The cheap mixer lasts 3 years, the expensive one 6: which costs less per year? The equivalent annual cost method

Q: When does the method work, and when not?

When both options do the same job. If revenue differs, compare net cash flow, not just cost (the revenue version of this method is called the "equivalent annual annuity"). When the equipment is replaced by the same kind at the end of its life. If the bakery itself is planned to close after 3 years, B's next 3 years are not needed and the comparison is made over the 3-year period only. When prices and technology do not change sharply. In a field where technology gets cheaper quickly, short-lived equipment may be more convenient: after 3 years a new, better machine is bought. When the discount rate is set before the calculation. The higher the rate, the heavier a large payment today (B's price) looks, so test the result at two or three rates. Nodira's conclusion: if B's 6-year life is reliable, it saves about 4.2 million soums a year. If the life drops to 5 years, the advantage disappears.

Nodira, the process engineer at the "Oq Bug'doy" bakery, has to choose one of two dough mixers. Both knead the same volume of dough, and the bakery's revenue does not change whichever one is bought. So there is only one question: which one costs less?

All amounts are hypothetical, compiled by the author, in million soums. Taxes are ignored for simplicity, and we assume the machines have no resale value at the end of their useful life.

	Machine "A"	Machine "B"
Purchase price	90	150
Useful life	3 years	6 years
Annual running cost (electricity, repairs, spare parts)	12	8

First attempt: adding up total costs

Simply adding up, A costs 90 + 3 × 12 = 126 over 3 years, and B costs 150 + 6 × 8 = 198 over 6 years. 126 is smaller, but these figures relate to different periods: A's 126 is the price of 3 years of service, B's 198 is the price of 6 years of service.

Adding the time value of money does not remove the problem either. Nodira took the firm's cost of money as a hypothetical 14% and brought each year's cost to today's value (factor = 1 / 1.14 ^ year):

Year	Factor	A: discounted cost	B: discounted cost
1	0.8772	10.53	7.02
2	0.7695	9.23	6.16
3	0.6750	8.10	5.40
4	0.5921	–	4.74
5	0.5194	–	4.15
6	0.4556	–	3.64
Total running costs		27.86	31.11
Including the price (present value)		117.86	181.11

At first sight A is 63 million cheaper. But after three years A breaks down and the bakery buys a machine again. That cost is not in the table.

Second step: bringing each machine to a "price per year"

Equivalent annual cost answers this question: if, instead of all the costs over the machine's life, the same amount were paid every year, how much would it be? To get it, the present value is divided by the annuity factor:

annuity factor = (1 − (1 + r) ^ −n) / r, where n is the useful life.

It is really the sum of the factor column over n years: - for A (3 years): 0.8772 + 0.7695 + 0.6750 ≈ 2.3216 - for B (6 years): the sum of the six factors ≈ 3.8887

	Machine "A"	Machine "B"
Present value	117.86	181.11
Annuity factor	2.3216	3.8887
Equivalent annual cost	≈ 50.77	≈ 46.57

The result is reversed: B is about 4.2 million soums cheaper every year. A's price is low, but it serves for a short time and its running cost is high.

Check: putting both on a 6-year horizon

You can check that the method works in a simple way. Assume that over 6 years the bakery buys A twice: now and at the end of year 3, at the same price and with the same costs. - Present value of the second A: 117.86 × 0.6750 ≈ 79.55 - Total for 6 years: 117.86 + 79.55 = 197.41 - B's 6-year present value: 181.11

The difference is about 16.3 million soums, again in B's favour. Multiplying the annual difference (4.2) by the 6-year factor (3.8887) gives the same amount (allowing for rounding).

Which assumption is the result most sensitive to?

B's advantage depends on it really lasting 6 years. Nodira changed B's life while keeping the other figures the same:

B's actual useful life	Present value	Annuity factor	Equivalent annual cost	Compared with A (50.77)
6 years	181.11	3.8887	46.57	B cheaper
5 years	177.46	3.4331	51.69	A slightly cheaper
4 years	173.31	2.9137	59.48	A much cheaper

So before choosing B, check its real useful life: the manufacturer's warranty, whether there is a service centre, and the experience of other bakeries using this machine.

When does the method work, and when not?

When both options do the same job. If revenue differs, compare net cash flow, not just cost (the revenue version of this method is called the "equivalent annual annuity").
When the equipment is replaced by the same kind at the end of its life. If the bakery itself is planned to close after 3 years, B's next 3 years are not needed and the comparison is made over the 3-year period only.
When prices and technology do not change sharply. In a field where technology gets cheaper quickly, short-lived equipment may be more convenient: after 3 years a new, better machine is bought.
When the discount rate is set before the calculation. The higher the rate, the heavier a large payment today (B's price) looks, so test the result at two or three rates.

Nodira's conclusion: if B's 6-year life is reliable, it saves about 4.2 million soums a year. If the life drops to 5 years, the advantage disappears.