Question

Air conditioning for a college dormitory will cost $2.4 million to install and $155,000 per year to operate. The system should last 16 years. The real cost of capital is 6%, and the college pays no taxes. What is the equivalent annual cost? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Equivalent annual cost            $

Answer 1

In order to compute the equivalent annual cost, we shall use the undermentioned formula:

= (Net Present value of project / Present value annuity factor of the project)

We first shall compute the net present value of project by using the following function:

= Initial cost + Year 1 cash outflow / (1 + cost of capital )¹ + Year 2 cash outflow / (1 + cost of capital )² + Year 3 cash outflow / (1 + cost of capital )³ + Year 4 cash outflow / (1 + cost of capital )⁴ + Year 5 cash outflow / (1 + cost of capital )⁵ + Year 6 cash outflow / (1 + cost of capital )⁶ + Year 7 cash outflow / (1 + cost of capital )⁷ + Year 8 cash outflow / (1 + cost of capital )⁸ + Year 9 cash outflow / (1 + cost of capital )⁹ + Year 10 cash outflow / (1 + cost of capital )¹⁰ + Year 11 cash outflow / (1 + cost of capital )¹¹ + Year 12 cash outflow / (1 + cost of capital )¹² + Year 13 cash outflow / (1 + cost of capital )¹³ + Year 14 cash outflow / (1 + cost of capital )¹⁴ + Year 15 cash outflow / (1 + cost of capital )¹⁵ + Year 16 cash outflow / (1 + cost of capital )¹⁶

Initial cost = $ 2.4 million i.e. 2,400,000

Cost of capital = 6% or 0.06

Cash outflows each year = $ 155,000

Now we shall plug the above figures in the net present value function to get the net present value:

= 2,400,000 + 155,000 / 1.06¹ + 155,000 / 1.06² + 155,000 / 1.06³ + 155,000 / 1.06⁴ + 155,000 / 1.06⁵ + ​​​​​155,000 / 1.06⁶​​​​​​ + 155,000 / 1.06⁷ + 155,000 / 1.06⁸ + ​​​​​​​155,000 / 1.06⁹ + ​​​​​​​155,000 / 1.06¹⁰​​​​​​​ + 155,000 / 1.06¹¹​​​​​​​ + 155,000 / 1.06¹²​​​​​​​ + 155,000 / 1.06¹³​​​​​​​ + 155,000 / 1.06¹⁴​​​​​​​ + 155,000 / 1.06¹⁵​​​ + ​​​​155,000 / 1.06¹⁶​​​​​​

= $ 3,966,413.767 (Approximately)

Now we shall compute the present value annuity factor of 6% for 16 years

Year	Present Value Factor Computation	Present Value Factor
1	= 1 / 1.061	0.9434
2	= 1 / 1.062	0.8899
3	= 1 / 1.063	0.8396
4	= 1 / 1.064	0.7921
5	= 1 / 1.065	0.7472
6	= 1 / 1.066	0.7050
7	= 1 / 1.067	0.6650
8	= 1 / 1.068	0.6274
9	= 1 / 1.069	0.5919
10	= 1 / 1.0610	0.5584
11	= 1 / 1.0611	0.5268
12	= 1 / 1.0612	0.4970
13	= 1 / 1.0613	0.4688
14	= 1 / 1.0614	0.4423
15	= 1 / 1.0615	0.4173
16	= 1 / 1.0616	0.3936
	Total	10.1057

So as we can see from the above table the Present value annuity factor comes by adding all the present value factors i.e. 10.1057 Approximately

Now the equivalent annual cost will be determined by using the below formula i.e.:

= (Net Present value of project / Present value annuity factor of the project)

= 3,966,413.767 / 10.1057

= $ 392,492.7268 Approximately.

Feel free to ask in case of any queries regarding this question.