Evaluation of projects with unequal terms of implementation...

Evaluation of projects with unequal time frames

Many companies find themselves in a situation where they have to compare projects with different terms of implementation.

Example

There are two independent projects with different implementation times. It is necessary to rank them by priority. The discount rate is set at 15%. The cash flows of the projects (in million rubles) are presented in Table. 4.11.

Table 4.11

 Project Year 0th 1st 2nd 3rd A -200 250 - - In -100 60 80 20

The values ​​of NPV for these projects are equal to:

Project A: million rubles.

Project B: million rubles.

Since the net present value of project B is greater than that of project A, at first glance it can be concluded that project B is more preferable. But if we compare these projects by the NPV, criterion, then we automatically sort them by duration, not taking into account the difference in the terms of implementation. Project A is completed two years ahead of project B, and therefore funds that can be used in the new project are released earlier.

Suppose that project A can be implemented sequentially several times. Eliminate the temporary incompatibility of projects by replicating a shorter one. Each implementation of Project A will ensure its income, and their amount (in the example considered, 3 implementations), taking into account the time factor, is already comparable in time criterion with the implementation of Project B. Following this logic, we move from Project A to conditional At project of 3 years duration (Table 1). 4.12).

Table 4.12

Forecast of cash flows of conditional project A 1, million rubles.

 Project Year 0th 1st 2nd 3rd A -200 250 First repeat of project A -200 250 The second repetition of project A -200 250 A1 -200 50 50 250

Define the NPV of project A,:

A similar result could be obtained by reasoning as follows:

The current value of cash flow, due to the implementation of Project A, is 17.4 million rubles. (NPV of project A). The current value of cash flow is determined at the time of the start of the project. Thus, the cash flow generated by the project can be replaced by a single cash flow of NPV of this project, assigned at the time the project was launched. Following this logic, the cash flow of Project A (in million rubles) can be presented in the following form (Table 4.13):

Table 4.13

 Project Year 0th 1st 2nd 3rd A ( 17.4 17.4 17.4

Define the NPV project A,: million rubles.

Project A, has the value NPV, equal to 45.7 million rubles., Which is more than the value NPV of Project B (25.8 million rubles). Therefore, according to the NPV criterion, project A is preferable to project B. Therefore, the choice between the initial projects A and B in favor of the latter is no longer indisputable.

The need to compare each other on the effectiveness of investment projects of different duration in practice arises quite often. Special methods have been developed that allow investment in projects of different duration to be comparable in time to a comparable factor. The methods of chain repeat, infinite chain repeat and equivalent annuity are singled out. Let's consider them in more detail.

The chain repeat method was used in the example above. In this example, the project implementation dates were multiples, but not in all cases, this condition is met. Often the task is to compare the effectiveness of projects of different duration, the implementation time of which is not a multiple. To bring projects into a comparable kind by time factor, it is important that they start and end simultaneously. To achieve this, they determine the least total duration of projects, in which each of them can be repeated an integer number of times. Consider the sequence of actions in this case:

1) find the smallest common multiple of project implementation time N ;

2) consider each of the projects as a recurring one, calculate the total NPV of each of them, implemented the required number of times, taking into account the time factor. If we take the duration of the first project for , the duration of the second project for t, the number of repetitions will be and.

The total NPV of a repeating stream can be determined by the formula

(4.11)

where: NPV (l) is the net present value of the initial project;/- duration of this project; N - the least common multiple of project implementation time; and - the number of repetitions of the original project (n = N/G); r - discount rate;

3) from the source projects choose the one for which the total NPV of the repeating stream has the greatest value.

There are two independent projects with different implementation times. It is necessary to rank them by priority. The discount rate is set at 15%. The cash flows of projects (in million rubles) are presented in Table 4.14.

Table 4.14

 Project Year 0th 1st 2nd 3rd A -200 150 120 In -100 60 80 20

Rank the efficiency projects taking into account the time factor, using the above algorithm.

1. The duration of the project A is not a multiple of the duration of the project. The smallest common multiple of the implementation time of the projects under consideration is 6 years. During this period, project A should be repeated 3 times, project B - 2 times.

2. We calculate the NPV of the source projects:

Project A: million rubles.

Project B: million rubles.

The total NPV of projects A and B, realized respectively 3 and 2 times, will be:

Project A: million rubles.

Project B: million rubles.

3. The total NPV of project A, repeated 3 times, more has the total NPV of Project B, repeated 2 times (49.4 & gt; 42.8), therefore, it is more preferable .

The method of infinite chain repeat. The chain repeat method can be laborious if the least common multiple of compared projects is too large. In this case, the number of repetitions of the project can be significant. The method of infinite chain repeat makes it possible to simplify the previous method in terms of calculations. In this case, it is assumed that each of the projects can be repeated an infinite number of times. In this case, formula 4.11 is simplified and takes the following form:

(4.12)

The preferred design is a project having a larger value of NPV when it repeats an infinite number of times .

Example

Rank the efficiency of projects, the cash flows of which are presented in Table. 4.14, taking into account the time factor, using the method of infinite chain repeat.

Project A: million rubles.

Project B: million rubles.

The results obtained using the infinite chain repeat method coincide with the results obtained using the chain repeat method: project A is preferable to project B.

Equivalent annuity method. Equivalent annuity is a standard (unified) annuity that has the same duration as the project in question, and the same present value as the net present value of the initial project. The definition of the present value of the annuity was considered in Ch. 3. Presented value of annuity postnumerando is determined by the formula 3.22:

Assuming the present value of the annuity equal to the NPV of the project and the annuity period equal to the project term, calculate the value of the regular annual income (equivalent annuity):

(4.13)

where NPV is the net present value of the initial project; r - the discount rate; n - the duration of the project.

The annuity is replaced with an unlimited annuity, the amount of annuity payment of which coincides with the value of the regular annual income, found by the formula 4.13, and calculate the present value of this perpetual annuity .

This procedure is repeated for all projects under consideration. A project with a larger value is preferred.

The present value of an unlimited annuity is determined by the formula

(4.14)

Obviously, the higher the value of the equivalent annuity EA, , the higher the value of the present value of the corresponding perpetual annuity.

Example

Rank the efficiency of projects, the cash flows of which are presented in Table. 4.14, taking into account the time factor, using the equivalent annuity method.

1. The net present value of a one-time implementation of projects A and B is defined above.

Project A: NPV = 21.2 million rubles.

Project B: NPV = 25.8 million rubles.

2. For each project, we find the equivalent urgent annuity:

3. Replace the found annuity with an unlimited annuity with the same amount of annuity payment and calculate the present value of the perpetual annuity:

The results obtained using the equivalent annuity method coincide with the results obtained using the chain repeat and infinite chain repeat methods: project A is preferable to project B.

Note that when making a decision, execution of item 3 is not necessary, since the decision can be made on the basis of comparing the equivalent annuity indicators (paragraph 2).

To summarize, we note that the methods based on the repetition of the original projects have a certain conventionality. First, when using these methods, it was assumed that investments could be replicated by reinvesting cash receipts to achieve the same time horizon for all projects. Secondly, it was assumed that infinite reinvestment of cash receipts is possible within the framework of the firm's activities (at least for one of the investment projects).

In practice, these conditions may not be met, which is due to the following reasons:

- it is not always possible to make an accurate estimate of the duration of the original project;

- it's not obvious that the project will be repeated several times;

- the conditions for its implementation in the case of repetition may change;

- the calculations in the methods considered are absolutely formalized, and various factors that are either not formalized or have a general economic nature (inflation, technology change, etc.) are not taken into account.

For these reasons, the use of such methods must be approached consciously, and if the initial parameter of the compared projects is characterized by high uncertainty, one can not take into account the difference in the duration of their action.

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