Question

Your company is buying a new piece of equipment to reduce emissions in your factory. It will cost​ $2,000 to run next year​ (Year 1), and that operating cost will increase by​ 4% every year until Year​ 20, at which point you will discard it​ (after paying the operating cost​ - there is a cash flow in Year​ 20). You will also have to pay to maintain the​ equipment, which you estimate will cost​ $400 next year​ (Year 1) and increase by​ 5% per year until Year 20. Assume an interest rate of​ 9%.

a. What is the equivalent present value in Year 0 of the total costs of the​ equipment?

b. What is the equivalent future value in Year 20 of the total costs of the​ equipment?   

Answer 1

Answer a. Let's calculate present value (PV) of operating cost and maintainance cost. PV = C / (1+r)^n
Where, c is cashflow in a year, r is rate of interest and n is time period.
Note that operating expense in first year is $2000 and increases by 4% every year. Maintainance cost is $400 and increases by 5% per year. Interest rate is 9%. Following table shows operating and maintainance expenses for entire time period in column 1 and column 2 respectively. column 3 and column 4 shows the present value of operating and maintainance expenses respectively. Sum of column 3 and column 4 will give equivallent present value of total cost of equipment in year 0.

Year	Operating cost (OC)	Maintainance cost (MC)	PV of OC	PV of MC
1	2000	400	1834.86	366.97
2	2080	420	1750.69	353.51
3	2163.2	441	1670.39	340.53
4	2249.73	463.05	1593.76	328.04
5	2339.72	486.20	1520.66	316.00
6	2433.31	510.51	1450.90	304.40
7	2530.64	536.04	1384.35	293.23
8	2631.86	562.84	1320.84	282.47
9	2737.14	590.98	1260.25	272.10
10	2846.62	620.53	1202.44	262.12
11	2960.49	651.56	1147.29	252.50
12	3078.91	684.14	1094.66	243.23
13	3202.06	718.34	1044.45	234.31
14	3330.15	754.26	996.53	225.71
15	3463.35	791.97	950.82	217.43
16	3601.89	831.57	907.21	209.45
17	3745.96	873.15	865.59	201.76
18	3895.80	916.81	825.89	194.36
19	4051.63	962.65	788.00	187.23
20	4213.70	1010.78	751.85	180.35

Total equivallent present cost = 29627.14 (sum of all present values calculated in column 3 and column 4).

Answer B. Let's now calculate future value of total cost of equipment. Future value = amount * (1+r)^n

Where, r is rate of interest and n is time period.

Here, amount is $29627.14 (from answer A).

Future value = 29627.14 * 1.09^20 = 166042.6.