Application of compound interest in calculations. The use of compound interest in economic calculations

In financial practice, a significant part of calculations is carried out using a compound interest scheme.

The use of a compound interest scheme is advisable in cases where:

Interest is not paid as it accrues, but is added to the original amount owed. Adding accrued interest to the amount of debt, which serves as the basis for their calculation, is called capitalization of interest;

loan term is more than a year.

If interest money is not paid immediately as it accrues, but is added to the original amount of debt, then the debt is thus increased by the unpaid amount of interest, and subsequent interest accrual occurs on the increased amount of debt:

FV = PV + I = PV + PV i = PV (1 + i)

– for one accrual period;

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 kn,

where FV is the accumulated amount of debt;

PV – initial amount of debt;

i – interest rate in the accrual period;

n – number of accrual periods;

kн – coefficient (multiplier) of compound interest accumulation.

This formula is called the compound interest formula.

As stated above, the difference between the calculation of simple and compound interest is in the basis for their calculation. If simple interest is always calculated on the same original amount of debt, i.e. Since the accrual base is a constant value, compound interest is calculated on a base that increases with each accrual period. Thus, simple interest is inherently an absolute increase, and the formula for simple interest is similar to the formula for determining the level of development of the phenomenon being studied with constant absolute increases. Compound interest characterizes the process of growth of the initial amount with a stable growth rate, while increasing it in absolute value with acceleration; 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 the base growth rate, it is necessary to multiply the chain growth rates. Since the interest rate for the period is the chain growth rate, the chain growth rate is equal to:

Then the basic growth rate for the entire period, based on a constant growth rate, has the form:

Basic growth rates or coefficients (multipliers) of increase, depending on the interest rate and the number of periods of increase, are tabulated and presented in Appendix 2. The economic meaning of the increase 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 graphic illustration of the ratio of the accrued amount for simple and compound interest is presented in Figure 4.

Rice. 4. Increase in simple and compound interest.

As can be seen from Figure 4, for short-term loans, simple interest is preferable to compound interest; for a period of one year there is no difference, but for medium-term and long-term loans the accumulated amount calculated using compound interest is significantly higher than using simple interest.

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 persons providing credit:

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 one-time interest charge.

Example 8. An amount of $2,000 is loaned for 2 years at an interest rate of 10% per annum. Determine the interest and the amount to be repaid.

Accrued amount

FV = PV (1 + i)n = 2"000 (1 + 0"1)2 = 2"420 dollars

FV = PV kn = 2"000 1.21 = 2"420 dollars,

where kн = 1.21 (Appendix 2).

Amount of accrued interest

I = FV - PV = 2"420 - 2"000 = $420. 6>>>

Thus, after two years, a total amount of $2,420 must be repaid, of which $2,000 is debt and $420 is the “cost of debt.”

Quite often, financial contracts are concluded for a period other than a whole number of years.

In cases where the term of a financial transaction is expressed in a fractional number of years, interest can be calculated using two methods:

The general method is to directly calculate using the compound interest formula:

FV = PV (1 + i)n,

where n is the transaction period;

a – integer number of years;

b – fractional part of the year.

the mixed calculation method involves using the compound interest formula for an integer number of years of the interest accrual period, and for the fractional part of the year - the simple interest formula:

FV = PV (1 + i)a (1 + bi).

Since b< 1, то (1 + bi) >(1 + i)a, therefore, the accumulated amount will be greater when using a mixed scheme.

Example. A loan was received from the bank at 9.5% per annum in the amount of $250 thousand with a repayment period of two years and 9 months. Determine the amount that must be repaid at the end of the loan term in two ways, given that the bank uses German interest calculation practices.

General method:

FV = PV (1 + i)n = 250 (1 + 0.095)2.9 = 320.87 thousand dollars.

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 using a mixed method

I = S - P = 321.11 - 250.00 = 71.11 thousand dollars.

As you can see, the mixed scheme is more beneficial to the lender.

When using financial tables, you must ensure that the length of the period and the interest rate match.

Compare the result obtained with the result of example 1. It is not difficult to notice that a complex rate gives a large amount of interest.

When calculating using a mixed method, the result is always greater.

1 slide

2 slide

INTRODUCTION 1. Relevance 2. History of origin. 3. Origin of the designation. 4. Recruitment rules. 5. Comparison of percentages 6. Types of percentages. 7. Factors taken into account in financial and economic calculations. 8. Conclusion.

3 slide

Modern life makes problems on percentages relevant, as the scope of practical application of percentage calculations is expanding. Relevance.

4 slide

The word "percent" comes from the Latin word pro centum, which literally translates to "per hundred" or "per hundred." Percentages are very convenient to use in practice, since they express parts of whole numbers in the same hundredths. Origin story.

5 slide

The % sign was due to a typo. In manuscripts, pro centum was often replaced by the word “cento” (one hundred) and was written as abbreviated as cto. In 1685, a book was printed in Paris - a manual on commercial arithmetic, where the typesetter mistakenly typed % instead of cto. Origin of the designation.

6 slide

In the text, the percent sign is used only for numbers in digital form, from which, when typed, they are separated by a non-breaking space (income 67%), except in cases where the percent sign is used to abbreviate complex words formed using the numeral and the adjective percent. Recruitment rules.

7 slide

Sometimes it is convenient to compare two values not by the difference in their values, but as a percentage. Comparison of percentage values

8 slide

There are simple and compound types of interest. When using simple interest, interest is accrued on the initial amount of the deposit (loan) throughout the entire accrual period. Types of interest

Slide 9

Methods of financial mathematics are used in calculating the parameters, characteristics and properties of investment operations and strategies, parameters of government and non-government loans, loans, credits, in calculating depreciation, insurance contributions and bonuses, pension accruals and payments, in drawing up debt repayment plans, and assessing the profitability of financial transactions . Factors taken into account in financial and economic calculations.

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;font-family:"Times New Roman"">CONTENTS

;font-family:"Times New Roman"">Introduction…………………………………………………………………………………1

">Percent………………………………………………………………...2
">Use of simple and compound interest;color:#000000">………………………………………………………………………………6
;color:#000000">Application of simple interest…………………………………………...7
;color:#000000">Application of compound interest……………………………………………………….…….9
">Comparison of simple and compound interest methods;color:#000000">…………………………………………………………………………………..14
">Combined interest calculation schemes;color:#000000">………………………………………………………………..…16
">Nominal interest rate……………………………………………................................. ...............18
;color:#000000">The concept of nominal interest rate…………………………….…19
;color:#000000">Effective interest rate……………………………………………………….…20
;color:#000000">Continuous compounding……………………..……21
">INTEREST ACCRUALS……………………………………………...22

">Bibliography………………………………………....25

">CONCLUSION……..……………………………………………………......26

">PRACTICAL PART………………………………………………….....27

INTRODUCTION

;font-family:"Times New Roman"">In any developed market economy, the interest rate in the national currency is one of the most important macroeconomic indicators, closely monitored not only by professional financiers, investors and analysts, but also by entrepreneurs and ordinary citizens. The reason for this attention is clear: the interest rate is the most important price in the national economy: it reflects the price of money over time. In addition, the cousin of the interest rate is the inflation rate, also measured in percentage points and recognized in accordance with the monetarist paradigm as one of the main ones. guidelines and results of the state of the national economy (the lower the inflation, the better for the economy, and vice versa. The relationship here is simple: the level of the nominal interest rate should be higher than the inflation rate, while both indicators are measured as a percentage per annum. In modern economic theory, a general term. "interest rate" is used in the singular. Here it is considered as an instrument with which the state, represented by the monetary authorities, influences the country’s economic cycle, signaling a change in monetary policy and changing the volume of money supply in circulation.

;font-family:"Times New Roman"">The variety of specific interest rates in national currency is a topic that is very useful practical knowledge, the accumulation of which in the life of any person occurs empirically. Thanks to the media, or in one’s professional activities, or when managing personal savings and investments, we have all heard of or regularly come across different interest rates on a variety of products.

;font-family:"Times New Roman"">1. PERCENTAGE

;font-family:"Times New Roman"">Interest is the amount paid for the use of money. This is the absolute amount of income.

;font-family:"Times New Roman"">The ratio of interest money received per unit of time to the amount of capital is called the interest rate, or rate. With respect to the moment of payment or accrual of income for the use of the provided funds, interest is divided into ordinary and advance.

;font-family:"Times New Roman"">Regular (decursive,;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">postnumerando;font-family:"Times New Roman"">) interest is calculated at the end of the period relative to the original amount of funds. Interest income is paid at the end of the financial transaction periods.

;font-family:"Times New Roman"">The interest accrual period should be understood as the period of time between two successive procedures for charging interest or the period of a financial transaction if interest is accrued once (Fig. 1). As the name implies, These percentages (ordinary) are used more often in most deposit and lending transactions, as well as in insurance.

;font-family:"Times New Roman"">Interest calculation scheme

;font-family:"Times New Roman"">If income, determined by interest, is paid at the time the loan is granted, then this form of payment is called advance, or accounting, and the interest applied is advance (anticipatory,;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">prenumerando;font-family:"Times New Roman"">), which are accrued at the beginning of the period relative to the final amount of money.

;font-family:"Times New Roman"">Interest income is paid at the beginning of the period, at the time the debt is issued. This is how interest is calculated in some types of lending, for example, when selling goods on credit, in international payments, transactions with discounted securities. When In this case, the basis for calculating interest is the amount of money with interest (the amount of debt repayment), and the interest calculated in this way is charged in advance and is an advance.

;font-family:"Times New Roman"">The following types of interest rates exist:

;font-family:"Times New Roman"">Decursive rate,;font-family:"Times New Roman"">rate of return;font-family:"Times New Roman""> which is calculated based on the initial loan amount. Interest income is paid along with the loan amount.

;font-family:"Times New Roman"">An anticipatory rate, the rate of return of which is calculated based on the final amount of debt. Interest income is paid at the time the loan is granted.

;font-family:"Times New Roman"">The effective rate, the rate of return of which corresponds to receiving interest income once a year.

;font-family:"Times New Roman"">A nominal rate whose interest income increases by a multiple of times per year.

;font-family:"Times New Roman"">The practice of paying interest is based on the theory of increasing funds in an arithmetic or geometric progression.

;font-family:"Times New Roman"">Arithmetic progression corresponds to simple percentages, geometric progression corresponds to complex ones, i.e. depending on whether the calculation base is a variable or constant value.

;font-family:"Times New Roman"">Percentages are divided into:

;font-family:"Times New Roman""> - simple ones, which accrue on the original amount throughout the entire period of the obligation;

;font-family:"Times New Roman""> - complex, the calculation base for which is constantly changing due to the addition of previously accrued interest.

;font-family:"Times New Roman";color:#000000">Increase can be carried out according to the scheme of simple and compound interest.

;font-family:"Times New Roman"">Formula for compounding simple interest (simpleinterest). The compounding of simple interest means that the invested amount increases annually by PV r. In this case, the amount of invested capital after n years can be determined by the formula:

;font-family:"Times New Roman"">FV = PV (1 + r n).

;font-family:"Times New Roman"">Compound interest compounding formula. Compound interest compounding means that the next annual income is calculated not from the original amount of invested capital, but from the total amount, which also includes previously accrued and not interest demanded by the investor. In this case, the amount of invested capital after n years can be determined by the formula:

;font-family:"Times New Roman"">FV = PV (1 + r);font-family:"Times New Roman";vertical-align:super">n;font-family:"Times New Roman">.

;font-family:"Times New Roman"">For the same interest rate:

;font-family:"Times New Roman"">1) the rate of increase in compound interest is higher than the rate of increase in simple interest, if the period of increase exceeds the standard income accrual interval;

;font-family:"Times New Roman"">2) the rate of increase of compound interest is less than the rate of increase of simple interest, if the period of increase is less than the standard interval for accrual of income.

;font-family:"Times New Roman"">Areas of application of simple and compound interest. Simple and compound interest can be applied both in separate transactions and simultaneously. The areas of application of simple and compound interest can be divided into three groups:

;font-family:"Times New Roman";color:#000000">1. operations using simple interest;

;font-family:"Times New Roman";color:#000000">2. operations using compound interest;

;font-family:"Times New Roman";color:#000000">3. operations with the simultaneous application of simple and compound interest.

;font-family:"Times New Roman"">2 USING SIMPLE AND COMPOUND INTEREST

">From an economic point of view, the compound interest method is more justified, since it expresses the possibility of continuous reinvestment (re-investment) of funds. However, for short-term (lasting less than a year) financial transactions, the simple interest method is most often used. There are several reasons for this reasons:

;font-family:"Times New Roman"">Firstly, and a few decades ago this was quite relevant, calculations using the simple interest method are much simpler than calculations using the compound interest method.
;font-family:"Times New Roman"">Secondly, for small interest rates (within 30%) and short periods of time (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” means anything to you, then you will understand why this is so.
;font-family:"Times New Roman"">Third, and perhaps this is the main reason, debt found using the simple interest method for a period of time less than a year is always;font-family:"Times New Roman">more;font-family:"Times New Roman""> than the 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.

;font-family:"Times New Roman"">2.1 Application of simple interest

The scope of application of simple interest is most often short-term transactions (with a period of up to one year) with one-time interest accrual (short-term loans, bill credits) and less often long-term transactions.

;font-family:"Times New Roman"">For short-term transactions, the so-called intermediate interest rate is used, which is understood as the annual interest rate adjusted to the term of investment of funds. Mathematically, the intermediate interest rate is equal to a fraction of the annual interest rate. Formula for compounding simple interest using the intermediate interest rate is as follows:

;font-family:"Times New Roman";color:#000000">FV = PV (1 + f r),

;font-family:"Times New Roman";color:#000000">or

;font-family:"Times New Roman";color:#000000">FV = PV (1 + t r / T),

;font-family:"Times New Roman";color:#000000">where f=t/T;

;font-family:"Times New Roman";color:#000000">t period of investment of funds (in this case, the day of investment and the day of withdrawal of funds are taken as one day); T estimated number of days in a year.

;font-family:"Times New Roman"">For long-term transactions, the accrual of simple interest is calculated using the formula:

;font-family:"Times New Roman";color:#000000">FV = PV (1 + r n),

;font-family:"Times New Roman";color:#000000">where n is the term of investment of funds (in years). ,

;font-family:"Times New Roman"">2.2 Application of compound interest

;font-family:"Times New Roman";color:#000000">The scope of application of compound interest is long-term transactions (with a period exceeding a year), including those involving intra-annual interest accrual.

;font-family:"Times New Roman";color:#000000">In the first case, the usual formula for calculating compound interest is applied:

;font-family:"Times New Roman";color:#000000">FV = PV (1 + r);font-family:"Times New Roman";vertical-align:super;color:#000000">n;font-family:"Times New Roman";color:#000000">.

;font-family:"Times New Roman";color:#000000">In the second case, the formula for calculating compound interest is applied, taking into account intra-annual accrual. Intra-annual interest accrual means payment of interest income more than once a year. Depending on the number of income payments per year (m) intra-annual accrual can be:

;font-family:"Times New Roman";color:#000000">1) semi-annual (m = 2);

;font-family:"Times New Roman";color:#000000">2) quarterly (m = 4);

;font-family:"Times New Roman";color:#000000">3) monthly (m = 12);

;font-family:"Times New Roman";color:#000000">4) daily (m = 365 or 366);

;font-family:"Times New Roman";color:#000000">5) continuous (m -" ?).

;font-family:"Times New Roman"">The compounding formula for semi-annual, quarterly, monthly and daily compound interest is as follows:

;font-family:"Times New Roman";color:#000000">FV = PV (1 + r / m);font-family:"Times New Roman";vertical-align:super;color:#000000">nm;font-family:"Times New Roman";color:#000000">,

;font-family:"Times New Roman";color:#000000">where PV original amount;

;font-family:"Times New Roman";color:#000000">g annual interest rate;

;font-family:"Times New Roman";color:#000000">n number of years;

;font-family:"Times New Roman";color:#000000">m number of intra-annual accruals;

;font-family:"Times New Roman";color:#000000">FV accrued amount.

;font-family:"Times New Roman";color:#000000">Interest income with continuous compounding is calculated using the following formula:

;font-family:"Times New Roman";color:#000000">FV;font-family:"Times New Roman";vertical-align:sub;color:#000000">n;font-family:"Times New Roman";color:#000000"> = P e;font-family:"Times New Roman";vertical-align:super;color:#000000">rn;font-family:"Times New Roman";color:#000000">,

;font-family:"Times New Roman";color:#000000">or:

;font-family:"Times New Roman";color:#000000">where: e = 2, 718281 transcendental number (Euler number);

;font-family:"Times New Roman";color:#000000">e;font-family:"Times New Roman";vertical-align:super;color:#000000">?n;font-family:"Times New Roman";color:#000000"> increment multiplier, which is used for both integer and fractional values of n;

;font-family:"Times New Roman";color:#000000">? special designation of the interest rate for continuous compounding (continuous interest rate, “growth force”);

;font-family:"Times New Roman";color:#000000">n number of years.

;font-family:"Times New Roman"">With the same initial amount, the same investment period and interest rate, the returned amount turns out to be greater when using the intra-annual compounding formula than when using the usual compounding formula:

;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">FV = PV (1 + r / m);font-family:"Times New Roman";vertical-align:super;color:#000000" xml:lang="en-US" lang="en-US">nm;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">> FV = PV (1 + r);font-family:"Times New Roman";vertical-align:super;color:#000000" xml:lang="en-US" lang="en-US">n;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">.

;font-family:"Times New Roman"">If the income obtained using intra-annual compounding is expressed as a percentage, then the resulting interest rate will be higher than that used with ordinary compounding.

;font-family:"Times New Roman"">Thus, the initially stated annual interest rate for compounding, called nominal, does not reflect the actual efficiency of the transaction. The interest rate that reflects the actual income received is called effective. Classification of interest rates for intra-annual The calculation of compound interest is clearly illustrated in the figure.

;font-family:"Times New Roman"">The nominal interest rate is set initially. For each nominal interest rate and based on it, you can calculate the effective interest rate (r;font-family:"Times New Roman";vertical-align:sub">e;font-family:"Times New Roman">).

;font-family:"Times New Roman"">From the compound interest formula, you can get the effective interest rate formula:

;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">FV = PV (1 + r);font-family:"Times New Roman";vertical-align:super;color:#000000" xml:lang="en-US" lang="en-US">n;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">;

;font-family:"Times New Roman"">Here is the formula for increasing compound interest with intra-annual accruals, at which r/m interest is accrued every year:

;font-family:"Times New Roman";color:#000000">FV = PV (1 + r / m);font-family:"Times New Roman";vertical-align:super;color:#000000">nm;font-family:"Times New Roman";color:#000000">.

;font-family:"Times New Roman";color:#000000">Then the effective interest rate is found by the formula:

;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">(1 + r;font-family:"Times New Roman";vertical-align:sub;color:#000000" xml:lang="en-US" lang="en-US">e;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">) = (1 + r/m);font-family:"Times New Roman";vertical-align:super;color:#000000" xml:lang="en-US" lang="en-US">m;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">,

;font-family:"Times New Roman";color:#000000">or

;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">r;font-family:"Times New Roman";vertical-align:sub;color:#000000" xml:lang="en-US" lang="en-US">e;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US"> = (l + r/m);font-family:"Times New Roman";vertical-align:super;color:#000000" xml:lang="en-US" lang="en-US">m;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">- 1,

;font-family:"Times New Roman";color:#000000">where r;font-family:"Times New Roman";vertical-align:sub;color:#000000">e;font-family:"Times New Roman";color:#000000"> effective interest rate; r nominal interest rate; m number of intra-annual payments.

;font-family:"Times New Roman"">The effective interest rate depends on the number of intra-annual accruals (m):

;font-family:"Times New Roman";color:#000000">1) for m = 1, the nominal and effective interest rates are equal;

;font-family:"Times New Roman";color:#000000">2) the greater the number of intra-annual accruals (the value of m), the greater the effective interest rate.

;font-family:"Times New Roman"">The area of simultaneous application of simple and compound interest is long-term transactions, the term of which is a fractional number of years. In this case, interest can be calculated in two ways:

;font-family:"Times New Roman";color:#000000">1) calculation of compound interest with a fractional number of years;

;font-family:"Times New Roman";color:#000000">2) accrual of interest according to a mixed scheme.

;font-family:"Times New Roman"">In the first case, the compound interest formula is used for calculations, which includes raising to a fractional power:

;font-family:"Times New Roman";color:#000000">FV = PV (1 + r);font-family:"Times New Roman";vertical-align:super;color:#000000">n+f;font-family:"Times New Roman";color:#000000">,

;font-family:"Times New Roman";color:#000000">where f is the fractional part of the investment period.

;font-family:"Times New Roman"">In the second case, the so-called mixed scheme is used for calculations, which includes a formula for calculating compound interest with an integer number of years and a formula for calculating simple interest for short-term operations:

;font-family:"Times New Roman";color:#000000">FV = PV (1 + r);font-family:"Times New Roman";vertical-align:super;color:#000000">n;font-family:"Times New Roman";color:#000000"> (1 + f r),

;font-family:"Times New Roman";color:#000000">or

;font-family:"Times New Roman";color:#000000">FV = PV (1 + r);font-family:"Times New Roman";vertical-align:super;color:#000000">n;font-family:"Times New Roman";color:#000000"> (1 + t r / T);font-family:"Times New Roman";color:#52594f;display:none">;font-family:"Times New Roman";color:#52594f">.

;font-family:"Times New Roman";color:#000000">
;font-family:"Times New Roman"">3 COMPARISON OF SIMPLE AND COMPOUND INTEREST METHODS

">Let's take a closer look at the second and third reasons (since the first is obvious). If we combine the debt growth graphs given in the previous paragraph, we get the following picture:

;color:#000000">
">Comparison of debt growth charts using simple and compound interest methods.

">Thus, if the same interest rate is used, then:

;font-family:"Times New Roman";color:#000000">for periods of time less than a year, the debt found using the simple interest method will always be greater than the debt found using the compound interest method;
;font-family:"Times New Roman";color:#000000">for periods of time greater than a year, on the contrary, the debt found using the compound interest method will always be greater than the debt found using the simple interest method;
;font-family:"Times New Roman";color:#000000">well, and, of course, for a period of time equal to one year, the results are the same.

">At the same time, if the interest rate is low and the time period is less than a year, then S;vertical-align:sub">sl ">(t) and S ;vertical-align:sub">pr ">(t) are quite close to each other. However, one must always remember that if these conditions are not met, then the discrepancies in the results can be significant!

">Example
In the early 90s, during a period of strong inflation, Russian banks offered very high interest rates on ruble deposits and loans, amounting to hundreds of percent.

">As an example, let's see what discrepancies can result from using simple interest for a semi-annual deposit, when the interest rate is 300% per annum. If the deposit size is S rubles, then after six months the depositor's account will have the amount

" xml:lang="en-US" lang="en-US">\

">If the bank used compound interest, the total amount would be

" xml:lang="en-US" lang="en-US">\

">The difference in results is ½S, or 25% relative to the complex result.

;font-family:"Times New Roman"">4 COMBINED INTEREST CALCULATION SCHEMES

">In practice, for long, but not entire periods of time, particularly scrupulous lenders sometimes use a combined interest calculation scheme. In this case, for a whole number of years, the compound interest method is used, and for a non-integer “remainder”, the simple interest method. For example, if a loan of size 1 million rubles issued for 3 years and 73 days (73 days this is 0.2 non-leap years) at 10% per annum, then the total debt can be found in the following way:

;color:#000000" xml:lang="en-US" lang="en-US">$S(3,2) = (1+0,1)^3 \cdot (1+0,1 \ cdot 0.2) \cdot 1\ 000\ 000 = 1\ 357\ 620$;color:#000000">rubles ;color:#000000" xml:lang="en-US" lang="en-US">.

">The combination of simple and compound interest can also naturally arise when the same short-term operation is repeated many times. For example, banks offer their clients short-term deposits for periods ranging from a month to a year. During the validity period of the deposit agreement, an increase in the amount by the depositor's account follows a simple scheme: at the end of the deposit period, capitalization occurs (interest money is added to the original amount). If the client does not withdraw the money, the deposit agreement is extended for a new period and the increased amount becomes the basis for calculating interest. From the point of view of a bank client, the amount of a deposit left for several periods will grow according to the compound interest scheme:

">where t the duration of that very “basic” contribution, and n the number of periods.

">Example
A certain bank offers its clients time deposits for a period of six months at a simple interest rate of 10% per annum. If a client of this bank deposited 200,000 rubles, and then extended the deposit agreement twice, then after a year and a half he withdrew from his account

;color:#000000" xml:lang="en-US" lang="en-US">$S(1,5) = (1+0,1 \cdot \frac(1)(2))^ 3 \cdot 200\ 000 = 231\ 525$;color:#000000">rubles ;color:#000000" xml:lang="en-US" lang="en-US">.

;font-family:"Times New Roman"">5 NOMINAL INTEREST RATE

">From this paragraph we begin to consider the compound interest method, which is not as often used in lending as the simple interest method, but is widespread in other areas of finance. In particular, the compound interest method is used to calculate interest on long-term deposits (lasting more than a year ).

"> Let me remind you that the meaning of this method is expressed by the phrase “accrual of interest on interest.” This means that the debt of the borrower at the previous point in time serves as the basis for calculating interest at the next moment. In this case, the amount of debt increases exponentially (or in accordance with the exponential function, if we consider time continuous). For example, if a depositor deposited 100 thousand rubles in a bank at a compound interest rate of i = 6%, then after, say, five months there will be an amount in his account

;color:#000000">S(5/12) = (1 + i);vertical-align:super;color:#000000">5/12;color:#000000">S ;vertical-align:sub;color:#000000">0;color:#000000"> = 1.06 ;vertical-align:super;color:#000000">5/12;color:#000000"> · 100,000 ≈ 102,458 rubles.

;font-family:"Times New Roman"">5.1 The concept of nominal interest rate

">It is clear that without special equipment it is not very convenient to make such calculations, and until recently this was only possible with the help of special tables with taboo multipliers. To avoid the need to extract cumbersome roots when calculating using compound interest, to set compound interest In practice, so-called nominal interest rates are used. Their essence is as follows.

">If you deposited money in a bank, then interest on the deposit will not be accrued continuously, but with some frequency - once a year, quarter, month or even day. This process of accruing interest money and adding it to the deposit amount is called “interest capitalization” So, let’s say that interest capitalization occurs m times a year. Then, if j the nominal interest rate on the deposit is known, then each time interest is accrued, the amount in the depositor’s account will increase by (1 + \dfrac(j)(m). )\) once.

">It is clear that in essence we are talking here about the use of a combined scheme of simple and compound interest.

">Example
The depositor deposited an amount of 200 thousand rubles into a bank account. If the nominal interest rate on a deposit is 8%, and the interest is capitalized once a quarter (the bank, of course, uses compound interest), then after six months (that is, after two interest charges) the amount in the depositor’s account will be

;color:#000000">200,000 · (1 + 0.08/4);vertical-align:super;color:#000000">2;color:#000000"> = 208,080 rubles.

;font-family:"Times New Roman"">5.2 Effective interest rate

">If a nominal interest rate is specified, and interest capitalization is carried out m times a year, then over the year the deposit amount will increase by

" xml:lang="en-US" lang="en-US">$\left(1+ \dfrac(j)(m) \right)^m$

">times.

">Since, on the other hand, the relation for a compound interest rate must always be satisfied:

" xml:lang="en-US" lang="en-US">S(1) = (1+ i) S;vertical-align:sub" xml:lang="en-US" lang="en-US">0

">then

" xml:lang="en-US" lang="en-US">\[\tag(15.1) i = \left(1+ \frac(j)(m) \right)^m - 1\]

">The compound interest rate found in this way is called “effective”, since it, unlike the nominal rate, characterizes the real profitability (efficiency) of the loan operation.

">Example
If the nominal interest rate on a deposit is 18%, and interest is compounded every month, then the effective interest rate will be

;color:#000000" xml:lang="en-US" lang="en-US">$i = \left(1+ \dfrac(0.18)(12) \right)^(12) - 1\approx 0.1956 = 19.56\%$;color:#000000">per annum;color:#000000" xml:lang="en-US" lang="en-US">,

">that is, one and a half percent more than stated.

">Generally speaking, the effective interest rate is always greater than the nominal interest rate. This is easy to verify by expanding the right-hand side of relation (15.1) using the Newton binomial formula.

;font-family:"Times New Roman"">5.3 Continuous compounding

">As is known, for the number x tending to infinity there is a limit

" xml:lang="en-US" lang="en-US">\[\lim_(x \to \infty) \left(1 + \frac(1)(x) \right)^x = e, \]

">where e = 2.718281828... the base of natural logarithms. This formula is called the second remarkable limit. It follows, in particular, that the relation is true

">\[\ " xml:lang="en-US" lang="en-US">lim">_{ " xml:lang="en-US" lang="en-US">m"> \ " xml:lang="en-US" lang="en-US">to"> \ " xml:lang="en-US" lang="en-US">infty">} \ " xml:lang="en-US" lang="en-US">left">(1 + \ " xml:lang="en-US" lang="en-US">frac">{ " xml:lang="en-US" lang="en-US">j">}{ " xml:lang="en-US" lang="en-US">m">} \ " xml:lang="en-US" lang="en-US">right">)^ " xml:lang="en-US" lang="en-US">m"> = " xml:lang="en-US" lang="en-US">e">^ " xml:lang="en-US" lang="en-US">j">\]

">This means that if interest capitalization is carried out quite often, for example, daily, then the effective interest rate can be approximately found as follows:

">\[\ " xml:lang="en-US" lang="en-US">tag">{15.2} " xml:lang="en-US" lang="en-US">i"> \ " xml:lang="en-US" lang="en-US">approxe">^ " xml:lang="en-US" lang="en-US">j"> - 1\]

">Example
Again, we will assume that the nominal interest rate on the deposit is 18%, but interest is capitalized daily (m = 365). The exact value of the effective interest rate, found using formula (15.1), will be equal to

">If you use the approximate formula (15.2), you can get the following result:

;color:#000000">i ≈ e ;vertical-align:super;color:#000000">0.18;color:#000000"> 1 = 0.197217...

">As you can see, the discrepancy is quite small.

6 Interest charges

;font-family:"Times New Roman";color:#000000">To calculate interest on deposits, and loans too, the following interest formulas are used:

;font-family:"Times New Roman";color:#000000">simple interest formula,
;font-family:"Times New Roman";color:#000000">compound interest formula.

;font-family:"Times New Roman";color:#000000">The procedure for calculating interest on formulas is carried out using a fixed or floating rate.

;font-family:"Times New Roman";color:#000000">A fixed rate is when the interest rate established on a bank deposit is fixed in the deposit agreement and remains unchanged for the entire period of investment, i.e. is fixed. Such a rate can change only at the time of automatic extension of the contract for a new term or upon early termination of the contractual relationship and payment of interest for the actual period of the investment at the “on demand” rate, which is stipulated by the conditions.

;font-family:"Times New Roman";color:#000000">A floating rate is when the interest rate initially established under the agreement can change throughout the entire investment term. The conditions and procedure for changing rates are stipulated in the deposit agreement. Interest rates can change : due to changes in the refinancing rate, changes in the exchange rate, the transfer of the deposit amount to another category, and other factors.

;font-family:"Times New Roman";color:#000000">To calculate interest using formulas, you need to know the parameters for investing funds in a deposit account, namely:

;font-family:"Times New Roman";color:#000000">deposit amount,
;font-family:"Times New Roman";color:#000000">interest rate on the selected deposit),
;font-family:"Times New Roman";color:#000000">cyclical interest calculation (daily, monthly, quarterly, etc.),
;font-family:"Times New Roman";color:#000000">deposit term,
;font-family:"Times New Roman";color:#000000">sometimes the type of interest rate used is also required - fixed or floating.

;font-family:"Times New Roman";color:#000000">The simple interest formula is applied if the interest accrued on the deposit is added to the deposit only at the end of the deposit period or is not added at all, but is transferred to a separate account, i.e. the calculation of simple interest does not provide for the capitalization of interest. When choosing the type of deposit, it is worth paying attention to the procedure for calculating interest. When the deposit amount and the placement period are significant, and the bank uses the simple interest formula, this leads to an underestimation of the amount of interest income of the depositor.

;font-family:"Times New Roman";color:#000000">The formula for simple interest on deposits looks like this:

;font-family:"Times New Roman";color:#000000">S the amount of funds due to be returned to the depositor at the end of the deposit period. It consists of the original amount of funds placed, plus accrued interest.

;font-family:"Times New Roman";color:#000000">

;font-family:"Times New Roman";color:#000000" xml:lang="en-US" lang="en-US">t;font-family:"Times New Roman";color:#000000"> - the number of days for accrual of interest on the attracted deposit.

;font-family:"Times New Roman";color:#000000">

;font-family:"Times New Roman";color:#000000">P the initial amount of funds attracted to the deposit.

;font-family:"Times New Roman";color:#000000">

;font-family:"Times New Roman";color:#000000">If interest accrued on a deposit is added to the deposit at regular intervals (daily, monthly, quarterly), then in these cases the amount of interest is calculated using the compound interest formula. Compound interest provides for the capitalization of interest (accrual of interest on interest). To calculate compound interest, you can use two formulas for compound interest on deposits, which look like this:

;font-family:"Times New Roman";color:#000000">I annual interest rate.

;font-family:"Times New Roman";color:#000000">t number of days for accrual of interest on the attracted deposit.

;font-family:"Times New Roman";color:#000000">K number of days in a calendar year (365 or 366).

;font-family:"Times New Roman";color:#000000">P the amount of funds attracted to the deposit.

;font-family:"Times New Roman";color:#000000">Sp amount of interest (income).

;font-family:"Times New Roman";color:#000000">n number of interest periods.

;font-family:"Times New Roman";color:#000000">S amount of the deposit (deposit) with interest.

;font-family:"Times New Roman";color:#000000">However, when calculating interest, it is easier to first calculate the total amount of the deposit with interest, and only then calculate the amount of interest (income).;font-family:"Times New Roman"">
REFERENCES

;font-family:"Times New Roman"">Techniques of financial and economic calculations: Textbook. M.: Finance and Mathematics, 2000. 80 pp.: ill.
;font-family:"Times New Roman"">John C. HullChapter 4. Interest Rates // Options, Futures and Other Derivatives = Options, FuturesandOtherDerivatives. 6th ed. M.:;font-family:"Times New Roman"">"Williams";font-family:"Times New Roman"">, 2007. P. 133-165.
;font-family:"Times New Roman"">http://forexaw.com/Cont-Economy/
;font-family:"Times New Roman"">http://www.bibliotekar.ru/
;font-family:"Times New Roman"">http://ru.wikipedia.org/

;font-family:"Times New Roman"">
CONCLUSION

;font-family:"Times New Roman"">Currently, in conditions of economic stabilization, the niche of bank lending services for the Russian market has not yet been filled, i.e. lending can be identified as the most promising means of generating income for banks.

;font-family:"Times New Roman"">In the conditions of stabilization of the economy, there has been a tendency to increase the volume of borrowings in industry and banks to attract potential borrowers. It is necessary to determine the value of the lending interest rate as the most important factor influencing the borrower’s choice of a particular bank , and, therefore, it is necessary to consider in more detail the components that form the interest rate and affect the cost of loans.

;font-family:"Times New Roman"">Also, in conditions of stabilization of the economy, it becomes possible to expand such a promising direction, which has enormous potential lending to the consumer sector. And here the interest rate also plays a decisive role in attracting private borrowers.

;font-family:"Times New Roman"">
PRACTICAL PART

;font-family:"Times New Roman"">Task 1

;font-family:"Times New Roman"">The bank offers 17% per annum for placing funds on deposit accounts it opens. Using the discounting formula, calculate the size of the initial deposit so that after 4 years you will have 180 thousand rubles in the account.

;font-family:"Times New Roman"">Solution

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">S = P * (1+i);font-family:"Times New Roman";vertical-align:super" xml:lang="en-US" lang="en-US">n

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">180,000 = P * (1+0.17);font-family:"Times New Roman";vertical-align:super" xml:lang="en-US" lang="en-US">4

;font-family:"Times New Roman"">180;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 =;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">P;font-family:"Times New Roman""> * 1.8738

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">P;font-family:"Times New Roman""> = 96;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">061rub.

;font-family:"Times New Roman"">Answer: in order to have 180 thousand rubles on your deposit after 4 years, it is necessary that the size of the initial deposit be 96,061 rubles.

;font-family:"Times New Roman"">Task 2

;font-family:"Times New Roman"">A citizen received a mortgage loan from a bank in the amount of 1.5 million rubles for a period of 8 years on the following conditions: for the first year, the compound interest rate is 14% per annum; for the next two years the margin is set at 0.5% and for subsequent years the margin is 0.7%. Find the amount that the citizen must return to the bank at the end of the loan term.

;font-family:"Times New Roman"">Solution

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">S = P×((1+i1)*n1 +(1+i2)*n2 + … +(1+ik)*nk)

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">S;font-family:"Times New Roman""> = 1;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">500;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 × ((1+0.14) + (1+0.145)*2 + (1+0.152)*5)) = 1;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">500;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 *9.19 = 13;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">785;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 rubles.

;font-family:"Times New Roman"">Answer: at the end of the loan term, the citizen must return 13.785 million rubles to the bank.

;font-family:"Times New Roman"">Task 3

;font-family:"Times New Roman"">The organization, having available funds in the amount of 2 million rubles, intends to invest them for a period of 5 years. There are two investment options, determine the more profitable one:

;font-family:"Times New Roman"">a) funds are deposited into a deposit account in a bank with interest accrued every 6 months at a rate of 18% per annum;

;font-family:"Times New Roman"">b) the funds are transferred to another organization as a loan with an interest rate of 24% annually.

;font-family:"Times New Roman"">Solution

;font-family:"Times New Roman">a);font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">S;font-family:"Times New Roman""> = 2,000;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 * (1+0.18/2);font-family:"Times New Roman";vertical-align:super">10;font-family:"Times New Roman">= 2;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 * 2.37= 4,740,000 rub.

;font-family:"Times New Roman">b);font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">S;font-family:"Times New Roman""> = 2;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 * (1+0.24);font-family:"Times New Roman";vertical-align:super">5;font-family:"Times New Roman">= 2;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 * 2.93 = 5;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">860;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 rub.

;font-family:"Times New Roman"">Answer: the second option is more profitable.

;font-family:"Times New Roman"">Task 4

;font-family:"Times New Roman"">Determine the required deposit amount in the present in order to have savings in the amount of 150 thousand rubles in two years. The annual interest rate is 11%, interest is calculated once a quarter according to the compound interest scheme.

;font-family:"Times New Roman"">Solution

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">S = P * (1+i/m);font-family:"Times New Roman";vertical-align:super" xml:lang="en-US" lang="en-US">m*n

;font-family:"Times New Roman"">150;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 =;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">P;font-family:"Times New Roman">*;font-family:"Times New Roman";vertical-align:super">;font-family:"Times New Roman">(1+0.11/4);font-family:"Times New Roman";vertical-align:super">4*2

;font-family:"Times New Roman"">150;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">000 =;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">P;font-family:"Times New Roman">* (1+0.0275);font-family:"Times New Roman";vertical-align:super">8;font-family:"Times New Roman"">

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">P;font-family:"Times New Roman""> = 120;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">;font-family:"Times New Roman"">968

;font-family:"Times New Roman"">Answer: the required deposit amount is 120,968 rubles.

;font-family:"Times New Roman"">Task 5

;font-family:"Times New Roman"">Six months after concluding a financial agreement to receive a loan, the debtor is obliged to pay 317 thousand rubles. What is the initial amount of the loan if it is issued at 18% per annum and simple interest is calculated with an approximate number of days?

;font-family:"Times New Roman"">Solution

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">S =P × (1+n×i)

;font-family:"Times New Roman"">where;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">S;font-family:"Times New Roman""> - accumulated amount,

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">P;font-family:"Times New Roman""> - amount of debt,

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">n;font-family:"Times New Roman""> - period (fraction of a year),

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">i;font-family:"Times New Roman""> - interest rate.

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">P;font-family:"Times New Roman"> =;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">S;font-family:"Times New Roman">/ (1+;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">n;font-family:"Times New Roman"">×;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">i;font-family:"Times New Roman">)

;font-family:"Times New Roman"" xml:lang="en-US" lang="en-US">n;font-family:"Times New Roman""> = 180/360 = 0.5.

;font-family:"Times New Roman"">Р = 317,000 / (1 + 0.5×0.18) = 317,000 /1, 09 = 290,826 rubles.

;font-family:"Times New Roman"">Answer: the initial loan amount was 290,826 rubles.

Brief theoretical background

In medium- and long-term financial and commercial transactions, interest may not be paid immediately after it accrues, but may be added to the amount of debt. In this case, compound interest is used for growth.

When accruing compound interest (compound interest) a method is adopted in which the interest calculation base is taken as the amount received at the previous stage of accumulation or discounting. In this case, it is often said that interest accrues on interest.

Unlike simple interest, the basis for calculating compound interest does not remain constant, but increases with each step in time. Compound interest accumulation is the sequential reinvestment of funds invested at simple interest for one accrual period.

The accrued amount at compound interest is calculated using the formula S=P( 1+r) t, Where t – number of accrual periods.

Example 2.1. What value will the amount of debt reach, equal to 1 million rubles? in five years with growth at a compound rate of 15.5% per annum?

S=1,000,000(1+0.155) 5 =2,055,464.22 rub. 

Contracts usually specify an annual rate r and the number of interest accruals m during a year. This means that the base period is one year divided by m, and the compound interest rate for the period is equal to r/ m. The formula for compound interest, taking into account the signs of Excel financial functions, will take the form: S+P( 1+ r/ m) t = 0. Parameter t measured in periods. If accrual occurs k years, then the formula takes the form S+P( 1+r/ m) km =0.

In addition to interest rates fixed over time, "floating" rates (floating rate). The amount of increase with variable rates is determined by the formula: , where
– time-consistent interest rates;
– periods of validity of the corresponding rates.

Example 2.2. The loan was issued for 5 years. The fixed part of the interest rate is set at 12% per annum plus premiums (margin) of 0.5% in the first two years and 0.75% in the remaining years. Find the growth factor.

The growth multiplier will be:

q= (1+0.125) 2 (1+0.1275) 3 =1.81407 

Often the interest period is not a whole number of years. In this case, two methods are used for calculation. With the general method, the calculation is carried out using the compound interest formula. With the mixed method, interest is calculated for an integer number of years using the compound interest formula, and for the fractional part of the period - using the simple interest formula:
, Where a+ b= t; a – integer number of periods; b– fractional part of the period t.

Work order

To calculate problems of calculating compound interest, we use the same algorithm and financial functions as for simple interest.

In cell B1 we place the value of the initial contribution value. In cells B2:G2 we will place the numbers 0, 1,..., 5, in cells AZ:A7 we will place the values 10%, 20%,..., 50% (these numbers are entered using techniques for generating arithmetic progressions). It is necessary to tabulate a function of two variables (interest rate and number of years), depending on the parameter - the initial deposit. Let's enter the formula =BS ($AZ, B$2, -$B$1) into cell ВЗ. The formula is copied to the remaining cells of the interval B3:G7. 

Example 2.4. A loan of $20,000 was given for a year and a half at an interest rate of 28% per annum with quarterly accrual. Determine the amount of the final payment.

Here the base period is a quarter. The loan term is 6 periods (4 quarters per year, term one and a half years), 7% = 28%/4 is charged per period. Then the formula that gives the solution to the problem is: = BC (28% / 4.4 * 1.5, 20000). It returns the result -$30014.61. 

Tasks

4. The bank accepts deposits for a period of 3 months with an announced annual rate of 100% or for 6 months at 110%. What is more profitable to invest money for six months: twice for three months or once for 6 months?

5. Amount 2000 rub. placed at 9% per annum for 3 years. Interest is calculated quarterly. What amount will be in the account?

6. What is the amount of debt after 26 months if its original amount is $500,000, compound interest, 20% per annum, compounded quarterly? Carry out calculations using general and mixed methods.

7. A loan in the amount of 250 million rubles was received from the bank. The annual interest rate is 9.5% with an assumed year length of 360 days. Calculate the amount of accumulated debt using general and mixed methods for different lending periods, the duration of which is:

equal to the whole number of years (without fractional part) – 3 years;

equal to one year;

equal to less than a year – 0.25 years;

equal to an integer number of years + fraction of a year – 2 years and 270 days.

Compare the obtained values according to the options and identify patterns in the differences in results.

The scope of application of simple interest is most often short-term transactions (with a period of up to one year) with one-time interest accrual (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 adjusted to the term of investment of funds. Mathematically, the intermediate interest rate is equal to a fraction of the annual interest rate. The formula for increasing simple interest using an intermediate interest rate is as follows:

FV = PV (1 + f * r),

FV = PV (1 + t * r / T),

t -- the period for investing 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 accrual of simple interest is calculated using the formula:

FV = PV (1 + r * n),

where n is the period of investment of funds (in years). ,

Application of compound interest

The scope of application of compound interest is long-term transactions (with a period exceeding a year), including those involving intra-annual interest accrual.

In the first case, the usual formula for calculating compound interest is applied:

FV = PV (1 + r)n.

In the second case, the formula for calculating compound interest is applied, taking into account intra-annual accrual. Intra-annual compounding refers to 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:

1) six-monthly (m = 2);
2) quarterly (m = 4);
3) monthly (m = 12);
4) daily (m = 365 or 366);
5) continuous (m -» ?).

The compounding 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;

r -- annual interest rate;

n -- number of years;

m -- number of intra-annual accruals;

FV -- accumulated amount.

Interest income with continuous compounding is calculated using the following formula:

where: e = 2, 718281 -- transcendental number (Euler number);

e?n -- increment multiplier, which is used for both integer and fractional values of n;

A special designation for the interest rate with continuous compounding (continuous interest rate, “growth force”);

n -- number of years.

With the same initial amount, the same investment period and interest rate, the returned amount turns out to be greater when using the intra-annual accrual formula than when using the usual compound interest formula:

FV = PV (1 + r / m)nm> FV = PV (1 + r)n.

If the income obtained using intra-annual compounding is expressed as a percentage, the resulting interest rate will be higher than that used with conventional compounding.

Thus, the initially stated annual interest rate for compounding, called the nominal rate, does not reflect the actual performance of the transaction. The interest rate that reflects the actual income received is called effective. The classification of interest rates for intra-annual compounding 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 increasing compound interest, you can obtain the formula for the effective interest rate:

FV = PV (1 + r)n;

(1 + re) = FV / PV.

Here is the formula for increasing compound interest with intra-annual accruals, in which r/m interest 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 -- number of intra-annual payments.

The effective interest rate depends on the number of intra-annual accruals (m):

1) when m = 1, nominal and effective interest rates are equal;
2) the greater the number of intra-annual accruals (the value of m), the greater the effective interest rate.

The area of simultaneous application of simple and compound interest is long-term operations, the term of which is a fractional number of years. In this case, interest can be calculated in two ways:

1) calculation of compound interest with a fractional number of years;
2) interest accrual according to a mixed scheme.

In the first case, the compound interest formula is used for calculations, which involves raising 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 formula for calculating compound interest with an integer number of years and a formula for calculating simple interest for short-term operations:

FV = PV (1 + r)n * (1 + f * r),

FV = PV (1 + r)n * (1 + t * r / T) .