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 formula compound interest:
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. In order to distinguish continuous interest rates from 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.
Compound interest is the amount of income that is accrued in each interval and is added to the principal amount of the capital and participates as a base 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 following formula is used:
Sc = P * (1 + i)n.
Accordingly, the amount of compound interest is determined by: Ic = Sc - P,
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. economic sense 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: Рс = 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.
Estimating the value of money in an annuity is associated with the use of the most complex algorithms and the determination of the method of calculating interest - preliminary (prenumerando) or subsequent (postnumerando) .1. When calculating the future value of an annuity on the terms of pre-payments (prenumerando), the following formula is used: SA pre =R * ([(1 + i) n -1] / i) * (1 + i)
where SA pre is the future value of the annuity carried out on the terms of advance payments (prenumerando); R is an annuity member characterizing the size of a separate payment; i - used interest rate, expressed as a decimal fraction; n is the number of intervals over which each payment is made in the total stipulated period of time. 2. When calculating the future value of an annuity carried out on the terms of subsequent payments (postnumerando), the following formula is applied: SA post = R * ([(1 + i) n-1] / i)
3. When calculating the present value of an annuity carried out on the terms of advance payments (prenumerando), the following formula is used: PA pre = R * ([(1 + i) - n - 1] / i) * (1 + i)
4. When calculating the present value of an annuity carried out on the terms of subsequent payments (postnumerando), the following formula is applied: PApost = R * ([(1 + i) - n - 1] / i)
5. When calculating the amount of a separate payment for a given future value of an annuity, the following formula is used: R = SA post * (i / [(1 + i) n - 1])
The concept of accounting for the inflation factor is the need for a real reflection of the value of assets and cash flows and ensuring compensation for income losses caused by inflationary processes in the implementation of long-term financial transactions.
Inflation is a process of constantly exceeding growth rates money supply over commodity (including the cost of work and services), resulting in an overflow of circulation channels with money, which leads to their depreciation and an increase in prices for goods.
Let us consider the most important terms and concepts used in assessing inflationary processes.
The nominal interest rate is the rate set without taking into account changes in the purchasing value of money due to inflation.
The real interest rate is the rate that takes into account changes in the purchasing value of money due to inflation.
The inflation premium is additional income, paid (or expected to be paid) to a creditor or investor in order to compensate for losses from the depreciation of money associated with inflation.
To predict the annual inflation rate, the following formula is used: TIg \u003d (1 + TIm) ^ 12 - 1,
where TIg is the projected annual inflation rate, in fractions of a unit; TIm is the expected average monthly inflation rate in the coming period, in fractions of a unit.
To estimate the future value of money, taking into account the inflation factor, a formula based on the Fisher model is used: S = P x [(l + Ip) x (1 + T)]n - 1,
where S is the nominal future value of the deposit, taking into account the inflation factor; P is the initial cost of the deposit; Iр - interest rate, in fractions of a unit; T is the predicted rate of inflation, in fractions of a unit; n is the number of intervals for which interest is calculated.
The Fisher model has the form: I = i + a + i * a ,
where I is the real interest premium; i - nominal interest rate; a is the rate of inflation.
This model assumes that in order to assess the expediency of investments in conditions of inflation, it is not enough just to add up the nominal interest rate and the projected inflation rate, it is necessary to add to them the amount that is their product i * a.
It should be noted that forecasting inflation rates is a rather complicated and time-consuming process, the results of which are probabilistic in nature and are subject to a significant influence of subjective factors. In practice, to simplify calculations and avoid the need to take inflation into account, calculations are made in hard currencies.
Risk factor concept consists in assessing its level in order to ensure the formation of the required level of profitability of financial and economic operations and the development of a system of measures to minimize its negative financial consequences. Return is understood as the ratio of the income generated by a certain asset to the amount of investment in this asset. Entrepreneurial activity is always associated with risk. At the same time, there is usually a clear relationship between the risk and return of this activity: the higher the required or expected return (i.e. return on invested capital), the higher the degree of risk associated with the possibility of not receiving this return, and vice versa. When making management decisions, various tasks can be set, including: maximizing profitability or minimizing risk, but, as a rule, more often it is about achieving a reasonable balance between risk and profitability. As part of financial management risk category has importance when making decisions on the capital structure, forming an investment portfolio, justifying dividend policy and etc.
To assess the risk, qualitative and quantitative methods are used, including: sensitivity analysis, scenario analysis, Monte Carlo method, etc.
To assess the level of financial risk (UR), an indicator that characterizes the likelihood of a certain type of risk and the amount of possible financial losses in its implementation, the following formula is used: UR \u003d VR x RP , where VR is the probability of occurrence of this financial risk; RP - the amount of possible financial losses in the realization of this risk.
The concept and methodology for accounting for the liquidity factor:
1) The value of own working capital: WC=CA-CL, where CA - current assets, CL - short-term liabilities.
2) Coefficient current liquidity: Ktl = current assets/short-term liabilities.
The ratio reflects the company's ability to repay current (short-term) liabilities at the expense of only current assets. The higher the indicator, the better the solvency of the enterprise. Taking into account the degree of liquidity of assets, it can be assumed that not all assets can be sold on an urgent basis. The normal value of the coefficient is from 1.5 to 2.5, depending on the industry. A value below 1 indicates a high financial risk associated with the fact that the company is not able to consistently pay current bills. A value greater than 3 may indicate an irrational capital structure.
3) Quick liquidity ratio: Kbl = Short-term accounts receivable+ Short-term financial investments+ Cash) / (Short-term liabilities - Deferred income - Reserves for future expenses) or Kbl = (Current assets - Inventories) / Current liabilities (the indicator must be<1. 1 – низкий показатель). Коэффициент отражает способность компании погашать свои текущие обязательства в случае возникновения сложностей с реализацией продукции.
4) Absolute liquidity ratio \u003d (Cash + short-term financial investments) / Current liabilities or Cash / (Short-term liabilities - Deferred income - Reserves for future expenses).
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INTRODUCTION 1. Relevance 2. History of origin. 3. Origin of the designation. 4. Set rules. 5. Comparison of values in percent 6. Types of percent. 7. Factors taken into account in financial and economic calculations. 8. Conclusion.
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Modern life makes interest tasks relevant, as the scope of practical application of interest calculations is expanding. Relevance.
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The word "percent" comes from the Latin word pro centum, which literally translates as "per hundred", or "from a hundred". Percentages are very convenient to use in practice, since they express parts of integers in the same hundredths. History of origin.
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The % sign is due to a typo. In manuscripts, pro centum was often replaced by the word "cento" (one hundred) and abbreviated - cto. In 1685, a book was printed in Paris - a guide to commercial arithmetic, where by mistake the compositor typed % instead of cto. The origin of the designation.
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In the text, the percent sign is used only for numbers in digital form, from which, when typing, it is separated by a non-breaking space (67% income), except when the percent sign is used to abbreviate compound words formed using the numeral and adjective percent. Set rules.
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Sometimes it is convenient to compare two quantities not by the difference between their values, but by percentage. Percent comparison
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Distinguish between simple and compound interest. When using simple interest, interest is charged on the initial amount of the deposit (loan) throughout the entire period of accrual. Interest types
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Methods of financial mathematics are used in calculating the parameters, characteristics and properties of investment operations and strategies, parameters of state and non-state loans, loans, credits, in calculating depreciation, insurance premiums and premiums, pension accruals and payments, in drawing up debt repayment plans, assessing the profitability of financial transactions . Factors taken into account in financial and economic calculations.
In financial practice, a significant part of the calculations are carried out using the compound interest scheme.
The use of the compound interest scheme is advisable in cases where:
interest is not paid as it accrues, but is added to the original 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;
loan term for more than a year.
If interest money is not paid immediately as it accrues, but is added to the original amount of the debt, then the debt is thus increased by the unpaid amount of interest, and the subsequent accrual of interest occurs on the increased amount of debt:
FV = PV + I = PV + PV i = PV (1 + i)
- for one period of accrual;
FV = (PV + I) (1 + i) = PV (1 + i) (1 + i) = PV (1 + i)2
- for two accrual periods;
hence, for n accrual periods, the formula will take the form:
FV = PV (1 + i)n = PV kн,
where FV is the accumulated amount of debt;
PV - the initial amount of debt;
i - interest rate in the accrual period;
n is the number of accrual periods;
kn is the coefficient (multiplier) of accruing compound interest.
This formula is called the compound interest formula.
As mentioned above, the difference between the calculation of simple and compound interest in the basis of their calculation. If simple interest is charged all the time on the same initial amount of the debt, i.e. the accrual base is a constant value, then compound interest is accrued on the base increasing with each accrual period. Thus, simple interest is inherently absolute growth, and the simple interest formula is similar to the formula for determining the level of development of the phenomenon under study with constant absolute growth. Compound interest characterizes the process of growth of the initial amount with stable growth rates, while accelerating its absolute value, therefore, the compound interest formula can be considered as determining the level based on stable growth rates.
According to the general theory of statistics, to obtain a basic growth rate, it is necessary to multiply the chain growth rates. Since the interest rate for the period is a chain growth rate, the chain growth rate is:
Then the basic growth rate for the entire period, based on a constant growth rate, is:
The basic growth rates or accrual factors (multipliers) depending on the interest rate and the number of accrual periods are tabulated and presented in Appendix 2. The economic meaning of the accrual multiplier is that it shows what one monetary unit will be equal to (one ruble, one dollar etc.) after n periods at a given interest rate i. 5>>>
A graphical illustration of the ratio of the accrued amount for simple and compound interest is shown in Figure 4.
Rice. 4. Accretion of simple and compound interest.
As can be seen from Figure 4, for short-term loans, the accrual of simple interest is preferable to that of compound interest; with a term of one year, there is no difference, but with medium-term and long-term loans, the accumulated amount calculated for compound interest is much higher than for simple ones.
For any i,
if 0< n < 1, то (1 + ni) >(1 + i)n ;
if n > 1 then (1 + ni)< (1 + i)n ;
if n = 1, then (1 + ni) = (1 + i)n .
Thus, for lenders:
the simple interest scheme is more profitable if the loan term is less than a year (interest is charged once at the end of the year);
the compound interest scheme is more profitable if the loan term exceeds one year;
both schemes give the same result with a period of one year and a single calculation of interest.
Example 8. An amount of 2,000 dollars is lent for 2 years at an interest rate of 10% per annum. Determine the interest and the amount to be repaid.
Accrued amount
FV \u003d PV (1 + i) n \u003d 2 "000 (1 + 0" 1) 2 \u003d 2 "420 dollars
FV \u003d PV kn \u003d 2 "000 1.21 \u003d 2" 420 dollars,
where kn = 1.21 (Appendix 2).
Amount of accrued interest
I \u003d FV - PV \u003d 2 "420 - 2" 000 \u003d $ 420. 6>>>
Thus, after two years it is necessary to return total amount in the amount of $2,420, of which $2,000 is debt and $420 is the "price of debt".
Quite often, financial contracts are concluded for a period other than a whole number of years.
In the event that the term financial transaction expressed as a fractional number of years, interest can be calculated using two methods:
the general method is to calculate directly using the compound interest formula:
FV = PV (1 + i)n,
where n is the transaction period;
a is an integer number of years;
b is the fractional part of the year.
The mixed calculation method involves using the compound interest formula for a whole number of years of the interest calculation period, and the simple interest formula for the fractional part of the year:
FV = PV (1 + i)a (1 + bi).
Because b< 1, то (1 + bi) >(1 + i)a, therefore, the accumulated amount will be larger when using a mixed scheme.
Example. The bank received a loan at 9.5% per annum in the amount of 250 thousand dollars with a maturity of two years and 9 months. Determine the amount to be repaid at the end of the loan term in two ways, given that the bank uses the German practice of calculating interest.
General method:
FV = PV (1 + i)n = 250 (1 + 0.095)2.9 = $320.87 thousand.
mixed method:
FV = PV (1 + i)a (1 + bi) =
250 (1 + 0,095)2 (1 + 270/360 0,095) =
321.11 thousand dollars.
Thus, according to the general method, the interest on the loan will be
I = S - P = 320.87 - 250.00 = 70.84 thousand dollars, 7>>>
and by mixed method
I = S - P = 321.11 - 250.00 = 71.11 thousand dollars.
Apparently, the mixed scheme is more favorable to the creditor.
When using financial tables, it is necessary to monitor the correspondence between the length of the period and the interest rate.
Compare the result with the result of example 1. It is not difficult to see that the compound rate gives a large amount of interest.
When calculating by the mixed method, the result is always greater.
The area of application of simple interest is most often short-term operations (with a period of up to one year) with a single accrual of interest (short-term loans, bill credits) and less often long-term operations.
For short-term transactions, the so-called intermediate interest rate is used, which is understood as the annual interest rate reduced to the investment period Money. 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, simple interest is calculated using 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 for both integer and fractional values of n;
Special designation of the interest rate with 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 stated 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) .
