The power of growth with continuous compounding. Relationship between discrete and continuous interest rates

Federal Agency for Education and Science

State educational institution of higher education

vocational education

Tambov State University named after G.R. Derzhavina

on the topic: “Actions with continuous interest”

Performed

5th year student, group 502

Full-time education Geghamyan M.A.

Tambov 2013

1.Consistent growth force<#"justify">1. Constant growth force

When using a discrete nominal rate<#"55" src="doc_zip1.jpg" />

When moving to continuous percentages we get:

Growth multiplier<#"20" src="doc_zip4.jpg" />, we get:

because discrete and continuous rates are functionally related to each other, then we can write the equality of the increment multipliers

For initial capital 500 thousand rubles. compounded interest - 8% per annum for 4 years. Determine the accrued amount if interest accrues continuously.

Discounting based on continuous interest rates

In formula (4.21) we can determine the modern value

The continuous interest rate used in discounting is called the discount rate. It is equal to the growth force, i.e. used to discount discount force or growth force<#"justify">Example

Determine the modern cost of payment, provided that discounting is carried out at a growth rate of 12% and at a discrete complex discount rate of the same size.

Variable growth force

Using this characteristic, processes of increasing amounts of money with a changing interest rate are modeled. If the growth force is described by some continuous function of time, then the formulas are valid.

For the accrued amount:<#"47" src="doc_zip13.jpg" />

Modern cost:

) Let the power of growth<#"25" src="doc_zip15.jpg" />at time intervals, then upon expiration of the loan term the accrued amount will be:

If the growth period is equal to n, and the average growth value is: , then

Determine the accrual multiplier for continuous compounding of interest for 5 years. If the growth force changes discretely and corresponds to: 1 year - 7%, 2 and 3 - 8%, last 2 years - 10%.

2)The growth force changes continuously over time and is described by the equation:

where is the initial growth force (at)

a - annual increase or decrease.

Let's calculate the degree of the increase multiplier:

The initial value of the growth force is 8%, the interest rate is continuous and changes linearly.

The increase per year is 2%, the growth period is 5 years. Find the growth factor.

) The growth force changes exponentially, then

Growth multiplier:<#"50" src="doc_zip29.jpg" />

Determine the growth multiplier for continuous compounding of interest for 5 years, if the initial growth rate is 10% and the interest rate increases annually by 3%.

The loan term is determined by the formulas:

when increasing at a constant rate

when increasing at a changing rate, when it changes in geometric progression

Determine the period required to increase the initial rate by 3 times when accruing at a continuous interest rate changing with a constant growth rate, if the initial rate is 15% and the annual growth rate is 1.05

Interest rate equivalence

Rates that ensure equivalence of financial consequences are called equivalent or relative.

Equivalence of financial consequences can be ensured if there is equality of increase multipliers<#"23" src="doc_zip36.jpg" />;

2) increased amount<#"41" src="doc_zip37.jpg" />

If, then the increment factors are equal

If the loan term is less than a year, then equivalence is determined for two cases of equal time bases and different time bases.

If the time bases are the same (), then the formulas look like:

If interest is calculated at rate i with a base of 365, and at rate d with a base of 360, then the following is true:

The bill was discounted in the bank at a discount rate of 8% on the day of expiration of its circulation period = 200 (k=360). Determine the profitability of this operation at the simple interest rate (k=365).

Equivalence of simple and compound interest rates

When interest is calculated once a year, it is determined by the formulas:

Simple bet:

complex bet:

What compound annual rate can replace the simple rate of 18% (k=365) without changing the financial consequences. The duration of the operation is 580 days.

Equivalence of a simple interest rate and a compound rate.

When calculating m times a year, it is determined by the formula:

When developing the terms of the contract, the parties agreed that the loan yield should be 24%. What should be the size of the nominal rate when interest is calculated monthly, quarterly.

The equivalence of the simple discount rate and the compound interest rate is determined by the formula:

The equivalence of the nominal compound interest rate when interest is calculated m times a year and the simple discount rate is determined by the formulas:

The equivalence of complex bets is determined by the formulas:

The equivalence of the compound discount rate and the nominal compound interest rate when interest is calculated m times a year is determined by the formulas:

Equivalence of continuous and discrete rates:

Equivalence of growth force and nominal rate:

With a discrete and linear change in force growth, as well as if it changes at a constant rate, the equivalent relationship with compound interest rates can be expressed by the formulas:

Strength equivalence<#"41" src="doc_zip68.jpg" />

For a complex discount rate:

Comment. Using formulas for the equivalence of discrete and continuous rates, it is possible to present the results of applying continuous interest in the form of generally accepted characteristics.

Average values in financial calculations

For multiple interest rates<#"63" src="doc_zip72.jpg" />

During the year, the company received 2 equal-sized loans of 500 thousand rubles. every. 1 loan for 3 months at 10% per annum. 2 loan - for 9 months at 16% per annum. Determine the average interest rate, check the result by calculating the accrued amounts.

When receiving loans of different sizes issued at different interest rates, the average rate is also calculated using the weighted average formula with weights equal to the products of the amounts of loans received and the terms they were issued.

Calculation of the average simple discount rate<#"67" src="doc_zip78.jpg" />

Average compound interest rate<#"37" src="doc_zip79.jpg" />

When analyzing the work of credit institutions, the following indicators are calculated: the average loan size, its average duration, the average number of loan turnovers and other indicators.

The average size of one loan, excluding the number of turnovers per year, is calculated using the formula:

Taking into account the number of revolutions per year according to the formula:

where is the number of revolutions,

Length of period

K is the number of clients who received loans.

The average size of all loans, taking into account the number of turnovers per year, shows the outstanding balance of all loans for the year. It is equal to the average size of one loan, taking into account turnover per year, multiplied by the number of clients who received the loan:

where is the total turnover, i.e. the amount of repaid loans repaid during the period.

The average balance of all loans, taking into account the number of turnovers per year, is determined by the formula of the average chronological moment series according to the monthly balance sheets of the credit institution that issued the loan according to the formula:

where is the monthly balance of issued loans.

The number of turnovers of individual loans, subject to their continuous turnover during the period under study, is determined as the quotient of dividing the duration of the period by the term of the loan.

The average number of turnovers of all loans for the period, provided that their continuous turnover occurs, is calculated using a formula based on the availability of data.

The average loan term of individual loans or all loans as a whole is calculated using various formulas

equivalence conversion discounting rate

Financial equivalence of obligations and conversion of payments

Replacing one monetary obligation with another or combining several payments into one is based on the principle of financial equivalence of obligations.

Equivalent payments are considered to be payments that, when brought to the same point in time, turn out to be equal. It follows from the accumulation and discounting formulas. Two amounts are considered equal if their modern values at one point in time are the same; with an increase in the interest rate, the sizes of modern values decrease. The rate at which is called critical or barrier. It is derived from equality.

In the case of a compound interest rate, the barrier rate is calculated using the formulas:

The principle of financial equivalence applies to various changes in the conditions for payment of monetary amounts. A general method for solving such problems is to develop an equivalence equation in which the amount of replaced payments reduced to a certain point in time is equated to the amount of payments under the new obligation reduced to the same date. For short-term obligations, simple is used, for medium and long-term - complex.

One of the common cases of changing the terms of contracts is consolidation, i.e. consolidation of payments. There are 2 possible formulations of the problem:

)A deadline is given and you need to find the amount of payment;

)The amount of the consolidated payment is given; it is necessary to determine its term.

When consolidating several payments into one, provided that the term of the new payment is longer than the previously established term, the equivalence equation is written as:

Where is the accumulated amount of the consolidated payment,

Payments subject to consolidation

Time intervals between and:

In general, the amount of the consolidated payment will look like:

Amounts of combined payments, terms of repayment of which are less than the first term; - amounts of combined payments with terms exceeding the new term.

When consolidating bills<#"27" src="doc_zip115.jpg" />

When consolidating payments using a compound interest rate, the consolidated amount is found using the formulas:

If the amount of the consolidated payment is known and it is necessary to determine the period of its consolidation, maintaining the principle of equivalence:

where is the consolidated amount of the modern payment. If the partners agree to consolidate payments without changing the total amount of payments, then the term of the consolidated payment:

To calculate the deadline for payment of consolidated payments, discount rates may be used,<#"45" src="doc_zip122.jpg" />

When using compound interest, the formulas look like:

Bibliography

1.Kochovic E. Financial mathematics: Theory and practice of financial and banking calculations. - M.: Finance and Statistics, 2004

2.Krasina F.A. Financial calculations - Financial calculations: textbook / F. A. Krasina. - Tomsk: El Content, 2011.

3.Selezneva N.N., Ionova A.F. Financial management. Tasks, situations, tests, schemes: Proc. manual for universities. - M.: UNITY-DANA, 2004. - 176 p.

When using a discrete nominal rate, the accrued amount is determined by the formula:

When moving to continuous percentages we get:

Increase multiplier for continuous interest capitalization.

Denoting the growth force through, we get:

because discrete and continuous rates are functionally related to each other, then we can write the equality of the increment multipliers

To the initial capital 500 thousand rubles. compounded interest - 8% per annum for 4 years. Determine the accrued amount if interest accrues continuously.

Discounting based on continuous interest rates

In formula (4.21) we can determine the modern value

The continuous interest rate used in discounting is called the discount rate. It is equal to the growth force, i.e. used for discounting, discount forces or growth forces lead to the same result.

Define the modern cost of payment, provided that discounting is carried out at a growth rate of 12% and at a discrete complex discount rate of the same size.

Variable growth force

For the accrued amount:

Modern cost:

1) Let the growth force change discretely and take the values: at time intervals, then at the end of the loan period the accumulated amount will be:

If the growth period is equal to n, and the average growth value is: , then

Determine the accrual multiplier for continuous compounding of interest for 5 years. If the growth force changes discretely and corresponds to: 1 year - 7%, 2 and 3 - 8%, last 2 years - 10%.

2) The growth force changes continuously over time and is described by the equation:

where is the initial growth force (at)

a - annual increase or decrease.

Let's calculate the degree of the increase multiplier:

Initial value growth force is 8%, the interest rate is continuous and changes linearly.

The increase per year is 2%, the growth period is 5 years. Find the growth factor.

3) The growth force changes exponentially, then

A discretionary interest rate is a rate at which interest is compounded over predetermined, or specified, periods. If you reduce the interest accrual period to an infinitesimal value (the period for which accruals will be made tends to zero, and the number of interest accruals tends to infinity), then interest will accrue continuously. In this case the interest rate is called continuous rate or force of growth .

In theoretical studies and in practice, when payments are made repeatedly, it is convenient to use the continuous method of calculating interest. The transition to the limit can be carried out similarly to how it was done in paragraph 2.2 when deriving formula (2.12) or in the following way.

A continuous rate can be constant or variable. Consider the case when the continuous interest rate is different at different points in time.

Let a(t) be a function that describes the dependence of the continuous rate (growth force) on time t. The increase in capital S(t) at moment t over a period of time Δt is equal to:

S(t + Δt) – S(t) = a(t) Δt S(t)

Then, we have:

When Δt →0 we find that the rate of change of capital is proportional to capital. Then, the payment amount (capital) S(t) satisfies the linear homogeneous differential equation of the first order:

, (2.28)

– rate of change of payment (rate of change of capital);

S(t) - payment amount (capital);

a(t) – continuous accrual percentage or growth force.

In another form the equation will be written:

dS = a(t) S dt, (2.29)

i.e., the payment increment is proportional to the payment S itself and the time increment dt. The proportionality coefficient a(t) is the force of growth or the percentage of accrual.

Another possible representation of the differential equation is:

, (2.30)

i.e., the relative increment in the payment amount dS/S is proportional to the time increment dt. Moreover, as before, a(t) is determined by the accrual percentage and in the general case may depend on time. All three equations for capital (2.28), (2.29), (2.30) are equivalent.

Let's consider some of the simplest properties of capital, described by the differential equation (2.28)-(2.30). If the function a(t)>0 is positive, then with positive capital S>0 the derivative of capital dS/dt >0 is also positive and, therefore, capital S(t) grows. In this case a(t) is called continuous accrual percentage or growth force .

Otherwise, if the function a(t)<0 отрицательна, то при положительном капитале S>0 derivative of capital dS/dt<0 отрицательна и, следовательно, капитал S(t) убывает. В этом случае абсолютная величина |a(t)| называется continuous discount .

The solution to a linear differential equation is well known. Indeed, equation (2.30) is a separable equation and can be integrated:

Having calculated the integral, we get:

Where - indefinite integral of a(t),

C 1 is an arbitrary constant.

Hence, we have:

Finally, the general solution of the differential equation will be written as:

, (2.31)

where is a new arbitrary constant.

To determine an arbitrary constant WITH you need to know the capital at least at one point in time. If it is known that at time t=t 0 capital is equal to S = S 0 (i.e. S(t 0) = S 0), then an arbitrary constant WITH is easily determined from (2.31):

Substituting the result obtained into (2.31), we have:

Using the classical formula for the connection between the definite and indefinite integral (Newton–Leibniz formula):

we obtain a solution to the differential equation with initial conditions S(t 0)=S 0 in the form:

Often, time can be counted from the initial moment, then t 0 = 0 and the solution to the linear differential equation is written in the form:

, (2.32)

S(0) – initial amount at time 0;

S(t) – payment amount at time t.

Obviously, the given formulas for a(t)>0 correspond to the calculation of lending, and for a(t)<0 – расчету дисконтирования.

If the growth force is constant over the entire period of time under consideration, i.e. a(t)= r, then for the final payment at time t we have:

. (2.33)

Obviously, this formula coincides with the previously obtained limiting formula for continuous percentages (2.12).

Let's look at some examples of using these formulas.

Example 28.

Loan 200 thousand rubles. given for 2.5 years at an interest rate of 20% per annum with quarterly accrual. Find the final payment amount. Calculate using discrete and continuous percentages.

Solution.

The final payment amount satisfies the differential equation, where r=20%=0.2 in accordance with the annual accrual percentage and time t is measured in years. The solution to the linear equation is known:

Then the final payment amount is:

Thousand rub.

Calculation for the discrete case using formulas (2.11) gives:

Thousand rub.

It can be seen that with repeated calculations of small interest, the results of calculating the final payment amounts are close.

Let us now consider an example of calculating discounting in the continuous case.

Example 29.

Bill of exchange for 3 million rubles. with an annual discount rate of 10% and discounting 2 times a year, issued for 2 years. Find the initial amount that should be lent against this bill. Calculate using discrete and continuous percentages.

Solution.

The payment amount borrowed against the bill satisfies a linear differential equation, the solution of which is known:

Calculating the amount borrowed against a bill using discrete formulas (2.24) gives similar results:

million rubles

Thus, theoretical and practical calculations using continuous formulas give results close to the results of calculations using discrete formulas if the number of accruals is large and the accrual percentage is small.

Continuous interest is a term in theoretical economics that implies constant, systematic compounding of interest. If you delve into the basics of economic theory, then continuous interest is accrued at intervals that tend to the smallest number. That is, continuous interest is accrued continuously, but for the convenience of calculation, entrepreneurs or economists say that this or that amount is accrued per second, per hour or day. For example, Bill Gates' income can be called continuous interest income. Theoretic economists have calculated that Bill Gates, one of the richest people in the world, earns approximately $6,600 every minute - this is the amount of continuous interest from his business and investments that is converted into.

The meaning of continuous interest in theoretical and practical economics

Speaking about the importance of continuous interest, it should first be noted that they are a key form of passive income. In essence, passive income consists of two theoretical components: an asset that works without the intervention of an entrepreneur, and the continuous interest that it gives on the amount invested in it. For example, I bought an apartment for 10,000,000 rubles and rents it out at a price of 40,000 rubles per month - this is passive income. The annual income will be 480,000 rubles, from ten million this is 4.8 percent. It turns out that the entrepreneur continuously receives 4.8 percent per annum of the invested amount, this is his annual interest.

The second meaning is that continuous percentages indicate a stable situation in the development of a particular company. If it constantly brings interest, then it is working normally. If the receipt of interest is suspended, it can be judged that problems have occurred in the company. If interest rates rise and fall, this also indicates internal problems of the enterprise. Therefore, in the theory of economic analysis, continuous interest is very important.

The third value we will pay attention to is return on investment. The summation of continuously incoming interest will ultimately lead to the fact that investments in a business will pay off one hundred percent, that is, the entrepreneur will receive back the invested funds and will only have to receive. In economic theory there are many calls for analyzing various factors of economic life (inflation rates and so on) and comparing the results with continuous percentages. It may turn out that the income from the company, expressed as a percentage, will be lower than the percentage of depreciation of money and the like. If, for example, a person receives five percent per year from a deposit in a bank, and the amount is equal to eight percent, then ultimately the depositor loses three percent of his capital. Most people do not pay attention to this, which is a major economic mistake and the cause of many bankruptcies. This is especially important during periods of economic restructuring and disasters.

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Posted on http://www.allbest.ru/

Federal Agency for Education and Science

State educational institution of higher education

vocational education

Tambov State University named after G.R. Derzhavina

on the topic: “Actions with continuous interest”

Performed

5th year student, group 502

Full-time education Geghamyan M.A.

Tambov 2013

1. Constant growth force

2. Variable growth force

6. References

1. Constant growth force

When using a discrete nominal rate, the accrued amount is determined by the formula:

When moving to continuous percentages we get:

Increase multiplier for continuous interest capitalization.

Denoting the growth force through, we get:

because discrete and continuous rates are functionally related to each other, then we can write the equality of the increment multipliers

Example

To the initial capital 500 thousand rubles. compounded interest - 8% per annum for 4 years. Determine the accrued amount if interest accrues continuously.

Discounting based on continuous interest rates

In formula (4.21) we can determine the modern value

The continuous interest rate used in discounting is called the discount rate. It is equal to the growth force, i.e. used for discounting, discount forces or growth forces lead to the same result.

Example

Define the modern cost of payment, provided that discounting is carried out at a growth rate of 12% and at a discrete complex discount rate of the same size.

2. Variable growth force

For the accrued amount:

Modern cost:

1) Let the growth force change discretely and take the values: at time intervals, then at the end of the loan period the accumulated amount will be:

If the growth period is equal to n, and the average growth value is: , then

Example

Determine the accrual multiplier for continuous compounding of interest for 5 years. If the growth force changes discretely and corresponds to: 1 year - 7%, 2 and 3 - 8%, last 2 years - 10%.

2) The growth force changes continuously over time and is described by the equation:

where is the initial growth force (at)

a - annual increase or decrease.

Let's calculate the degree of the increase multiplier:

Example

Initial value growth force is 8%, the interest rate is continuous and changes linearly.

The increase per year is 2%, the growth period is 5 years. Find the growth factor.

3) The growth force changes exponentially, then

Growth multiplier:

Example

Determine the increment multiplier with continuous compounding of interest for 5 years, if the initial growth rate is 10%, and the interest rate increases annually by 3%.

The loan term is determined by the formulas:

When increasing at a constant rate

When increasing at a changing rate, when it changes in geometric progression

Example

Determine the time required for increasing the initial rate by 3 times when accrued at a continuous interest rate changing with a constant growth rate, if the initial rate is 15% and the annual growth rate is 1.05

3. Equivalence of interest rates

Rates that ensure equivalence of financial consequences are called equivalent or relative.

Equivalence of financial consequences can be ensured if there is equality in the increase multipliers.

If in expressions

1) simple interest rate

2) accrued amount at the discount rate

If, then the increment factors are equal

If the loan term is less than a year, then equivalence is determined for two cases of equal time bases and different time bases.

If the time bases are the same (), then the formulas look like:

If interest is calculated at rate i with a base of 365, and at rate d with a base of 360, then the following is true:

Example

The bill was accounted for by the bank discount rate of 8% on the expiration date of its circulation = 200 (k=360). Determine the profitability of this operation at the simple interest rate (k=365).

Equivalence of simple and compound interest rates

When interest is calculated once a year, it is determined by the formulas:

Simple bet:

Complex bet:

Example

What compound annual rate can replace the simple rate of 18% (k=365) without changing the financial consequences. The duration of the operation is 580 days.

Equivalence of a simple interest rate and a compound rate.

When calculating m times a year, it is determined by the formula:

Example

When developing contract terms The parties agreed that the loan yield should be 24%. What should be the size of the nominal rate when interest is calculated monthly, quarterly.

The equivalence of the simple discount rate and the compound interest rate is determined by the formula:

The equivalence of the nominal compound interest rate when interest is calculated m times a year and the simple discount rate is determined by the formulas:

The equivalence of complex bets is determined by the formulas:

The equivalence of the compound discount rate and the nominal compound interest rate when interest is calculated m times a year is determined by the formulas:

Equivalence of continuous and discrete rates:

Equivalence of growth force and nominal rate:

With a discrete and linear change in force growth, as well as if it changes at a constant rate, the equivalent relationship with compound interest rates can be expressed by the formulas:

The equivalence of the growth force and discount rates for a constant discount rate is determined by the formulas:

For a complex discount rate:

Comment. Using formulas for the equivalence of discrete and continuous rates, it is possible to present the results of applying continuous interest in the form of generally accepted characteristics.

4. Average values in financial calculations

For several interest rates, their average is the equivalent value. If the amounts of loans received are equal to each other, then the average interest rate for simple interest is calculated using the weighted average formula with weights equal to the time periods during which this rate was in effect.

Comment. Replacing all averaged rate values with the average interest rate does not change the results of compounding or discounting:

Example

The average simple discount rate of the discount rate is calculated using the formula:

The average compound interest rate is determined by the formula:

When analyzing the work of credit institutions, the following indicators are calculated: the average loan size, its average duration, the average number of loan turnovers and other indicators.

The average size of one loan, excluding the number of turnovers per year, is calculated using the formula:

Taking into account the number of revolutions per year according to the formula:

where is the number of revolutions,

Length of period

K is the number of clients who received loans.

where is the total turnover, i.e. the amount of repaid loans repaid during the period.

where is the monthly balance of issued loans.

The average number of turnovers of all loans for the period, provided that their continuous turnover occurs, is calculated using a formula based on the availability of data.

The average loan term of individual loans or all loans as a whole is calculated using various formulas

equivalence conversion discounting rate

5. Financial equivalence of obligations and conversion of payments

Replacing one monetary obligation with another or combining several payments into one is based on the principle of financial equivalence of obligations.

In the case of a compound interest rate, the barrier rate is calculated using the formulas:

One of the common cases of changing the terms of contracts is consolidation, i.e. consolidation of payments. There are 2 possible formulations of the problem:

1) A deadline is given and you need to find the amount of payment;

2) The amount of the consolidated payment is given; its term must be determined.

When consolidating several payments into one, provided that the term of the new payment is longer than the previously established term, the equivalence equation is written as:

Where is the accumulated amount of the consolidated payment,

Payments subject to consolidation

Time intervals between and:

In general, the amount of the consolidated payment will look like:

Amounts of combined payments, terms of repayment of which are less than the first term; - amounts of combined payments with terms exceeding the new term.

When consolidating bills, the discount rate is taken into account and the amount of the consolidated payment is determined by the formula:

When consolidating payments using a compound interest rate, the consolidated amount is found using the formulas:

If the amount of the consolidated payment is known and it is necessary to determine the period of its consolidation, maintaining the principle of equivalence:

where is the consolidated amount of the modern payment. If the partners agree to consolidate payments without changing the total amount of payments, then the term of the consolidated payment:

To calculate the deadline for payment of consolidated payments, discount rates can be used, then calculations are made using the formula:

When using compound interest, the formulas look like:

Bibliography

1. Kochovic E. Financial mathematics: Theory and practice of financial and banking calculations. - M.: Finance and Statistics, 2004

2. Krasina F.A. Financial calculations - Financial calculations: textbook / F. A. Krasina. -- Tomsk: El Content, 2011.

3. Selezneva N.N., Ionova A.F. Financial management. Tasks, situations, tests, schemes: Proc. manual for universities. - M.: UNITY-DANA, 2004. - 176 p.

Posted on Allbest.ru

The power of growth with continuous compounding. Relationship between discrete and continuous interest rates

Variable growth force

The meaning of continuous interest in theoretical and practical economics

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