economic element- this is an economically homogeneous type of cost for the production and sale of products (works, services), which cannot be decomposed into its component parts within a given enterprise.
"Rules accounting» (PBU 10/99, p. 8) regulate a single list economic elements, which form the cost of production:
1) material costs : a) the cost of purchasing raw materials, materials used in the production of goods (performance of work, provision of services); b) the cost of acquiring tools, fixtures, inventory, instruments, laboratory equipment, overalls and other means of individual and collective protection and other property that is not depreciable property; c) the cost of purchasing components, semi-finished products undergoing additional processing; d) the cost of acquiring fuel, water and energy of all types spent for technological purposes, the production of all types of energy, heating buildings, as well as the costs of transformation and transmission of energy; e) the cost of acquiring works and services of an industrial nature performed by third parties;
2) labor costs: any accruals to employees in cash and (or) in kind, incentive accruals and allowances, compensation accruals, etc.;
3) contributions for social needs: in the form of a single social tax(ESN). The UST scale is regressive, the rate decreases with the growth of the wage fund .;
4) depreciation: depreciation deductions for the full restoration of fixed assets. Depreciation is an estimated amount that reflects the part of the cost of fixed assets transferred to the finished product and accumulated for the intended use on capital investments;
5) other costs: a very large group, which includes costs with different ways of attributing them to the cost.
71. Profit: approaches to the definition
Profit as the final financial result is a key indicator in the system of enterprise goals. Due to the great complexity of this economic category in economics There are many definitions and interpretations of profit. Among a number of approaches, economic and accounting approaches can be singled out as basic ones.
Economic approach considers profit as an increase in the capital of owners for reporting period(and, accordingly, the loss - as a decrease in capital). Profit, interpreted from the standpoint of this approach, is usually called economic.
Calculus economic profit possible in two ways:
1) based on the dynamics of market valuations of capital - this path is possible only if securities companies are listed on the stock exchange;
2) based on the data contained in the liquidation balance sheets at the beginning and end of the reporting period. But the result of either of these two calculations is extremely conditional (in particular, because not every change in capital is an element of profit).
Accounting approach many authors regard it as more realistic and justified. Here, profit is considered as a positive value of the difference between the income of the enterprise and its expenses (a negative value, respectively, is regarded as a loss). The income of the enterprise is an increase in the total valuation assets; this increment is accompanied by an increase in the capital of the owners. Expenses – decrease in the total value of assets.
Fundamental differences between sets:
1. The accounting approach contains a clear definition of the elements of profit - the types of income and expenses for which separate accounting is carried out. This creates an objective, verifiable basis that allows you to calculate the final financial result.
2. These approaches interpret realized and unrealized earnings differently. AT economic approach there is no distinction between these types of income, and in the accounting approach, unrealized income can be recognized as profit only if it is realized.
It is a well-known situation that the same amount of money is not equivalent in different periods of time. Accounting for the time factor in financial transactions is carried out by accruing interest.
Interest money (interest) is the amount of income from lending money in any form (loans, opening deposit accounts, buying bonds, renting equipment, etc.).
The amount of interest depends on the amount of the debt, the term of its payment and interest rate characterizing the intensity
interest calculation. The amount of debt with accrued interest is called the accrued amount. The ratio of the accrued amount to the initial amount of debt is called the accrual multiplier (coefficient). The time interval for which interest is calculated is called the accrual period.
Using simple bets interest, the amount of interest money is determined based on the initial amount of the debt, regardless of the number of accrual periods and their duration according to the formula:
The above formula is used to determine the value of the accumulated cost of capital for short-term financial investments.
If the term of the debt is given in days, the following expression must be inserted into the above formula:
where 5 is the duration of the accrual period in days;
The number of days in a year can be taken exactly - 365 or 366 (exact interest) or approximately - 360 days (ordinary interest). The number of days in each whole month during the term of the debt can also be taken exactly or approximately (30 days). In world banking practice, the use of:
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approximate number of days in each whole month and ordinary interest is called "German practice";
the exact number of days in each month and ordinary interest - "French practice";
the exact number of days and exact percentages - "English practice".
Depending on the use of a particular practice of calculating interest, their amount will vary.
Consider examples of financial and economic calculations for securities.
Example 7.1.
Savings certificate with a face value of 200 thousand rubles. issued on 20.01.2005 due on 05.10.2005 at 7.5% per annum.
Determine the amount of accrued interest and the redemption price of the certificate when using various methods of interest calculation.
Let's determine the exact and approximate number of days until the certificate is redeemed.
tT04H = 11 days of January + 28 days of February + 31 days of March + 30 days of April + 31 days of May + 30 days of June + 31 days of July + 31 days of August + 30 days of September + 5 days of October = 258 days.
Iapprox \u003d 11 days of January + 30 x 8 days (February - September) + 5 days of October \u003d 256 days.
Certificates earn income at an interest rate. There are three ways to calculate interest:
1) exact interest, loan term - the exact number of days:
Іfinal \u003d 0.075 x 200 x 258/365 \u003d 10.6 thousand rubles; certificate redemption price:
51 \u003d 200 + 10.6 \u003d 210.6 thousand rubles;
2) ordinary interest, loan term - the exact number of days, the redemption price of the certificate:
52 \u003d 200 + 10.8 \u003d 210.8 thousand rubles;
3) ordinary interest, loan term - an approximate number
Іbіkn = 0.075 х 200 х 256/360 = 10.7 thousand rubles, certificate redemption price:
53 \u003d 200 + 10.7 \u003d 210.7 thousand rubles.
Example 7.2.
The Bank accepts deposits for 3 months at a rate of 4% per annum, for 6 months at a rate of 10% per annum and for a year at a rate of 12% per annum. Determine the amount that the owner of the deposit will receive 50 thousand rubles. in all three cases.
The amount of the deposit with interest will be:
1) for a period of 3 months:
S \u003d 50 x (1 + 0.25 x 0.04) \u003d 50.5 thousand rubles;
2) for a period of 6 months:
S \u003d 50 x (1 + 0.5 x 0.1) \u003d 52.5 thousand rubles;
3) for a period of 1 year:
S \u003d 50 x (1 + 1 x 0.12) \u003d 56 thousand rubles.
When deciding on the placement of funds in a bank, an important factor is the ratio of the interest rate and the inflation rate. The inflation rate shows how many percent prices have increased over the period under review, and is defined as:
The inflation index shows how many times prices have risen over the period under review. The inflation rate and the inflation index for the same period are related by the ratio:
where Ju is the inflation index for the period;
N is the number of periods during the period under consideration.
The inflation rate for the period.
Example 7.3.
Determine the expected annual inflation rate at a monthly inflation rate of 6% and 12%.
Ju = (1 + 0.06)12 = 2.01.
Therefore, the expected annual inflation rate will be = 2.01 - 1 = 1.01, or 101%.
Ju = (1 + 0.12)12 = 3.9.
The expected inflation rate will be:
3.9 - 1 = 2.9, or 290%.
Inflation affects the profitability of financial transactions.
The real value of the accumulated amount with interest for the deadline, given by the time the money is loaned, will be:
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Example 7.4.
The bank accepts deposits for six months at a rate of 9% per annum. Determine the real results of the deposit operation for a deposit of 1000 thousand rubles. with a monthly inflation rate of 8%.
The amount of the deposit with interest will be:
S \u003d 1 x (1 + 0.5 x 0.09) \u003d 1045 thousand rubles.
The inflation index for the term of the deposit is equal to:
Ju = (1 + 0.08)6 = 1.59.
The accumulated amount, taking into account inflation, will correspond to the amount:
1045 / 1.59 \u003d 657 thousand rubles.
When using compound interest rates, interest money accrued after the first accrual period, which is part of the general term debt are added to the amount of debt. In the second accrual period, interest will accrue based on the original amount of the debt, increased by the amount of interest accrued after the first accrual period, and so on for each subsequent accrual period. If compound interest is accrued at a constant rate and all accrual periods have the same duration, then the accrued amount will be equal to:
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where P is the initial amount of the debt;
in - interest rate in the accrual period;
n is the number of accrual periods during the term.
Example 7.5.
Deposit 50 thousand rubles. deposited in the bank for 3 years with compound interest at the rate of 8% per annum. Determine the amount of accrued interest.
The amount of the deposit with accrued interest will be equal to:
S \u003d 50 x (1 + 0.08) 3 \u003d 63 thousand rubles.
The amount of accrued interest will be:
I \u003d S - P \u003d 63 - 50 \u003d 13 thousand rubles.
If interest were accrued at a simple rate of 8% per annum, their amount would be:
I \u003d 3 x 0.08 x 50 \u003d 12 thousand rubles.
Thus, the calculation of interest at a compound rate gives a large amount of interest money.
Compound interest can be compounded several times a year. At the same time, the annual interest rate, on the basis of which the amount of interest in each accrual period is determined, is called
nominal annual rate percent. With a debt term of n years and compound interest accrual m times a year, the total number of accrual periods will be equal to:
The accumulated amount will be equal to:
1) term of debt:
Example 7.6.
The depositor makes a deposit of 40 thousand rubles. for 2 years at a nominal rate of 40% per annum with monthly accrual and interest capitalization. Determine the accumulated amount and the amount of accrued interest.
The number of accrual periods is equal to:
Therefore, the accumulated amount will be:
A bill of exchange or other monetary obligation before the maturity date on it can be bought by the bank at a price less than the amount that must be paid on them at the end of the term, or, as they say, discounted by the bank. In this case, the bearer of the obligation receives money earlier than the period specified in it, minus income
bank in the form of a discount. The Bank, upon the due date of payment of a bill or other obligation, receives the full amount indicated in it.
If the period from the moment of accounting to the moment of repayment of the obligation will be some part of the year, the discount will be equal.
The scope of simple interest is most often short-term transactions (with a period of up to one year) with a single interest calculation (short-term loans, bill credits) and less often long-term transactions.
For short-term transactions, the so-called intermediate interest rate is used, which is understood as the annual interest rate, reduced to the term of investment of funds. Mathematically, the intermediate interest rate is equal to the percentage of the annual interest rate. The formula for accruing simple interest using an intermediate interest rate is as follows:
FV=PV(1+f*r),
FV = PV (1 + t * r / T),
t -- the term of investment of funds (in this case, the day of investment and the day of withdrawal of funds are taken as one day); T is the estimated number of days in a year.
For long-term transactions, the calculation of simple interest is calculated by the formula:
FV=PV(1+r*n),
where n is the period of investment of funds (in years). ,
The scope of compound interest is long-term transactions (with a period exceeding a year), including those involving intra-annual interest.
In the first case, the usual compound interest formula is applied:
FV = PV (1 + r)n.
In the second case, the compound interest formula is applied, taking into account intra-annual accrual. Intra-annual interest is the payment of interest income more than once a year. Depending on the number of income payments per year (m), the intra-annual accrual can be:
The accrual formula for semi-annual, quarterly, monthly and daily compound interest is as follows:
FV = PV (1 + r / m)nm,
where PV is the original amount;
g -- annual interest rate;
n is the number of years;
m -- the number of intra-annual accruals;
FV -- accumulated amount.
Interest income in the case of continuous interest calculation is calculated according to the following formula:
where: e \u003d 2, 718281 is a transcendental number (Euler number);
e?n is the incremental factor, which is used both for integer and fractional values of n;
Special designation of the interest rate for continuous interest calculation (continuous interest rate, “growth force”);
n is the number of years.
With the same amount of the initial amount, the same term of investment of funds and the value of the interest rate, the amount returned is greater in the case of using the intra-annual accrual formula than in the case of using the usual compound interest formula:
FV = PV (1 + r / m)nm> FV = PV (1 + r)n.
If the income received when using intra-annual accruals is expressed as a percentage, then the interest rate received will be higher than that used in the usual calculation of compound interest.
Thus, the initially declared annual interest rate for calculating compound interest, called nominal, does not reflect the real efficiency of the transaction. The interest rate that reflects the actual income received is called effective. The classification of interest rates for intra-annual compound interest is clearly illustrated in the figure.
The nominal interest rate is set initially. For each nominal interest rate and based on it, the effective interest rate (re) can be calculated.
From the formula for accruing compound interest, you can get the formula for the effective interest rate:
FV = PV (1 + r)n;
(1 + re) = FV / PV.
Here is the formula for accruing compound interest with intra-annual accruals, in which r / m percent is accrued every year:
FV = PV (1 + r / m)nm.
Then the effective interest rate is found by the formula:
(1 + re) = (1 + r/m)m,
re = (l + r/m)m- 1,
where re is the effective interest rate; r -- nominal interest rate; m -- the number of intra-annual payments.
The value of the effective interest rate depends on the number of intra-annual accruals (m):
The area of simultaneous application of simple and compound interest are long-term operations, the term of which is a fractional number of years. In this case, interest can be calculated in two ways:
In the first case, the compound interest formula is used for calculations, in which there is an exponentiation to a fractional power:
FV = PV (1 + r)n+f,
where f is the fractional part of the investment period.
In the second case, the so-called mixed scheme is used for calculations, which includes a compound interest formula with an integer number of years and a simple interest formula for short-term transactions:
FV = PV (1 + r)n * (1 + f * r),
FV = PV (1 + r)n * (1 + t * r / T) .
1.2. dividend payment methods.
Dividend payment methods:
Method of constant percentage distribution of profits. This method assumes a stable percentage of net profit for a long time directed to the payment of dividends on ordinary shares (for example, 40% of net profit annually).
Advantages: the presence of a direct relationship between dividend payments and the financial result of the enterprise.
Flaw consists in a possible significant fluctuation in the market value of the company's shares, with a change in dividend payments in monetary terms attributable to one ordinary share.
2) Methodology of fixed dividend payments. This method implies the regular payment of dividends per share in the same amount over a long period of time, regardless of changes in the financial condition of the enterprise. This amount of dividend payments can be adjusted for the inflation index.
Advantage is a sense of reliability, which gives shareholders a sense of confidence in the invariability of current income, regardless of various circumstances. In addition, this technique allows avoiding significant fluctuations in the market value of shares.
Flaw consists in the absence of a relationship between dividend payments and the financial results of the enterprise, therefore, in periods that are unfavorable for the enterprise, it may not have enough money not only for development, but also to ensure its core activities.
3) Guaranteed Minimum Payment Methodology and Extra Dividends. This method provides for regular payments of a fixed amount of dividends, in case of favorable market conditions and a large amount of net profit received, extra dividends are paid to shareholders. Thus, the annual income of shareholders consists of dividends fixed at a minimum level and extra dividends paid periodically, depending on the financial result.
Advantage lies in the sense of security that shareholders receive in connection with the payment of dividends in the minimum established amount, regardless of financial results. In addition, there is a high relationship between dividend payments and the financial results of the enterprise, which allows you to increase the amount of dividend payments (extra dividends) in favorable periods for the enterprise without reducing its invested activity.
Flaw is that with a long-term payment of minimum fixed dividends, the investment attractiveness of the company's shares decreases, otherwise, with regular payments of extra-dividends, their stimulating effect on shareholders decreases.
4) The method of constant increase in the amount of dividends. This method provides for a stable increase in the level of dividend payments per share, the increase in the amount of dividends is made, as a rule, in a fixed percentage of the level of dividends in the previous period.
Advantage is to provide high market value shares of the enterprise and their attractiveness, both for shareholders and for potential investors.
Flaw lies in its inflexibility and the constant increase in financial tension, as well as lagging behind the growth rate of profits from the growth rate of dividend payments, which means a reduction in the amount of reinvested profit, a decrease in the financial stability of the enterprise.
5) Residual Dividend Methodology. This technique implies the payment of dividends as a last resort after the financing of all effective investment projects. Dividend payments are determined after a sufficient amount of financial resources has been generated from the profit of the reporting year to ensure the implementation of the most profitable investment projects of the enterprise.
Advantages are to ensure high rates of development of the enterprise, increase its market value and maintain financial stability.
Flaws:
1) payment of dividends is not guaranteed and regular;
2) the amount of dividends is not fixed and varies depending on the financial results and the amount of own funds allocated for investments;
3) dividends are paid only if the company has net profit not in demand for the development of the enterprise.
6) Methodology for paying dividends by shares. This method provides for the issuance of an additional block of shares to shareholders in the form of dividend payments instead of cash. A small amount of dividends paid in this way does not have a significant impact on the market value of shares, but if dividends are significant, then the market price of shares after an additional issue may significantly decrease. Enterprises are most often forced to use this technique in an unstable financial situation and the absence of highly liquid assets for settlements with shareholders, or if it is necessary to reinvest profits in a highly effective project.
Flaw consists in significant fluctuations in the market price of shares, due to the appearance on the market of an additional volume of shares of this enterprise.
2. Method of calculation and scope of compound interest
Compound interest- this is the amount of income that accrues in each interval and is added to the principal amount of capital and participates as a basis for accrual in subsequent periods. Compound interest is usually used for long-term financial transactions (for example, investing).
When calculating the amount of future value (Sc), the formula is applied:
Sc = P * (1 + i) n .
Accordingly, the amount of compound interest is determined by:
where Ic - the amount of compound interest for a specified period of time; P is the initial cost of money; n is the number of periods for which interest payments are calculated; i - used interest rate, expressed in fractions of a unit.
Formulas for calculating compound interest are basic in financial calculations. The economic meaning of the factor (1 + i)n is that it shows what one ruble will be equal to in n periods at a given interest rate i. To simplify the calculation procedure, special financial tables have been developed for calculating compound interest, which allow you to determine the future and present value of money.
The present value of money (Rc) when calculating compound interest is:
Pc = Sc / (1 + i) n
The discount amount (Dc) is determined by:
D c \u003d Sc - Rc.
When calculating the time value of money in terms of compound interest, it must be borne in mind that the results of the assessment are affected not only by the interest rate, but also by the number of payment intervals during the entire payment period, which leads to the fact that in some cases it is more profitable to invest money under a lower rate, but with more payouts during the pay period.
From an economic point of view, the compound interest method is more reasonable, since it expresses the possibility of continuous reinvestment (re-investment) of funds. However, for short-term (less than a year) financial transactions, the simple interest method is most often used. There are several reasons for this:
First, and it was quite relevant a few decades ago, calculations using the simple interest method are much simpler than calculations using the compound interest method.
Secondly, at low interest rates (within 30%) and short time intervals (within one year), the results obtained using the simple interest method are quite close to the results obtained using the compound interest method (discrepancy within 1% ). If the phrase "Taylor formula" tells you something, then you will understand why this is so.
Thirdly, and perhaps this is the main reason, the debt found using the simple interest method for a period of time less than a year is always more than debt found using the compound interest method. Since the rules of the game are always dictated by the creditor, it is clear that in this case he will choose the first method.
Comment: short-term transactions (less than a year) make up the bulk of all financial transactions. Why? Because long-term loans that are repaid in installments once a month or once a quarter (or even once every six months) are not one big financial transaction, but the totality of a large number of short-term operations (lasting a month, quarter or half a year). That is why in Russia the simple interest method is used to calculate interest on any loans.
The use of the compound interest scheme is advisable in cases where:
- interest is not paid as they accrue, but are added to the initial amount of the debt. The addition of accrued interest to the amount of debt, which serves as the basis for their calculation, is called interest capitalization;
- the term of the loan is more than a year.
Option 3.
The company's balance sheet looks like this:
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Sum |
Sum |
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Fixed assets |
Authorized capital |
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Long-term credits and loans |
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Accounts receivable over 12 months |
Short-term credits and loans |
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Accounts receivable less than 12 months |
Accounts payable |
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Cash |
Other current liabilities |
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Sales proceeds in the reporting period amounted to 14,500 rubles; the cost of goods sold is 10,100 rubles. Conduct an analysis of the business activity of the enterprise.
The business activity of the enterprise in the financial aspect is manifested primarily in the speed of turnover of its funds. The profitability of an enterprise reflects the degree of profitability of its activities. The analysis of business activity and profitability consists in researching the levels and dynamics of various financial turnover and profitability ratios, which are relative indicators of the financial performance of an enterprise.
Analysis of business activity allows you to identify how efficiently the company uses its funds.
1.Asset turnover ratio= Revenue / Assets (p. 10 form No. 2 / p. 300 form No. 1)
Cob.=14500/23250=0.62
The coefficient shows that from one ruble of assets, an enterprise receives an average of 0.62 rubles. revenues or, on average, assets make 0.62 turnovers per year.
2.Duration of one turn in days= Number of days of the analyzed period / turnover ratio
PO \u003d 365 days / 0.62 \u003d 180 (days)
The higher the turnover rate, the faster you can sell inventory and, if necessary, pay off the debt.
3.The indicator of the turnover of the enterprise's own funds= Sales proceeds / Equity
K about. sob. avg. = 10100/5000 = 2.02
The turnover rate of own funds reflects the activity of their use. In this case, it is high, which means that the level of sales significantly exceeds the invested capital.
4.Profitability indicators characterize the profitability of the company.
To rent. = Balance sheet profit/Revenue *100% (F2(140)/F2(010))
To rent. = (4400/14500) * 100% \u003d 30.35
The coefficient shows how much profit falls on a unit of sold products.
An entrepreneur can get a loan in one of three ways:
on the terms of quarterly interest accrual at the rate of 35% per annum;
on the terms of semi-annual interest at the rate of 40% per annum;
on the terms of monthly interest at the rate of 30% per annum.
Which option is more preferable?
The relative costs of an entrepreneur for servicing a loan can be determined by calculating the effective annual interest rate, the higher it is, the greater the level of expenses, according to the formula:
re \u003d (1 + r / m) m -1
re- effective rate (depends on intra-annual accruals)
1.On the terms of quarterly accrual (35% per annum):
re = (1+0.35/4) 4 -1=(1+ 0.0875) 4 -1=1.9567-1=0.9567
2. On the terms of semi-annual accrual (40% per annum):
re = (1+0.4/2) 2 -1=(1+ 0.02) 2 -1=1.440-1=0.440
3.On the terms of monthly accrual (30% per annum):
re = (1+0.30/12) 12 -1=(1+ 0.025) 12 -1=1.3449-1=0.3449
Thus, option 3 is more preferable for the entrepreneur. It should be noted that the decision does not depend on the size of the loan, since the criterion is a relative indicator - the effective rate, and, as follows from the formula, it depends only on the nominal rate and the number of accruals.
In the first year of operation of the enterprise, sales revenue amounted to 12,000 rubles, variable costs 9,000 rubles, fixed costs 1,300 rubles. Next year, it is planned to increase the proceeds from sales to 14,000 rubles.
Determine how to change the profit of the enterprise:
a) in the traditional way;
b) using the operating lever.
Production leverage effect (EPR) is a potential opportunity to influence sales profit by changing the cost structure, namely the ratio between variable and fixed costs.
The essence of the production leverage effect6 any change in sales revenue leads to an even greater change in profits.
1.Traditional way:
CR= Sales revenue – Variable costs – Fixed costs
PR = 12000-9000-1300 = 1700
K=14000/12000=1.167 (coefficient of change in sales revenue)
(14000/12000)*100% -100=16.7% (by this percentage the sales revenue increased)
PR1 \u003d 14000-1300-9000 * 1.167 \u003d 2197
% PR \u003d (2197/1700) * 100% -100 \u003d 129.23% -100% \u003d 29.23% - growth
2. With operating lever:
% PR =% in * EPR
EPR \u003d BM / profit \u003d (Revenue - variable costs) / profit
EPR \u003d (12000-9000) / 1700 \u003d 1.76 (production leverage effect)
Finding the percentage change in profit
% PR = 16.7 * 1.76 = 29.39% - growth
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Price rub./pc. |
Volume of sales |
Revenue, rub. |
Unit Variable Costs |
General variable costs, rub. |
Unit fixed costs |
General fixed costs, rub. |
Specific total costs |
Total costs, rub. |
Profit (loss) per unit |
Profit (loss) for the entire volume |
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In practical calculations, discrete percentages are mainly used, i.e. interest accrued for fixed equal time intervals (year, half year, quarter, etc.). Time is a discrete variable. In some cases, in proofs and calculations related to continuous processes, it becomes necessary to use continuous percentages. Consider the compound interest formula:
S= P(1 + i) n .(6.16)
Here P is the initial amount, i is the interest rate (as a decimal fraction), S is the amount formed by the end of the loan term at the end n th year. Compound interest growth is a process that develops exponentially. The addition of accrued interest to the amount that served as the basis for their determination is often called interest capitalization. In financial practice, they often face a problem that is the opposite of determining the accumulated amount: for a given amount S, which should be paid after some time n, it is necessary to determine the amount of the received loan P. In this case, we say that the amount S discounted, and percentages in the form of the difference S - P are called discount. The value P found by discounting S is called modern, or given, value S. We have:
P = ; P == 0.
Thus, with very long payment terms, the present value of the latter will be extremely insignificant.In practical financial and credit operations continuous processes of accumulation of sums of money, i.e., accumulations over infinitely small periods of time, are rarely used. Continuous growth is of much greater importance in the quantitative financial and economic analysis of complex production and economic objects and phenomena, for example, when choosing and justifying investment decisions. The need to use continuous accruals (or continuous interest) is determined primarily by the fact that many economic phenomena are inherently continuous, therefore an analytical description in the form of continuous processes is more adequate than based on discrete ones. We generalize the compound interest formula for the case when interest is charged m once a year:
S = P (1 + i/m) mn .
The accumulated amount in discrete processes is found by this formula, here m- the number of accrual periods in a year, i- annual or nominal rate. The more m, the shorter the time intervals between the moments of interest calculation. In the limit for m→ ∞ we have:
S = P (1 + i/m) mn = P ((1 + i/m) m) n .
Since (1 + i/m) m = e i , then ` S = P e in .
With a continuous increase in interest, a special type of interest rate is used - strength of growth, which characterizes the relative increase in the accumulated amount in an infinitely small period of time. With continuous capitalization of interest, the accrued amount is equal to the final amount, which depends on the initial amount, the accrual period and the nominal interest rate. To distinguish between continuous interest rates and discrete interest rates, we denote the former by d , then S = Pe .
Growth strength d is the nominal interest rate at m→ ∞. The multiplier is calculated using a computer or according to function tables.
Contracts, transactions, commercial and production and business operations often provide not for separate one-time payments, but for many payments and receipts distributed over time. Individual elements of such a series, and sometimes the series of payments as a whole, is called payment flow. Payment stream members can be either positive (receipts) or negative (payments) values. A stream of payments in which all members are positive and time slots between two successive payments are constant, called financial rent . Annuities are divided into annual and R- urgent, where R characterizes the number of payments during the year. These are discrete rents. In financial and economic practice, there are also sequences of payments that are made so often that in practice they can be considered as continuous. Such payments are described by continuous annuities.
Example 3.13.Suppose that at the end of each year for four years, 1 million rubles are deposited in the bank, interest is accrued at the end of the year, the rate is 5% per annum. In this case, the first installment will turn to the end of the annuity period in the amount of 10 6´
1.05 3 since the corresponding amount has been in the account for 3 years, the second installment will increase to 10 6´
1.05 2 , as it was on the account for 2 years. Last installment earns no interest. Thus, at the end of the annuity period, contributions with accrued interest represent a series of numbers: 10 6´ 1.05 3 ; 10 6 ´ 1.05 2 ; 10 6' 1.05; 10 6. The value accumulated by the end of the annuity period will be equal to the sum of the members of this series. To summarize what has been said, we derive the corresponding formula for the accumulated amount of the annual annuity. Denote: S - the accumulated amount of the annuity, R - the size of the annuity member,
i - interest rate (decimal fraction), n - annuity term (number of years). The annuity members will bear interest for n - 1, n - 2,..., 2, 1 and 0 years, and the accumulated value of the annuity members will be
R (1 + i) n - 1 , R (1 + i) n - 2 ,..., R (1 + i), R.
Let's rewrite this series in reverse order. It is a geometric progression with the denominator (1+i) and the first term R. Let's find the sum of the terms of the progression. We get: S = R´ ((1 + i) n - 1)/((1 + i) - 1) = R´ ((1 + i) n - 1)/ i. Denote S n; i = ((1 + i) n - 1)/ i and will call it rent accumulation factor. If interest is charged m once a year, then S = R´ ((1 + i/m) mn - 1)/((1 + i/m) m - 1), where i is the nominal interest rate.
The value a n; i = (1 - (1 + i) - n)/ i is called rent reduction factor. Annuity reduction coefficient at n → ∞shows how many times the present value of the annuity is greater than its term:
a n; i =(1 - (1 + i) - n) / i \u003d 1 / i.
Example 3.14.Under eternal annuity is understood as a sequence of payments, the number of members of which is not limited - it is paid for an infinite number of years. Perpetual annuity is not a pure abstraction - in practice it is some type of bonded loans, an assessment of the ability pension funds meet its obligations. Based on the essence of perpetual annuity, we can assume that its accumulated amount is equal to an infinitely large value, which is easy to prove by the formula:
R×´ ((1 + i) n - 1)/ i → ∞ as n →∞.
Reduction coefficient for perpetual annuity a n; i →1/i, whence A = R/i, i.e. the present value depends only on the value of the annuity term and the accepted interest rate.
