# Equivalent annual costs and benefits

### Equivalent annual costs

At some stage you have probably bought an asset such as a car, a washing machine or a computer and you may have considered how long you should keep that asset prior to replacing it. If the asset is kept for a longer period its initial cost, less any residual value, is spread over more years which is likely to reduce your cost per year of ownership. However, as the asset ages it is likely to require more and more maintenance and may operate less effectively which will increase your costs per year. Determining the optimal time to replace the asset (the optimal replacement cycle) is difficult.

As a general rule you and I don’t worry too much about this. Indeed most of us will make a decision based on our ‘gut feel’ and other factors such as image for example. Indeed I tend to keep my car until such time as I have lost confidence in its ability to get me reliably from A to B or it has deteriorated so much I no longer want to be seen in it!

Companies also face exactly the same asset replacement decisions. However as the amounts involved can often be very significant, making decisions based on ‘gut feel’ is not really accurate enough. The calculation of equivalent annual costs is a tool that can be used to assist in this decision-making process.

The equivalent annual cost method involves the following steps:

• Step 1 – Calculate the net present value (NPV) of cost for each potential replacement cycle.
• Step 2 – For each potential replacement cycle an equivalent annual cost is calculated.
• The decision – The replacement cycle with the lowest equivalent annual cost may then be chosen, although other factors may also have to be considered.

This will now be demonstrated and explained further through the use of an example.

EXAMPLE 1

A machine has a cost of \$3,500. The annual maintenance costs of the machine are forecast to be \$900 in the first year, \$1,000 in the second year and \$1,200 in the third year of ownership.

The residual value of the machine is expected to be \$2,100 after two years and \$1,600 after three years.

The cost of capital of the company is 11% per year.

Calculate the optimal replacement cycle for the machine.

SOLUTION 1

Step 1 – Calculate the NPV of cost for each potential replacement cycle.

As we have not been given the residual value after one year of ownership, we cannot calculate an NPV of cost for a one-year replacement cycle. Hence, our decision here will be between a two- or three-year replacement cycle.

NPV of cost – two-year replacement cycle:
Here we evaluate all the cash flows associated with buying and keeping the asset for two years.

Time

0

1

2

Initial cost

(3,500)

Maintenance

(900)

(1,000)

Residual value

______

_____

2,100

Net cash flows

(3,500)

(900)

1,100

11% Discount factors

1

0.901

0.812

Present values

(3,500)

(811)

893

Net present value

(3,418)

Please note that the normal assumptions with regard to the timings of the cash flows continue to be made. Hence, the maintenance costs are shown at the end of each year, whereas in reality they will arise throughout the year.

One complication that arose in a past question was that the maintenance was an annual overhaul required at each year end rather than ongoing maintenance occurring throughout each year. Logically the maintenance/overhaul cost was not incurred in the year of disposal as a company would be unlikely to overhaul an asset just prior to selling it. Hence, in the two-year replacement cycle above, if the maintenance had been an annual overhaul the \$1,000 cost at time 2 would be excluded.

NPV of cost – three-year replacement cycle:
We now evaluate all the cash flows associated with buying and keeping the asset for three years.

Time

0

1

2

3

Initial cost

(3,500)

Maintenance

(900)

(1,000)

(1,200)

Residual value

______

______

______

1,600

Net cash flows

(3,500)

(900)

(1,000)

400

11% Discount factors

1

0.901

0.812

0.731

Present values

(3,500)

(811)

(812)

292

Net present value

(4,831)

Whilst there is an element of repetition in these calculations I would still advise using the above simple and logical format or something similar. Although I have seen formats which try to combine the calculations, they are more complex and tend to lead to mistakes being made.

A classic mistake to be avoided is including the residual value after two years in the calculation of the NPV of cost for the three-year replacement cycle. For the three-year replacement cycle, the sale will occur at the end of the three years. Please remember if you buy the asset once you can only sell it once!

The two NPVs calculated should not be compared as quite obviously buying and keeping an asset for a longer period is likely to cost more than buying and keeping it for a shorter period as there is less benefit to the owner. This has proved to be the case here. In order to make a fair comparison we must calculate the equivalent annual costs.

Step 2 – For each potential replacement cycle an equivalent annual cost is calculated.

The costs calculated in Step 1 are spread over the period for which they will give benefit. Hence, the NPV of cost for the two-year cycle is spread over two years and the NPV of cost for the three-year cycle is spread over three years.  This is done by using annuity factors to turn each NPV of cost into an equivalent annual cost (EAC) at the end of each year of ownership.

Remember if you have equal annual cash flows for a number of years and want to calculate a present value (PV) you must multiply the annual cash flow by an annuity factor: so to calculate the equivalent annual cost or EAC from an NPV of cost we must divide by the relevant annuity factor.

EAC – two-year cycle:
As the NPV of cost of \$3,418 will give the benefit of ownership for two years, we divide by the two-year annuity factor at the 11% cost of capital to get the EAC.

EAC = \$3,418/1.713 = \$1,995 per year

This is the equivalent annual cost at time 1 and time 2 which equates to an NPV of cost of \$3,418.

EAC – three-year cycle:
As the NPV of cost of \$4,831 will give benefit for three years, we divide by the three-year annuity factor at the 11% cost of capital to get the EAC.

EAC = \$4,831/2.444 = \$1,977 per year

This is the cost at time 1, time 2 and time 3 which equates to an NPV of cost of \$4,831.

While some textbooks will continue to put brackets around these cost figures, I am content to show them as positive as we are describing them as costs.

The decision:

As the calculated equivalent annual costs are both annual costs, they can be compared to come to a decision.

Hence, as an annual cost of \$1,977 is less than an annual cost of \$1,995, the three-year replacement cycle is said to be the optimal replacement cycle.

Weaknesses

Having worked through an example we should now consider the weaknesses of the approach we have used. These include the following:

1. Our analysis has ignored the impact of taxation.
Both buying an asset and incurring a maintenance cost will cause tax cash flows. While these cash flows could be included they would add to the complexity of the calculation. Past exam questions have specifically excluded the impact of taxation on the cash flows.

2. Our analysis assumes that we can replace like with like.
Our analysis has assumed that the asset can be replaced by exactly the same asset in perpetuity. In reality, this will not be possible as assets are constantly developing. Even if you replace your car with exactly the same model after a number of years the new car will undoubtedly have improvements and other differences to the old one. In our worked example above, if we were to imagine that the asset was a computer then although the calculated optimal replacement cycle is three years, the difference in cost between the two- and three-year replacement cycles is small. Hence, we might decide to use a two-year replacement cycle as we would then benefit from having a new, more up-to-date computer with more functionality on a more regular basis.

3. Our analysis assumes that we will want to replace like with like.
Additionally the analysis assumes we will want to replace the asset with the same asset in perpetuity. In reality, business needs develop and when it becomes time to replace an asset a company may want to acquire a different asset with different functionality. For instance, a company may want an asset with greater capacity due to growth in their business. You and I face exactly the same issue. Over my lifetime I have had a variety of different cars as my need has developed – my two-seater sports car proved less than useful when my first child was born!

4. Our analysis has ignored inflation.
Different cash flows may suffer from different specific inflation rates and as a result our analysis may not be correct. For instance, the initial cost of assets often inflates quite slowly as manufacturers find more efficient ways of production. However, maintenance costs often inflate much more quickly as maintenance is often labour-intensive and labour costs often grow quickly. This differential between the inflation rates of different cash flows means that an alternative method, which you are not required to know, should be used. If all the cash flows inflate at one rate then the EAC method can be used with real cash flows and a real cost of capital.

Without going into great detail it is worth being aware that a similar technique can be used in other circumstances. These include:

1. Evaluating the best time to replace an existing asset with a new asset.
2. Deciding between assets which would have the same functionality but have different lives. For instance when you or I are buying a car we could buy a cheaper car of lower quality or a more costly car of higher quality. It would be unfair to simply compare the costs directly, as the higher quality car is likely to last longer.

### Equivalent annual benefit

If a company is faced with mutually exclusive projects, where only one out of a number of projects can be accepted, then the general rule is that the company should choose the project that generates the highest NPV as this creates the biggest increase in shareholder wealth. However, if the situation is such that it is anticipated that the same projects could be repeated in perpetuity and  the projects have different lives then the equivalent annual benefit approach can be used. This is simply a further variation on the equivalent annual cost approach and is demonstrated in the following example.

EXAMPLE 2

Two mutually exclusive projects are being considered:

• Project A has an NPV of \$47m and is expected to last three years.
• Project B has an NPV of \$58m and is expected to last four years.

It is anticipated that if either project is chosen it will be possible to repeat it for the foreseeable future.

The cost of capital of the company is 13% per year.

Calculate which project the company should accept.

SOLUTION 2

Step 1 – Calculate the NPV for each potential project.
This would involve calculating the NPV of each project as normal. I have already done this for us to save time!

Project A – \$47m
Project B – \$58m

Step 2 – Calculate the equivalent annual benefit for each potential project.
This is calculated using annuity factors in exactly the same way as an EAC is calculated. Hence, the NPV of Project A is divided by the 3-year annuity factor at the cost of capital of 13% as the project life is three years. For Project B the 4-year annuity factor is used to reflect the four-year life of the project.

Project A – equivalent annual benefit = \$47m/2.361 = \$19.9m per year
Project B – equivalent annual benefit = \$58m/2.974 = \$19.5m per year

The decision:

As Project A has the highest equivalent annual benefit it should be chosen instead of Project B, which has the higher NPV, so long as the project can be repeated for the foreseeable future. This result arises because although the shorter project produces the lower NPV that NPV will be obtained more frequently than the NPV of the longer project.

The equivalent annual benefit technique suffers similar weaknesses to the EAC technique.

### Conclusion

Although this topic is a relatively small one within your Financial Management syllabus, it is a topic well worth mastering as when it has been examined in the past those with the necessary knowledge have been able to earn very good marks. Equally, I would not expect any significant question on this topic to be wholly calculative and hence students should be ready to discuss the reasons for the approach used and the weaknesses or limitations of that approach.

William Parrott, freelance tutor and senior FM tutor, MAT Uganda