Continuous interest. Constant growth force

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 the 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.

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.

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%.

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.

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, 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.