## Simple and complex percentage

The above are ways to assess the profitability and risk of investment in the event that the investment is carried out at a single calculation step. Meanwhile, as a rule, the investor invests his funds in one or another investment object for several periods (in particular, years). This refers to bank deposits, bonds, investment projects in the field of capital investments, etc. Such investment raises many questions about the evaluation of the return on investment, taking into account the time factor in calculating future amounts of income and current costs. The answers to most of these questions are given in later chapters.

For now, let us turn to the very common problem of the following property: suppose the investor decided to invest some sum of money for several periods (for example, for * t * years). What amount of Scon he can receive in * t * years? Like the nature of the problem is usually attributed to the task of finding the future value * FV * of the invested funds. To calculate the initial sum Sv (which is equivalent to finding the value Scon), we should answer the following question: does the investing method provided by the investor provide reinvestment of the sums received at the end of each step of calculation (in this case in our case) ?

What do you mean? Let Snach = 1000 rubles., And the investor decides to lend this amount to the borrower for 5 years. Taking into account the planned level of inflation and the risk of investing, both sides conclude that the yields on this loan should change as follows: in the coming year, the yield will be the rate * r * 1 = 18%, for the second year of the loan * r * 2 = 16%, and the subsequent rates will be equal to * r * 3 = 14%; * r * 4 = 12%; * r * 5 = 10%. How much will Scon provide this way of investing? It all depends on how these yields will be calculated. There are two main options.

1. The accrual of profitability occurs on the initial loan amount. This means that within the first year the loan will provide the investor with an income of 180 rubles. in the form of 18% charged on the initial amount. In the next year, 16% will be re-accrued for SNach = 1000 rubles. As a result, at the end of the second year the investor will receive another 160 rubles. interest income. The same will happen in each year of borrowing. Obviously, under this scheme of accrual of interest payments received in five years, the total income of the investor will be (180 rubles. + 160 rubles. + 140 rubles. + 120 rubles. + 100 rubles.) = = 700 rubles. interest payments plus 1000 rubles. the nominal value of the debt. Total - 1700 rubles. Such accrual of interest amounts is called a * simple interest scheme. *

2. The accrual of profitability occurs on subsequent amounts. In this case, after the end of the first year of borrowing, when the investor has 1,180 rubles, the next 16% are not accrued for Snach = 1,000 rubles, but for 1,180 rubles. As a result, at the end of the second year, the total income of the investor should increase to S2 = 1180 rubles. • (1 + 0.16) = S | start • (1 + 0.18) • (1 + 0.16) = 1368.8 rubles. At the end of the third year, the next 14% will be accrued already for the amount of 52, resulting in S3 = 1368.8 • (1 + 0.14) = Snach • (1.18) • • (1.16) (1.14) = 1560.43 rubles.

As a result, in five years

Scon = Sinch • (1,18) (1,16) • (1,14) (1,12) • (1,1) = 1,922.45 rubles.

A similar charge of interest is called a * compound interest scheme. *

Very often when investing for several periods, the yield for each step of the calculation (the interest rate) does not change. This is typical of bank deposits, debt securities, etc. In this case, the calculation of the Scon values is simplified. If the investment is carried out at * t * steps

(years) and the yield (interest rate) for each calculation step is * r *%, then using the scheme:

• a simple percentage:

(1.8)

• compound interest:

(1.9)

As a rule, the yields (interest rates) are expressed on an annual basis, therefore formulas (1.8) and (1.9) should be used if interest rates are accrued per year once. Meanwhile, there are funds that accrue interest on which occurs several times a year. For example, for most corporate bonds of domestic issuers coupon payments are made twice a year, and individual bonds provide a compound interest charge quarterly. It should be borne in mind that more frequent accrual of interest allows you to receive a larger final amount.

Suppose, for example, that the annual interest rate on a security is 12% and payments are made twice a year. Then, investing 1 thousand rubles in a security, for the first six months the investor will receive 6%, i.е. 60 rubles., And will have at the end of the first half of 1060 rubles. This money, he can invest for six months at the same 6% and at the end of the year have the amount

If the investor invested 1 thousand rubles. under 12%, which were charged once a year, then at the end of the year he would have received 1200 rubles. Consequently, investing at 12% per annum, charged every six months, is equivalent to investing at 12.36% per annum, charged once a year.

If the interest is charged twice a year, then the final amount at the end of the year can be found by the formula

In the general case, if within a year the percentage is charged * t * once a year, then the final amount

and this is equivalent to the case if the interest was calculated once a year

Sometimes in calculations resort to a continuous accrual of interest, i.e. suppose . The limit of with equals * e * r, where * e * = 2, 71828 ... is the basis of natural logarithms. Therefore, if the investor directs 1000 rubles. under 7% per annum, charged continuously, at the end of the year he will receive the following amount:

i.e. investing at 7% per annum, accrued continuously, is equivalent to investing money at 7.25%, charged once a year.

If * continuous interest accrual * occurs

*n*years, the original amount will increase to . For example, if at 1000 rubles. 7% per annum will be accrued continuously for three years, then at the end of the investment period the investor will receive the amount

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