The concept of the present present value of the project and the calculation procedure. Net present value

NPV (abbreviation in English - Net Present Value), in Russian this indicator has several variations of the name, among them:

net present value (abbreviated NPV) is the most common name and abbreviation, even the formula in Excel is called exactly that;
net present value (abbreviated NPV) - the name is due to the fact that cash flows are discounted and only then summed up;
net present value (abbreviated NPV) - the name is due to the fact that all income and losses from activities due to discounting are, as it were, reduced to the current value of money (after all, from the point of view of economics, if we earn 1,000 rubles and then actually receive less than if we received the same amount, but now).

NPV is an indicator of the profit that participants in an investment project will receive. Mathematically, this indicator is found by discounting the values of net cash flow (regardless of whether it is negative or positive).

Net present value can be found for any period of time of the project since its beginning (for 5 years, for 7 years, for 10 years, and so on) depending on the need for calculation.

What is it needed for

NPV is one of the indicators of project efficiency, along with IRR, simple and discounted payback period. It is needed to:

understand what kind of income the project will bring, whether it will pay off in principle or is it unprofitable, when it will be able to pay off and how much money it will bring at a particular point in time;
to compare investment projects (if there are a number of projects, but there is not enough money for everyone, then projects with the greatest opportunity to earn money, i.e. the highest NPV, are taken).

Calculation formula

To calculate the indicator, the following formula is used:

CF - the amount of net cash flow over a period of time (month, quarter, year, etc.);
t is the time period for which the net cash flow is taken;
N is the number of periods for which the investment project is calculated;
i is the discount rate taken into account in this project.

Calculation example

To consider an example of calculating the NPV indicator, let's take a simplified project for the construction of a small office building. According to the investment project, the following cash flows are planned (thousand rubles):

Article	1 year	2 year	3 year	4 year	5 year
Investments in the project	100 000
Operating income		35 000	37 000	38 000	40 000
Operating expenses		4 000	4 500	5 000	5 500
Net cash flow	- 100 000	31 000	32 500	33 000	34 500

The project discount rate is 10%.

Substituting into the formula the values of net cash flow for each period (where negative cash flow is obtained, we put it with a minus sign) and adjusting them taking into account the discount rate, we get the following result:

NPV = - 100,000 / 1.1 + 31,000 / 1.1 2 + 32,500 / 1.1 3 + 33,000 / 1.1 4 + 34,500 / 1.1 5 = 3,089.70

To illustrate how NPV is calculated in Excel, let's look at the previous example by entering it into tables. The calculation can be done in two ways

Excel has an NPV formula that calculates the net present value, to do this you need to specify the discount rate (without the percent sign) and highlight the range of the net cash flow. The formula looks like this: = NPV (percent; range of net cash flow).
You can create an additional table yourself where you can discount the cash flow and sum it up.

Below in the figure we have shown both calculations (the first shows the formulas, the second the calculation results):

As you can see, both calculation methods lead to the same result, which means that depending on what you are more comfortable using, you can use any of the presented calculation options.

Both concepts from the title of this section, discounted (present) value, PS (presentvalue, or PV ), And net present value, NPV (netpresentvalue, or NPV ), denote current the value of expected future cash receipts.

As an example, consider valuing an investment that promises an income of $100 per year at the end of this year and the next four years. We assume that this series of five payments of $100 each is guaranteed and the money will certainly arrive. If a bank were to pay us 10% annual interest on a five-year deposit, then that 10% would be the opportunity cost of the investment—the benchmark rate of return against which we would compare the benefits of our investment.

You can calculate the value of an investment by discounting its cash flows using opportunity cost as the discount rate.

Calculation formula inExceldiscounted (present) value (PV)= NPV(C1,B5:B9)

Present value(PS) in the amount of $379.08 is the current value of the investment.

Suppose this investment were to sell for $400. Obviously, it would not be worth the asking price, since - assuming an opportunity return (discount rate) of 10% - the real value of this investment would be only $379.08. it is appropriate to introduce the concept net present value(NPS). Denoted by the symbol r discount rate for this investment, we get the following NPV formula:

Where CF t is the cash flow from the investment at time t; CF 0 – flow of funds (receipt) at the current moment.

Calculation formula inExcel net present value (NPV)= NPV(C1,B6:B10)+B5

Excel terminology for discounted cash flows differs slightly from standard financial terminology. In Excel, the abbreviation MUR (NPV) denotes present value (not chiI'm standing present value) of a series of cash receipts.

To calculate in Excel net present value series of cash receipts in the usual sense of financial theory, you must first calculate present value future cash flows (using an Excel function such as NPV), and then subtract the cash flow at the initial point in time from this number. (This value is often the same as the value of the asset in question.)

Publications

Textbook “Assessing the effectiveness of investment projects”
Calculation and analysis of investment projects, preparation of business plans

Textbook “Financial calculation techniques using Excel”
Basic concepts of financial mathematics and recommendations for performing calculations

Discussions

Note! Discussions use the reverse order of messages (i.e., the last message on top), and the beginning of the discussion is often located in archives, links to which are located at the beginning of the page

Forum section: Investments, business plan, business valuation
In this section you can ask your questions or express your opinion on this term.

Determining the project lifespan
Determination of the forecasting horizon used in calculating project efficiency

Finance for dummies. NPV, IRR, Break-even-point, taxes etc.
A variety of issues related to assessing the effectiveness of investments are discussed, with many references

Evaluation of investment projects in Russia: NPV vs. ROV
An alternative to using NPV when evaluating investment projects

Related sections and other sites

Analysis of investment projects »»
Efficiency, risk, discounting, selection of projects for investment

See also:

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The present value of future cash flows of an investment project, calculated taking into account discounting, minus investments.

Net present value is calculated using the projected cash flows associated with the planned investment using the following formula:

where NCFi is the net cash flow for the i-th period,
Inv - initial investment
r is the discount rate (the cost of capital raised for the investment project).

With a positive NPV value, it is considered that this investment of capital is effective.

The concept of net present value (NPV) is widely used in investment analysis to evaluate various types of investments. The formula presented above is only valid for a simple case of cash flow structure, when all investments occur at the beginning of the project. In more complex cases, the analysis may require a more complex formula to account for the distribution of investments over time. Most often, for this purpose, investments lead to the start of a project similar to income.

In MS Excel, the function =refinery() is used to calculate NPV.

Terms used in the calculator

Investments— placement of capital in order to make a profit. Investments are an integral part of the modern economy. Investments differ from loans in the degree of risk for the investor (lender) - the loan and interest must be repaid within the agreed time frame, regardless of the profitability of the project, investments (invested capital) are returned and generate income only in profitable projects. If the project is unprofitable, the investment may be lost in whole or in part.

Flow, cash free- the cash flow that the company has after financing all the investments that it finds appropriate to make; is defined as operating profit after tax plus depreciation minus investment.

Discount rate— this parameter reflects the rate of change in the value of money in the current economy. It is taken to be equal to either the refinancing rate, or the interest on long-term government bonds considered risk-free, or the interest on bank deposits.

To calculate investment projects, this parameter can be taken equal to the planned profitability of the investment project.

Net present value (NPV) is the balance of all operating and investment cash flows, additionally taking into account the cost of capital used. The NPV of the project will be positive, and the project itself will be effective, if calculations show that the project covers its internal costs and also brings capital owners an income not lower than what they demanded (not lower than the discount rate).

Investment Profitability Index (PI)— The indicator illustrates the ratio of return on capital to the amount of invested capital; the return on investment indicator shows the relative profitability of the project or the discounted value of cash receipts from the project per unit of investment. The profitability index is calculated using the formula: PI = NPV / I, where I is investment.

Internal rate of return (IRR)- the interest rate at which the project is neither profitable nor unprofitable. For projects lasting more than two years, there is no formula for calculating this indicator; it can only be determined by the iteration method (or using a computer program that uses this method, for example, Excel). It can be determined graphically.

IMPORTANT: Not one of the listed investment efficiency indicators is sufficient to accept the project for implementation. At the same time, the ratio and distribution of own and borrowed funds, as well as other factors (presence of preliminary agreements for the sale of project products; cash flow and the ability to repay obligations according to your business plan; payback period and loan repayment period; debt coverage ratio, etc.) are important. .).

Net present value NPV (Net Present Value).

Advantages and disadvantages of use

Net present value (NPVNet Present Value) is one of the most important criteria for investment evaluation of projects.

Formula for Calculating Net Present Value

where: CF t - cash flows; r is the discount rate; CF 0 - initial investment (negative).
Cash flows, which in the formula, as a rule, are formed for the periods under consideration: year, quarter, month. As a result, the cash flow, for example monthly, will be equal to all cash receipts for the month.
CF=CF 1 +CF 2 +… + CF n

Net present value (NPV) allows you to compare different investment projects with each other. A positive NPV value indicates that this investment is effective and attractive. If NPV<0, то доходы от инвестиций не могут покрыть риск по данному проекту. Чем выше значения чистой текущей стоимости, тем инвестиционно привлекательнее проект.

For calculating the discount rate, As a rule, they take a risk-free investment rate, for example, in government securities (GKOs, OFZs), supplement it with compensation for the risk (the risk of the project not being implemented). Also, the discount rate can be determined by the market based on the return on the stock market for a project with the same level of risk.

Advantages and disadvantages of the net present value indicator (NPV)
The advantages of net present value include:

clarity of the indicator for management decisions when choosing an investment object;
the use of a discount rate reflects the property of the value of money to depreciate;
The discount rate may include additional project risks.

Disadvantages of net present value include:

the complexity of calculating the discount rate can distort the results of assessing the NPV indicator.
This is typical for complex projects involving many risks;
difficulty in forecasting cash flows. Although the cash flows of the enterprise are determined, these are only forecast values that may change in the process;
failure to take into account the intangible advantages and values of the enterprise.

Since cash flows can change over time and are probabilistic in nature, simulation modeling is used to set the possible probabilities of receiving a particular cash flow. The probabilities for each cash flow are determined by experts. To solve the shortcomings of net present value (NPV), a mixed approach is used, where intangible capital and future cash flows are assessed by experts or an expert group.

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Future and present values

Future value is a development of the concept of compound interest - this is the amount to which the current deposit will increase over the period from the moment it is placed in the account, subject to the accrual of compound interest.

Future value is the amount to which a current deposit will grow over the period from the moment it is placed in an account on which compound interest is calculated (future value is sometimes called accrued value). For example, a deposit of 10,000 rubles, yielding annually 6% calculated using the compound interest method, at the end of the first year will increase to 10,600 rubles (10,000 * 1.06 = 10,600). If the money had been left for another year, 6% would have been charged on the account balance of 10,600 rubles. Thus, by the end of the second year the account would have 11,236 rubles (10,600 * 1.06 = 11,236). To determine the future value by the end of year n, the considered procedure must be performed n times or 10,000 * (1+ 0.06) n. To simplify the procedure for calculating the future value of any initial investment amount, there are tables of growth factors. A set of such tables is presented in Appendix B.

The future value of the annuity.

Annuity is a flow of equal amounts of cash occurring at regular intervals.

A sum of 10,000 rubles received at the end of each year annually for 10 years is an example of an annuity. Cash flows can be inflows of income received from investing or outflows of funds invested for future income. Investors are sometimes interested in determining the future value of an annuity. As a rule, this applies to the so-called ordinary annuity - one in which regular cash flow occurs at the end of each year. Future value can be determined mathematically using a calculator, computer, or appropriate financial spreadsheets. Here we use tables of accrual factors, or future value factors, for the annuity. A complete set of growth factor tables for annuities is included in Appendix D. The growth factors represent the amount to which regular contributions of 1 monetary unit made at the end of the year would have grown under various combinations of periods and interest rates.

For example, a ruble deposited on a bank deposit that pays 8% at the end of each year, for a period of 6 years, would grow to 7.3359 rubles. If you invest 10,000 rubles at the end of each year for 6 years at 8%, the final future value will be 73,359 rubles (7.3359 * 10,000).

Present value– the flip side of future value. Present value, instead of measuring the value of a current amount at some point in the future, allows us to determine how much a future amount of money is worth today. Using present value techniques, you can calculate the present value of an amount that will be received in the future.

When determining the present value of a future amount of money, the key question is: How much money would have to be deposited today into an account that pays n interest to equal some amount that will be received in the future? The interest rate used to find the present value is usually called the discount rate (or opportunity cost). It represents the annual rate of return that could now be obtained from a similar investment. Basic present value calculations are best illustrated by a simple example. Imagine that you have the opportunity to receive 10,000 rubles exactly one year from today. If you could earn 7% on similar types of investments, what is the highest possible amount of money you would pay for this opportunity? In other words, what is the present value of 10,000 rubles to be received in a year, discounted at a rate of 7%? Let X be the present value. To describe this case, the following equality is used:

X * (1 + 0.07) = 10,000 rubles. Solving the equation for X, we get:

X = 10,000/(1 + 0.07) = 9345.79 rubles.

From these calculations it should be clear that the present value of 10,000 rubles, which will be received in a year and discounted at a rate of 7%, is 9345.79 rubles. In other words, the RUB 9,345.79 placed in an account that pays 7% will increase to RUB 10,000 over the course of the year. To check this conclusion, let us multiply the factor of increase in future value for 7% and one year, or 1.07 by 9345.79 rubles. This amount will bring a future value of 10,000 rubles (1.07 * 9345.79).

Since calculations of the present value of amounts that will be received in the distant future are more complex than for investments for a year, it is recommended to use present value tables in this case. A set of these tables is included in Appendix A. The discount factors in such tables represent the present value of 1 monetary unit calculated for various combinations of periods and discount rates. For example, the present value of 1 ruble, which is expected to be received in a year and discounted at a rate of 7%, is 0.9346 rubles. Based on this factor (0.9346), the present value of 10,000 rubles, which is expected to be received in a year at a 7% discount rate, can be found by multiplying this factor by 10,000 rubles. The resulting present value of 9346 rubles (0.9346 * 10000) corresponds (with the exception of a small difference due to rounding) to the value calculated earlier.

Another example will help you understand how present value tables are used.

The present value of 500 rubles expected to be received in 12 years, discounted at a rate of 5%, can be calculated as follows:

Present value = 0.5568* 500 = 278.4 rubles.

The number 0.5568 represents the discount or discount factor for 12 periods and the discount rate of 5%.

Present value of annuity can be found in the same way using financial tables. A complete set of such present value discount factors for annuities is included in Appendix B. The factors in such tables represent the present value of a 1 currency unit annuity associated with various combinations of years and discount rates. For example, the present value of 1 ruble, which will flow every year for the next five years, discounted at a rate of 9%, will be 3.8897 rubles. If we use this discount factor, then the present value of a 500-ruble annuity for 5 years at a discount rate of 9% can be found by multiplying the annual income by this factor. In this case, the final present value will be 1944.85 rubles (3.8897 * 500).

The concept of present value can be used to select an appropriate investment vehicle. Ignoring risk for the moment, it can be determined that an investor would be satisfied with an investment instrument in which the present value of all future earnings (discounted at the appropriate rate) would be equal to or greater than the present value of the costs of acquiring it. Since investment costs (or acquisition price) arise at the initial stage (at time zero), costs and their present value are considered one and the same. If the present value of revenues equaled costs, the investor would receive a rate of return equal to the discount rate. If the present value of the earnings exceeded the costs incurred, the investor would receive a rate of return on the investment greater than the discount rate. Finally, if the present value of earnings were less than costs, the investor would receive a return on investment that is less than the discount rate. Therefore, an investor would prefer only those investments for which the present value of earnings equals or exceeds costs; in these cases, the yield would match or exceed the discount rate.

Income measurement

In the investment process, the problem arises of comparing income from various instruments, for which it is necessary to apply appropriate meters. One of these meters is income during the period of ownership of the asset. Asset holding period is the period over which a person wishes to measure the income from any investment instrument. When comparing returns from different instruments, using holding periods of the same length makes the analysis more objective.

Earnings in the form of capital gains may not be realized, becoming " paper income. Capital gains are realized only when the investment instrument is actually sold at the end of the holding period of the asset. Realized income – This is the income received by an investor during a certain period of holding an asset. Although capital gains may not be realized during the period over which total return is measured, they must be taken into account when calculating return.

When calculating, you should also take into account that both current income and capital gains can be negative numbers. In addition, you need to keep in mind that capital losses can result from any investment instrument.

One of the key and most used, especially in international practice, methods for assessing the quality of investment projects is the method net present value (NPV), based on the calculation of the possible increase in the value of the company as a result of the implementation of the corresponding investment project. The formula for calculating net present value is

where – cash receipts (cash flow) for the period V, r – the desired rate of profitability (profitability), i.e. the level of return on invested funds that can be ensured when they are placed in publicly accessible financial institutions and instruments. In other words, r – opportunity costs (opportunity cost) of capital raised for investment in the project under consideration; – initial investment of funds, or the amount of initial investment.

In reality, however, an investor may be faced with a situation where the project does not involve one-time capital expenditures, but multiple ones, when investments are carried out over several time increments. In this case, the formula for calculating net present value takes a slightly different form:

where – investment costs for the period t.

Obviously, if the present value of the cash inflow from a project exceeds the present value of the sum of all capital investments, the project in question will have a positive net present value. Positive value NPV for the project means that investment costs generate net cash flows with a return greater than alternative options in the market with the same level of risk, i.e. the project's profitability exceeds the required return of capital owners. In this case, the project can be accepted for implementation, since the costs of it will be reimbursed and, in addition, its implementation will provide some income that increases the value of the company and the welfare of its shareholders.

Obviously, in the case of analyzing several alternative projects, the project with a higher value should be accepted NPV Projects with NPV = 0 do not change the position of capital owners, since the company’s valuation in this case does not change and the share price remains unchanged. But the adoption of such projects increases the company’s assets by the amount of investment, which may be of interest to management (increasing prestige, power, etc.).

Negative value NPV shows that the desired rate of profitability is not achieved and the project is unprofitable; it is usually rejected. Among several alternative projects, the one with the higher value should be accepted NPV

When calculating NPV Discount rates that vary from year to year may be used. If the value of r is not constant and will change from period to period, then it is necessary to apply individual discount factors to each cash flow that will correspond to a given calculation step. In this case NPV It is recommended to calculate using the formula

Where .

At the same time, it is quite possible that a project that is acceptable at a constant discount rate may become unacceptable at a variable one.

It is also important to note that net present value is an additive criterion in the spatiotemporal aspect, i.e. . Consequently, the net present value of a set of projects, for example an entire company, is equal to the sum of the present values of the projects that make it up. This important property makes it possible to use this criterion when analyzing the optimality of an investment portfolio of projects. Besides, in NPV both the lifespan of the project and all income (expenses) at all its stages are taken into account.

When using the method in practice NPV The choice (justification) of the discount rate remains a rather difficult problem.

Since a company may have a large number of shareholders, the discount rate must satisfy the minimum return on capital requirements of most of these individuals. Moreover, in companies with some degree of leverage, the discount rate must represent a return that satisfies all types of investors (shareholders and creditors) of the company. Therefore, for such a company, an acceptable discount rate would be the weighted average cost of capital

where is the price of the company’s source of funds; – the share of the th source in their total amount.

It should be noted that the validity of using this indicator in analytical calculations is associated with some reservations and conventions. In particular, its value is influenced not only by the internal conditions of the company’s activities, but also by the external conditions of the financial market. Thus, when interest rates change, the rate of return required by shareholders on invested capital also changes, which affects the value WACC.

For completeness of presentation of information necessary for calculation NPV Let's look at typical cash flows at an enterprise.

Typical input cash flows:

additional sales volume and increase in product price;
reduction of average gross costs (reduction of production costs);
the residual value of the equipment at the end of the last year of the investment project (since the equipment can be sold or used for another project);
release of working capital at the end of the last year of the investment project (closing accounts receivable, selling remaining inventory, selling shares and bonds of other enterprises).

Typical weekend cash flows:

initial investment in the first year(s) of the investment project;
increase in working capital needs in the first year(s) of the investment project (increase in accounts receivable to attract new customers, purchase of raw materials and components to start production);
equipment repair and maintenance costs;
additional non-production costs (social, environmental, etc.).

Previously, we noted that the resulting net cash flows are designed to ensure the return of the invested amount of funds and obtain the maximum (if possible) income for investors. Let's consider how cash flows are divided into input (output) by estimating using the method NPV specific investment project.

Example. The Multihvat company needs to make a choice between two models of new equipment, which it plans to use to increase its own production volumes in order to enter the world market. Investments in equipment of the type A amount to 30 thousand dollars, for equipment of the type IN - 45 thousand dollars with the same period of operation of 5 years.

Let's calculate (Table 6.3) the net present value for both models of equipment for the discount rate r= 20%.

Table 6.3

Cash flows, dollars
Cash flows, dollars
Fixed Assets

Payments for business activities

Depreciation deductions
Taxable result
Income tax
Net result
Depreciation deductions
Net cash flow
Discount coefficient					1,2-*

The same on a cumulative basis

According to the calculation results presented in table. 6.3, for equipment model A net present value will be

Similar calculations for the equipment of the model IN are presented in table. 6.4.

Table 6.4

Net present value of the modelIN

Cash flows, dollars
Cash flows, dollars
Fixed Assets
Income from economic activities
for economic activities
Cash flow before taxes
Depreciation deductions
Taxable result
Income tax
Net result
Depreciation deductions
Net cash flow
Coefficient discounting
Discounted Cash Flow
The same on a cumulative basis

Net present value for model equipment IN

A comparison of the net present values for both models shows that the model IN – preferable (47,895 > 28,620).

To take inflation into account when assessing the effectiveness of investments, the discount rate (yield) must be adjusted to the inflation rate i in accordance with the conclusions from the famous Fisher effect:

Example. The Default company plans to purchase new equipment at a price of $40,000, which, according to the company's administration, will provide $20,000 in cost savings (in the form of input cash flow over the next three years). During this period, the equipment will be completely worn out. The company's cost of capital is the expected rate of inflation per year.

Let us first evaluate the project without taking into account inflation (Table 6.5).

Now let’s take into account the effect of inflation in the calculation scheme (Table 6.6).

Table 6.5

CalculationNPV excluding inflation

Cash flows, dollars
Cash flows, dollars
Fixed Assets
Annual fund inflow
Net cash flow
Discount coefficient
Discounted Cash Flow
The same on a cumulative basis

Table 6.6

CalculationNPV taking into account inflation

Cash flows, dollars
Cash flows, dollars
Fixed Assets
Annual real inflow of funds
Inflation index
Indexed (nominal) inflow of funds
Net cash flow

Discounted Cash Flow
The same on a cumulative basis

The answers for both options are exactly the same, which is completely natural in the case of the same inflation rate for all components of expenses and income. For this reason, and also taking into account the relatively low level of inflation in developed countries, most companies in Western countries, as a rule, do not take inflation into account when calculating the effectiveness of investment projects.

Criterion NPV has, as follows from the above, both advantages and disadvantages. The obvious advantage of this approach is that this criterion is absolute, and therefore takes into account the scale of investment. This allows you to calculate the increase in the value of the company, which is its main goal. However, advantages also come with disadvantages. The first is that the value NPV difficult, and in some cases impossible, to standardize. For example, NPV of some project is 200 thousand dollars. Is this a lot or a little? It is very difficult to answer this question, especially if a non-alternative project is being considered.

The second disadvantage is due to the fact that NPV does not explicitly show by what investment efforts a particular result is achieved. Although in calculation NPV the size of the investment is taken into account; a relative comparison of investment costs with the results obtained is not carried out. And finally, the third drawback of the criterion NPV is explained by the fact that for the investor (and, of course, not only for him), it is important to have information about the payback period of the investment costs made.

Considering the above, criteria calculated as relative values, in particular such as the profitability index and internal rate of return, are widely used in financial management.

NPV is an abbreviation for the first letters of the phrase “Net Present Value” and stands for net present value (to date). This is a method of evaluating investment projects based on the discounted cash flow methodology. If you want to invest money in a promising business project, then it would be a good idea to first calculate the NPV of this project. The calculation algorithm is as follows:

you need to estimate the cash flows from the project - the initial investment (outflow) of funds and the expected receipts (inflows) of funds in the future;
determine the cost of capital Cost of Capital) for you - this will be the discount rate;
discount all cash flows (inflows and outflows) from the project at the rate that you estimated in step 2);
Fold. The sum of all discounted flows will be equal to the NPV of the project.

If NPV is greater than zero, then the project can be accepted; if NPV is less than zero, then the project should be rejected.

The rationale behind the NPV method is very simple. If the NPV is zero, this means that the cash flows from the project are sufficient to:

recoup invested capital and
provide the necessary income on this capital.

If the NPV is positive, it means that the project will bring profit, and the higher the NPV value, the more profitable the project is for the investor. Since the income of the creditors (who you borrowed money from) is fixed, all income above this level belongs to the shareholders. If the company approves a project with zero NPV, the shareholders' position will remain unchanged - the company will become larger, but the share price will not increase. However, if the project has a positive NPV, the shareholders will become richer.

NPV calculation. Example

The formula for calculating NPV looks complicated to a person who does not consider himself a mathematician:

Where

n, t — number of time periods;
CF - cash flow Cash Flow);
R is the cost of capital, also known as the discount rate. Rate).

In fact, this formula is just a correct mathematical representation of the summation of several quantities. To calculate NPV, let's take two projects as an example A And B, which have the following cash flow structure for the next 4 years:

Table 1. Cash flow of projects A and B.

Year	Project A	Project B
0	($10,000)	($10,000)
1	$5,000	$1,000
2	$4,000	$3,000
3	$3,000	$4,000
4	$1,000	$6,000

Both projects A And B have the same initial investment of $10,000, but the cash flows in subsequent years are very different. Project A assumes a faster return on investment, but by the fourth year the cash flow from the project will drop significantly. Project B, on the contrary, in the first two years shows lower cash inflows than the proceeds from the Project A, but in the next two years the Project B will bring more money than the project A. Let's calculate the NPV of the investment project.

To simplify the calculation, let's assume:

all cash flows occur at the end of each year;
the initial cash outflow (investment of money) occurred at time “zero”, i.e. Now;
The cost of capital (discount rate) is 10%.

Let us recall that in order to bring the cash flow to today, you need to multiply the amount of money by the coefficient 1/(1+R), while (1+R) must be raised to a power equal to the number of years. The value of this fraction is called the factor or discount factor. In order not to calculate this factor every time, you can look it up in a special table called the “discount factor table.”

Let's apply the NPV formula for the Project A. We have four annual periods and five cash flows. The first flow ($10,000) is our investment at time zero, that is, today. If we expand the NPV formula given just above, we get a sum of five terms:

If we substitute the data from the table for the Project into this amount A instead of CF and a rate of 10% instead R, then we get the following expression:

What is in the divisor can be calculated, but it is easier to take the ready-made value from the table of discount factors and multiply these factors by the amount of cash flow. As a result, the present value of cash flows for the project A equal to $788.2. NPV calculation for a project A can also be presented in table form and as a time scale:

Year	Project A	Rate 10%	Factor	Sum
0	($10,000)	1	1	($10,000)
1	$5,000	1 / (1.10) 1	0.9091	$4,545.5
2	$4,000	1 / (1.10) 2	0.8264	$3,305.8
3	$3,000	1 / (1.10) 3	0.7513	$2,253.9
4	$1,000	1 / (1.10) 4	0.6830	$683.0
TOTAL:	$3,000			$788.2

Figure 1. NPV calculation for project A.

Let's calculate NPV for the project in a similar way B.

Because discount factors decrease over time, the contribution to the present value of the project from large ($4,000 and $6,000) but distant (years 3 and 4) cash flows will be less than the contribution from cash flows in the early years of the project. Therefore, it is expected that for the project B the net present value of the cash flows will be less than for the project A. Our NPV calculations for the project B gave the result - $491.5. Detailed NPV calculation for the project B shown below.

Table 2. NPV calculation for project A.

Year	Project B	Rate 10%	Factor	Sum
0	($10,000)	1	1	($10,000)
1	$1,000	1 / (1.10) 1	0.9091	$909.1
2	$3,000	1 / (1.10) 2	0.8264	$2,479.2
3	$4,000	1 / (1.10) 3	0.7513	$3,005.2
4	$6,000	1 / (1.10) 4	0.6830	$4,098.0
TOTAL:	$4,000			$491.5

Figure 2. NPV calculation for project B.

Conclusion

Both of these projects can be accepted, since the NPV of both projects is greater than zero, which means the implementation of these projects will lead to an increase in the income of the investor company. If these projects are mutually exclusive and you need to choose only one of them, then the project looks preferable A, since its NPV=$788.2, which is greater than the NPV=$491.5 of the project B.

Subtleties of NPV calculation

Applying a mathematical formula is not difficult if all the variables are known. Once you have all the numbers - cash flows and cost of capital - you can easily plug them into the formula and calculate NPV. But in practice it is not so simple. Real life differs from pure mathematics in that it is impossible to accurately determine the magnitude of the variables that enter into this formula. As a matter of fact, this is why in practice there are many more examples of unsuccessful investment decisions than successful ones.

Cash flows

The most important and most difficult step in analyzing investment projects is assessing all the cash flows associated with the project. Firstly, this is the amount of the initial investment (outflow of funds) today. Secondly, these are the amounts of annual cash inflows and outflows that are expected in subsequent periods.

Making an accurate forecast of all the costs and revenues associated with a large, complex project is incredibly difficult. For example, if an investment project is associated with the launch of a new product on the market, then to calculate NPV it will be necessary to make a forecast of future sales of the product in units and estimate the sales price per unit of product. These forecasts are based on an assessment of the general state of the economy, the elasticity of demand (the dependence of the level of demand on the price of a product), the potential effect of advertising, consumer preferences, and the reaction of competitors to the launch of a new product.

In addition, it will be necessary to make a forecast of operating expenses (payments), and for this to evaluate future prices for raw materials, employee salaries, utilities, changes in rental rates, trends in changes in exchange rates, if some raw materials can only be purchased abroad, etc. Further. And all these assessments need to be made several years in advance.

Discount rate

The discount rate in the NPV calculation formula is the cost of capital for the investor. In other words, this is the interest rate at which the investing company can attract financial resources. In general, a company can obtain financing from three sources:

borrow (usually from a bank);
sell your shares;
use internal resources (for example, retained earnings).

The financial resources that can be obtained from these three sources have their own costs. And she is different! The most clear is the cost of debt obligations. This is either the interest on long-term loans that banks require, or the interest on long-term bonds if the company can issue its debt instruments in the financial market. It is more difficult to estimate the cost of financing from the other two sources. Financiers have long developed several models for such an assessment, among them the well-known CAPM(Capital Asset Pricing Model). But there are other approaches.

The company's cost of capital (and therefore the discount rate in the NPV formula) will be the weighted average of the interest rates from these three sources. In English financial literature this is referred to as WACC(Weighted Average Cost of Capital), which translates as the weighted average cost of capital.

Dependence of the NPV of the project on the discount rate

It is clear that obtaining absolutely accurate values of all cash flows of the project and accurately determining the cost of capital, i.e. discount rate is not possible. In this regard, it is interesting to analyze the dependence of NPV on these values. It will be different for each project. The most frequently performed analysis is the sensitivity of the NPV indicator to the cost of capital. Let's calculate NPV for projects A And B for different discount rates:

Cost of capital, %	NPV A	NPV B
0	$3,000	$4,000
2	$2,497.4	$3,176.3
4	$2,027.7	$2,420.0
6	$1,587.9	$1,724.4
8	$1,175.5	$1,083.5
10	$788.2	$491.5
12	$423.9	($55.3)
14	$80.8	($562.0)
16	($242.7)	($1,032.1)
18	($548.3)	($1,468.7)

Table 3. Dependence of NPV on the discount rate.

The tabular form is inferior to the graphical form in terms of information content, so it is much more interesting to look at the results on the graph (click to enlarge the image):

Figure 3. Dependence of NPV on the discount rate.

The graph shows that the NPV of the project A exceeds the NPV of the project B at a discount rate of more than 7% (more precisely 7.2%). This means that an error in estimating the cost of capital for the investing company could lead to an erroneous decision regarding which of the two projects to choose.

In addition, the graph also shows that Project B is more sensitive to the discount rate. That is, the NPV of the project B decreases faster as this rate increases. And this is easy to explain. In project B Cash receipts in the first years of the project are small, but they increase over time. But discount rates for longer periods of time decrease very significantly. Therefore, the contribution of large cash flows to the net present value also drops sharply.

For example, you can calculate what $10,000 will be equal to in 1 year, 4 years and 10 years at discount rates of 5% and 10%, you can clearly see how much the present value of a cash flow depends on the time of its occurrence.

Table 4. Dependence of NPV on the time of its occurrence.

Year	Rate 5%	Rate 10%	Difference, $	Difference, %
1	$9,524	$9,091	$433	4.5%
4	$8,227	$6,830	$1,397	17.0%
10	$6,139	$3,855	$2,284	37.2%

The last column of the table shows that the same cash flow ($10,000) at different discount rates differs after a year by only 4.5%. Whereas the same cash flow, only 10 years from today at a discount rate of 10%, will be 37.2% less than its present value at a discount rate of 5%. The high cost of capital “eats up” a significant part of the income from an investment project in distant annual periods, and nothing can be done about it.

That is why, when evaluating investment projects, cash flows that are more than 10 years distant from today are usually not used. In addition to the significant impact of discounting, the accuracy of estimating distant cash flows is significantly lower.