Definitions of the real (current) value of money

In financial calculations, there is a need to compare different amounts of money at different points in time. That is why quite often there is a need to determine the real (current) value (Present Value - PV) money, which serves as the basis for comparing the profitability of various projects and investments over a certain period.

Present value is the monetary value of future receipts or earnings adjusted for discount rates (capitalization).

From a formal point of view, the discount rate - it is the interest rate used to reduce future earnings (cash flows and profits) to present value. The discount rate is expressed as a percentage or fraction of a unit. The upper level of the discount rate can theoretically be greater than 100% (greater than 1), and the lower level is determined by economic factors. From an economic point of view, the discount rate- a measure of the costs of attracting capital to invest in a specific investment project.

In other words, the discount rate is the rate of return expected by investors on invested capital in the presence of alternative opportunities for investing it in investment objects with a similar level of risk. In connection with this, the lower level of the discount rate is the so-called "risk-free" rate. Essentially, it is the rate of interest at which investors would lend money if there was no danger of paying it back, or at which they would borrow money if their collateral were so strong that lenders would consider the chances of non-payment to be negligible.

In developed market economies, the "risk-free" rate is the interest on US government-guaranteed securities or the current yield on Treasury bills and bonds. In some large projects that involve financing from both domestic and foreign capital, the level of “risk-free” rate is taken at the LIBOR rate (the interest rate at which banks offer deposits to each other on the European foreign exchange market). For the conditions of Ukraine, the question of establishing the level of the “risk-free” rate cannot be determined unambiguously. One of the main reasons for this is the lack of an established capital market within the country.

To calculate the current value, you must determine a discount rate that takes into account the riskiness of a particular project or investment. There is a simple rule:

risk means a high discount rate (capitalization), low risk means a low discount rate.

In general, the following are used to estimate discount rates: principles:

of two future receipts, the higher discount rate will be the one that arrives later;

the lower a certain level of risk, the lower the discount rate should be; if general interest rates in the market rise, discount rates also rise; the risk may decrease if there is a prospect of business growth, a decrease in other...

Calculation of the present value of money is carried out using the discounting process, which is the opposite of compounding.

Discounting - this is finding the initial or current amount of debt (PV) according to a known final amount (FV)% which must be given back after some time (P). That is, discounting is the process of raising economic indicators of different years to a form comparable over time.

They say: the amount FV is discounted, and the difference FV - PV is called a discount and is denoted D. Discount is interest money (interest) accrued and collected in advance.

In market conditions, the problem of discounting arises very often when developing the terms of contracts between two enterprises, various business entities, and when determining the current market value of bills, shares, bonds and other securities.

The practical application of discounting to determine the present value of cash flows requires an appropriate financial and mathematical formalization of the discounting model - determination of the absolute value of the discount. Depending on the needs of analyzing cash flows and changes in their value over time, the following can be used: discounting models: simple discounting of annuities (deferred or advance annuity) - will be discussed in detail in paragraph 4.4.

simple discounting(single discounting) understand the financial and mathematical model for calculating the present value of a future cash flow, the receipt of which is expected to take place once over a clearly defined period. The result of simple discounting is the present value (PV) of an individual future cash flow.

Compounding and discounting processes closely interconnected together. Determining the current value (discounting) is the direct opposite of compounding, that is, these quantities are characterized by an inverse relationship:

Thus, if we know the indicator of the future value of money (RU), then using discounting we can calculate their present value (RU).

Discounting is carried out using discount factor (discount factor, a1).

Let's determine the discount rate d, like the following relation:

The real value of money can be determined based on a simple or complex interest calculation scheme.

Using relation (4.15) and taking into account the relationship between the compounding and discounting functions, we present a formula for determining the current value of money in the case of using a discount rate for the simple interest scheme:

Where RU- present value of future cash flow; RU- absolute value of future cash flow; P- number of intervals in the planning period; r - discount rate (expressed as a decimal fraction); Kao - discount factor when applying simple interest (expressed as a decimal fraction).

Example. What amount must the investor put into the deposit account in order to receive 25,000 games at the end of the fourth year, if the interest is set at 16% and they are calculated according to a simple scheme?

When taking inflation into account, as in the case of determining future value, the result is adjusted by taking into account its forecast level (IPR):

where / is the forecast inflation rate;

Discounting using compound interest is a fairly common way of determining the current value of money, which is used not only in financial management, but also in investment design and in determining the value of a business.

The problem of determining present value according to the compound interest scheme solve using formula 4.19:

where ---- is the discount factor. Economic discount factor

is that its value corresponds to the current value of one monetary unit, which will be received at the end of period n at compound interest r. Its value depends on the duration of the entire period and the required discount rate.

Example. Let’s say that someone would like to have 1000 games in 4 years, missing the amount of payment for a child’s education at a prestigious university. If the average deposit rate is 15%, how much should he deposit into the bank?

You can determine the present value of future cash flow using a financial spreadsheet (Appendix B), which contains the absolute value of the discount rate, based on the level of the interest rate and the number of intervals in the planning period. A present value table saves a lot of effort in calculating its various factors. This table, for example, shows that the cost decreases when the time period increases and also when the discount rate increases.

Appendix B Only those factor values ​​are given that, if multiplied by the future value, give the present value. Accordingly, taking into account the data from the financial table with Appendix B the present value is calculated using formula 4.20:

Where PVIF- current value factor (multiplier), the standard values ​​of which are given in the table of current value factor values (Appendix B).

Example. Let's say you want to determine the present value of $1,000 years from now; you are hoping for a 10% annual risk level associated with project implementation.

As can be seen from Appendix A, the value of factors increases with time and the growth of compound interest. Therefore, if these factors are substituted into the denominator of the last equation, the present value of $1000 in 3 years will be:

$ 1000/(1+0,10)* = $ 751

How did this cost come about? By simply multiplying (1.10 x 1.10 x 1.10= 1.33) and using this factor for discounting: $1000/1.33 = $751

In the last example, where the task was to determine the present value of $1000 in 3 years, it was enough to look at the number of years and the corresponding Present Value Interest Factor (PVIF) according to the applied discount rate. As shown in Appendix B, this factor is 0.751. To get the present value of $1000 in 3 years, with a 10% discount, multiply the factor values ​​by the sum of the present value ($1000*(0751 = $751). You got the same result by doing long calculations.

If Interest is scheduled to accrue more than once a year, then the calculation is carried out according to formula 4.21:

Where T- number of accruals per year, units.

interest accrued continuously the cost of funds is determined by formula 4. 22:

Present value is the amount of money that, if invested in the current year at a given interest rate, G will grow // years in the future to the required or desired level. Present value is the only correct way to convert future payment streams into today's money.

Obviously, if you have two different projects with the same implementation period and costs, but different risk factors, then you can determine their real cost and compare which one is more appropriate to choose. Assessing the feasibility of investing in certain projects or investments is based on the concept of present value. It all comes down to discounting future returns based on the level of risk and uncertainty of the future. The present value method allows you to do this.

Example. Let's say the company hopes to receive the following amounts of money over the next four years: 1st year - 1000 thousand games; 2nd year -1200 thousand UAH; 3rd year -1500 thousand UAH; 4th year - 900 thousand UAH.

The present value of the total cash flow is the simple sum of the value of the cash flows for each year. If the discount rate is 10%, the current value of cash flows for 4 years is equal to UAH 3642.43 thousand:

The present value of cash flow for 4 years is 3642.43 thousand UAH.

Present Value Table (Appendix B) obviously saves financiers a lot of time. Please note that when the discount rate decreases, the value of the present value increases; when rates rise, the value falls. Therefore, it should be clear that the concept of present value is an important factor in making investment decisions and investments.

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:

1. 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;
2. 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

1. 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).
2. 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.

where PV is the current value of money,

FV – future value of money,

n – number of time intervals,

i – discount rate.

Example. What amount must be deposited into the account in order to receive 1000 rubles in five years? (i=10%)

PV = 1000 / (1+0.1)^5 = 620.92 rub.

Thus, to calculate the current value of money, we must divide its known future value by the value (1+i) n. The present value is inversely related to the discount rate. For example, the present value of a monetary unit received in 1 year at an 8% interest rate is

PV = 1/(1+0.08) 1 = 0.93,

And at a rate of 10%

PV = 1/(1+0.1) 1 = 0.91.

The current value of money is also inversely related to the number of time periods before it is received.

The considered procedure for discounting cash flows can be used when making investment decisions. The most common decision rule is the net present value (NPV) rule. Its essence is that participation in an investment project is advisable if the present value of future cash receipts from its implementation exceeds the initial investment.

Example. It is possible to buy a savings bond with a nominal value of 1000 rubles. and a repayment period of 5 years for 750 rubles. Another alternative investment option is to place your money in a bank account with an interest rate of 8% per annum. It is necessary to evaluate the feasibility of investing in the purchase of a bond.

To calculate NPV as an interest rate, or more generally as a rate of return, it is necessary to use the opportunity cost of capital. The opportunity cost of capital is the rate of return that can be obtained from other investment avenues. In our example, an alternative type of investment is to place money on a deposit with a yield of 8%.

A savings bond provides cash receipts of RUB 1,000. after 5 years. The present value of this money is

PV = 1000/1.08^5 = 680.58 rub.

Thus, the current value of the bond is 680.58 rubles, while they offer to buy it for 750 rubles. The net present value of the investment will be 680.58-750=-69.42, and investing in the purchase of a bond is not advisable.

The economic meaning of the NPV indicator is that it determines the change in the financial condition of the investor as a result of the implementation of the project. In this example, if the bond is purchased, the investor’s wealth will decrease by 69.42 rubles.

The NPV indicator can also be used to evaluate various options for borrowing funds. For example, you need to borrow $5,000. to purchase a car. The bank offers you a loan at 12% per annum. Your friend can borrow $5,000 if you give him $9,000. After 4 years. It is necessary to determine the optimal borrowing option. Let's calculate the current value of $9,000.

PV = 9000/(1+0.12)^4 = $5719.66

Thus, the NPV of this project is 5000-5719.66= -719.66 dollars. In this case, the best borrowing option is a bank loan.

To calculate the effectiveness of investment projects, you can also use the internal rate of return (IRR). The internal rate of return is the value of the discount rate that equalizes the present value of future revenues and the present value of costs. In other words, IRR is equal to the interest rate at which NPV = 0.

In the above example of purchasing a bond, the IRR is calculated from the following equation

750 = 1000/(1+IRR)^5

IRR = 5.92%. Thus, the yield on the bond at maturity is 5.92% per year, which is significantly less than the yield on a bank deposit.

As we have already found out, today's money is more expensive than future money. If we are offered to purchase a zero-coupon bond, and in a year they promise to redeem this security and pay 1000 rubles, then we need to calculate the price of this bond at which we would agree to buy it. In fact, for us the task comes down to determining the current value of 1000 rubles that we will receive in a year.

Present value is the flip side of future value.

Present value is the discounted value of future cash flow. It can be derived from the formula for determining future value:

where RU is the current value; V- future payments; G - discount rate; discount coefficient; P - number of years.

In the example above, we can calculate the price of the bond using this formula. To do this, you need to know the discount rate. The discount rate is taken as the yield that can be obtained in the financial market by investing money in any financial instrument with a similar level of risk (bank deposit, bill, etc.). If we have the opportunity to place funds in a bank that pays 15% per year, then the price of the bond offered to us

Thus, having purchased this bond for 869 rubles. and having received 1000 rubles in a year when it is repaid, we will earn 15%.

Let's consider an example where an investor needs to calculate the initial deposit amount. If after four years an investor wants to receive an amount of 15,000 rubles from the bank. If market interest rates are 12% per annum, what amount should he place in a bank deposit? So,

To calculate present value, it is advisable to use discount tables showing the current value of a monetary unit that is expected to be received in a few years. A table of discount factors showing the present value of a monetary unit is presented in Appendix 2. A fragment of this table is given below (Table 4.4).

Table 4.4. The present value of a monetary unit that will be received after and years

	Annual interest rate

For example, you want to determine the present value of $500 expected to be received in seven years at a discount rate of 6%. In table 4.4 at the intersection of the row (7 years) and column (6%) we find a discount factor of 0.665. In this case, the present value of $500 is equal to 500 0.6651 = $332.5.

If interest is paid more often than once a year, then the formula for calculating the present value is modified in the same way as we did with the calculations of the future value. When interest is accrued multiple times during the year, the formula for determining the current value has the form

In the considered example with a four-year deposit, we assume that interest on the deposit is accrued quarterly. In this case, in order to receive $15,000 in four years, the investor must deposit the amount

Thus, the more often interest is accrued, the lower the current value for a given final result, i.e. The relationship between compounding frequency and present value is the opposite of that for future value.

In practice, financial managers are constantly faced with the problem of choosing options when it is necessary to compare cash flows at different times.

For example, there are two options for financing the construction of a new facility. The total construction period is four years, the estimated cost of construction is 10 million rubles. Two organizations are participating in the competition for the contract, offering the following terms of payment for work by year (Table 4.5).

Table 4.5. Estimated cost of construction, million rubles.

	Organization A	Organization IN

The estimated cost of construction is the same. However, the costs over the time of their implementation are distributed unevenly. Organization A the main amount of costs (40%) is incurred at the end of construction, and the organization IN - in the initial period. Of course, it is more profitable for the customer to attribute payment costs to the end of the period, since funds depreciate over time.

In order to compare cash flows at different times, it is necessary to find their value reduced to the current point in time and sum up the resulting values.

Present value of a stream of payments (RU) calculated by the formula

where is cash flow per year; t - serial number of the year; G - discount rate.

If in the example under consideration r = 15%, then the results of calculating the present values ​​for the two options look as follows (Table 4.6).

Table 4.6.

Based on the present value criterion, the financing option proposed by the organization A, turned out to be cheaper than the organization's offer IN. In these conditions, the customer will certainly prefer to outsource the contract to the organization A (other things being equal).

Net present value is the sum of the current values ​​of all predicted cash flows, taking into account the discount rate.

The net present value (NPV) method is as follows.
1. The current cost of costs (Io) is determined, i.e. The question of how much investment needs to be reserved for the project is decided.
2. The current value of future cash receipts from the project is calculated, for which the income for each year CF (cash flow) is reduced to the current date.

The calculation results show how much money would need to be invested now to receive the planned income if the income rate were equal to the barrier rate (for an investor, the interest rate in a bank, in a mutual fund, etc., for an enterprise, the price of total capital or through risks). Summing up the current value of income for all years, we obtain the total current value of income from the project (PV):

3. The present value of investment costs (Io) is compared with the present value of income (PV). The difference between them is the net present value of income (NPV):

NPV shows the investor's net gains or net losses from investing money in a project compared to keeping the money in a bank. If NPV > 0, then we can assume that the investment will increase the wealth of the enterprise and the investment should be made. At NPV

Net present value (NPV) is one of the main indicators used in investment analysis, but it has several disadvantages and cannot be the only means of evaluating an investment. NPV measures the absolute value of the return on an investment, and it is likely that the larger the investment, the greater the net present value. Hence, it is not possible to compare multiple investments of different sizes using this indicator. In addition, NPV does not determine the period over which the investment will pay off.

If capital investments associated with the upcoming implementation of the project are carried out in several stages (intervals), then the NPV indicator is calculated using the following formula:

, Where

CFt - cash inflow in period t;

r - barrier rate (discount rate);
n is the total number of periods (intervals, steps) t = 1, 2, ..., n (or the duration of the investment).

Typically for CFt the t value ranges from 1 to n; in the case where CФо > 0 is considered a costly investment (example: funds allocated for an environmental program).

Defined by: as the sum of the current values ​​of all predicted, taking into account the barrier rate (discount rate), cash flows.

Characterizes: investment efficiency in absolute values, in current value.

Synonyms: net present effect, net present value, Net Present Value.

Acronym: NPV

Flaws: does not take into account the size of the investment, the level of reinvestment is not taken into account.

Eligibility Criteria: NPV >= 0 (the more the better)

Comparison conditions: To correctly compare two investments, they must have the same investment costs.

Example No. 1. Calculation of net present value.
The investment amount is $115,000.
Investment income in the first year: $32,000;
in the second year: $41,000;
in the third year: $43,750;
in the fourth year: $38,250.
The size of the barrier rate is 9.2%

Let's recalculate cash flows in the form of current values:
PV 1 = 32000 / (1 + 0.092) = $29304.03
PV 2 = 41000 / (1 + 0.092) 2 = $34382.59
PV 3 = 43750 / (1 + 0.092) 3 = $33597.75
PV 4 = 38250 / (1 + 0.092) 4 = $26899.29

NPV = (29304.03 + 34382.59 + 33597.75 + 26899.29) - 115000 = $9183.66

Answer: The net present value is $9,183.66.

The formula for calculating the NPV (net present value) indicator taking into account the variable barrier rate:

NPV - net present value;
CFt - inflow (or outflow) of funds in period t;
It is the amount of investments (costs) in the t-th period;
ri - barrier rate (discount rate), fractions of a unit (in practical calculations, instead of (1+r) t, (1+r 0)*(1+r 1)*...*(1+r t) is used, because . the barrier rate can vary greatly due to inflation and other components);

N is the total number of periods (intervals, steps) t = 1, 2, ..., n (usually the zero period implies the costs incurred to implement the investment and the number of periods does not increase).

Example No. 2. NPV with a variable barrier rate.
Investment size - $12800.

in the second year: $5185;
in the third year: $6270.

10.7% in the second year;
9.5% in the third year.
Determine the net present value for the investment project.

n =3.
Let's recalculate cash flows in the form of current values:
PV 1 = 7360 / (1 + 0.114) = $6066.82
PV 2 = 5185 / (1 + 0.114)/(1 + 0.107) = $4204.52
PV 3 = 6270 / (1 + 0.114)/(1 + 0.107)/(1 + 0.095) = $4643.23

NPV = (6066.82 + 4204.52 + 4643.23) - 12800 = $2654.57

Answer: The net present value is $2,654.57.

The rule according to which, from two projects with the same costs, the project with a large NPV is selected does not always apply. A project with a lower NPV but a short payback period may be more profitable than a project with a higher NPV.

Example No. 3. Comparison of two projects.
The cost of investment for both projects is 100 rubles.
The first project generates a profit equal to 130 rubles at the end of 1 year, and the second 140 rubles after 5 years.
For simplicity of calculations, we assume that barrier rates are equal to zero.
NPV 1 = 130 - 100 = 30 rub.
NPV 2 = 140 - 100 = 40 rub.

But at the same time, the annual profitability calculated using the IRR model will be equal to 30% for the first project, and 6.970% for the second. It is clear that the first investment project will be accepted, despite the lower NPV.

To more accurately determine the net present value of cash flows, the modified net present value (MNPV) indicator is used.

Example No. 4. Sensitivity analysis.
The investment amount is $12,800.
First year investment income: $7,360;
in the second year: $5185;
in the third year: $6270.
The size of the barrier rate is 11.4% in the first year;
10.7% in the second year;
9.5% in the third year.
Calculate how the net present value would be affected by a 30% increase in investment income?

The initial value of the net present value was calculated in example No. 2 and is equal to NPV ex = 2654.57.

Let's recalculate cash flows in the form of current values, taking into account sensitivity analysis data:
PV 1 ah = (1 + 0.3) * 7360 / (1 + 0.114) = 1.3 * 6066.82 = $7886.866
PV 2 ah = (1 + 0.3) * 5185 / (1 + 0.114)/(1 + 0.107) = 1.3 * 4204.52 = $5465.876
PV 3 ah = (1 + 0.3) * 6270 / (1 + 0.114)/(1 + 0.107)/(1 + 0.095) = 1.3 * 4643.23 = $6036.199

Let's determine the change in net present value: (NPV ach - NPV out) / NPV out * 100% =
= (6036,199 - 2654,57) / 2654,57 * 100% = 127,39%.
Answer. A 30% increase in investment income resulted in a 127.39% increase in net present value.

Note. Discounting cash flows with a time-varying barrier rate (discount rate) corresponds to “Methodological guidelines No. VK 477...” clause 6.11 (p. 140).

Net present value

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