Nominal interest rate is the interest rate that is set. Types of interest rates, nominal and real interest rates. Distinctive features of nominal and real rates

When people talk about interest rates, they usually mean real interest rates as opposed to nominal interest rates. However, actual rates cannot be directly observed. When concluding a loan agreement or viewing financial bulletins, we receive information primarily about nominal interest rates.
Nominal interest rate- This is a percentage in monetary terms. For example, if a $1,000 annual loan pays $120 in interest, the nominal interest rate would be 12% per annum. Having received an income of $120 on a loan, will the lender become richer? It depends on how prices have changed during the year. If prices rose by 8%, then the lender's income actually increased by only 4% (12%-8% = 4%). The real interest rate is the increase in real wealth, expressed as an increase in the purchasing power of the investor or lender, or the exchange rate at which today's goods and services, real goods, are exchanged for future goods and services. Essentially, the real interest rate is the nominal rate adjusted for price changes.
The above definitions enable us to consider the relationship between nominal and real interest rates and inflation. It can be expressed by the formula
i = r + i, (1)
where i is the nominal interest rate; r-real interest rate; it is the inflation rate.
Equation (1) shows that the nominal interest rate can change for two reasons: due to changes in the real interest rate and/or due to changes in the inflation rate. Real interest rates change very slowly over time because changes in nominal interest rates are caused by changes in the inflation rate. An increase in the inflation rate by 1% causes an increase in the nominal rate by 1%."
When the borrower and lender agree on a nominal rate, they do not know what rate inflation will take at the end of the contract. They are based on expected inflation rates. The equation becomes

r + i[*. (2)

Equation (2) is known as the Fisher equation, or the Fisher effect. Its essence is that the nominal interest rate is determined not by the actual rate of inflation, since it is not yet known, but by the expected rate of inflation (e). The dynamics of the nominal interest rate follows the movement of the expected inflation rate.
Since it is impossible to accurately determine the future rate of inflation, rates are adjusted according to the actual level of inflation. Expectations match current experience. If the inflation rate changes in the future, there will be deviations in the actual rate from the expected rate. They are called the unexpected inflation rate and can be expressed as the difference between the future actual rate and the expected inflation rate (ts-ts).
If the unexpected rate of inflation is zero (it = iG), then neither the lender nor the borrower loses or gains anything from inflation. If unforeseen inflation occurs (i -i(gt; 0), then the borrowers benefit at the expense of the creditors, since they repay the loan with depreciated money. In the case of unforeseen deflation, the situation will be the opposite: the lender will benefit at the expense of the borrower.
1 The given formula is an approximation that gives satisfactory results only at low values of the inflation rate. The higher the inflation rate, the greater the error in equation (1). The exact formula for determining the real interest rate is more complex: i = r + i + m or r = (i - i)/ 1 + i.
From the above, three can be distinguished important points: 1) nominal interest rates include a markup or premium on expected inflation; 2) due to unforeseen inflation, this premium may turn out to be insufficient; 3) as a result, there will be an effect of redistribution of income between lenders and borrowers.
This problem can be looked at from the other side - from the point of view of real interest rates. In this regard, two new concepts arise:

expected real interest rate - the real interest rate that the borrower and lender expect when granting a loan. It is determined by the expected level of inflation (g- i - ts*);
actual real interest rate. It is determined by the actual level of inflation (r = g - l).

Since the lender expects to earn income, the nominal interest rate on new borrowings must be at a level that will provide good prospects for real income consistent with current estimates of future inflation. Deviations of the actual real rate from the expected one will depend on the accuracy of the forecast of future inflation rates.
At the same time, along with the accuracy of forecasts, there is difficulty in measuring the real rate. It consists of measuring inflation and choosing a price index. In this matter, one must proceed from how the funds received will ultimately be used. If loan proceeds are intended to finance future consumption, then the appropriate measure of income is the consumer price index. If a company needs to estimate the real cost of borrowed funds to finance working capital, then the wholesale price index will be adequate.
When the rate of inflation exceeds the rate of increase in the nominal rate, the real interest rate will be negative (less than zero). Although nominal rates typically rise when inflation rises, real interest rates have been known to fall below zero."
Negative real rates are holding back lending. At the same time, they encourage borrowing because the borrower gains what the lender loses.
Under what conditions and why does a negative real rate exist in financial markets? Negative real rates may be established for some time:

during periods of runaway inflation or hyperinflation, lenders provide loans even if real rates are negative because earning some nominal income is better than holding cash;
during an economic downturn, when demand for loans falls and nominal interest rates fall;

1 And, Fischer noted: “The real interest rate in the United States from March to April 1917 fell to -70%) In Germany, during the peak of inflation in August - September 1923, it fell to the absurd level of -99.9%. This meant that creditors lost not only interest, but also almost all capital; suddenly, unexpectedly, prices were deflated, and the real interest rate jumped to 100%” (cited by: Sonny J. Financial management in commercial banks. M., 1994. P. 255).

at high inflation, to provide income to creditors. Borrowers will not be able to borrow at such high rates, especially if they expect inflation to slow soon. At the same time, rates on long-term loans may be lower than the inflation rate, since financial markets will expect a fall in short-term rates;
if inflation is not sustainable. Under the gold standard, the actual rate of inflation may be higher than expected, and nominal interest rates will not be high enough: “inflation takes merchants by surprise.”

Positive real interest rates mean higher income for lenders. However, if interest rates rise or fall in line with inflation, then the lender suffers a potential capital gain loss. This happens in the following cases:
a) inflation reduces the real cost of a loan (loan received). A homeowner who takes out a mortgage loan will find that the amount of debt they owe decreases in real terms. If the market value of his home rises but the face value of his mortgage remains the same, the homeowner benefits from the decreasing real value of his debt. The lender will suffer a capital loss;
b) the market value of securities, such as government bonds or corporate bonds, falls if the market nominal interest rate rises, and, conversely, rises if the interest rate falls.
For example, if a government issues a long-term 25-year bond with a coupon interest rate of, say, 10%, and the market par interest rate is also 10%, then the market value of the bond will be equal to its par value, or $100 for every $100 of par value . Now, if the par rate rises to 14%, the market value of the bond will fall to $71.43 ($100 x 10%: 14% = $71.43 per $100 par value). The bondholder will incur a capital loss of $28.57 for every $100 if he decides to sell the bonds on the stock exchange. Capital loss is caused by rising interest rates.
You can look at this problem differently. For example, the holder of a $100 loan obligation will receive $100 at the end of the loan term. But with the $100 he previously spent on the liability, he can buy a liability that pays 14% rather than the 10% he is earning now. Thus, an increase in the interest rate causes the lender to lose part of the value of the capital lent.
Continuing with the example, consider a drop in the interest rate to 8%, then the resale value of the bond will increase to $125. The bondholder can sell this asset for an increase in capital of $25 per hundred.
The lender faces constant changes in market interest rates due to changes in expected inflation rates. Moreover, if a creditor sells securities, he either incurs losses or increases capital. If he continues to hold these securities, then his real income changes in accordance with the rate of expected inflation.

What is the nominal interest rate on a deposit?

The nominal deposit rate is the value of the nominal income that the depositor will receive after the period established by the agreement. This is what banks indicate when attracting clients to place deposits. It does not reflect the investor’s real income, which he will receive taking into account the depreciation of money (or inflation) and other expenses. Thus, the nominal interest on the deposit is determined by several components:

Real interest rate.
Expected inflation rate.
Other expenses of the depositor, including personal income tax for the difference in the excess rate from the refinancing rate increased by 5 percentage points), etc.

Of all the components, the rate of annual inflation shows the greatest fluctuations. Its expected value depends on historical fluctuations. If inflation consistently shows low values (0.1-1%, as in the West or the USA), then in future periods it is set at approximately the same level. If the state experienced high inflation rates (for example, in the 90s in Russia this figure reached 2500%), then bankers set a high value for the future.

What is the real deposit rate?

The real interest rate is interest income adjusted for inflation. Its value is usually not indicated anywhere by banks. The client can calculate it independently or rely on the bank’s honest attitude towards him.

The real income from investing money on a deposit is always less than the nominal one, since it takes into account the amount that will be obtained after adjusting for inflation. The real rate reflects the purchasing power of money upon expiration of the deposit term (i.e., more or fewer goods can be purchased for the final amount compared to the initial amount).

Unlike nominal interest, real interest can also have negative values. The client will not only not save his savings, but will also receive a loss. Developed countries They deliberately keep real rates negative to stimulate economic development. In Russia, real rates change from positive to negative, especially recently.

How to calculate the real interest rate on a deposit?

To start the calculation, you need to determine all the investor's expenses. These include:

Tax. For deposits there is a 13% personal income tax. It applies if the nominal interest on ruble deposits is 5 percentage points higher than the SR. (until December 31, 2015, the conditions apply that personal income tax will be levied on deposits with a rate higher than 18.25%). The accrued tax will be automatically deducted by the bank when issuing the accumulated amount to the depositor.
Inflation. As the amount of savings increases, the price of goods and services also increases. As of May 2015, inflation was estimated at 16.5%. At the end of the year, its predicted value is estimated at 12.5% (taking into account the stabilization of the economic situation).

Let's look at example 1.

The investor managed to place 100 thousand rubles at the beginning of the year. at 20% per annum for 1 year without capitalization with interest payment at the end of the term. Let's calculate his real income.

Nominal income (NI) will be:

100,000+(100,000*20%) = 120,000 rub.

Real income:

RD = ND - Tax - Inflation

Tax = (100,000 * 20% - 100,000 * 18.25%) * 13% = 227.5 rubles.

Inflation=120,000*12.5% = 15,000 rubles.

Real income = 120,000 -227.5-15,000 = 104,772.5 rubles.

Thus, the depositor actually increased his wealth by only 4,772 rubles, and not by 20,000 rubles, as stated by the bank.

Let's look at example 2.

The investor placed 100 thousand rubles. at 11.5% per annum for 1 year with interest payment at the end of the deposit term. Let's calculate his real profit.

The nominal profit will be:

100,000+(100,000*11.5%) = 111,500 rub.

Tax=0, because interest rate below SR+5 pp.

Inflation = 111,500 * 12.5% = 13,937.5 rubles.

Real income = 111,500 - 13,937.5 = 97,562.5 rubles.

Loss = 100,000 - 97,562.5 = 2437.5 rubles.

Thus, under these conditions, the purchasing power of the depositor's savings turned out to be negative. Not only was he unable to increase his savings, but he also lost some.

a) an interest rate established without taking into account changes in the purchasing value of money due to inflation (or a general interest rate in which its inflationary component is not eliminated);

B) the interest rate on a fixed income security that refers to the par value rather than the market price of the security.

Was the page helpful?

More found about the nominal interest rate

Explanation of the algorithm for calculating the free cash flow of a company and free cash flow to owners using the example of public financial statements According to the explanations to financial statements nominal interest rates on loans do not exceed the limits established in the Tax Code of the Russian Federation. Detailed information
Real interest rate The real interest rate is equal to the nominal interest rate minus the inflation rate Next is the nominal interest rate The page was helpful
Nominal interest rate Nominal interest rate Nominal interest rate is the bank interest rate in numerical terms Shows the increase in the nominal value
Nominal value Next nominal value nominal interest rate Synonyms Face value The page was helpful
Bond coupon Income from coupons is set in the form of an interest rate on the face value of the security, which can be permanently guaranteed or fixed for all
Effective interest rate Next is the nominal interest rate This page was helpful
Coupon rate Represents the ratio of the coupon rate to the market value of the bond, expressed as a percentage of the par value of the bonds Amount of interest for
The need to take into account other income and expenses in marginal analysis In this case, the effective rate is only slightly higher than the nominal rate of payment for the loan provided for in the loan agreement When applying for a loan on terms of added interest
Nominal Income If a stock or bond is purchased at par value, nominal income is equal to real income because market prices for fixed income securities fall when market interest rates rise Nominal income refers to the total cash receipts over a given period Nominal
Estimation of the cost of factoring services of a company VAT at the current tax rate and is determined by the following formula D D D 18% 1 D Tsnom PRfin... Tpl 365, where Tsnom is the nominal amount of the monetary claim in rubles PRfin the percentage of financing the monetary claim from the amount of the claim STpp
Certificate of deposit The following interest payment methods can be established for interest certificates: flat rate interest fluctuating interest rate the value of which is tied to some financial indicator refinancing rate, etc. Primary placement of discount
Preferred share The size of the dividend on preferred shares is fixed in the charter, usually expressed as a percentage of the company’s net profit or the par value of the share. Quite widespread in Russia.
Depreciation charges and their role in the formation of the investment potential of an enterprise. To take it into account, it is necessary in this formula, instead of the nominal interest rate E, to take into account the real interest rate E p taking into account inflation. Therefore, the depreciation coefficient
Valuation of receivables of municipal unitary enterprise housing and communal services in the process of bankruptcy proceedings. Depreciation of receivables depends on 2 factors of inflation and interest on the use of other people's funds by a bank loan, that is, indirect losses of the creditor due to diversion... Rp 1 In In where Rn is the nominal rate with taking into account inflation Rp - real rate excluding inflation - as
Bond discount Let's assume that the normal rate is close to 7% per annum It's profitable for an investor to buy this security; there's a lot of demand for it... Most likely, such a bond will be sold at a premium of 3% of the face value. The opposite is also true. Let's say the bond is issued with a coupon of only 3%, that is with... Let's say a bond is issued with a coupon of only 3%, that is, with an interest payment that is obviously lower than the market one, for such an income investors will be uninterested in investing money And then
LLC rub for 10 years at the rate of 15% per annum and sells them for 95% of the nominal value. If interest on bonds is allowed by law to be attributed to the cost of production, then the real cost... SP coupon interest rate on a bond % Ze level of issue costs in relation to the volume of issue
Risk-free rate of return The risk-free rate of return is the interest rate on highly liquid assets, that is, it is a rate that reflects the actual market opportunities for investing money... During the assessment process, it is taken into account that nominal and real risk-free rates can be both ruble and foreign currency. Analysis of the risk-free rate
Share price Initially, when a share is issued, its nominal rate is formed, which is indicated on the share itself. In the process of purchase and sale, the market price of shares or... The market price of shares is determined by market considerations, the emerging relationship between the dividend rate and bank interest on long-term loans, the reputation of the joint-stock company and the results its financial and economic
Market return For example, a bond with a nominal value of 100 rubles and a 5 percent interest rate will bring an annual income of 5 rubles However... However, if this paper can be bought on the open market for 50 rubles then the actual interest rate on it will increase to 10% and the income will be 10 RUB for invested 50
Capitalization Regardless of the nominal price of shares on the stock market, they are sold at the market price or rate located in... N number of capitalization periods % - the interest rate is identical for each of the capitalization periods Next, profit capitalization profit capitalization ratio capitalization

Compound interest can be calculated several times a year

(for example, by month, by quarter, by half-year). To consider this case, we introduce the concept of a nominal rate.

Nominal rate is the annual rate at which interest is calculated m once a year ( m > 1). Let us denote it by j . Therefore, for one period interest is accrued at the rate j/m.

Example. If at nominal rate j= 20% is accrued 4 times a year, then the rate for one period (quarter) will be equal to

20 % : 4 = 5%.

Formula (8) can now be represented as follows:

S = P ( 1+j/m) N , (10)

Where N- total number of accrual periods, N= m×t, t - number of years. With increasing frequency m accruals per year, the accumulation coefficient and, consequently, the absolute annual income grow.

Effective interest rate

To compare the real relative income for the year when calculating interest of one and m Once again, let's introduce the concept of the effective interest rate.

Effective annual interest rate i ef - This is the rate that measures the real relative income that is received for the year as a whole from interest, i.e. i ef - is the annual compound interest rate which gives the same result as m- one-time interest accrual at the rate for the period i = j/m .

The effective rate is found from the condition of equality of the two corresponding growth rates for one year:

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

It follows that

i ef = ( 1+ j / m) m - 1(11)

Example. Determine the effective compound interest rate to obtain the same compounded amount as using the nominal rate j=18%, with quarterly interest accrual ( m=4).

Solution . From formula (11) we obtain:

ief = (1 + 0.18 / 4) 4 - 1 = 0.1925 (or 19.25%).

Example. Find the effective rate if the nominal rate is 25% compounded monthly.

Solution . i eff = (1 + 0.25 / 12) 12 - 1 = 0.2807 or 28.07%.

It makes no difference to the parties to the transaction whether to apply a rate of 25% (for monthly calculations) or an annual rate of 28.07%.

Example. Find the nominal interest rate, compounded semiannually, that is equivalent to the nominal rate of 24% compounded monthly.

Solution. Let j 2 - interest rate corresponding to half-year accrual, j 12 - by month.

From the equality of the growth coefficients we obtain:

(1 + j 2 / 2) 2 = (1 + j 12 / 12) 12 ,

1 + j 2 / 2 = (1 + j 12 / 12) 6 Þ j 2 = 2[(1 + j 2 / 12) 6 - 1] =

2 [(1 + 0.24/12) 6 - 1 ] = 0.25 or j 2 = 25 %.

Continuous accrual of interest

The amount increased for t years according to formula (10) at a constant interest rate j m with increasing number m increases, but with unlimited growth m sum S = Sm tends to the final limit.

Really

This fact gives grounds to use continuous interest accrual at an annual rate d. At the same time, the accumulated amount over time t is determined by the formula

S = Pe d t . (12)

Interest rate d called growth force.

Example . The bank charges interest at a continuous rate of d=8% on the amount of 20 thousand rubles. within 5 years. Find the accrued amount.

Solution . From formula (12) it follows that the accumulated amount

S= 20,000 e 0.08 × 5 = 20,000 × e 0.4 = 20,000 × 1.49182 = 29,836.49 rub.

Tasks

3.1. Amount 400 thousand rubles. invested for 2 years at 30% per annum. Find the accrued amount and compound interest for this period.

3.2. A loan of 500 thousand rubles. issued at compound interest for 1 year at a rate of 10% per month. Calculate the total amount owed at the end of the term.

3.3. Determine the compound interest for one and a half years accrued on 70 thousand rubles. at a rate of 5% per quarter.

3.4. A time deposit in the bank was credited with $200 at a rate of 6% per annum. Find the amounts accumulated on the account after 2, 3, 4 and 5 years, subject to accrual of: a) simple interest; b) compound interest; c) continuous interest.

3.5. Calculate the effective interest rate equivalent to the nominal rate of 36%, compounded monthly. Answer: 42.6%.

3.6. For a nominal rate of 12% compounded twice a year, calculate the equivalent rate compounded monthly.

ACCOUNTING FOR INFLATION

IN modern conditions Inflation often plays a decisive role, and without taking it into account, the final results are a very relative value. IN real life Inflation is manifested in a fall in the purchasing power of money and a general level of price increases. Therefore, it must be taken into account when conducting financial transactions. Let's consider ways to take it into account.

Inflation rates are measured using the system inflation indices, which characterize the average change in the price level for a certain fixed set (basket) of goods and services over a certain period of time. Let the value of the basket at a point in time t equal to S(t) .

Price index or inflation index JP for the time from t 1 to t 2 called dimensionless quantity

JP = S(t 1 ) / S(t 2 ),

A inflation rate during this period is called the relative price increase:

h = = JP- 1.

Hence the price index

J P = 1+ h .

If the inflation consideration period includes n periods, in each of which the average inflation rate is h, That

J P = ( 1+h)n.

In the case where the inflation rate is i- th period is equal to h i , inflation index for n periods is calculated by the formula

J P = ( 1+ h 1 ) ( 1+ h 2 )…( 1+ h n).

Inflation index JP shows how many times and the inflation rate h - by what percentage did prices increase during the period under review?

Money Purchasing Power Index JD equal to the reciprocal of the price index:

J D = 1 /JP= 1/ ( 1+ h).

Example. You have an amount of 140 thousand rubles. It is known that over the previous two years prices have doubled, i.e. price index JP= 2. In this case, the purchasing power index of money is equal to JD= 1/2. This means that the real purchasing power is 140 thousand rubles. at the time of receipt will be only 140 × 1/2 = 70 thousand rubles. in money from two years ago.

If h is the annual inflation rate, then the annual price index is equal to 1+ h , therefore the increased amount taking into account inflation

S and = P ( 1+ i) n = P(13)

Obviously, if the average annual inflation rate h equal to the interest rate i, That S and = P, those. the real amount will not grow: the increase will be absorbed by inflation. If h > i , then the real amount is less than the original. Only in a situation h< i real growth is happening.

Example. A constant inflation rate of 10% per month over the year leads to an increase in prices of JP= 1.1 12 = 3.14. Thus, the annual inflation rate h = JP- 1 = 2.14 or 214%.

In order to reduce the impact of inflation and compensate for losses from a decrease in the purchasing power of money, interest rate indexation is used. In this case, the rate is adjusted in accordance with the inflation rate.

The adjusted rate is called gross rate. Let us calculate this rate, denoting it by r.

If inflation is compensated in the amount gross rates in the presence of simple interest, then the amount r we find from the equality of the increment factors:

1+n×r = ( 1+ n × i) J P = ( 1+ n × i)( 1+ h)n,

(14)

The value of the gross rate for increasing the compound interest rate is found from the equality ( n = 1):

1+ r = ( 1+ i)( 1+ h),

r = i + h + h×i(15)

Formulas (14), (15) mean the following: to ensure real profitability in i%, at an inflation rate h you need to set a rate of r %.

Example . The bank issued a loan for 6 months - 5 million rubles. The expected monthly inflation rate is 2%, the required real return on the operation is 10% per annum. Determine the interest rate on the loan taking into account inflation, the amount of the increased amount and the amount of the interest payment.

Solution . Inflation index JP= (1 + 0.02) 6 = 1.1262. From (14) we obtain the gross rate:

r = =0.365 (or 36.5%).

Amount of accrued amount

S= P( 1+ n r)= 5 (1 + 0.5×0.365) = 5.9126 million rubles.

Amount of interest payment (loan fee)

I= 5.9126 - 5.0 = 0.9126 million rubles.

Example . Loan of 1 million rubles. issued for two years. The real return should be 11% per annum (compound interest). Estimated inflation rate is 16% per year. Determine the interest rate when issuing a loan, as well as the increased amount.

Solution . From formula (15) we have:

r = 0.11+0.16+ 0.11×0.16 = 0.2876;

S= 1.0 (1 + 0.2876) 2 = 1.658 million rubles.

Tasks

4.1. Loan 500 thousand rubles. issued from June 20, 1998. to 09/15/98 When issuing a loan, it is assumed that the price index at the time of repayment will be 1.3. Determine the gross rate and the amount to be repaid.

Answer: R = 134% ; S R= 658,194 rub.

4.2. Loan in the amount of 5 million rubles. issued for 3 years. The real profitability of the operation should be 3% per annum at a compound rate. The estimated inflation rate is 10% per year. Calculate the gross rate and the repayable amount. Answer : R = 13,3 % ; S to R= 7,272,098 rub.

4.3. A deposit in the amount of 100 thousand rubles was placed in the bank. at 100% per annum for a period of 5 years. Expected inflation rate during this period h= =50% per year. Determine the real amount that the client will have after five years: a) taking into account inflation; b) without taking into account inflation.

4.4. What rate should the bank set so that, with annual inflation of 11%, the real return is 6%.

FINANCIAL RENTS

Regular annuity

Financial transactions often involve not one-time payments, but some sequence of them over time. An example would be loan repayment, rent, etc. Such sequences of payments are called flow of payments.

Let the financial transaction under the contract begin at the moment t 0, and ends at the moment tn . Payments Rk (k = 1,2,..,n) occur at moments tk . It is usually believed t 0 = 0 (Fig. 1).

Financial rent called a sequence of periodic payments Rk, Rk > 0 carried out at regular intervals.

Payments Rk called members of the annuity . If all payments are the same, i.e. Rk = R , then the rent is called constant.

Let d - annuity period, and n - the number of payments, then the product of the period by the number of payments nd represents calendar period of annuity. If payment is made at the end of each period (Fig. 1), then the annuity is called ordinary, and if at the beginning of the period, then given(Fig. 2).

Choosing base unit of time , let's ask annuity interest rate(complicated). We'll find increased amount S ordinary annual annuity, consisting of n payments, i.e. the sum of all members of the payment stream with interest accrued on them by the end of the term. To do this, let's look at a specific problem. Let within n years, at the end of each year, deposits are made into the bank R rubles Contributions are subject to compound interest at the rate i% per annum (Fig. 3).

Accrued amount S consists of n terms. Exactly

S = R + R( 1+ i) + R( 1+ i) 2 + ...+ R( 1+i)n- 1

On the right is the amount n terms of a geometric progression with the first term R and denominator 1+ i . Using the formula for the sum of a geometric progression, we get

(16)

s(n;i) and is called increase factor ordinary annuity. Formula (16) can be rewritten as

S = R  s(n; i)

Present value of annuity A is the sum of all annuity terms discounted at the beginning of the annuity term. From the equivalence condition for the current and increased values of ordinary annuity, we find modern meaning annuities A:

S = A( 1 +i)n or A = S( 1 + i) -n .

Thus,

. (17)

The expression is indicated by the symbol a(n;i) and is called discount factor ordinary annuity or reduction coefficient annuities. Thus, the modern meaning of rent

A = R × a(n; i) .

Example. Find the current and increased value of the annuity with payments of 320 thousand rubles. at the end of every month for two years. Interest is calculated monthly at a nominal rate of 24% per annum.

Solution . The effective monthly rate is 24% : 12 = 2% The current value is calculated using formula (17):

A= 320 = 6052.4619 thousand rubles.

The accrued value is calculated using formula (14):

S= = 9734.9952 thousand rubles.

Example . The company decided to create an investment fund. For this purpose, for 5 years, at the end of each year, 100 thousand rubles are deposited into the bank. at 20% per annum with their subsequent capitalization, i.e. adding to the already accumulated amount. Find the amount of the investment fund.

Solution . Here we consider a regular annuity with annual payments R= 100 thousand rubles. for n= 5 years. Interest rate i= 20%. From formula (16) we find:

S= 100 = 744.160 thousand rubles.

Reduced rent

The difference between a regular annuity and a reduced annuity is that all payments R for the reduced annuity are shifted to the left by one period relative to the payments of a regular annuity (compare Fig. 4a and 4b).

It is easy to understand that for each member of the reduced annuity, interest is accrued for one period more than in a regular annuity.

Hence the increased amount of reduced rent S P more in (1 + i) times the increased amount of ordinary annuity:

S P = S (1 + i) And sP(n; i) = s(n; i) (1 + i).

Exactly the same dependence is associated with the modern values of ordinary annuity A and reduced rent A P :

A P=A (1 + i), A P(n; i) = a( n; i) (1 + i) . (18)

Example . Loan in the amount of 5 million rubles. repayable in 12 equal monthly payments. The interest rate on the loan is set at i =3% per month. Find the monthly payment amount R upon payment:

A ) postnumerando(regular annuity),

b) prenumerando(adjusted rent).

Solution. A) R× a(12;0.03) = 5 million rubles.

Reduction coefficient a(12; 0.03) = = 9,95400 .

From here R= 5 million rubles / 9.95400 = 502311 rubles.

b) Similar to the previous one: R× a(12;0.03) = 5 million rubles. From formula (18):

A P(12;0.03) = a(12;0.03) × (1+ i) = 9.954 × 1.03 = 10.25262;

R= 5 million rubles/10.25262 = 487680 rubles.

Deferred annuity

If the term of the annuity begins at some point in the future, then such an annuity is called postponed or delayed. We will consider deferred annuity as ordinary. The length of the time interval from now to the beginning of the annuity is called period from deferment. Thus, the period of deferment of annuity with payments in half a year and the first payment in two years is equal to 1.5 years (Fig. 5).

In Fig. 5 figure 3 (1.5 years) means the beginning of the annuity. The beginning of payments for a deferred annuity is shifted forward relative to a certain point in time. It is clear that the time shift does not in any way affect the amount of the accumulated amount. The present value of rent is a different matter. A .

Let the rent be paid later k years (or periods) after the initial period of time. In Fig. 5, the initial period is indicated by the number 0, and the modern value of ordinary annuity is A . Then the modern value deferred by k years of annuity A k equal to the discounted value A , that is

A k = A( 1+ i)-k= R a (n;i) ( 1+i)-k. (19)

Example . Find the current value of deferred annuity with payments of 100 thousand rubles. at the end of each half-year, if the first payment occurs after two years and the last after five years. Interest is calculated at a rate of 20% per six months.

Solution. Rent starts in three months. The first payment is made at the end of the fourth half of the year, and the last at the end. There are 7 payments in total. From formula (18) at k= 3; n = 7; i= 0.2, we get:

A 3 = 100 = 208599 rub.

Example. Find the amount of annual payments of annuity deferred for two years for a period of 5 years, the current value of which is 430 thousand rubles. Interest is charged at a rate of 21% per annum.

Solution. From formula (19) we find:

R = A k(1+ i)k/A( n;i) .

At k= 2; n = 5; i= 0.21, we get:

R= 430 ·1.21 2 = 215163 rub.

We examined the method of calculating the accrued amount and the modern value, when annuity payments are made once a year and interest is also calculated once a year. However, in real situations (contracts) may provide for other conditions for the receipt of rental payments, as well as the procedure for calculating interest on them.

5.4. Annual rent with interest calculation m once a year

In this case, rent payments are made once a year. Interest will be calculated at the rate j/m , Where j - nominal (annual) compound interest rate. The value of the accumulated amount will be obtained from formula (16), if we put in it

i = (1+ j/m)m- 1 (see (11)).

As a result we get:

(20)

Example. An insurance company that has entered into an agreement with the company for 3 years, annual insurance premiums in the amount of 500 thousand rubles. deposits it in the bank at 15% per annum with interest accrued semi-annually. Determine the amount received by the insurance company under this contract.

Solution. Assuming in formula (20) m = 2; n = 3; R = 500; j = 0.15, we get:

S= 500 = 1,746,500 rub.

5.5. P- fixed-term annuity

Rent payments are made P once a year in equal amounts, and interest is calculated once at the end of the year ( m = 1). In this case, the rent term will be equal to R/P , and the formula for the accumulated amount is obtained from formula (16), in which the rate for the period iP is found from the condition of financial equivalence (total periods P· n ):

(1 + i) = (1 + iP)P , iP = (1+ i) 1/P – 1.

Substituting the resulting rate for the period iP in (16), we have:

(21)

Example . The insurance company accepts the established annual insurance premium of 500 thousand rubles. twice a year for 3 years. The bank servicing the insurance company charges it compound interest at the rate of 15% per annum once a year. Determine the amount received by the company at the end of the contract.

Solution . Here R = 500; n = 3; P = 2; m= 1. Using formula (21) we find:

S = · = 1779 thousand rubles.

Perpetual annuity

Perpetual annuity means an annuity with an infinite number of payments. Obviously, the accumulated amount of such an annuity is infinite, but the modern value of such an annuity is equal to A = R/i. To prove this fact, we use formula (17) for final rent:

A = R/i.

Passing in this formula to the limit at n® ¥, we get that A = R/i.

Example: The company rents the building for $5,000 a year. What is the redemption price of the building at an annual interest rate of 10%?

Solution . The redemption price of the building is the current value of all future rental payments and is equal to A = R/i= $50,000

Consolidation and replacement of annuities

General rule combining rents: the modern values of rents (components) are found and added, and then the rent is selected - the amount with such a modern value and the necessary other parameters.

Example . Find the combination of two annuities: the first with a duration of 5 years with an annual payment of 1000, the second with 8 and 800. Annual interest rate

Solution . Modern values of annuities are equal to:

A 1 = R 1× a(5;0.08)= 1000 × 3.993 = 3993; A 2 = R × a(8;0.08) = =800×5.747=4598.

A= A 1 + A 2 = 3993 + 4598 = 8591.

Consequently, the combined annuity has a modern value A= 8591. Next, you can set either the duration of the combined annuity or the annual payment, then we determine the second of these parameters from the formulas for annuities.

Tasks

5.1. Amounts of 500 thousand rubles will be deposited annually into a deposit account with compound interest at a rate of 80% per annum for 5 years. at the beginning of each year. Determine the accumulated amount.

5.2. At the end of each quarter, amounts of 12.5 thousand rubles will be deposited into the deposit account, on which compound interest will also be accrued quarterly at a nominal annual rate of 10% per annum. Determine the amount accumulated over 20 years. Answer: RUB 3,104,783.

5.3. Calculate the amount that needs to be deposited into the account of a private pension fund so that it can pay its participants 10 million rubles monthly. The fund can invest its funds at a constant rate of 5% per month.

(Hint: use the perpetual annuity model).

5.4. A businessman rented a cottage for $10,000 a year. What is the redemption price of the cottage at an annual rate of 5%. Answer: $200,000.

5.5. During the court hearing, it turned out that Mr. A underpaid taxes by 100 rubles. monthly. The tax office wants to recover unpaid taxes for the last two years along with interest (3% monthly). How much should Mr. A pay?

5.6. For reclamation work, the state transfers $1000 per year to the farmer. The money goes into a special account and is accrued every six months at 5% according to the compound interest scheme. How much will accumulate in the account after 5 years.

5.7. Replace a five-year annuity with annual payments of $1,000 with an annuity with semi-annual payments of $600. Annual rate 5%.

5.8. Replace the 10-year annuity with an annual payment of $700 with a 6-year annuity. Annual rate 8%.

5.9. What amount should the parents of a student studying at a fee-paying institute deposit in the bank so that the bank transfers $420 to the institute every six months for 4 years? Bank rate 8% per annum.

REPAYMENT OF DEBT (LOAN)

This section provides an application of the theory of annuities to planning the repayment of a loan (debt).

Developing a loan repayment plan involves drawing up a schedule of periodic payments by the debtor. The debtor's expenses are called debt service costs or loan amortization. These costs include both current interest payments, as well as funds intended for principal repayment.Exist various ways debt repayment. Participants in a credit transaction stipulate them when concluding a contract. In accordance with the terms of the contract, a debt repayment plan is drawn up. The most important element of the plan is determining the number of payments during the year, i.e. definition of number urgent payments

D) a rate that decreases as the object of taxation decreases