Question

How long will it take to recover an investment of $5,000 at 3% interest rate under the following independent scenarios?

a. $800 anually

b. $100 monthly

c. An increasing gradient of $400 per year

Answer 1

a. For first scenario, we obtain the following excel:

Time (in years)	Interest rate factor	Loan amount	Loan + Interest	Amount paid	Remaining loan amount
0	1	5000	5000		5000
1	1.03	5000	5150	800	4350
2	1.03	4350	4480.5	800	3680.5
3	1.03	3680.5	3790.915	800	2990.915
4	1.03	2990.915	3080.642	800	2280.64245
5	1.03	2280.642	2349.062	800	1549.061724
6	1.03	1549.062	1595.534	800	795.5335752
7	1.03	795.5336	819.3996	800	19.39958246
8	1.03	19.39958	19.98157	800	-780.01843

Clearly, the investment is recovered in 8 years

b. For second scenario, we have the following excel:

Time (in months)	Interest rate factor	Loan amount	Loan + Interest	Amount paid	Remaining loan amount
0	1	5000	5000		5000
1	1.0025	5000	5012.5	100	4912.5
2	1.0025	4912.5	4924.781	100	4824.781
3	1.0025	4824.781	4836.843	100	4736.843
4	1.0025	4736.843	4748.685	100	4648.685
5	1.0025	4648.685	4660.307	100	4560.307
6	1.0025	4560.307	4571.708	100	4471.708
7	1.0025	4471.708	4482.887	100	4382.887
8	1.0025	4382.887	4393.844	100	4293.844
9	1.0025	4293.844	4304.579	100	4204.579
10	1.0025	4204.579	4215.09	100	4115.09
11	1.0025	4115.09	4125.378	100	4025.378
12	1.0025	4025.378	4035.442	100	3935.442
13	1.0025	3935.442	3945.28	100	3845.28
14	1.0025	3845.28	3854.893	100	3754.893
15	1.0025	3754.893	3764.281	100	3664.281
16	1.0025	3664.281	3673.441	100	3573.441
17	1.0025	3573.441	3582.375	100	3482.375
18	1.0025	3482.375	3491.081	100	3391.081
19	1.0025	3391.081	3399.558	100	3299.558
20	1.0025	3299.558	3307.807	100	3207.807
21	1.0025	3207.807	3215.827	100	3115.827
22	1.0025	3115.827	3123.616	100	3023.616
23	1.0025	3023.616	3031.176	100	2931.176
24	1.0025	2931.176	2938.503	100	2838.503
25	1.0025	2838.503	2845.6	100	2745.6
26	1.0025	2745.6	2752.464	100	2652.464
27	1.0025	2652.464	2659.095	100	2559.095
28	1.0025	2559.095	2565.493	100	2465.493
29	1.0025	2465.493	2471.656	100	2371.656
30	1.0025	2371.656	2377.585	100	2277.585
31	1.0025	2277.585	2283.279	100	2183.279
32	1.0025	2183.279	2188.738	100	2088.738
33	1.0025	2088.738	2093.959	100	1993.959
34	1.0025	1993.959	1998.944	100	1898.944
35	1.0025	1898.944	1903.692	100	1803.692
36	1.0025	1803.692	1808.201	100	1708.201
37	1.0025	1708.201	1712.471	100	1612.471
38	1.0025	1612.471	1616.503	100	1516.503
39	1.0025	1516.503	1520.294	100	1420.294
40	1.0025	1420.294	1423.845	100	1323.845
41	1.0025	1323.845	1327.154	100	1227.154
42	1.0025	1227.154	1230.222	100	1130.222
43	1.0025	1130.222	1133.048	100	1033.048
44	1.0025	1033.048	1035.63	100	935.6303
45	1.0025	935.6303	937.9694	100	837.9694
46	1.0025	837.9694	840.0643	100	740.0643
47	1.0025	740.0643	741.9145	100	641.9145
48	1.0025	641.9145	643.5193	100	543.5193
49	1.0025	543.5193	544.8781	100	444.8781
50	1.0025	444.8781	445.9903	100	345.9903
51	1.0025	345.9903	346.8552	100	246.8552
52	1.0025	246.8552	247.4724	100	147.4724
53	1.0025	147.4724	147.8411	100	47.84105
54	1.0025	47.84105	47.96065	100	-52.0393

Hence, investment is recovered in 54 months.

c. For this scenario, we have the following excel:

Time (in years)	Interest rate factor	Loan amount	Loan + Interest	Amount paid	Remaining loan amount
0	1	5000	5000		5000
1	1.03	5000	5150	400	4750
2	1.03	4750	4892.5	800	4092.5
3	1.03	4092.5	4215.275	1200	3015.275
4	1.03	3015.275	3105.733	1600	1505.73325
5	1.03	1505.733	1550.905	2000	-449.094752

Here, investment is recovered in 5 years.