Continuous rate (growth power) and continuous discount. Continuous interest: accumulation, discounting, connection between discrete and continuous interest rates when accruing at a compound annual rate of %

Federal Agency for Education and Science

State educational institution of higher education

vocational education

Tambov State University named after G.R. Derzhavina

on the topic: “Actions with continuous interest”

Performed

5th year student, group 502

Full-time education Geghamyan M.A.

Tambov 2013

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

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

When moving to continuous percentages we get:

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

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

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

Discounting based on continuous interest rates

In formula (4.21) we can determine the modern value

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

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

Variable growth force

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

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

Modern cost:

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

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

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

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

where is the initial growth force (at)

a - annual increase or decrease.

Let's calculate the degree of the increase multiplier:

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

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

) The growth force changes exponentially, then

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

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

The loan term is determined by the formulas:

when increasing at a constant rate

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

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

Interest rate equivalence

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

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

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

If, then the increment factors are equal

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

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

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

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

Equivalence of simple and compound interest rates

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

Simple bet:

complex bet:

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

Equivalence of a simple interest rate and a compound rate.

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

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

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

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

The equivalence of complex bets is determined by the formulas:

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

Equivalence of continuous and discrete rates:

Equivalence of growth force and nominal rate:

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

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

For a complex discount rate:

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

Average values in financial calculations

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

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

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

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

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

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

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

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

where is the number of revolutions,

Length of period

K is the number of clients who received loans.

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

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

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

where is the monthly balance of issued loans.

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

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

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

equivalence conversion discounting rate

Financial equivalence of obligations and conversion of payments

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

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

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

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

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

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

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

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

Where is the accumulated amount of the consolidated payment,

Payments subject to consolidation

Time intervals between and:

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

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

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

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

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

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

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

When using compound interest, the formulas look like:

Bibliography

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

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

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

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.

2.2.3. Variable interest rate

It should be noted that the basic compound interest formula assumes constant interest rate throughout the entire interest accrual period. However, when providing a long-term loan, compound interest rates that vary over time, but are fixed in advance for each period, are often used. In case of use variables interest rates, the accumulation formula is as follows:

Where ik– time-consistent interest rates;

nk– the duration of the periods during which the corresponding rates are used.

Example. The company received a bank loan in the amount of $100,000 for a period of 5 years. The interest rate on the loan is set at 10% for the 1st year, for the 2nd year an increase in the interest rate is provided in the amount of 1.5%, for subsequent years 1%. Determine the amount of debt to be repaid at the end of the loan term.

Solution:

We use the formula for variable interest rates:

FV = PV (1 + i 1)n 1 (1 + i 2)n 2 … (1 + ik)nk =

100"000 (1 + 0,1) (1 + 0,115) (1 + 0,125) 3 =

174"632.51 dollars

Thus, the amount to be repaid at the end of the loan term will be $174,632.51, of which $100,000 is the direct amount of the debt, and $74,632.51 is interest on the debt.

2.2.4. Continuous accrual of interest

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:

kn = (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 ej:

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:

F.V. = PV e j n = P e δ n

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.

Graphically, the change in the accrued amount depending on the accrual frequency has the following form:

With discrete accrual, each “step” characterizes the increase in the principal amount of the debt as a result of the next accrual of interest. Please note that the height of the “steps” is increasing all the time.

Within one year, one “step” on the left chart corresponds to two smaller “steps” on the middle chart, but in total they exceed the height of the “step” of a single accrual. The increase occurs at an even faster pace with the continuous accrual of interest, which is what the graph on the right shows.

Thus, depending on the frequency of interest accrual, the initial amount is increased at different rates, and the maximum possible increase is carried out with an infinite division of the annual interval.

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

2.2.5. Determining the loan term and interest rate

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.

2.3. Rate equivalence and payment replacement

2.3.1. Interest rate equivalence

Quite often in practice a situation arises when it is necessary to compare the profitability of the conditions of various financial transactions and commercial transactions. The conditions of financial and commercial transactions can be very diverse and not directly comparable. To compare alternative options, the rates used in the terms of the contracts result in a uniform figure.

Equivalent interest rate- this is the rate that for the financial transaction in question will give exactly the same monetary result (accumulated amount) as the rate applied in this transaction.

A classic example of equivalence is the nominal and effective interest rates:

i = (1 + j / m)m - 1.

j = m[(1 + i) 1 / m - 1].

The effective rate measures the relative income that can be received for the year as a whole, i.e. it makes absolutely no difference whether to apply the rate j when calculating interest m once a year or annual rate i, – both rates are financially equivalent.

Therefore, it does not matter at all which of the given rates is indicated in the financial conditions, since using them gives the same accrued amount. In the United States, the nominal rate is used in practical calculations, while in European countries they prefer the effective interest rate.

If two nominal rates determine the same effective interest rate, then they are said to be equivalent.

Example. What would be the equivalent nominal interest rates with semiannual compounding and monthly compounding if their corresponding effective rate were to be 25%?

Solution:

We find the nominal rate for semi-annual interest accrual:

j = m[(1 + i) 1 / m - 1] = 2[(1 + 0,25) 1/2 - 1] = 0,23607.

Find the nominal rate for monthly interest calculation:

j = m[(1 + i) 1 / m - 1] = 4[(1 + 0,25) 1/12 - 1] = 0,22523.

Thus, nominal rates of 23.61% compounded semiannually and 22.52% compounded monthly are equivalent.

When deriving equalities connecting equivalent bets, the increment multipliers are equated to each other, which makes it possible to use the equivalence formulas for simple and complex bets:

simple interest rate:

i = [(1 + j / m)m n - 1] / n;

compound interest rate:

Example. It is proposed to place capital for 4 years either at a compound interest rate of 20% per annum with semi-annual compounding, or at a simple interest rate of 26% per annum. Find the best option.

Solution:

We find the equivalent simple rate for the compound interest rate:

i = [(1 + j / m)m n - 1] / n = [(1 + 0,2 / 2) 2 4 - 1] / 4 = 0,2859.

Thus, the simple interest rate equivalent to the compound rate under the first option is 28.59% per annum, which is higher than the proposed simple rate of 26% per annum under the second option, therefore, it is more profitable to place capital under the first option, i.e. at 20% per annum with semi-annual compounding.

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Federal Agency for Education and Science

State educational institution of higher education

vocational education

Tambov State University named after G.R. Derzhavina

on the topic: “Actions with continuous interest”

Performed

5th year student, group 502

Full-time education Geghamyan M.A.

Tambov 2013

1. Constant growth force

2. Variable growth force

6. References

1. Constant growth force

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

When moving to continuous percentages we get:

Increase multiplier for continuous interest capitalization.

Denoting the growth force through, we get:

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

Example

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

Discounting based on continuous interest rates

In formula (4.21) we can determine the modern value

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

Example

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

2. Variable growth force

For the accrued amount:

Modern cost:

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

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

Example

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

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

where is the initial growth force (at)

a - annual increase or decrease.

Let's calculate the degree of the increase multiplier:

Example

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

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

3) The growth force changes exponentially, then

Growth multiplier:

Example

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

The loan term is determined by the formulas:

When increasing at a constant rate

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

Example

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

3. Equivalence of interest rates

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

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

If in expressions

1) simple interest rate

2) accrued amount at the discount rate

If, then the increment factors are equal

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

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

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

Example

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

Equivalence of simple and compound interest rates

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

Simple bet:

Complex bet:

Example

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

Equivalence of a simple interest rate and a compound rate.

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

Example

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

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

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

The equivalence of complex bets is determined by the formulas:

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

Equivalence of continuous and discrete rates:

Equivalence of growth force and nominal rate:

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

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

For a complex discount rate:

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

4. Average values in financial calculations

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

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

Example

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

The average compound interest rate is determined by the formula:

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

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

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

where is the number of revolutions,

Length of period

K is the number of clients who received loans.

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

where is the monthly balance of issued loans.

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

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

equivalence conversion discounting rate

5. Financial equivalence of obligations and conversion of payments

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

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

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

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

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

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

Where is the accumulated amount of the consolidated payment,

Payments subject to consolidation

Time intervals between and:

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

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

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

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

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

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

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

When using compound interest, the formulas look like:

Bibliography

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

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

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

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