Annuity Method
Description of the method
The annuity method (AM) uses the same discounted cash flow model as the NPV method. The only change is a different target measure, the annuity:
Key Concept:
An annuity is a series of cash flows of equal amounts in each period of the total planning period.
The annuity can be regarded as an amount that an investor can withdraw in every period when undertaking the investment project. The annuity of an investment project is equivalent to the NPV of that project, i.e. it is possible to equate both measures mathematically.
A limitation of this approach is that the annuity method is not (completely) suitable for the assessment of relative profitability. This will be further outlined below. Setting this issue aside for now, however, the following profitability criteria can be applied:
Key Concept:
Absolute profitability is achieved if an investment project's annuity is greater than zero.
Relative profitability: An investment project is preferred if it has a higher annuity than the alternative investment project(s).
When calculating an annuity, the cash flows are normally allocated to the end of each period (deferred cash flows) and this is assumed in the following notes. Initially, the time span used is the economic life of the project.
The annuity (ANN) of an investment project can be determined by multiplying the net present value (NPV) by the capital recovery factor (see Section 3.1). This is dependent on the uniform discount rate (i) and the economic life (T). The annuity is calculated as follows:
As can be deduced from the formula, the annuity method leads to the same assessment of absolute profitability as the NPV method. Since any meaningful i and T are higher than zero, the capital recovery factor is higher than zero as well. Thus, a positive (negative) annuity is achieved as the result of a positive (negative) NPV. Similar reasoning applies to relative profitability assessments where the projects under comparison have identical economic life spans, as the recovery factors are identical. If this is not the case, and the assumptions of the NPV model are applied to subsequent projects used for balancing economic life differences, then the annuity method should be applied in modified form, e.g. using identical time spans. Then the assessment of profitability is again identical to that obtained by the NPV method. However, the annuity method can also be applied if a different assumption is made in regard to subsequent projects. This is illustrated in the following example.
Example 3-2
Here Example 3-1 is reconsidered using the annuity method. Investment project A's annuity can be calculated as:
1 085 0 08
For project B a different capital recovery factor arises on account of its different economic life. The annuity amounts to:
1 084 0 08
Both projects have a positive annuity and, thus, are profitable in absolute terms. For the assessment of relative profitability it should be noted that the annuities refer to different time spans - due to different economic lives - and therefore encompass different numbers of cash flows. The annuity of project B is higher, but it runs for a shorter time span. If the assumptions of the NPV method are maintained (in particular, that future investments yield at the uniform discount rate) then the annuity method should be replaced by the NPV method or, alternatively, the annuity method should be applied in a modified form, relating only to project annuities for the same number of periods. If the annuity of project B is recalculated for 5 years it becomes:
1 085 0 08
Now, as might be expected, project A regains relative profitability. The obvious question of why the annuity method is used despite the results for absolute and relative profitability being identical, is taken up later.
If assumptions are modified so that a repetition of the investment project will follow, then a problem can arise with the NPV method. In particular, where a project will have unlimited, identical repetitions the annuity method should be used since, owing to the unlimited time horizon, no NPV can be determined. Using the annuity method, the NPV of an unlimited chain of identical projects can be calculated as follows:
Net present value = Annuity (3.13)
Interest rate
In the example it amounts to:
In this situation, investment project B is relatively profitable on account of the higher NPV of the continuously repeated project: an effect due to the shorter economic life.
This argument and the calculation of limited and unlimited chains of investment projects are explored in more detail in Section 5.3.
Assessment of the method
The assessment of the annuity method is similar to that of the NPV method, since identical model assumptions and data requirements exist. The calculation of the annuity is only slightly more difficult than that of the NPV.
However, use of the annuity method is unnecessary in many situations. In the analysis of absolute profitability, the NPV method leads to identical results. When the economic lives of the projects under consideration are identical, or when the calculations are modified in the ways described, the assessments of relative profitability are identical as well. Also, annuities, in contrast to NPVs, can be calculated only approximately if the uniform discount rates change during the course of the project's life. Only two arguments remain after these reservations: annuities are needed to calculate the NPV of an unlimited identical investment chain; and the target measure for the annuity method can be interpreted more easily (esp. by the inexperienced user). As an annuity is a measure related to a period, it represents a specific form of 'average profit' and, thus, can be interpreted more easily than a NPV result.
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