Continuous interest rate formula. Continuous rate (growth power) and continuous discount

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 = PJ p(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).

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

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 for discounting, discount forces or growth forces lead to the same result.

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.

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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.

The accrued amount at discrete percentages is determined by the formula

S=P(1+j/m) mn ,

Where j is the nominal interest rate, and m– the number of interest periods per year.

The more m, the shorter the time intervals between the points of interest accrual. Increasing the frequency of interest calculations ( m) at a fixed value of the nominal interest rate j leads to an increase in the accrual multiplier, which, with continuous interest accrual ( m) reaches its limit value

It is known that

Where e– the base of natural logarithms.

Using this limit in expression (2.5), we finally obtain that the accrued amount at the rate j equal to

S=Pe jn .

The continuous interest rate is called the force of growth and is denoted by the symbol  . Then

S=Pe  n . (2.6)

The power of growth represents the nominal interest rate at m.

The law of accumulation for the continuous calculation of interest (2.6) coincides in form with (2.2) with the difference that in (2.2) time changes discretely with a step of 1/ m, and in (2.6) – continuously.

It is easy to show that discrete and continuous increment rates are functionally dependent. From the equality of the increment multipliers, we can obtain a formula for the equivalent transition from one bet to another:

(1+i) n =e  n ,

from which follows:

=ln(1+ i), i=e  -1.

Example 20 . The amount on which continuous interest is accrued for 5 years is 2000 den. units, growth force 10%. The increased amount will be S=2000· e 0.1·5 =2000·1.6487=3297.44 den. units

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

i=e 0,1 -1=1,10517-1=0,10517.

As a result we get S=2000·(1+0.10517) 5 =3297.44 den. units

Discounting based on growth force is carried out according to the formula

P=Se -  n

Example 21. Let us determine the modern cost of payment from example 17, provided that discounting is carried out according to the growth rate of 15%.

Solution. The amount received for the debt (modern value) is equal to

P=5000· e-0.15·5 =5000·0.472366=2361.83 den. units

When applying a discrete complex discount rate of the same size, we obtained the value (see example 17) P=2218.53 den. units

2.5. Calculation of loan term and interest rates

In a number of practical problems, the initial (P) and final (S) amounts are specified by the contract, and it is necessary to determine either the payment period or the interest rate, which in this case can serve as a measure of comparison with market indicators and a characteristic of the profitability of the operation for the lender. The indicated values can be easily found from the initial formulas for compounding and discounting (for simple interest, these problems are discussed in paragraph 1.8.).

Loan term. Consider the calculation problem n for various conditions of interest accrual and discounting.

i from the original growth formula (2.1) it follows that

where the logarithm can be taken to any base, since it is present in both the numerator and the denominator.

j m

d f m

;

When increasing by constant growth force, based on formula (2.6) we obtain:

Example 22. For what period in years is the amount equal to 75 thousand den. units, will reach 200 thousand den. units when interest is calculated at a compound rate of 12% once a year and quarterly?

Solution. Using the formulas for calculating the period for accrual at complex accrual rates, we obtain:

n=(log(200/75)/log(1+0.12))=3.578 years;

n=(log(200/75)/(4·log(1+0.12/4))=3.429 years;

Calculation of interest rates. From the same initial formulas as above, we obtain formulas for calculating rates under various conditions for increasing interest and discounting.

When increasing at a complex annual rate i from the original growth formula (2.1) it follows that

i=(S/P) 1/ n –1=
.

When accrued at the nominal interest rate m once a year from formula (2.2) we obtain:

j=m((S/P) 1/ mn –1)=
.

When discounted at a compound annual discount rate d and at the nominal discount rate f m once a year from formulas (2.3) and (2.4), respectively, we obtain:

d =1– (P/S) 1/ n =
;

f = m(1– (P/S) 1/ mn =
.

When increasing by constant growth force, based on formula (2.6), we obtain:

Example 23. The savings certificate was purchased for 100 thousand den. units, its redemption amount is 160 thousand den. units, period 2.5 years. What is the rate of return on the investment expressed as annual compound interest?

Solution. Using the resulting formula for the annual rate i, we get: i=(160/100) 1/2.5 –1=1.2068–1=0.20684, i.e. 20.684%.

Example 24. The maturity of the bill is 2 years. The discount when taking it into account was 30%. What compound annual discount rate corresponds to this discount?

Solution. According to the task P/S=0.7. Then d=1–
=0.16334, i.e. 16.334%.

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.

When simple interest is calculated multiple times, the calculation is made in relation to the original amount and represents the same amount each time. In other words,

P - original amount;

S - accrued amount (original amount plus accrued interest);

i - interest rate expressed in shares;

n is the number of accrual periods.

In this case, we talk about a simple interest rate.

When charged multiple times compound interest each time the accrual is made in relation to the amount with interest already accrued earlier. In other words, S= (1 + i) n P

In this case they talk about compound interest rate.

The following situation is often considered. The annual interest rate is j, and interest is calculated m times a year at a compound interest rate of j/m (for example, quarterly, then m = 4, or monthly, then m = 12). Then the formula for the accumulated amount will look like:

In this case they talk about nominal interest rate.

Sometimes they consider the situation of the so-called continuously accrued interest, that is, the annual number of accrual periods m tends to infinity. The interest rate is denoted by δ, and the formula for the accrued amount is:

In this case, the nominal interest rate δ is called growth force.

Real and nominal rates

There is a distinction between nominal and real interest rates.

Real interest rate is the interest rate adjusted for inflation. The relationship between real, nominal rates and inflation is generally described by the following (approximate) formula:

i r = i n − π

i n - nominal interest rate; i r - real interest rate;

π - expected or planned inflation rate.

Irving Fisher proposed a more accurate model of the relationship between real, nominal rates and inflation, expressed by the Fisher formula named after him:

For small values of the inflation rate π, the results differ little, but if inflation is high, then the Fisher formula should be used.

Compound Interest Formula

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 percent.

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

S= P+ I = P + P i = P (1 + i) – for one accrual period;

S = (P + I) (1 + i) = P ( 1 + i) ( 1 + i) = P (1 + i) 2

– for two accrual periods; from here, beyond n accrual periods, the formula will take the form: S=P (1 + i)n= P kn, Where

S– increased amount of debt;

P– initial amount of debt;

i– interest rate in the accrual period;

n– number of accrual periods;

k n– coefficient (multiplier) of compound interest accumulation.

This formula is called the compound interest formula.

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. The accrual base is a constant value, then compound interest is accrued 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 a chain growth rate, the chain growth rate is equal to: (1 + i).

Then the basic growth rate for the entire period, based on a constant growth rate, has the form: (1 + i)n.

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.) through n periods at a given interest rate i.

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:

A 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);

A 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 1. An amount of 2,000 rubles is loaned for 2 years at an interest rate of 10% per annum. Determine the interest and the amount to be repaid.

Solution:

Accrued amount

S=P (1 + i)n= 2"000 (1 + 0.1) 2 = 2"420 rub.

S=Pk n= 2"000 1.21 = 2"420 rub.,

Where k n = 1,21

Amount of accrued interest

I =S-P= 2"420 - 2"000 = 420 rub.

Thus, after two years it is necessary to return the total amount of 2,420 rubles, of which 2,000 rubles. is a debt, and 420 rubles. - "price 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:

-general The method consists of direct calculation using the compound interest formula:

S=P (1 + i)n, n=a+b,

Where n– transaction period;

a– an integer number of years;

b– fractional part of the year.

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

S=P (1 + i)a (1 + bi).

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

Example 2. A loan was received from the bank at 9.5% per annum in the amount of 250 thousand rubles. maturing in two years and 9 months. Determine the amount that must be repaid at the end of the loan term in two ways.

Solution:

General method:

S= P (1 + i)n= 250 (1 + 0.095) 2.9 = 320.87 thousand rubles.

Mixed method:

S= P (1 + i)a (1 + bi) =

250 (1 + 0,095) 2 (1 + 270/360 0,095) =

321.11 thousand rubles.

Thus, according to the general method, the interest on the loan will be

I = S - P= 320.87 - 250.00 = 70.84 thousand rubles,

and using a mixed method

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

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