Question

A young couple borrow $5,000 on the 1st September, agreeing to repay it as well as the interest charged over 36 months with 36 equal repayments, at the interest rate of 3%. No interest is charged for their 3 interest-free months, but their first repayment is required on the 1st of October.

a) Show all mathematical workings and formulas to calculate the repayment required to repay the loan and interest.

b) Calculate the individual monthly repayments required for all 36 months, displaying them in a table with headings: "Month", "Amount Owing", "Interest" and "Repayment". Show that this calculation method yields the same result as part a).

Answer 1

Assume Equal Monthly Repayments =X

First three months are interest Free.

Hence Principal Balance after 3 months (1^st December)=5000-3X

Number of balance instalments =(36-3)=33

Interest rate per month =(3/12)%=0.25%=0.0025

Capital Recovery Factor (CAF)=(A/P,i,N)=(i*((1+i)^N))/(((1+i)^N)-1)

N=number of Payments =33

i=Interest Rate =0.25%=0.0025

CAF=(A/P,0.25%,33)= (0.0025*(1.0025^33))/((1.0025^33)-1)=0.031608

Amount of Monthly Equal Repayment =X=0.031608*(5000-3X)

X=0.031608*5000-0.031608*3*X=158.0403-0.094824X

X=158.0403-0.094824X

1.094824X=158.0403

X=158.0403/1.094824=144.35

Amount to be paid per month =$144.35(Rounded off to2 decimal place)

REPAYMENT SCHEDULE(Ending Balance of $0.08 is because of rounding off effect)

	A	B	C=A*0.25%	D=B-C	E=A-D
	Beginning	Monthly	First Three Months=$0		Ending
Month	Balance	Payment	Interest	Principal	Balance
1	$5,000	$144.35	$0	$144.35	$4,855.65
2	$4,855.65	$144.35	$0	$144.35	$4,711.30
3	$4,711.30	$144.35	$0	$144.35	$4,566.95
4	$4,566.95	$144.35	$11.42	$132.93	$4,434.02
5	$4,434.02	$144.35	$11.09	$133.26	$4,300.75
6	$4,300.75	$144.35	$10.75	$133.60	$4,167.15
7	$4,167.15	$144.35	$10.42	$133.93	$4,033.22
8	$4,033.22	$144.35	$10.08	$134.27	$3,898.96
9	$3,898.96	$144.35	$9.75	$134.60	$3,764.35
10	$3,764.35	$144.35	$9.41	$134.94	$3,629.41
11	$3,629.41	$144.35	$9.07	$135.28	$3,494.14
12	$3,494.14	$144.35	$8.74	$135.61	$3,358.52
13	$3,358.52	$144.35	$8.40	$135.95	$3,222.57
14	$3,222.57	$144.35	$8.06	$136.29	$3,086.28
15	$3,086.28	$144.35	$7.72	$136.63	$2,949.64
16	$2,949.64	$144.35	$7.37	$136.98	$2,812.66
17	$2,812.66	$144.35	$7.03	$137.32	$2,675.35
18	$2,675.35	$144.35	$6.69	$137.66	$2,537.68
19	$2,537.68	$144.35	$6.34	$138.01	$2,399.68
20	$2,399.68	$144.35	$6.00	$138.35	$2,261.33
21	$2,261.33	$144.35	$5.65	$138.70	$2,122.63
22	$2,122.63	$144.35	$5.31	$139.04	$1,983.59
23	$1,983.59	$144.35	$4.96	$139.39	$1,844.20
24	$1,844.20	$144.35	$4.61	$139.74	$1,704.46
25	$1,704.46	$144.35	$4.26	$140.09	$1,564.37
26	$1,564.37	$144.35	$3.91	$140.44	$1,423.93
27	$1,423.93	$144.35	$3.56	$140.79	$1,283.14
28	$1,283.14	$144.35	$3.21	$141.14	$1,142.00
29	$1,142.00	$144.35	$2.85	$141.50	$1,000.50
30	$1,000.50	$144.35	$2.50	$141.85	$858.65
31	$858.65	$144.35	$2.15	$142.20	$716.45
32	$716.45	$144.35	$1.79	$142.56	$573.89
33	$573.89	$144.35	$1.43	$142.92	$430.98
34	$430.98	$144.35	$1.08	$143.27	$287.70
35	$287.70	$144.35	$0.72	$143.63	$144.07
36	$144.07	$144.35	$0.36	$143.99	$0.08