# Net present value, Net present value of long-term projects - Investments

## Net Present Value

The feasibility of implementing a project in terms of its effectiveness can be estimated using the net present value ( NPV ), which is understood as the increase in net assets of the firm through the implementation of the project. In other words, NPV is defined as the difference between the present value of the PV funds and the sum of the initial C0 investments:

In this case, the values ​​ C 1 and C 0 should be considered as financial indicators. So, for the machine in question

Since in the future when the values ​​of NPV are used, the money flows and money inflows have the plus sign and the outflows are the minus sign, then in the case of using cash flows, the net present value is calculated by the formula

In particular, in this example

Then, when determining the expediency of investing money in a real facility, investors should use the following rule NPV:

• If NPV & gt; 0, then you need to buy a real tool;

• If NPV & lt; 0, then you do not need to acquire it;

• If NPV = 0, then you can buy or not buy.

## Net present value of long-term projects

As a rule, investment projects are implemented during several calculation steps (for the sake of simplicity, let's assume that the calculation step is equal to one year). Let the managers of the firm forecast the flow of future revenues from the investment project in the following amounts: С 0 - initial investment costs, С 1 - income at the end of the first year, С 2 - income at the end of the second year, etc. How to estimate the present value of IP in this case?

Let at the moment t 0 in the market there is a best alternative means worth PV 1 rub., yield r 1, which in a year will provide income , equal to C 1. Then, as stated above:

If C 1 = 12 million rubles, then PV 1 = 11.21 million rubles.

Let at the moment t 0 in the market there is another best alternative means with a value PV 2 rubles, yield r 2 per annum, which in two year will provide an income equal to C 2. Obviously, the best alternative means should ensure the accumulation of a compounded interest. Then

Reasoning similarly, it can be shown that

etc ...

One of the main advantages of the reduced value category is that the sums PV 1 , PV 2, ..., PV n are expressed in the same (current, current) rubles, reduced to the initial moment t 0 of the investment decision. In this connection, the sums PV 1 , PV 2 , PV 3, ... can be summed up to obtain the value of the total present value of the project:

where n is the total number of years of project operation.

This formula is called the formula for the discounted cash flow stream.

In order to find the net present value of a long-term project, it is necessary to take into account the initial investment cost C 0 at the time t 0:

(2.1)

When determining the net present value of projects, it is necessary to take into account a number of circumstances and rules. Let's mention some of them.

1. Ways of estimating discount rates r t . In real life, the values ​​ r 1 , r 2 , r 3, ... can differ from each other. Dependence of the yield of securities from maturity to term is called the terminal structure of the interest rate. In practice, in order to facilitate calculations, NPV resort to a fundamental simplification and believe that the terminal structure of the interest rate is horizontal and all r t are the same:

With this simplification, the formula for computing NPV long-term projects will take the form

To calculate the present values ​​in the case of multiple discount rates (ie, r = 1%, > 2%, 3%, etc.), you can use special tables that give the values discount factors for different values ​​of the discount rate and the number of years n income stream income.

2. Accounting for the flow of costs. In the formula (2.1), the sums C 1 , C 2, C3, ... reflect only the absolute values ​​of the money flows, according to ns their focus. In other words, C t can reflect the costs of the firm and have a negative sign. If the flow of expenditure occurs within several calculation steps, then these negative sums C t must be discounted on a general basis.

Consider the following example: the firm decides to implement the project, and the company's managers believe that future cash flows have the following form (Table 2.1).

If the acceptable discount rate is 11%, the net present value of the project will be

and such a project is worth taking.

Table 2.1

Conditional data for calculating NPV

 Cost element and proceeds, million rubles. Periods (in our case years) t = 0 t = 1 t = 2 Purchase of land -70 - - Workshop Building -120 -80 - Buying machines - -20 -20 Revenue - - +450 Total C 0 = -190 C 1 = -100 C 2 = +430

3. Specific cases of calculating present value . Sometimes the investor may have the following question: what is the present value of the money that will be provided annually (or after the holding periods) the same sums of money C for infinitely long. In other words, what amount of money should be spent today to purchase a means that gives a continuous stream of income (in the UK, for example, there are government bonds, so-called "perpetuities" that have no maturity and provide its owner with an annual the same amount of interest for an unlimited number of years).

The present value of such a facility with a fixed and unchanged discount rate should be equal to the following value:

(2.2)

So, if the discount rate is 0.1 (10%) and the promised annual amount of C = 1 million rubles, the present value of РV of such a funds will be 10 million rubles.

However, more often than not any means ensure the same time flows of the same money flows at regular intervals, but a certain number of periods. Such income is provided, for example, by a regular coupon bond with a fixed coupon or an apartment leased out. Such funds are called annuities . The present value of an annuity generating an income stream C for n periods (years) for unchanged discount rate, calculated by the formula

(2.3)

The expression in parentheses is called the annuity factor.

The calculation of the present value of annuities is greatly simplified when the discount rate is a multiple, since in this case special tables for annuity factors can be used.

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