## Inflation and its accounting with currency conversion

Without taking into account inflation, the final results of cash flow calculations are very conditional. Let us consider some concepts related to inflationary processes [33].

The real value * C * of the amount * S, * devaluated due to inflation for a given period of time, is calculated according to the formula

where - the price index showing how many times the average price for a given group of products has increased over a given period of time.

The price index can be calculated, for example, according to the Paasche formula:

where - the price of the/-th commodity in the analyzed and base periods, respectively; - the number of goods sold y in the period under study; * T - * the total number of goods examined.

The rate of inflation (* H *) is the relative increase in prices for the period:

(2.14)

The price index for several periods * n, * following one after another, is calculated by the formula

(2.15)

where * t * is the period number; - the price index in the period * t, * the rate of inflation in the period * t. *

If the expected rate of inflation is a constant for * n * periods, then formula (2.15) takes the form

(2.16)

The average for the period of the price index and the rate of inflation are found by formulas

where * n * is the number of periods (years).

For simple interest, the real value of the amount, taking into account inflation, is determined by the expression

(2.17)

It follows from (2.17) that an increase in the accumulated sum occurs when the relation

The reverse inequality testifies to the erosion of capital - the depreciation of money over time through inflation.

The rate , in which the increase is equal to the losses due to inflation, is determined from the equation . Comparing this with the expression (2.17), we find:

(2.18)

Example 17. The monthly inflation rate is: a)

. For cases a) and b), it is required to find the price index and the inflation rate for 12 and 3 months. respectively, and also to determine the depreciated accrued amount, if the amount of 10,000 rubles. within the specified periods a simple interest rate of 50% per annum was charged * (K * = 360). It is also necessary to determine the rate at which the increase is equal to the losses due to inflation.

*Solution *

The solution of the example uses the formulas (2.14) - (2.18).

In option a) there was an erosion of capital, and to increase it, the interest rate should exceed 60.1%. In option b), capital increased by 10294.54/10000 = 1.029454 times, or by about 2.94%.

For compound interests, the inflation-taking amount is determined by the expression

(2.19)

The dependence of the monetary amount on time, taking into account inflation, is shown in Fig. 2.19.

*Fig. 2.12. * ** Dependence of the amount of time against inflation **

To compensate for the depreciation of money, the rate is increased by the amount of the inflation premium, which is an additional yield compensating for inflationary losses. The final bet is called a gross rate, or a nominal rate. We express the size of the gross rate * r * through the profitability of the operation as a compound annual interest rate . Then the rate in the formula (2.17) and the bid a in the formula for compound percent should be considered equivalent, i.e. their relationship is determined by the equation

where - the index of the chain for * n * years. Hence we find:

(2.20)

For compound interests, the gross rate and yield are determined by the ratio

(2.21)

It follows from (2.21) that

(2.22)

For a constant inflation rate, when formula (2.16) is substituted into equation (2.21), we find:

From here we get the connection between the gross rate and yield:

(2.23)

With and we have

Thus, as follows from formula (2.23), it is possible to determine the gross rate by adding the profitability of the operation and the rate of inflation only for small values of these quantities.

**Example 18. ** Find the real profitability of a financial transaction for a period of 0.5 years, a simple gross rate of 10%, and a monthly rate of inflation growth of 1%.

*Solution *

The price index for six months is equal to

The real rate of return is , or -2.6%.

There was an erosion of capital by 2.6%.

**Example 19. ** Find a compound interest-rate-gross rate with a return of 15% per annum and an annual rate of inflation growth of 1.7 for the exact and approximate formulas.

*Solution *

For the exact formula or 95.5%; by the approximate formula , or 85%.

Thus, the error in the calculation was 10.5%. The relative error is . If the conditions and this error is less. We leave it to the readers to see for themselves.

Conversion (exchange) of currency and temporary increase of money can lead to both profit and loss. It depends on the size of the interest rate, exchange rates at the beginning and end of the operation, inflation. First of all, let's consider the conversion of currency through its purchase and sale. The analysis of profitableness at purchase and sale of currency it is possible to spend on the basis of a parity [32]

where * С * is the real value of the sum in rubles at the end of the transaction; * Р * - the sum in rubles at the beginning of the operation, and - exchange rate at the beginning and at the end of the operation, respectively, having, for example, the dimension of the ruble/dollar; - the price index for the operation time * n. * Ruble the amount P is exchanged for the currency (division by ), then exchanged for the period * n * in rubles (multiplying by ). To determine the real value of the amount received, it is divided by the price index for the operation time * n, * equal to . We introduce the notation . Then the resulting formula can be written in the form

(2.24)

To determine the yield a in the form of a compound interest rate of the financial transaction in question, the principle of financial equivalence of obligations is used. Equivalent are equal payments to each other when they are brought to one point in time. In accordance with the principle of financial equivalence of obligations, expression (2.24) can be written in the form

From here we find the formula for the profitability of the operation:

(2.25)

The profitability of the operation will be equal to zero if the condition is met. At the yield will be greater than zero, and at - less than zero. Since the price of buying a currency and the price of its sale differ at the same time, then when calculating the yield for it is necessary to accept the purchase chain, and for - the sale price.

**Example 20. ** Dollars were purchased at a rate of 31 rubles/dollar, and after 1.2 years sold for: 1) 32 rubles/dollar; 2) 33 rubles/dollar. The inflation rate for this period of time was 6%. It is required to determine the profitability of a financial transaction.

*Sieve *

For the given values, the ratio of the selling price to the purchase rate will be

The price index for 1.2 years is equal. Yield for the cases under consideration:

!, or -2.2% per annum:

1, whether it is 3.5% per annum.

In the first case, there was an erosion of capital, in the second case, capital increased.

If you increase interest with conversion, you can:

• rubles - SLE - increase in hard currency - rubles;

• SLE - ruble - the increase in rubles - SLE.

And the buildup can be carried out both in a simple way

and a difficult interest rate. Let's consider the first variant at accretion under the difficult interest rate. The designations used here are the same as before. If * r * is the compound annual rate of the accumulation of SLE, then the equivalence equation for the considered conditions takes the form

Hence, we find the profitability of a financial operation according to the first scheme of currency conversion with a percentage increase:

(2.26)

**Example 21. ** For the conditions of the previous example, let's put a complex rate of increasing the hard currency by 5% per annum.

*Solution *

, or 2.7% per annum; , or 5.4% per annum.

Now in both cases there was a capital increase.

For the second version of the currency conversion with the increase, the accumulated amount taking into account the inflation of the currency is determined by the expression

SCP shows that the value is measured in the currency units of the selected currency; well - the price index of the selected currency for the period in question; * r - * a complex ruble annual rate of increase. From this expression is the formula for the profitability of a financial transaction:

(2.27)

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