The difference between the PRES and PREMUMENDO streams is that financial calculations are shifted to one cycle and this leads to an additional one-time interest accrual (1 + r) in other words, the flow scheme is more profitable for accumulation money, the logic of estimating the streaming flow of the Penumrando is similar to the logic of the Flow estimate of the postnamerando stream, the direct task of the formula for calculating the future cost pre-stream will be a form FV \u003d PV Pre * (1 + R)
The inverse problem of the calculation formula of the reduced value of the PREMUMENDO will have a form PV Pre \u003d FV PST * (1 + R) so if in the previous example, the initial stream is a stream of PREMUMENDO, i.e. Regular income on the Ts B. will be paid not at the end and at the beginning of the period, then its discounted cost will be equal to PV Pre \u003d 44.97 * 1,12 \u003d 50.37
\u003d 6 \u003d Anniversary Evaluation
Annuitu is a private cash flow case in which cash receipts in each period are the same in size (a). Annicates may be urgent and indefinite if the number of equal time intervals is limited, then such an annuity is called urgent, if not limited to the perpetual. According to species cash streams There are two types of annuities and postgraduo and Penumrando. Examples of the urgent annuity of postnamerando can be the regular receipt of the rental fee in the same amount if the lease agreement provides for payment upon the expiration of each period. An example of urgent annuity of the Penumrando may be periodic cash contributions On the bank account at the beginning of each month to accumulate funds for large purchase
Because In the formulas for estimating cash flows previously considered the same cash receipts, and may be made over the amount of the formula for the assessment of annuities are significantly simplified.
formula 1 FV APST \u003d K (A) * (1 + R) T - K \u003d A * T-K
because Annuity as the type of cash flow is characterized by the same time intervals and the identical value of the cash flow elements, it is advisable to mathematically convert the second factor of this formula without allocating periods (K)
the future value of the annuity of postnamerando is determined by
formula 2.FA APST \u003d.
the second factor can be determined by the settlement, and you can use financial tables. In the tables, this multiplier is called a multiplicating multiplier for the annuity FM3 (R: T) formula 3. Fa APST \u003d A * FM3 (R; T)
the future value of the annuity of Penumerando is determined by
formula 4. FV Apre \u003d A * FM3 (R; T) * (1 + R)
an example of an organization was offered to rent equipment for 5 years and choose one of the payment options.
a) 12 tr. annually
b) 85 tr. at the end of the 5th term
which option is more profitable if the bank offers 20% per annum on deposits
condition A \u003d 12 T \u003d 5 R \u003d 0.2
A) Solution FV APST \u003d F * FV APST (0.2; 5)
FV APST \u003d 12 * 7,442 \u003d 89.3
2. Feedback (from the discounting position)
By similar transformation, the formula for the evaluation of the discounted cash flow is simplified into formulas for evaluating the discounted annuity in post-stagendo and Penumrando.
formula 5. PV APST \u003d A * or PV APST \u003d A * FM4 (R: T)
The second factor of this formula can be determined by calculation, but you can use financial tables. In the tables, this multiplier is called the discounted multitude of the annuity FM4 (R; T)
The discounted value of the annuity of the Penumerando is determined by formula 6. PV APRE \u003d PV APST * (1 + R) In a practical example, it is possible to estimate the discounted value of the annuity using the Deposit Book Method
\u003d 7 \u003d Deposit Book Method
Calculation of the current value of the annuity using the method of a deposit book is that the amount put on the deposit brings income in the form of a percentage and when removing some amount from the deposit, the base value with which interest is accrued decreases. The current value of the annuity is the amount of the deposit with the total amount due to interest annually on equal amounts annual payment It remains unchanged (and annuity) its structure is constantly changing. If interest in it is dominated by interest in it, then at the time of time, the share of interest payments is reduced and the proportion of the part of the fundamental debt is reduced. The logic and countable methods of the method will consider on the example.
Example: The company obtained a loan for a period of 5 years in the amount of 450 tr. Under 14% per annum which is charged according to the scheme of complex interest on the non-repayed residue to return debt it is necessary for equal amounts at the end of each year. Determine the magnitude of the annual payment of A.
Decision: For a better understanding of the method, it is advisable to argue from the creditor position for the bank this amount It is a cash outflow. In the future, within 5 years, the Bank will annually receive at the end of the year the amount of the annual payment (a) in this formulation of the problem we are dealing with the evaluation of the discounted value of the postnamerando annuity about which its current cost (PV APST) is known, interest rate (R) and duration of action (t) Substitut the data in the formula of the discounted value of the annuity of postnamerando
PV APST \u003d A * FM4 (0.14; 5) 450t.r. \u003d a * 3,433 hence the amount of annual payment is equal to A \u003d 450 / 3,433 \u003d 131.08
For clarity, pay the dynamics of payments in the table
To assess the movement financial flows In time, apply various formulas financial mathematics, including the calculation of the future value of urgent annuity Postnamerando.
Postsenrando - The receipts of payments occur at the end of the period.
cash flowconsisting of the same payout value and the existing certain time can be counted into the future value, summing up all extensive payouts, taking into account the condition of postnamerando.
Increment - financial operationat which there is a calculation of the future value of today's investment under a given period and interest rate.
Formula of the future value of urgent annuity Postsenrando:
FV - Future cost;
R - interest rate, shares of units;
N is the number of years.
FV \u003d 100 * ((1 + 0.12) 5 + (1 + 0.12) 4 + (1 + 0.12) 3 + (1 + 0.12) 2 + (1 + 0.12)) \u003d 635 rubles.
The amount planned for obtaining, under the above conditions, will be 635 rubles.
Fig. 1. Schedule of the future value of the urgent annuity of postnamerandoloan percent 4, 12, 20, 28% per annum
Future cost of urgent cancellando annuity
To assess the movement financial flows In time, apply various formulas financial mathematics, including the calculation of the future value of urgent annuity Penumrando.
Penumrando - Revenues of payments occur at the beginning of the period.
The essence of the calculation is that cash flowconsisting of the same amount of payments and the existing time can be counted into the future value, summing up all the extensive payouts, taking into account the conditions of the Penumrando.
The formula of the given value of urgent annuity Penumerando:
FV - Future cost;
A - the magnitude of uniform receipt;
R - interest rate, shares of units;
N is the number of years.
FV \u003d 100 * (1 + 0.12) * ((1 + 0.12) 5 + (1 + 0.12) 4 + (1 + 0.12) 3 + ...
+ (1 + 0.12) 2 + (1 + 0.12)) \u003d 711.51 rubles.
Fig. 2. Schedule of the future value of the urgent annuity of Penumrando; Final costs at annual receipts 1000 rubles. and bets loan percent 4, 12, 20, 28% per annum
Examples exist on this technique
Topic 8. Cash flows
1. Evaluation of the permanent annuity of the Penumrando.
2. Deposit Book Method.
3. Universal annuity.
4. Continuous annuity.
If only complex interests are accrued for cash receipts, the corresponding calculated formulas for the extensive amounts of the annuity of the PREMUMRADO can be easily removed from formulas (7.7), (7.11), (7.12), (7.14). Since the cash receipts in the annuity of Penumrando occur at the beginning of each period, this annuity differs from the annuity of the post-mando in the number of percentage of interest periods.
For example, for urgent annuity of the Penumrando with regular monetary receiptsequal BUT, and interest rate, extended cash flow has the form
therefore, given (7.7),
those. The outstanding amount (future value) of the annuity of the Penumrando is more than the incredible amount of the Annuity of Postsenorando.
Similarly, for the annuity of the Penumrando with interest accrual once during basic periodUsing (4.11), we get:
(7.32)
For r- Associated annuities, taking into account (4.12), (4.14) you can write the following ratios:
(7.33)
(7.34)
Of course, (7.31) - (7.33) are special cases (7.34). From formula (7.34) it follows that. The financial meaning of this inequality is obvious: For the recipient, the cash receipts of Penumrando is more profitable, as they begin for the period earlier than postnormando, i.e. The temporary value of the money is confirmed: money "now" is preferable than "then".
Some other will be the situation in r- Associate Annuit of PREMUMENDO, when the contributions arriving during the base period are accrued with simple interest. Unlike the annuity of postnamerando in this annuity in each period, any contribution "acts" still y) part of the period, thereby delivering an additional value by the end of the period. Consequently, by the end of each period, contributions, the number of which is r, deliver the magnitude.
After such arguments of a qualitative character, we derive analytically for the formula for the future value.
Last r-E-receipt are charged simple percentages for Y) part of the period, and it will be equal, the penultimate entrance will become equal, etc. Before the first admission, which will become equal. Therefore, the sum of these values \u200b\u200bforming arithmetic progression is equal to:
Thus, using (7.13), we get:
From a financial point of view, this formula follows from the above qualitative reasoning. Since by the end of each period, contributions are delivered to an additional amount, then by the future value of the initial annuity of the postnamerando, it is necessary to add an even future value of the annuity of the postnamerando with monetary receipts, and this is the second term in formula (7.35). Naturally, in this case.
In the case of calculating only complex percentage of formulas for calculating the above values \u200b\u200bof the absenteeism of Penumerando, are similar to formulas (7.31) - (7.34), i.e. There is a given value of the corresponding annuity of postnamerando and then the obtained value is multiplied by the corresponding factor of the increment. Thus, considering various annuities, you can write:
(7.37)
(7.38)
(7.39)
It's clear that . From the above formulas, it is clear why the financial tables do not specify which scheme is implied in the financial transaction - postnormando or Penumrando; The content of the financial table is invariant to this factor. However, when applying the calculated formulas or financial tables, it is necessary to strictly monitor the flow of cash payments.
Example:
Each year at the beginning of the year, the Bank makes another contribution of 10 thousand rubles. The bank pays 20% per annum. What amount will be on the score after three years?
In this case, we are dealing with the annuity of the Penimerando, the future value of which is proposed to evaluate. In accordance with formula (7.31), we will find the searched amount S:
Many practical tasks Can be solved in various ways, depending on which cash flow is highlighted by an analyst. Consider the simplest example.
Example:
You are invited to invest 100 thousand rubles. For a period of five years, subject to the return of this amount of parts (annually 20 thousand tenge). After five years, an additional remuneration of 30 thousand rubles is paid. Does this offer, if you can "safely" deposit money into a bank at the rate of 12% per annum?
thousand tenge
With regard to an alternative option, providing for the reimbursement of the invested amounts, it is assumed that annual receipts in the amount of 20 thousand tenge can be immediately allowed in turnover, receiving additional income. If there are no other alternatives to the efficient use of these amounts, they can be deposited to the bank. Cash flow in this case can be represented by two:
a) as an urgent annuntary of PostnoMendo C, and the one-time obtaining amount of 30 thousand tenge;
b) as an urgent annuntary of Penumrando C, and the one-time obtaining amounts of 20 and 30 thousand tenge.
In the first case, on the basis of formula (7.7), we have:
thousand tenge.
In the second case, on the basis of formula (7.31), we have:
thousand tenge.
Naturally, both options led to the same answer. Thus, the total amount of capital by the end of the five-year period will develop from income from deposit money in a bank (107.056 thousand tenge), returning the share from participation in a venture project for last year (20 thousand tenge) and one-time remuneration (30 thousand tenge). total amount will be, therefore, 157,056 thousand tenge. The offer is economically inappropriate.
In the event of an antisipative accrual of interest of the formula for estimating the annuity of the Penimerando, the formulas previously obtained are obtained in the same manner. Values \u200b\u200bwill be multiplied by the corresponding multiplier. For example, formulas like (7.31), (7.36) will be viewed:
(7.40)
(7.41)
If continuous interest is accrued, then, to obtain the formulas for determining the future or present value, the abnormation of Penumrando must be processed to the limit for, for example, in Forms (7.34), (7.39). So, in particular, from (7.34) it follows that for continuous interest
,
Concept and characteristics of cash flow
$1000 $1000 $1000 $1000
The cash flow element is customary CF K. (Cash Flow), where k. - The number of the period in which the cash flow is considered. Current value of cash flow is indicated PV (Present Value), and the future value - FV (Future Value).
Future cash flow value for all elements from 0 before m. We get:
Example 1.: After the implementation of the event to reduce administrative costs, the enterprise plans to obtain savings of $ 1,000 at the end of each year. Saved money is supposed to be placed on a deposit account (under 5% per annum) so that after 5 years accumulated money to use for investment. What amount will be on bank account Businesses?
 |
Thus, the company in 5 years will accumulate $ 5,526, which can invest.
Thus, cash flows are flows of payments (cash) under which the distribution in time, the movement of funds arising from the economic activity of the subject.
In addition, under cash flows are understood as a time-distributed sequence of payments and revenues generated by one or another asset, an assets portfolio or an operation of an investment project.
With every investment project It is customary to bind a cash flow (Cash Flow), the elements of which are either clean outflows (NET CASH OUTFLOW), or pure cash inflows (NET CASH INFLOW).
Under pure outflow B. k-m It is understood to be the excess of current cash expenditures on the project over current cash receipts (a clean influx takes place in the opposite ratio).
The flow of payments, all the elements of which are distributed in time so that the intervals between any two successive payments are constant, called financial Renta or Annuity (Annuity).
Annuitu has two important properties:
1) all of its N-elements are equal to each other: CF1 \u003d CF2 ... \u003d CFN \u003d CF;
2) Segments of time between the payment (obtaining amounts) CF are the same.
Future cost of simple annuity It is the sum of all components of its payments with interest accrued at the end of the operation.
Under current cash flow cost Understand the amount of all components of his payments discounted at the time of the start of the operation.
The current value of the annuity has the following form:
The expression in square brackets is a multiplier equal to the current value of the annuity of the same monetary unit.
Sharing the current value PVthe cash flow on the specified multiplier can be obtained by the amount of the periodic payment of the annuity equivalent to it.
Scheme of the discounting of a simple annuity.
Example 2.:
Pension Fund Must carry out annual payments of 100 monetary units For three years. What amount will provide the specified payments if the rate on urgent deposits is currently 8% per annum.
0 100 100 100
Total amount 257.7.
Estimation of the stream of the Penumrando
Annuity Redemarando - English Annuity Due is a series of payments that are periodically carried out. at the beginning each period (for example, month, quarter, half year or year). This type of tool may be an investment or credit, depending on the purpose and owner of the annuity. An example of the annuity can be savings accounts, insurance policies, Mortgage and other similar investments. The key feature of the annuity of the Penimerando is that all payments are carried out at the beginning of each period.
Predumerando cash flow elements
where a is the amount of payment;
i is the interest rate for the period;
N - the number of periods.
For example, The investor intends to post a monthly on a deposit on a deposit of 500 cu. For 2 years under 7% per annum, provided that each contribution will be carried out at the beginning of each month. To calculate the amount that will be at the disposal of the investor, we use the above formula. However, it is necessary to bring an annual interest rate to a month, which will be 0.583% (7% / 12). At the same time, the number of periods will be 24 (24 months).
Thus, at the disposal of the investor in two years will be a sum of 12914.87 cu.
For the inverse problem of the discount circuit, i.e., bringing all the elements of the source stream to point 0 can be represented in Fig.
Discount Discount Cash Flow Elements
For calculation present value of the annuity of Penumrando It is necessary to use the following formula.
This formula, for example, can be used to calculate the size of the annuity payment on the loan. Suppose the borrower intends to take a loan in the bank in the amount of 25,000 cu For a period of 5 years under 17% per annum, provided that the loan will be repaid monthly. To calculate the amount of payment, it is necessary to use the formula of the present value of the annuity of Penumrando, expressing the payment (A) from it.
To use the resulting formula to calculate the annuity payment, it is necessary to bring the source data.
1) The present value of the Annuita will be 25,000 USD.
2) An annual interest rate must be brought to a month, which will be 1.4167% (17% / 12).
3) the number of periods will be 60 (5 years by 12 payments.)
Thus, the size of the monthly annuity payment on the loan will be 621.31 cu.
Most often in financial calculations are used as the following under the terms of the generation of rent:
1st case: payments are made once a year, interest are accrued once at the end of the year, then the extensive amount is determined
S \u003d r * k n; I, (4.1)
where S is an extensive amount of rent,
R is the size of the member of the rent (one-time permanent payment),
K n; i - the inclusion coefficient with the parameters "N" (the term of the rent) and "I" (the rate of complex interest) is the sum of geometric progression - the first term of the geometric progression A \u003d 1, and the denominator of the geometric progression G \u003d (1 + I), then K n; i \u003d (1 + i) n -1 / i, and
2nd Case: Annual Rent with interest "M" once a year at the nominal rate "J"
or 
(4.3)
3D Case: R-urgent Renta, interest are accrued once at the end of the year (M \u003d 1)
or 
(4.4)
4th Case: R-urgent Renta, interest interest "M" once a year (m \u003d p)
5th case: R-urgent rent, (p.m)
(4.6)
Modern values \u200b\u200bof rent depending on the formation conditions are determined by the formulas (similar to the conditions listed above).
or 
(4.7)
- Rentation coefficient considered as the sum of geometric progression with parameters 
or 
(4.8)
3rd case: R-urgent rent with interest accrual once a year (m \u003d 1)
or 
(4.9)
4th Case: R-urgent Renta (P \u003d M)
or 
(4.10)
5th General: R-urgent Renta (R≈M)
or 
(4.11)
Some factor of increasing and bring to the tabulated and are presented in the form of tables.
If it is necessary to determine the members of the rent or the term of rent, they can be obtained by the transformation of the formulas of the increasing and discounting relatively interests for us.
4.3 Rent of Penumrando
In this rent, payments are made at the beginning of each period of accrual, that is, the amount of payments will be one more than in the render of Postsenrando.
1st Case: Annual Rent with interest accrual 1 time per year
or 
(4.16)
2nd Case: Annual Rent with interest "M" once a year
or 
(4.17)
3rd case: R-urgent rent with interest accrual once a year
or 
(4.18)
4th Case: R-urgent rent with interest "M" - times
or 
(4.19)
4.4 Universal Annuitu
Annuity is called indefinite if monetary revenues continue for a long time (50 or more). In this case, the direct task of meaning does not have, as for the inverse problem, its decision is also made according to formula 4.7.
Insofar as 
T. 
(4.20)
The above formula is used to assess the feasibility of acquiring an annuity.
4.5 Variable Payment Flows
There are payments flows, whose members are changed over time. These payment sequences can be represented as variables Payment flows.
A special case of such a stream is a rent variable, that is, the rent, whose members are changed in accordance with any given law of development.
If such a law is not specified, then the corresponding sequence is an irregular flow of payments.
4.6 irregular flow of payments
The time intervals between the two neighboring members in the irregular flow of payments can be any. Summarizing characteristics are obtained by direct account.
Extensive amount (interest accrual 1 time per year)
S \u003d. 
(4.21)
Modern value
A \u003d. 
(4.22)
where t \u003d time from the start of payment flow until the date of payment
R T - Payment Size (Rent Member)
4.7 Annutent conversion
Annutient's conversion is understood as a change in the initial parameters of the Annutient, after which the new annuntement would be equivalent to this.
Two cancelments are considered equivalent if their modern magnitude, conducted to the same point, are equal.
In practice, the need to calculate the parameters of the equivalent annuntement most often occurs when changes in the terms of debt payment, repayment of a loan or loan, etc. In this case, the conversion can occur at the time of the beginning of the annunciation and after the payment of some part of the annunciation. In the latter case, all calculations are made on the balance of the debt at the time of the conversion.
The most common case of conversion of permanent cancellations:
1) After a certain period of time (it can be equal to "0") after the beginning of the annunciation, the entire balance of debt can be paid at a time (renger redemption). Obviously, in the event that the amount paid will be equal to the modern value of the annuity, calculated for the period N 2 \u003d N 1 - n 0 (interest rates are consistent. The definitions of modern value are selected).
2) The task inverse the previous one may occur: the debt is repaid by parts, as a permanent cancellation payment, and it is required to determine one of the annuncation parameters with the remaining remained. Since the amount of debt is known here, that is, the modern annual value, the formulas are used to find an unknown parameter:
(4.23)
(4.24)
3) the period of debt payment can be changed while reducing the former interest rate. The value of R 2 of payment for the period N 2 is found using equations of the equivalent (equivalent annuncation values):
(4.25)
From here 
(4.26)
Obviously, if the annuncation period increases, the value of R 2 will be reduced and vice versa.
4) There may be a situation where the value of the R payment must be changed to the other side.
5) The start of debt payment at a given interest rate may be delayed:
a) when deciding the amount of payment;
b) with a reduction in the period of payment;
Obviously, in the first case, an annuent should increase, and in the second - the value of the payment.
If you designate through N 0, the flow period, then at the time of the start of payment, the amount of debt A 2, which should be the modern value of the new annuntement, will be on the formula complex percent
(4.27)
From here we obtain equivalence equation:
We find N 1 at R 1 \u003d R 2
Payment value R 2 at N 2 \u003d N 1 - N 0
6) In some cases, it may be necessary to combine several annuents in one (announcing consolidation).
At the same time, the combined annunciations can be any, and in the desired unifying annuity, one of the parameter is unknown with all other specified.
(4.29)
A - Modern cost replacing rent
AQ - Modern value Q - and replaceable rent
