## Payment Flows and Financial Rents

Financial transactions in most cases involve not separate payments, but a set of time-allocated receipts and payments. A ** payment flow ** refers to a series of time-divided payments and receipts.

Evaluation of the effectiveness of a management decision involves estimating the cash flow generated by the adoption of this decision.

There are positive (tributaries) and negative (outflows) cash flows. * Positive cash flows * arise when you receive money for an enterprise, * negative * - when you withdraw money from the enterprise. Internal transfers of money (for example, transfer of money from the cash desk to a settlement account, etc.) are not considered as cash flows.

Cash flows are classified into three types of activities, namely: operating, investment and financial.

Cash flow is a generalizing indicator and characterizes the amount of receipts and payments for a certain period of time. Cash flow for a certain time period is the difference between the amount of receipts and the amount of cash payments for this period.

Cash receipts and payments within a single period are not considered to be distributed within the period, but referred to one of its borders, i.e. all receipts and payments, according to the accepted assumption, take place either at the beginning or at the end of the time period to which they relate. In accordance with this, the cash flows * prenumerando * (receipts and payments are related to the beginning of the time period) and the flows * postnumerando * are distinguished (receipts and payments are referred to the end of the time period). The methodology for analyzing investment projects is based on the cash flow of post-numerando. It is this flow that has become more widespread in practice.

The summarizing characteristics of cash flows are the increased amount of the flow of payments and the current value of the flow of payments.

* The increased amount of the flow of payments * - the amount of all payments with compound interest accrued on them by the end of the term. When calculating the future value of the initial cash flow, formula 3.6 is applied to each element of the stream. The future value of the initial cash flow of post-indeterminate is determined by the formula

(3.18)

where * CF * k is the cash flow in the year * k, r * is the build rate.

* The current * (* current) value of the flow of payments * is the amount of payments discounted at the initial moment at a compound interest rate. When calculating the present value of the initial cash flow, formula 3.12 is applied to each element of the stream. In this case, a discount scheme is implemented: all elements are reduced to one point in time with the help of discount factors, which allows them to be summed up. The present value of the initial cash flow of post-indeterminate is determined by the formula

(3.19)

where * CF * k - the cash flow in the year * k, r - * the discount rate.

In the case when we are dealing with streams of prenumerando, this means that all cash receipts and payments take place one year earlier than in the case of postnumerando flows. Consequently, between the present and future value of the flows of pre-emer- gando and post-non-intermediate, there is the following relationship:

(3.20)

(3.21)

** Example **

Define the current and future value of cash flow postnumerando. Cash receipts by years were as follows: the 1st year - 40 thousand rubles, the second - 45 thousand rubles, the third - 50 thousand rubles, the 4th - 45 thousand rubles. The discount rate is 20%.

The future value of cash flow (at the end of the fourth year) in accordance with the formula 3.17 is

The current value of cash flow in accordance with the formula 3.18 is

If the revenues generated within the same time period were not concentrated at the end of the period (the flow of the post-non-run), but at the beginning of the period (the flow of the pre-emerent), then the present and future cost would be calculated as follows:

You can get the same result using formulas 3.20 and 3.21:

Consider a particular case of cash flow - an annuity. There are two options for determining the annuity:

- annuity - unidirectional cash flow, the elements of which take place at regular intervals;

- annuity - unidirectional cash flow, the elements of which are the same in value and occur at regular intervals.

The second approach is more common in practice.

The present value of an annuity (postnumerando) is made up of the present values of all future payments:

(3.22)

where * PVA * is the present value of the annuity; * R * - regular annual income; * n * - the number of years during which payments were received; * r * is the discount rate.

Sum up the series on the right side (geometric progression), we get

(3.23)

Often the formula 3.23 is written in the form

(3.24)

where - coefficient of rent reduction, tabulated function.

The future value of an annuity consists of the future values of all payments:

(3.25)

The incremental individual payments are members of a geometric progression with the first term equal to * R * and the denominator of the progression equal to . Summing up the series on the right side, we get

(3.26)

Often the formula 3.26 is written in the form

(3.27)

de - coefficient of increase in rent, tabulated function.

** Example **

Define the present and future cost of the annuity with payments of 100 thousand rubles. at the end of the year for 6 years. The discount rate is 20%. We use the formulas 3.23, 3.26:

Some calculations use the technique of perpetual annuity. An unlimited term is called an annuity, if the cash proceeds continue for a fairly long time. In Western practice, perpetual annuities are calculated for 50 years or more.

The present value of an indefinite annuity postnumerando is determined by the formula

(3.28)

The present value of an indefinite annuity of an ex ante bonus is determined by the formula

(3.29)

** Example **

Define the current cost of an unlimited annuity with payments of 100 thousand rubles. at the end of the year. The discount rate is 20%. By the formula 3.28 we obtain

Consider the task of replacing a series of payments with one payment.

** Example **

The organization, by virtue of the lease agreement, has to pay annually at the end of the year 8,000 rubles per year for 8 years. It is proposed to replace the series of lease payments with one payment in two years from the beginning of the contract. Determine the amount of payment if the money is 10% per annum.

Due to the lease agreement, the organization must pay the same amount once a year for 8 years at the end of the year. Hence, we are talking about the annuity (postnumerando). The annuity term is 8 years, the annual payment is 500 thousand rubles, the payment is made once a year (the annuity period is 1 year).

The current cost of an annuity will be

Since a series of rental payments are proposed to be replaced by one payment in two years from the beginning of the contract, the present value of one this payment should be equal to the present value of the annuity.

The incremental value of * PVA * at the end of the second year

Let's consider one more example, when it is necessary to compare proposals with which cash receipts do not coincide in time.

** Example **

The company is considering three proposals. The discount rate is 12%. The cash flow is distributed as follows:

Year |
Proposal 1 |
Proposal 2 |
Proposal 3 |

1st |
10 000 |
30 000 |
0 |

2nd |
10 000 |
20 000 |
0 |

3rd |
10 000 |
15 000 |
0 |

4th |
10 000 |
10 000 |
10 000 |

5th |
10 000 |
0 |
15 000 |

6th |
10 000 |
0 |
20 000 |

7th |
10 000 |
0 |
30 000 |

The 8th |
10 000 |
0 |
40 000 |

9th |
10 000 |
0 |
0 |

10th |
10 000 |
0 |
0 |

Total |
100 000 |
75,000 |
115 000 |

We calculate the current value of each offer at a discount rate of 12%.

So, the firm should accept the second offer, since the present value of this cash flow exceeds the present value of the first and third.

Let's pay attention to the fact that if you do not take into account the time value of money, the second proposal will seem the least interesting, since the amount of money received by the firm in this case is minimal. However, it must be remembered that the later the money is credited to the account, the less is their value. The distribution of the cash flow over time has a significant effect on the current value of the offer, as shown in this example.

Summarizing the above, we note that in all the above evaluation formulas the key parameter is the interest rate*g.*From the economic point of view, it is equal to the relative amount of income that the investor wants or can receive on the capital invested by him . Since the investment opportunities of different investors are not the same, the values of the discount rate, which they put into the valuation model, are different. As a consequence, the same financial asset is valued differently by different investors, which leads to the purchase and sale of financial instruments.

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