Actions with continuous interest. Variable interest rate

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.

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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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

Relationship between discrete and continuous interest rates
Discrete and continuous interest rates are in a functional relationship, thanks to which it is possible to transition from the calculation of continuous to discrete interest and vice versa. The formula for the equivalent transition from one bet to another can be obtained by equating the corresponding increase multipliers
(1+i)n=eSn.

Example 13.
The annual compound interest rate is 15%, which is the equivalent growth rate,
Solution.
Let's use formula (50)
d=N(1+^=N(1+0.15)=0.t76,
those. the equivalent growth force is 13.976%.
Calculation of loan term and interest rates
In a number of practical problems, the initial (P) and final (B) 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 accumulation or discounting. In fact, in both cases the inverse problem is solved in a certain sense.
Loan term
When developing the parameters of the agreement and assessing the time frame for achieving the desired result, it is necessary to determine the duration of the transaction (loan term) through the remaining parameters of the transaction. Let's consider this issue in more detail.
A) When increasing at a complex annual rate i. From the original growth formula
5=P(1+i)n
follows that
n = 1oi(B/R) (52)
1оё(1 +1) ’
where the logarithm can be taken to any base, since it is present in both the numerator and the denominator.

5=P(1+j/m)mn
we get
n =
t io§(1 + y I t)
B) When discounted at a compound annual discount rate d. From the formula
P=S(1d)n
we have n = 1ое(Р 15). (54)
1оё(1 – ^
D) When discounting at a nominal discount rate m times a year. From
P=S(1f/m)mn
we arrive at the formula
n = 1o8(P 15). (55)
t 1о§(1 – /1 t)
When building up by constant growth force. Based
B=Rv3p
we get
ip(B/P)=bp.
Interest Rate Calculation
From the same initial formulas as above, we obtain expressions for interest rates.
A) When building up at a complex annual rate I. From the original build-up formula
B=P(1+1)p
follows that
""i."1
B) When increasing at a nominal interest rate t times a year from the formula
B=P(1+]/t)tp
C) When discounted at a complex annual discount rate d. From the formula
Р=Б(1й)п
we have е = 1 – (§). (59)
D) When discounted at a nominal discount rate t once a year. From
P=B(1//t)tp
we arrive at the formula
1 /(tp)
D) When increasing by constant growth force. Based
we get
Interest and inflation
The consequence of inflation is a fall in the purchasing power of money, which over the period P is characterized by the index Jn. The purchasing power index is equal to the inverse of the price index Jp, i.e.
Jn 1/Jp¦
The price index shows how many times prices have increased over a specified period of time.
Increase at simple interest
If the amount of money accumulated over n years is S, and the price index is equal to Jp, then the actually accumulated amount of money, taking into account its purchasing power, is equal to
C=S/Jp.
Let the expected average annual inflation rate (characterizing the increase in prices over the year) be equal to b. Then the annual price index will be (1+b.).
If the increase is made at a simple rate over P years, then the real increase at the inflation rate b will be
c = p (1 + Ш)
where in general
P
JP =P (1+K),
g=1
and, in particular, with a constant price growth rate h,
Jp=(1+h)n. (66)
The interest rate that compensates for inflation when calculating simple interest is equal to
71
i =P1. (67)
P
One way to compensate for the depreciation of money is to increase the interest rate by the amount of the so-called inflation premium. The rate adjusted in this way is called the gross rate. The gross rate, which we will denote by the symbol G, is found from the equality of the inflation-adjusted increase multiplier for the gross rate to the increase multiplier for the real interest rate
1 + ng = 1 + n, (68)
-R
where
r = (1 + m)P 1. (69)
P
Compound interest compounding
The amount accrued at compound interest by the end of the loan term, taking into account the fall in the purchasing power of money (i.e. in constant rubles), will be
C = P (1+01, (70)
where the price index is determined by expression (65) or (66), depending on the volatility or constancy of the inflation rate.
In this case, the fall in the purchasing power of money is compensated at the rate i=h, ensuring equality C=P.
Two methods are used to compensate for losses from a decrease in the purchasing power of money when calculating compound interest.
A) Adjustment of the interest rate at which the increase is made by the amount of the inflation premium. The interest rate increased by the inflation premium is called the gross rate. We will denote it by the symbol r. Assuming that the annual inflation rate is equal to b, we can write the equality of the corresponding increment factors
- = 1 + /, (71)
1 + I
where i is the real rate.
From here we get the Fisher formula
r=i+h+ih. (72)
That is, the inflation premium is equal to h+ih.
B) Indexation of the original amount P. In this case, the amount P is adjusted according to the movement of a pre-agreed index. Then
S=PJp(1+i)n. (73)
It is easy to see that in both case A) and case B) we ultimately arrive at the same growth formula (73). In it, the first two factors on the right side reflect the indexation of the original amount, and the last two reflect the adjustment of the interest rate.
Measuring the real interest rate
In practice, we also have to solve the inverse problem - finding the real interest rate in conditions of inflation. From the same relationships between the multipliers of the increase, it is not difficult to derive formulas that determine the real rate i at a given (or announced) gross rate r.
When calculating simple interest, the annual real interest rate is equal to
(l\
1 + pg
1
R
When calculating compound interest, the real interest rate is determined by the following expression
1 + G G – I /YYYCH
I = 1 =. (75)
1+I 1+I
Practical applications of the theory
Let's look at some practical applications of the theory we've discussed. We will show how the formulas obtained above are used when solving real problems of calculating the efficiency of certain financial transactions, and compare different calculation methods.
Currency conversion and interest calculation
Let's consider the combination of currency conversion (exchange) and the increase in simple interest, compare the results from directly placing available funds in deposits or after a preliminary exchange for another currency. There are a total of 4 options for increasing interest:
1. No conversion. Currency funds are placed as a foreign currency deposit, and the initial amount is increased at the foreign exchange rate by directly applying the simple interest formula.
2. With conversion. The original currency funds are converted into rubles, the increase is at the ruble rate, and at the end of the operation the ruble amount is converted back into the original currency.
3. No conversion. The ruble amount is placed in the form of a ruble deposit, on which interest is accrued at the ruble rate using the simple interest formula.
4. With conversion. The ruble amount is converted into any specific currency, which is invested in a foreign currency deposit. Interest is calculated at the foreign exchange rate. The accrued amount is converted back into rubles at the end of the operation.?
Transactions without conversion are not difficult. In an accrual operation with double conversion, there are two sources of income: interest accrual and exchange rate changes. Moreover, interest accrual is an unconditional source (the rate is fixed, we are not considering inflation yet). A change in the exchange rate can be in either direction, and it can be either a source of additional income or lead to losses. Next, we will specifically focus on two options (2 and 4), which provide for double conversion.
Let us first introduce the following NOTATION:
Pv – deposit amount in foreign currency,
Pr – deposit amount in rubles,
Sv – accrued amount in currency,
Sr – accrued amount in rubles,
^ – exchange rate at the beginning of the operation (currency rate in rubles)
^ – exchange rate at the end of the transaction, P – deposit term,
I – accrual rate for ruble amounts (in the form of a decimal fraction),
j – growth rate for a specific currency.
OPTION: CURRENCIES RUBLES ^ RUBLES ^CURRENCY The operation consists of three stages: exchanging currency for rubles, increasing the ruble amount, converting the ruble amount back into the original currency. The accrued amount received at the end of the transaction in foreign currency will be
= RuK- (1 + pi)!.
k1
As you can see, the three stages of the operation are reflected in this formula in the form of three factors.
The growth multiplier taking into account double conversion is equal to
K0 „,h 1 + n 1 + n,
To
K o
where k=Kl/Ko is the growth rate of the exchange rate over the period of the operation.?
We see that the increase multiplier m is related linearly to the rate I and the inverse relationship to the exchange rate at the end of the operation K (or to the growth rate of the exchange rate k).
Let us theoretically study the dependence of the total profitability of an operation with double conversion according to the scheme CURRENCY ^ RUBLES ^ RUBLES ^ CURRENCY on the ratio of the final and initial exchange rates k.
The simple annual interest rate, characterizing the profitability of the operation as a whole, is equal to
/ = ^P,.
*,"")TMTM
* Rp
Let us substitute into this formula the previously written expression for Bu
-(1 + t)1
K1 1 (1 + t) 1?
CONCLUSION 1: If the expected values of k or K1 exceed their critical values, then the operation is clearly unprofitable
Zeff Now we determine the maximum allowable value of the exchange rate at the end of the transaction Ki, at which the efficiency will be equal to the existing rate on deposits in foreign currency, and the use of double conversion does not provide any additional benefit. To do this, let’s equate the growth factors for two alternative operations
To
1 + nj =tm(1 + ni)
K1
From the written equality it follows that
to to 1 + ni
max K1 = K 0
1 + nj
or
K, 1 + ni
max k = -L =
K o 1 + nj
CONCLUSION 2: A currency deposit through conversion into rubles is more profitable than a foreign currency deposit if the exchange rate at the end of the transaction is expected to be less than max K1.
OPTION: RUBLES ^ CURRENCY ^ CURRENCY ^ RUBLES
Let us now consider the option with double conversion, when the original amount is in rubles. In this case, the three stages of the operation correspond to three factors of the following expression for the accumulated amount
P K
S = K(1 + nj)K 1= Pr (1 + nj)L
K0 K0
Here, too, the increase multiplier linearly depends on the rate, but now on the foreign exchange interest rate. It also depends linearly on the final exchange rate.
Let's conduct a theoretical analysis of the effectiveness of this double conversion operation and determine the critical points.?
The profitability of the operation as a whole is determined by the formula
«¦ =.
1 „tmgm „
E Rgp
From here, substituting the expression for Sr, we get
TO
(1 + n])1. = Ko " = *(1 + p])1
"E11
P
The dependence of the efficiency indicator ieff on k is linear, it is presented in Fig. 3
For k=1 ізф=/", for k>1 ізф>;", for k Let us now find the critical value of k* at which bff=0. It turns out to be equal
k* =^^ or k *1 =K^~.
1 + p 1 + p
CONCLUSION 3: If the expected values of k or ^ are less than their critical values, then the operation is clearly unprofitable
(IZFF The minimum permissible value of k (the rate of growth of the exchange rate for the entire period of the operation), providing the same profitability as a direct deposit in rubles, is determined by
thus equating the increase multipliers for alternative operations (or from the equality ieff=i)
To
- L(1 + nj) = 1 + ni,
K 0
1 + ni 1 + ni whence mm k = or mm k = K
1 + nj 1 0 1 + nj
CONCLUSION 4: A deposit of ruble amounts through conversion into foreign currency is more profitable than a ruble deposit if the exchange rate at the end of the transaction is expected to be greater than min K1.
Now let's look at the combination of currency conversion and compound interest. Let's limit ourselves to one option.
OPTION: CURRENCY ^ RUBLES ^ RUBLES ^ CURRENCY
The three stages of the operation are written in one formula for the accumulated amount
sv = PVK 0(1+i) nK"
Ki
where i is the compound interest rate.
Growth multiplier
nKо _ (1 +i) n
K1 k
7 K
where k = is the growth rate of the exchange rate during the operation period. K 0
Let us determine the profitability of the operation as a whole in the form of the annual compound interest rate iе.
From the compound interest compounding formula
S=P(1+i)n
follows that
I.-n
]Pv
Substituting the BU value into this formula, we get
P (1 + Opgg,.
b = d, ^1 = 1+11.
From this expression it is clear that as the growth rate k increases, efficiency b decreases. This is shown in the graph in Fig. 4.
Rice. 4.
Analysis shows that for k = 1 1e = I, for k > 1 1e I.
The critical value of k, at which the efficiency of the operation is zero, i.e. b = 0,
is defined as k* = (1 + 1)p, which means that the average annual growth rate of the currency exchange rate is equal to the annual growth rate at the ruble rate: Vk = 1 + g.
CONCLUSION 5: If the expected values of k or K are greater than their critical values, then the operation in question with double conversion is clearly unprofitable (b The maximum permissible value of k, at which the profitability of the operation will be equal to the profitability of direct investment of foreign currency at the rate ] (i.e. in Fig. 4), is found from the equality of the corresponding increment factors
(1 +1)i
(1 + L)n =
ct?
where
P
1 +1
or max k = K
1 L(
1 +U, 1 "VI + y,
CONCLUSION 6: A currency deposit through conversion into rubles is more profitable than a foreign currency deposit if the exchange rate at the end of the transaction is expected to be less than
Repaying debt in installments Outline of a financial transaction
Financial or credit operations require a balance of investments and returns. The concept of balance can be explained in a graph. A)
IN
I,.
T
b)
Rice. 5.
Let a loan in the amount of Bo be issued for a period of T. During this period, for example, two intermediate payments K and Kg are made to repay the debt, and at the end of the term the balance of the debt K3 is paid, bringing up the balance of the operation.
At time interval i, the debt increases to the value Bb At the moment and the debt decreases to the value K1 = B1K1, etc. The operation ends with the creditor receiving the balance of the debt Kz. At this point, the debt is fully repaid.
Let's call the graph of type b) the outline of a financial transaction. A balanced operation necessarily has a closed loop, i.e. the last payment completely covers the balance of the debt. The transaction outline is usually used when repaying debt through partial interim payments.
Successive installment payments are sometimes used to pay off short-term obligations. In this case, there are two methods for calculating interest and determining the balance of debt. The first is called actuarial and is used mainly in transactions with a maturity of more than a year. The second method is called the merchant's rule. It is usually used by commercial firms in transactions with a maturity of no more than a year.
Note: When calculating interest, as a rule, ordinary interest is used with an approximate number of days of time periods.
Actuarial method
The actuarial method involves the sequential calculation of interest on the actual amounts of debt. The partial payment goes primarily to repay the interest accrued on the payment date. If the payment amount exceeds the amount of accrued interest, then the difference goes to repay the principal amount of the debt. The outstanding balance of the debt serves as the basis for calculating interest for the next period, etc. If the partial payment is less than the accrued
interest, then no offsets are made against the debt amount. This receipt is added to the next payment.
For the case shown in Fig. 5 b), we obtain the following calculation formulas for determining the debt balance:
K1=Bo(1+b1)K1; K2=Kb(1+b21)K2; K2(1+bz1)Kz=0,
where the time periods bb, b2, bz are specified in years, and the interest rate I is annual.
Merchant Rule
The merchant rule is another approach to calculating installments. There are two possible situations here.
1) If the loan term does not exceed, the amount of the debt with interest accrued for the entire period remains unchanged until full repayment. At the same time, partial payments are accumulated with interest accrued on them until the end of the term.
2) In the case where the period exceeds a year, the above calculations are made for the annual debt period. At the end of the year, the accumulated amount of partial payments is subtracted from the debt amount. The balance is repaid next year.
With a total loan term T m
S = D – K = P(l + L) – ? RJ (1 + tJi),
]=1
where E is the balance of the debt at the end of the term,
B – accumulated amount of debt,
K – increased amount of payments,
Ш – amount of partial payment,
b) is the time interval from the moment of payment to the end of the term, t is the number of partial (interim) payments.
Variable invoice amount and interest calculation
Let's consider a situation where a savings account is opened at a bank, and the account amount changes during the storage period: funds are withdrawn, additional contributions are made. Then, in banking practice, when calculating interest, a calculation method is often used to calculate the so-called percentage numbers. Each time the amount in the account changes, the percentage number Cj for the past period ], during which the amount in the account remained unchanged, is calculated using the formula
With. = R.,
at 100
where ^ is the duration of the period in days.
To determine the amount of interest accrued for the entire period, all interest numbers are added up and their sum is divided by a constant divisor D:
B = K,
where K is the time base (the number of days in a year, i.e. 360 or 365 or 366), i is the annual simple interest rate (in %).
When closing the account, the owner will receive an amount equal to the last amount in the account plus the amount of interest.
Example 14.
Let a demand account be opened on February 20 in the amount of P1=3000 rubles, the interest rate on the deposit was equal to r=20% per annum. The additional contribution to the account amounted to Rl=2000 rubles. and was done on August 15th. Withdrawal from the account in the amount of R2=4000 rubles. recorded on October 1, and the account was closed on November 21. It is required to determine the amount of interest and the total amount received by the depositor upon closing the account.
Solution.
We will carry out the calculation according to the scheme (360/360). There are three periods during which the amount in the account remained unchanged: from February 20 to August 15
^1 = 3000, and = 10 + 5*30 + 15 = 175),?
from August 15 to October 1
(P2 = P1 + R1 = 3000 + 2000 = 5000 rubles, b = 15 + 30 + 1 = 46), from October 1 to November 21
(Pz = P2 + R2 = 5000 – 4000 = 1000 rubles, bz = 29 + 21 = 50). Let's find the percentage numbers
R*D 3000 S. = -k = = 5250,
1 1LL 1LL
=2300,
Constant divisor
B=K/1=360/20=18.
The amount of interest is
I = (C, + C2 + C3)/ B = 5250 + 2300 + 500 = 447 rubles. 22 kopecks
18
The amount payable upon account closure is
Рз + I = 1000 + 447.22 = 1447 rub. 22 kopecks
Now we will show the connection of this technique with the simple interest formula. Let us consider the example presented above in algebraic form.
We find the amount paid upon closing the account as follows:
RL, + (P + O V 2 + (P + R. + 02 ^з /
P3 +1 = P + R1 + P2 +^-^ 1" 2 V 1 1 ^3 _
100 K
t1 +2 +13 I 1, o (, 2 +13 I 1, o (l, t3 I
= Р.1 1 +1 2 ^ 1 + О 1 + ^ ^ 1 + Р2| 1 +31 ^ K 100) ^ K 100) ^ K100
Thus, we have obtained an expression from which it follows that for each amount added or withdrawn
from the account, interest is accrued from the moment the corresponding transaction is completed until the account is closed. This scheme corresponds to the merchant rule discussed in Section 6.2.
Changing the terms of the contract
In practice, there is often a need to change the terms of the contract: for example, the debtor may ask for a deferment of the debt repayment period or, on the contrary, express a desire to repay it ahead of schedule; in some cases, there may be a need to combine (consolidate) several debt obligations into one, etc. In all these cases, the principle of financial equivalence of old (replaced) and new (replaced) obligations is applied. To solve problems of changing the terms of the contract, a so-called equivalence equation is developed, in which the amount of replaced payments, reduced to any one point in time, is equal to the amount of payments under the new obligation, reduced to the same date. For short-term contracts, simple interest rates are applied, and for medium- and long-term contracts, compound rates are applied.

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.