is $76.69 & Principal on the loan's 16th monthly payment is $342.52

d. Second method to find the Interest & Principal on the loan's sixteenth monthly payment:

To find the Interest & Principal on the loan's sixteenth monthly payment, lets first find the loan outstanding as on the end of 15th monthly payment.

Loan outstanding as on the end of 15th monthly payment.= PMT/(R)*(1-(1/(1+(R))^T)

where PMT = monthly payments ($419.21); R = monthly interest rate (5.4%/12); T = remaining tenure in the loan (45 months = 60 months total - 15 months elapsed).

=419.21/(5.4%/12)*(1-(1/(1+(5.4%/12))^45))=$17,042.55

Thus, Interest = Loan oustanding at the end of 15th monthly payment * monthly interest rate = $17,042.55*5.4%/12 = $76.69

Principal = Monthly payment - Interest =$419.21-$76.69= $342.52

Thus, Interest on the loan's 16th monthly payment is $76.69 & Principal on the loan's 16th monthly payment is $342.52