After studying this chapter, the student must:


• The concept of profitability;

• The concept of discount income;

• The concept of interest income;

• what is increment and discounting;

• what is the future and initial cost of money;

• what is an annuity;

be able to

• Calculate interest income at a simple and compound interest rate;

• calculate the profitability of operations with shares;

• calculate the yield: to redemption of discount bonds, sale of discount and interest bearing bonds, repayment of interest bonds;

• determine the market value of shares;

• Calculate the financial result of the operations (purchase and sale) of call and put options;


• Alternative yield methodology;

• a step-by-step methodology for solving tasks for calculating and comparing yields;

• the method of discounting cash flows;

• Graphical methodology for analyzing operations with options.

Specialists working on the stock market constantly have to evaluate the parameters that characterize transactions with securities. Often these tasks are formalized, programmed and to solve them it is enough to pick up the initial data, enter them into the personal computer and get the result. Nevertheless, when carrying out numerical analysis, it is necessary to understand how the final result is obtained, since only in this case it is possible to make qualified decisions on the implementation of a particular financial operation and only in this case it is possible to understand the limits of applicability of the estimates obtained.

By now, a lot of experience in training of stock market specialists has been accumulated and a certain approach to the solution of computational problems has been approved. Its peculiarity lies in the fact that the methodology for solving a large number of different computational tasks related to the numerical estimation of the parameters of operations in the stock market can be represented in the form of a "step-by-step" algorithm for their solution. In this chapter, the execution of this algorithm is illustrated by the solution of specific computational problems and by the completion of each step either to the final formula, or to a numerical result.

The practice of using this technique has shown, on the one hand, the ability to solve almost any computational tasks with which the expert may encounter transactions in the stock market, and on the other hand the need to supplement it with private methods and formulas for the rapid evaluation of parameters when solving special problems in a rapidly changing environment in the securities market.

Note that the successful solution of computing tasks involves not only the ability to perform numerical calculations, but also the knowledge of United States legislation regulating the stock market. Therefore, when solving problems, it is necessary to recall the contents of the previous chapters of this textbook, namely issues related to the taxation of operations with securities, their state registration, etc.

Basic concepts and formulas. Alternative yield method

Yield. The most significant parameter, the determination of which is necessary in the analysis of operations with stock valuables, is profitability. It is calculated by the formula


where d is the profitability of the operation,%;

D - income received by the owner of the financial instrument;

Z - the cost of its acquisition; τ is a coefficient that recalculates the yield for a given time interval.

The coefficient τ has the form


where Δ T is the time interval to which the profitability is recalculated;

Δ t is the time interval for which the income D was received.

Thus, if the investor received an income, say, for nine days (i.e., bought a financial instrument and then sold it in nine days at a profit, Δ t = 9), then calculation of profitability for the fiscal year (Δ T = 360), the numerical value of the coefficient τ will be equal to

It should be noted that usually the profitability of operations with financial instruments is calculated per one financial year, in which there are 360 ​​days.

As an illustration of the calculation of the yield of a financial instrument, consider the following example. Having carried out the operation of buying and selling a financial instrument, the broker received an income equal to D = 25,000 rubles for nine days, and the market value of this financial instrument at the time of purchase was Ζ = 10000000 rub. The yield of this operation in terms of a year

Revenue. The next important parameter used in calculating the efficiency of securities transactions is the income received from these transactions. It is calculated by the formula


where Δ d is the discount part of the income;

Δδ - the percentage of income.

Discount income. The formula for calculating the discount income is of the form


where Р пр - the selling price of the financial instrument with which transactions are performed;

Рпок is the purchase price of a financial instrument (note that in the expression for the yield, Р пок = Ζ ).

Interest income. Interest income is defined as the income received from interest accruals on this financial instrument. Two cases must be considered. First, when interest income is accrued at a simple interest rate, and the second, when interest income is charged at a compound interest rate.

The scheme of calculation of income at a simple interest rate. The first case is typical for calculating dividends on preferred shares, interest on bonds and simple interest on bank deposits. In this case, an investment of X0 rub. in a period of time equal to n interest payments, will lead to the fact that the investor will have an amount equal to


Thus, the interest income in the case of a simple interest calculation scheme will be equal to


where X n - the amount generated by the investor through interest payments n ;

X 0 - initial investment in the financial instrument under consideration;

α - the amount of the interest rate; n is the number of interest payments.

Scheme of calculating income at a compound interest rate. This case is typical for interest accrual on bank deposits under the scheme of compound interest. Such a scheme of payments assumes the accrual of interest on both the principal amount and the previous interest payments.

Investments in the amount of X0 rub. after the first interest payment will give an amount equal to

At the second interest payment, interest will be charged to the amount X 1. Thus, after the second interest payment, the investor will have an amount equal to

Therefore, after the n -th interest payment, the investor will have an amount equal to


Therefore, interest income in the case of accruing interest on a compound interest scheme will be equal to


Income taking into account taxation. The formula for calculating the income received by a legal entity in the course of transactions with corporate securities has the form


where σл - the tax rate for the discount part of the income; σп is the tax rate for the percentage of income. The discount income of legal entities (Δ d ) is subject to taxation in the general procedure. The tax is levied on the income recipient.

The tax on interest income (Δδ) is collected at the source of these revenues.

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