# The main types of problems solved in the stock market, Problem solving algorithms, Tasks for calculating yield, Tasks for comparing yields - Securities market. Theory and practice

## The main types of problems solved in the stock market

The tasks that are most often encountered when analyzing the parameters of operations in the stock market require an answer, as a rule, to the following questions:

• what is the yield of a financial instrument;

• the profitability of which financial instrument is higher;

• what is the market value of securities;

• what is the total income that the security brings (interest or discount);

• what should be the circulation period of securities that are issued at a given discount, in order to obtain an acceptable return?

The main difficulty in solving this type of problem is to compose an equation containing the parameter of interest to us as an unknown. The simplest tasks involve the use of formula (1) to calculate profitability.

However, the bulk of other, much more complex problems, with all the diversity of their formulations, surprisingly, has a common approach to solving. It consists in the fact that under a normally functioning stock market the yields of various financial instruments are approximately equal. This principle can be written as follows:

(10)

Using the principle of equality of returns, you can create an equation for solving the problem, revealing formulas for profitability (1) and reducing the factors. In this case, equation (10) takes the form

(11)

In more general form, using expressions (2) - (4) and (9), formula (11) can be transformed into an equation:

(12)

Converting this expression into an equation to calculate the unknown in the problem, you can get the final result.

## Algorithms for solving problems

The methodology for solving such problems is as follows:

1) determines the type of financial instrument for which you want to calculate the yield. Typically, the type of financial instrument with which transactions are performed is known in advance. This information is necessary to determine the nature of the income to be expected from this security (discount or interest) and the nature of the taxation of the income received (rate and availability of benefits);

2) the variables in formula (1) that are to be found are clarified;

3) If the result is an expression that allows you to compose an equation and solve it with respect to the unknown unknown, then the procedure for solving the problem practically ends here;

4) If the equation for the unknown unknown could not be compiled, then formula (1), successively using the expressions (2) - (4), (6), (8) and (9), leads to the form that allows calculate an unknown value.

The above algorithm can be represented by a scheme (Figure 10.1).

In solving problems of this type, the formula (11) is used as the initial one. The method of solving problems of this type is as follows:

1) determine financial instruments, the yield of which is compared with each other. It is assumed that with a normally functioning market, the yields of various financial instruments are approximately equal to each other;

2) then the algorithm for solving the problem repeats the previous one, namely:

• known and unknown variables are clarified in formula (11);

• If the result is an expression that allows you to compose an equation and solve it with respect to the unknown unknown, then the equation is solved and the procedure for solving the problem ends here;

• If the equation for the unknown unknown can not be compiled, then using (2) - (4), (6), (8) and (9), formula (11), successively leads to a form that allows us to calculate the unknown value.

The above algorithm is shown in Fig. 10.2.

Fig. 10.1. Algorithm for solving the problem of calculating profitability

Let's consider some typical computational problems solved using the proposed methods.

Example 1 ( yield calculation ). At the beginning of the year, the stock was bought for 1,000 rubles. At the end of the year it was sold. At the same time, the discount income from the operation was 200 rubles. Determine (excluding taxes) the annual profitability of this operation.

Step 1. A security is a stock. This security can bring to its owner both interest (dividends) and discount income. In the condition of the problem, only the discount income is mentioned.

Step 2. To solve the problem, we use formula (1). In this formula we know:

Fig. 10.2. Algorithm for solving the problem of comparing yields

- the cost of buying shares Z = 1000 rubles;

- income received by the shareholder, D = 200 rubles. Substituting the known variables into the formula (1), we obtain the expression

Step 3. It is known from the condition of the problem that the revenue was received for the year ( Δt = 1), and in the task it is necessary to calculate the annual yield ( ΔТ = 1). Using the formula (2), we obtain τ = 1.

Step 4. We substitute the computed value of τ into the formula obtained in step 2. As a result, we get the value of the annual profitability of the operation

Example 2 ( calculating the value of a financial instrument ). At what price an investor can buy a bond, if the annual profitability of operations with financial instruments on the market is 15%, and in a year the bond is repaid with a coupon payment of 300 rubles. Taxation is not taken into account.

Step 1 . A security is a bond. This security can bring to its owner both interest (coupon payments) and discount income. In the condition of the task, only interest income is mentioned.

Step 2. To solve the problem, we use formula (1). In this formula we know:

- the current profitability of operations with financial instruments d = 15%;

- income that the owner of the bond will receive in a year, D = 300 rubles.

Substituting known variables into formula (1), we obtain the expression

Step 3. From the condition of the problem, it is known that the revenue will be received for the year (Δ t = 1), and the problem knows the annual profitability of financial transactions ( ΔT = 1). Using the formula (2), we obtain τ = 1.

Step 4. Substitute the computed value of τ into the formula obtained in step 2, and obtain the expression

Step 5. We resolve the resulting expression to Ζ. As a result, we get the formula

from which it follows that Ζ = 2000 rubles.

Example 3 ( calculating the income from a transaction with a financial instrument ). The investor purchased for 20 183 rubles. a six-month maturity bond. The yield of this operation coincided with the average market and amounted to 18% per annum. On what income can the investor expect when the bond is redeemed?

Step 1. A security is a bond. This security can bring to its owner both interest (coupon payments) and discount income. In the condition of the problem, the type of income does not become thin, i.e. it can be either interest or discount.

Step 2. To solve the problem, we use formula (1). In this formula we know:

- the current profitability of operations with financial instruments d = 18%;

- the cost of acquiring a bond Z = 20 183.49 rubles.

Substituting the known variables into the formula (1), we obtain the expression:

Step 3. From the condition of the problem it is known that the income will be received for half a year ( Δt = 1/2), and the problem knows the annual profitability of financial transactions ( ΔT = 1). Using formula (2), we obtain τ = 2.

Step 4. Substitute the computed value of τ in the formula obtained in step 2, and obtain the expression

Step 5. We resolve the resulting expression to Z. As a result, we get the expression

which implies that D = 1816.51 rubles.

Example 4 ( yield calculation ). The deposit certificate was purchased six months before the maturity date at a price of 10,000 rubles. and sold two months before maturity at a price of 10,400 rubles. Determine (for a simple percentage of

the rate without taxes) profitability of this operation in terms of a year.

Step 1: The type of security is clearly indicated: the certificate of deposit. This security, issued by the bank, can bring to its owner both interest and discount income.

Step 2. From the formula (1) we obtain the expression

However, we have not yet received equations for the solution of the problem, since the values ​​of D and τ remained undefined in the expression obtained.

Step 3. We use the formula (2) for solving the problem, in which Δ T = 12 months and Δ t = 6 - 2 = 4 months. Thus, τ = 3. As a result, we obtain the expression

This equation can not be used to solve the problem, because the quantity D ns is defined.

Step 4. From the formula (3), taking into account that Δδ = 0, we obtain the expression

This expression also does not allow to solve the task, since the discount income ( Ad ) is not defined.

Step 5. From the formula (4), taking into account that Рпр = 10400 rub. and Р пок = 10000 rub., we obtain an expression that allows us to solve the task:

Example 5 ( yield comparison ). Determine the placement price Z by the bank of its bills (discount), provided that the bill is issued for the amount of 200,000 rubles. with a payment period Δ t 2 = 300 days, the bank interest rate is equal to β = 14% per annum. The year is equal to 360 days.

Step 1. The first financial instrument is a deposit account in a bank. The second financial instrument is a discount bill.

Step 2. In accordance with formula (10), the yields of financial instruments should be approximately equal to each other:

However, this formula does not represent an equation for an unknown quantity.

Step 3. We refine the equation, using the formula (11) to solve the problem. We take into account that Δ T 1 = AT 2 = 360 days, Δ t 1 = 360 days and Δ t 2 = 300 days. Thus, τ1 = 1 and τ2 = 360: 300 = 1,2. We also take into account that Z1 = Z 2 = Ζ. As a result we obtain the expression

This equation can not be used to solve the problem.

Step 4. From the formula (6) we determine the amount that will be received in the bank when paying income at a simple interest rate with one percent payment:

From the formula (4) we define the income that the owner of the bill will receive:

We substitute these expressions in the formula obtained in the previous step, and we get

We solve this equation with respect to the unknown Ζ and as a result we find the offering price of the bill, which will be equal to Ζ = 179104,48 rubles.

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