where PV is the present value of money,
FV is the future value of money,
n is the number of time intervals,
i - discount rate.
Example. How much money 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 rubles
Thus, to calculate the current value of money, we must divide their known future value by (1 + i) n . The present value is inversely related to the discount rate. For example, the present value of a currency received in 1 year at an interest rate of 8% is
PV \u003d 1 / (1 + 0.08) 1 \u003d 0.93,
And at a rate of 10%
PV \u003d 1 / (1 + 0.1) 1 \u003d 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 in making investment decisions. The most common decision rule is the net present value (NPV) rule. Its essence lies in the fact 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 face value of 1000 rubles. and a maturity of 5 years for 750 rubles. Another alternative investment option is to deposit 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 bonds.
To calculate NPV as an interest rate, or more broadly as a rate of return, the opportunity cost of capital must be used. The opportunity cost of capital is the rate of return that can be obtained from other avenues of investment. In our example, an alternative type of investment is to place money on a deposit with a yield of 8%.
Savings bond provides cash receipts in the amount of 1000 rubles. after 5 years. The present value of this money is
PV = 1000/1.08^5 = 680.58 rubles
Thus, the current value of the bond is 680.58 rubles, while the offer to buy it is 750 rubles. The net present value of the investment will be 680.58-750=-69.42, and it is not advisable to invest in the purchase of a bond.
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 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 money. 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. in 4 years. It is necessary to determine the optimal borrowing option. Calculate the current value of 9000 dollars.
PV = 9000/(1+0.12)^4 = $5719.66
Thus, the NPV of this project is 5000-5719.66= -719.66 USD. 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 discount rate that equalizes the present value of future receipts and the present value of costs. In other words, IRR is equal to the interest rate at which NPV = 0.
In the considered example of purchasing a bond, IRR is calculated from the following equation
750 = 1000/(1+IRR)^5
IRR = 5.92%. Thus, the yield on the bond at its redemption is 5.92% per year, which is significantly less than the yield on a bank deposit.
net present value (NPV, net present value, net present value, NPV, EnglishNet present value , accepted in international practice for the analysis of investment projects abbreviation - NPV) is the sum of the discounted values of the payment stream, reduced to today.
The net present value method has been widely used in capital investment budgeting and investment decision making. Also, NPV is considered the best selection criterion for making or rejecting a decision to implement an investment project, since it is based on the concept of the time value of money. In other words, the net present value reflects the expected change in the investor's wealth as a result of the project.
NPV formula
The net present value of a project is the sum of the present value of all cash flows (both incoming and outgoing). The calculation formula is as follows:
- CF t– expected net cash flow (difference between incoming and outgoing cash flow) for the period t,
- r- discount rate,
- N- the duration of the project.
Discount rate
It is important to understand that when choosing a discount rate, not only the concept of the time value of money must be taken into account, but also the risk of uncertainty in the expected cash flows! For this reason, it is recommended to use the weighted average cost of capital as the discount rate ( English Weighted Average Cost of Capital, WACC) involved in the implementation of the project. In other words, WACC is the required rate of return on capital invested in a project. Therefore, the higher the risk of cash flow uncertainty, the higher the discount rate, and vice versa.
Project Selection Criteria
The decision rule for selecting projects using the NPV method is quite straightforward. A threshold value of zero indicates that the project's cash flows can cover the cost of capital raised. Thus, the selection criteria can be formulated as follows:
- A single independent project must be accepted if the net present value is positive or rejected if it is negative. Zero value is the point of indifference for the investor.
- If an investor is considering several independent projects, those with a positive NPV should be accepted.
- If a number of mutually exclusive projects are being considered, the one with the highest net present value should be selected.
As we have already found out, today's money is more expensive than the future. 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 is to determine the current value of 1000 rubles, which we will receive in a year.
Present value is the flip side of future value.
The present value is the present value of the future cash flow. It can be derived from the formula for determining the 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 a bond using this formula. To do this, you need to know the discount rate. As a discount rate, they take 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, by purchasing this bond for 869 rubles. and having received 1000 rubles in a year when it is repaid, we will earn 15%.
Consider an example where an investor needs to calculate the initial deposit amount. If in four years the investor wants to receive the amount of 15,000 rubles from the bank. at market interest rates of 12% per annum, how much should he place in a bank deposit? So,
To calculate the present value, it is advisable to use discount tables showing the current value of the monetary unit, which is expected to be received in a few years. The table of discount coefficients showing the present value of the 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 the monetary unit, which will be received in and years
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Annual interest rate |
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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 the discount factor 0.665. In this case, the present value of $500 is 500 0.6651 = $332.5.
If interest is paid more 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. With multiple interest accruals during the year, the formula for determining the present value has the form
In the above example with a four-year deposit, let's assume that interest on the deposit is calculated quarterly. In this case, in order to receive $15,000 in four years, the investor must deposit an amount
Thus, the more often interest is calculated, the lower the current value for a given end result, i.e. the relationship between interest rate and present value is inverse to 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 tender for a contract, offering the following terms of payment for work by year (Table 4.5).
Table 4.5. Estimated cost of construction, million rubles
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Organization BUT |
Organization IN |
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The estimated cost of construction is the same. However, the costs of their implementation are distributed unevenly. Organization BUT the main amount of costs (40%) is carried out at the end of construction, and the organization IN - in the initial period. Of course, it is more profitable for the customer to attribute the payment costs to the end of the period, since over time the funds depreciate.
In order to compare multi-temporal cash flows, it is necessary to find their value reduced to the current point in time and sum the obtained values.
Present value of the payment stream (RU) calculated by the formula
where is the cash flow per year; t - serial number of the year; G - discount rate.
If in the example under consideration r \u003d 15%, then the results of calculating the reduced costs for the two options are as follows (Table 4.6).
Table 4.6.
According to the present value criterion, the financing option proposed by the organization BUT, turned out to be cheaper than the offer of the organization IN. The customer in these conditions will certainly prefer to give the contract to the organization BUT (ceteris paribus).
The Time Value of Money (TVM) is an important metric in the accounting and finance industry. The idea is that a ruble today is worth less than the same ruble tomorrow. The difference between these two financial values is the profit that can be made from one ruble or loss. For example, this profit can be received from interest accrued on a bank account or as dividends from investments. But there may also be a loss when paying interest on repaying a loan debt.
An example of calculating the current present value of an investment in Excel
Excel offers several financial functions for calculating the time value of money. For example, the PV (Present Value) function returns the present value of an investment. In simple terms, this function reduces the amount by the discount percentage and returns the fair value for that amount. If the investment project assumes to bring a profit of 10,000 in a year. Question: what is the maximum amount of rational risk to invest in this project?
For example, in Russia, the retail business sometimes makes a profit of up to 35% per annum, and the wholesale business does not exceed 15%. Given the small amount of investment, it is assumed that the investment object is not a wholesale business, which means that a profit of more than 15% per annum should be expected. The figure below shows an example of the formula for the percentage return on investment calculator:
As we see in the figure, the calculator displays to us, in order to get the amount of 10,000 for 1 year with a yield of 25%, we need to invest 8,000 financial resources. That is, if we had an amount of 8,000 and we invested it at 25% per annum, in a year we would have earned 10,000.
The PS function has 5 arguments:
- Rate - percentage discount rate. This is the percentage return that can be expected over the discount period. This value has the greatest impact on the calculation of the present value of the investment, but is the most difficult to accurately determine. Cautious investors most often underestimate the interest rate to the maximum realistically achievable level under certain conditions. If the funds are intended to repay the loan, then this argument is easily determined.
- Number of periods(Nper) - the period of time during which the future amount is discounted. In this example, 1 year is specified (recorded in cell B2). The interest rate and the number of years must be expressed in the appropriate units of measure. This means that you are using an annual rate, then the numeric value in this argument is the number of years. If the interest rate in the first argument is for months (for example, 2.5% monthly), then the number in the second argument is the number of months.
- Payment (Pmt) - the amount that is periodically paid during the discount period. If there is only one payment in the investment conditions, as in the above example, then this amount is the future value of money, and the payment itself is = 0. This argument must match the second number of periods argument. If the number of discount periods is 10 and the third argument is not<>0, then the PS function will count as 10 payments for the amount specified in the third argument (Pmt). The following example below shows how the present value of money is calculated with several installments in separate payments.
- Future value (FV) is the amount to be received at the end of the discount period. Excel financial functions are based on cash flow calculations. This means that the future value and present value of an investment have opposite signs. In this example, the future value is a negative number, so the formula evaluates to a positive number.
- Type - this argument must have the value 0 if the payment of the total amount falls at the end of the discount period, or the number 1 - if at its beginning. In this example, the value of this argument does not matter and will not affect the final result of the calculation in any way. Because the payment fee is zero and the type argument can be omitted. In this case, the function defaults to this argument with the value 0.
Formula for calculating the present value of money with inflation in Excel
In another example of applying the PV function, the future value of money is calculated for a whole series of future equal payments at once. If, for example, under an office lease, the tenant must pay 5,000 every month for one year, then the landlord can use the PV function to calculate how much he will lose in income, taking into account 6.5% annual inflation:
In this example, the fifth Type argument has a numeric value of 1 because the rent is paid at the beginning of each month.
If there is an amount of regular payments, the PS function actually calculates the current value of money separately for each payment and sums up the results. The figure shows the results of calculating the cost for each payment. The current value of the first payment is the same as the amount of the payment, as it is now paid after the fact. The next month's payment will be paid in a month and its current monetary value is already decreasing (depreciating). It is discounted to the amount of 4,973. The changes are not significant, but the last payment, which will be paid in 11 months, already has a value significantly lower - 4,712. All results of calculating the values of the present value of investments must be summed up. The PS function does all this work automatically without the need for a chronological payment schedule for the entire period.
08.03.2015 21:16 3473
BASICS OF THE THEORY OF THE VALUE OF MONEY IN TIME
Measuring the value of real estate in terms of money and the fact that its value is determined, as a rule, by the present value of future income from the ownership and use of real estate requires an appeal to the theory of the value of money over time, which explains the processes of determining the future value of money (accumulation) and bringing cash flows to their present value (discounting).
Given that these processes are based on the effect of compound interest, this chapter will focus on the application of standard compound interest functions in valuation procedures and explain their economic content. In particular, six main functions will be considered: the accumulated amount (future value) of the unit, the accumulation of the unit over the period, the contribution to the formation of the replacement fund, the present value of the unit (reversion), the present value of the ordinary annuity, and the contribution to the depreciation of the unit.
Processes of accumulation and discounting
As already noted, the value of real estate is expressed in monetary terms. In other words, money is the commodity for which real estate rights are exchanged. But, like any other commodity, money must have a value, i.e. in the relevant market, the capital market, you can borrow money for a certain period of time for a certain fee. In the same market, you can give your money for use for a while, expecting to receive a reward for this.
This is clearly illustrated by banking operations. When placing money on bank deposits, in fact, they are transferred for use, and the interest rate that the bank offers on the invested capital is a payment for this use. And, on the contrary, the money taken on credit must be returned to the bank in full, together with a certain percentage, as a payment for the use of this money.
In any case, the amount of money today, which is called the present value, and the amount of money tomorrow, which is called the future value, will differ by the amount of income at the interest rate:
where FV is the amount that reflects the future value;
PV - the amount reflecting the current value;
i - interest rate.
Arguing in a similar way, we can solve the inverse problem, how much PV must be invested today in order to receive a certain amount of FV in the future for a given level of remuneration i:
This task is called the task of discounting, that is, bringing the future value into the current value, and the coefficient DF=1/(1+i), which is used in this case, is called the discount factor.
Operations of accumulation and discounting
Thus, the most important operations that provide an opportunity to compare money at different times are the operations of accumulation and discounting.
Accumulation - the operation of bringing the current value into the future.
Discounting - bringing the future value into the current one.
Financial analysis is built on these two operations. One of his main criteria is the interest rate, or the ratio of net income to invested capital. When performing an accumulation operation, it is called the rate of return on capital, when discounting, it is called the discount rate.
Investing in real estate is very similar to using money. Investing money in the purchase and / or construction of real estate involves generating income in the future, and not today. Such a refusal of the current use of money also requires its payment - the receipt of income on invested capital. Thus, the future value of any property will be greater than the present value by the amount of this income.
EXAMPLE
An investment project for the construction of an office building is being considered. The forecast calculation showed that in a year the building could be sold for $400,000. It is necessary to determine how much it is worth investing in construction today, if the income level acceptable to the investor is 15%.
Naturally, the rate of return on capital that an investor can accept will be determined by the risk of earning that amount of return. The higher the risk of achieving a given value of income, the greater should be the rate of payment for capital invested in construction.
The above reasoning shows that the present value of the investment will be $347,826:
PV = FV × 1/(1 + i) = 400000 × 1/(1 + 0.15) = 347826
In this problem, one period was considered, at the end of which it was supposed to receive income, i.e. the rate was charged on the initial capital. If income will be received at the end of several periods (years, months), then the rate will be calculated from the amount accumulated in the previous period, i.e. by compound interest. In this case, the discount factor for the first period will be determined as
In subsequent periods, assuming that i = const, it should be calculated in this way:
It should be noted that many problems solved in real estate appraisal are based on the use of the compound interest effect. Typically, the interest rate is given as the nominal annual rate. If the number of periods is expressed not in years, but in months or quarters, then the interest rate must also be monthly or quarterly. In order to determine them, the nominal annual rate must be divided by the appropriate number of periods per year.
Different-time cash flows, reduced using a discount factor to the current value, have the property of additivity. This allows us to present in general terms the present value of the discounted cash flow for t periods, with the assumption of a constant value of i, as follows:
where Ct is the cash flow of the t-th period
This expression is called the discounted cash flow formula. The discounted cash flow formula can be greatly simplified under certain conditions. First of all, it concerns one of the main assumptions made in real estate valuation, about the infinity of income from land. If we assume that the amount of annual income will be constant, then the present value of an infinite stream of uniform constant receipts at a discount rate equal to i will be described by a geometric progression