Continuous rate. Compound and continuously compounded interest

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.

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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In practical financial and credit operations there is a continuous increase, i.e. growth over infinitesimal periods of time is used extremely rarely. Continuous growth is of much greater importance in the analysis of complex financial problems, for example, in the justification and selection of investment decisions, in financial design.

With a continuous increase in interest, a special type of interest rate is used - the power of growth.

The power of growth characterizes the relative increase in the accumulated amount over an infinitesimal period of time. It can be constant or change over time.

In order to distinguish a continuous rate from a discrete one, we denote the growth force as δ . Then the accrued amount at the continuous rate will be:

Discrete and continuous increment rates are functionally dependent. From the equality of the growth factors

follows: ,

Example: The amount on which continuous interest is accrued is equal to 2 million rubles, the growth rate is 10%, the term is 5 years. Determine the accrued amount.

Continuous increase at a rate = 10% is equivalent to an increase over the same period of discrete compound interest at an annual rate:

As a result we get:

Discount formula:

The discount factor is .

Example: Determine the current cost of payment if the accrued cost is equal to 5000 thousand rubles. subject to discounting based on growth rate of 12%. Payment term – 5 years.

For continuous interest there is no difference between the interest rate and the discount rate, since the growth rate is a universal indicator. However, along with a constant growth rate, a variable interest rate can be used, the value of which changes according to a given law (mathematical function).

Continuous compounding is used in the analysis of complex financial problems, such as the rationale and selection of investment decisions. When assessing the work of a financial institution where payments are received multiple times over a period, it is advisable to assume that the accumulated amount changes continuously over time and apply continuous interest calculation.

All the situations that we have considered so far relate to discrete interest, since they are calculated over fixed periods of time (year, quarter, month, day, hour). But in practice there are often cases when interest accrues continuously, for an arbitrarily short period of time. If interest were accrued daily, then the annual compounding coefficient (multiplier) would look like this:

k n = (1 + j / m)m = (1 + j / 365) 365

But since interest accrues continuously, then m tends to infinity, and the coefficient (multiplier) of the increase tends to e j:

Where e? 2.718281 is called Euler's number and is one of the most important constants in mathematical analysis.

From here we can write the formula for the accrued amount for n years:

FV = PV * e j * n = P * e d * n

The continuous interest rate is called force of interest and is designated by the symbol d, in contrast to the discrete interest rate ( j).

Example. A loan of $100 thousand was received for a period of 3 years at 8% per annum. Determine the amount to be repaid at the end of the loan term if interest accrues:

a) once a year;

b) daily;

c) continuously.

We use the formulas for discrete and continuous percentages:

accrual once a year

F.V.= 100"000 * (1 + 0.08) 3 = 125"971.2 dollars;

daily interest accrual

F.V.= 100"000 * (1 + 0.08 / 365) 365 * 3 = 127"121.6 dollars

continuous interest accrual

F.V.= 100"000 * e 0.08 * 3 = 127"124.9 dollars.

14. Loan term. Formulas necessary to calculate the duration of the loan in years and days

period in years

period in days (remember that n = t/K,Where K- temporary base)

The interest rate. The need to calculate the interest rate arises when determining the financial efficiency of an operation and when comparing contracts based on their profitability in cases where interest rates are not explicitly indicated. Having solved expressions (1.1) and (1.8) for i or d,we get

Payment term. Here are the calculation formulas P for various conditions of interest accrual and discounting. When increasing at a complex annual rate i and at a nominal rate j accordingly we get:

. (2.23) (2.24)

When discounted at a compound annual discount rate d and at the nominal discount rate f

. (2.25) (2.26)

When increasing by a constant growth force δ and by a growth force changing at a constant rate

The interest rate. Here are the formulas for calculating rates i, j, d, f, δ for various conditions of interest accrual and discounting. They were obtained by solving the equations that determine S And R, relative to the desired rates.

When accrued at a compound annual interest rate and at a nominal interest rate T we find it once a year

. (2.29) (2.30)

When discounting at a complex discount rate and at a nominal discount rate

. (2.31) (2.32)

When increasing by constant growth force

. (2.33)

When growing according to a growth force changing at a constant rate

15.Calculation of simple interest in conditions of inflation . Let's return to the problem of money depreciation as it increases. In general, we can now write:

If the increase is made at a simple rate, we have:

(2.43)

As we see, an increase in the accumulated amount, taking into account the preservation of the purchasing power of money, occurs only when 1 + ni > Jp.

Example. Let's say in the amount of 1.5 million rubles. for three months simple interest is accrued at a rate of 50% per annum ( K= 360). The increased amount is equal to 1.6875 million rubles. If monthly inflation is characterized by the rates given in example 2.22, b, then, taking into account depreciation, the accumulated amount will be only 1.6875/1.77 = 0.9534 million rubles.

16.Calculation of compound interest in conditions of inflation. Let us now turn to the compound interest compounding. Substituting into formula (2.42) the values S And Jp, we find

(2.44)

Quantities by which it is multiplied R in formulas (2.43) and (2.44), represent the growth factors taking into account inflation. Example. Let's find the real compound interest rate for the conditions: annual inflation 120%, gross rate 150%:

= 0.1364, or 13.68% (according to the simplified formula 30%).

Another method of compensating for inflation is to index the initial payment amount R. In this case, this amount is periodically adjusted using a pre-agreed index. This method is accepted in the UK. A-priory

C = PJp(1 + i)n.

17. Calculation of the real interest rate in conditions of inflation. Let us now move on to solving the inverse problem - to measuring real interest rate those. profitability taking into account inflation - definition i at a given gross rate value. If r- declared rate of return (gross rate), then the desired rate of return in the form of an annual interest rate i can be determined when calculating simple interest based on (2.43) as

. (2.48)

The real profitability, as we see, here depends on the period of interest accrual. Let us recall that the price index appearing in this formula covers the entire period of interest accrual.

An indicator similar in content, but with compound interest, can be found based on formula (2.44).

k n = (1 + j / m)m = (1 + j / 365) 365

But since interest accrues continuously, then m tends to infinity, and the coefficient (multiplier) of the increase tends to e j:

Where e≈ 2.718281, called the Euler number and is one of the most important constants in mathematical analysis.

From here we can write the formula for the accrued amount for n years:

FV = PV e j n = P e δn

The continuous interest rate is called force of interest and is designated by the symbol δ , in contrast to the discrete interest rate ( j).

Example. A loan of $100 thousand was received for a period of 3 years at 8% per annum. Determine the amount to be repaid at the end of the loan term if interest accrues:

a) once a year;

b) daily;

c) continuously.

Solution:

We use the formulas for discrete and continuous percentages:

accrual once a year

F.V.= 100"000 (1 + 0.08) 3 = 125"971.2 dollars;

daily interest accrual

F.V.= 100"000 (1 + 0.08 / 365) 365 3 = 127"121.6 dollars

continuous interest accrual

F.V.= 100"000 e 0.08 3 = 127"124.9 dollars.

12. Calculation of the loan term:

In any simple financial transaction there are always four values: the modern value ( PV), accumulated or future value ( F.V.), interest rate ( i) and time ( n).

Sometimes, when developing the terms of a financial transaction or analyzing it, it becomes necessary to solve problems related to determining missing parameters, such as the term of the financial transaction or the level of the interest rate.

As a rule, financial contracts necessarily specify terms, dates, and interest accrual periods, since the time factor plays an important role in financial and commercial calculations. However, there are situations when the term of a financial transaction is not directly specified in the terms of the financial transaction, or when this parameter is determined when developing the terms of the financial transaction.

Usually term of financial transaction determined in cases where the interest rate and the amount of interest are known.

If the period is determined in years, then

n = (FV - PV) : (PV i),

and if the transaction period must be determined in days, then the time base appears as a factor:

t = [(FV - PV) : (PV i)] T.

Just like for simple interest, for compound interest it is necessary to have formulas that allow you to determine the missing parameters of a financial transaction:

loan term:

n = / = / ;

compound interest rate:

Thus, increasing the deposit three times over three years is equivalent to an annual interest rate of 44.3%, so placing money at 46% per annum will be more profitable.

13. Calculation of the loan term:

14. Interest rate calculation:

- when increasing at a compound annual rate of %,

- when increasing at a nominal rate of % m times a year,

- when increasing by constant growth force.

15. Interest rate calculation:

- when discounted at a complex annual discount rate,

- when discounting at a nominal discount rate m times a year.