Question

A loan is amortized over five years with monthly payments at an annual nominal interest rate of 6% compounded monthly.

The first payment is 1000 and is to be paid one month from the date of the loan.

Each succeeding monthly payment will be 3% lower than the prior payment.

Calculate the outstanding loan balance immediately after the 40th payment is made.

Answer 1

Annual nominal interest rate = 6% compounded monthly

So monthly compounding rate = 6% / 12 = 0.5% = 0.005

First payment is made after 1 month from the date of loan (Payments are made on end of period, EOP)

• 1st payment is 1000 * (1-0.03)^0 = 1000 * 1 = 1000
• 2nd payment is reduced by 3% to 1000 * (1-0.03)^1 = 1000 * 0.97 = 970
• 3rd payment is reduced by 3% to 1000 *(1-0.03)^2 = 1000 * 0.9409 = 940.90 and so on...
• 59th payment is 1000 * (1- 0.03)^58 = 1000 * 0.1709073 = 170.91
• 60th payment is 1000 * (1-0.03)^59 = 1000 * 0.16578 = 165.78

Loan amount is the present value of all the monthly payments.

• PV(loan) = 1000/(1+0.005)^1 + 970/(1+0.005)^2 + 940.90/(1+0.005)^3 + ....+ 170.91/(1+0.005)^59 + 165.78/(1+0.005)^60
• PV(loan) = $25,165.21

Here discount factor, DF for each period is calculated as:

• 1st year: 1/(1+0.005)^1 = 0.995025
• 2nd year: 1/(1+0.005)^2 = 0.990075 and so on.

Below is the schedule with Period, Monthly payments, Discount Factor, PV(PMT) :

Year	PMT	DF	PV(PMT) = DF * PMT
1	1000	0.995025	995.0249
2	970.00	0.990075	960.3723
3	940.90	0.985149	926.9265
4	912.67	0.980248	894.6454
5	885.29	0.975371	863.4886
6	858.73	0.970518	833.4169
7	832.97	0.965690	804.3924
8	807.98	0.960885	776.3788
9	783.74	0.956105	749.3407
10	760.23	0.951348	723.2443
11	737.42	0.946615	698.0566
12	715.30	0.941905	673.7462
13	693.84	0.937219	650.2824
14	673.03	0.932556	627.6358
15	652.84	0.927917	605.7778
16	633.25	0.923300	584.6811
17	614.25	0.918707	564.319
18	595.83	0.914136	544.6661
19	577.95	0.909588	525.6977
20	560.61	0.905063	507.3898
21	543.79	0.900560	489.7195
22	527.48	0.896080	472.6646
23	511.66	0.891622	456.2036
24	496.31	0.887186	440.3159
25	481.42	0.882772	424.9816
26	466.97	0.878380	410.1812
27	452.97	0.874010	395.8963
28	439.38	0.869662	382.1088
29	426.20	0.865335	368.8016
30	413.41	0.861030	355.9577
31	401.01	0.856746	343.5612
32	388.98	0.852484	331.5964
33	377.31	0.848242	320.0483
34	365.99	0.844022	308.9023
35	355.01	0.839823	298.1445
36	344.36	0.835645	287.7614
37	334.03	0.831487	277.7398
38	324.01	0.827351	268.0673
39	314.29	0.823235	258.7316
40	304.86	0.819139	249.7211
41	295.71	0.815064	241.0243
42	286.84	0.811009	232.6304
43	278.24	0.806974	224.5289
44	269.89	0.802959	216.7095
45	261.79	0.798964	209.1624
46	253.94	0.794989	201.8781
47	246.32	0.791034	194.8475
48	238.93	0.787098	188.0618
49	231.76	0.783182	181.5124
50	224.81	0.779286	175.191
51	218.07	0.775409	169.0899
52	211.52	0.771551	163.2012
53	205.18	0.767713	157.5175
54	199.02	0.763893	152.0319
55	193.05	0.760093	146.7372
56	187.26	0.756311	141.627
57	181.64	0.752548	136.6947
58	176.19	0.748804	131.9342
59	170.91	0.745079	127.3394
60	165.78	0.741372	122.9047
	27973.11		25165.21

After 40th payment, the outstanding loan is:

L(40) = $241.0243 + $232.6304 + ....+ 127.3394 + 122.9047 = $3,514.6238

Answer = $3,514.62