The life cycle of projects is quite long, so there is a problem of comparing benefits and costs arising in a certain period. The concept of estimating money over time is based on the fact that the value of money changes over time, taking into account the norms of profit in the money market. The rate of return is often the loan interest, that is, the amount of income from the use of money in the capital market. In the process of comparing the cost of funds, two concepts are used: the future value (FV) and the present value (PV).
Future value (FV) is the amount of funds invested at the moment, in which they must be converted after a certain period of time, taking into account a certain interest rate.
The interest rate refers to the measurement of the time value of money, the amount of interest on an investment that can be received for a given period of time. If the investment is carried out in a short period of time, then use a simple percentage - the amount that is accrued on the initial value of the deposit at the end of one period. It is calculated by the formula:
I = r·i·n,where
I is the monetary expression of interest, the amount of interest money accrued during the investment period; p — the initial cost of the deposit; i — interest rate; n is the number of payment periods.
The future value is calculated as follows:
FV = PV + I,
where PV is the present value of money.
If the investment is carried out in a long period of time, then a complex percentage is used. This is the amount of income that is formed as a result of investing, provided that the amount of the accrued interest is not paid after each period, but joins the amount of the principal deposit and in the subsequent payment period itself generates income.
The process of transition from present value (PV) to future (FV) is called compounding.
Compounding (accrual) is a transaction that allows you to determine the value of the final future value using complex interest rates.
The equation for calculating future value by compounding is as follows:
FV = PV(1 + i)n,where
FV is the future value; PV — present value; i is the interest rate in current or real terms; n — number of years or term of service of the project; (1 + i)n — coefficient (factor) of future value for i and n.
The discounting process is an operation opposite to compounding (increasing complex interest) at the agreed final amount of funds.
Discounting is the process of determining the present value of a cash flow by adjusting future cash receipts using a discount ratio.
For illustration, let's give an example. Let's say you put 1000 Dollar in the bank. at 20% per annum. How much will you have at the end of the first year? To begin with, we determine that the present value or initial amount of your account is PV = 1000 Dollar, and the interest rate paid by the bank for one year is i = 20%. The future value at the end of one year (n=1) FV is equal to the initial rate multiplied by 1.0 plus the interest rate i=0.2.So at the end of the first year you will have 1200 Dollar. (1000+ +1000Ch0,2, or 1000Ch(1+0,2)).
Let's see what the result will be if you leave your 1000 Dollar. in a bank account for 3 years. The future value of the initial amount at the end of the third year can be determined using the equation:
FV = 1000 H (1 + 0.2)3 = 1728 Dollar.
Present value (PV) is the amount of future cash receipts given taking into account a certain interest rate to the present period.
Similarly, a simple and complex percentage can be applied in discounting, but in practice only a complex percentage is used. The calculation is as follows:
PV = FV / (1 + i)n = FV•1 / (1 + i)n,
where 1 / (1 + and)n is the percentage factor of the present value or discount factor.
Example 1. It is expected that the return on investment will be 5% per annum. According to the formula 100Dollar, nested now, in a year will cost:
FV1 = 100•(1 + 0.05) = 105.
If the investor wants to continue investing, then at the end of next year the deposit value will be:
FV2 = FV1•(1 + i) = 105•(1 + 0.05) = 110.2 or
by formula (2.2.1):
FV2 = PV•(1 + i)2 = 100•(1 + 0.05)2 = 110.25.
The process of increasing the cost of the initial 100 Dollar. can be submitted in the form of a table. 5.Example2
. Suppose that the investor wants to get 200 Dollar. in 2 years. How much should he put on an urgent deposit now if the interest rate is 5%?
To calculate, we use the formula:
PV = 200 : (1 + 0.05) = 181.40.
In this case, the value i is perceived as the discount rate (it is often called simply a discount).
The case discussed in example 2 can be interpreted as follows:
181,40Dollar. and 200 Dollar. — these are two ways to submit the same amount of funds at different points in time: 200 Dollar. after two years is equal to 181,40 Dollar. today.
To simplify the calculations of future and present values, you can use the tables of the value of the factor of future and present values (see Annexes 1, 2), which provide ready-made calculations using formulas (1+i)n and 1/(1+i)n for different values (i) and (n).
Calculations performed in the selection of projects quite often contain the need to determine the cost of equal payments (or receipts), which are made at the same intervals during a certain period of time. Such payments are called an annuite. There are the future value of the annuity (the cost of the annuity at the time of the last payment) and the present value of the annuity (discounted annuity amount at the date of the last payment).
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