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:

**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.

**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.

**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!

**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.

Additional applications of the technique

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

- Evaluating the best time to replace an existing asset with a new asset.
- 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**