**To improve the methods of calculation of stock exchange RTS index**

*Material posted: -*

*Publication date: 11-01-2009*

In the current (at the time of writing) the methodology of calculation of RTS index is the most vulnerable element is to identify the values of CI coefficients, restricting the share of capitalization of securities of the i-type (weighting factor).

It is not clear why the restriction is selected by the market capitalization of individual companies 15% of total capitalization, the iterative scheme of recalculating the values of capitalization is also quite controversial. The rest of the techniques are trivial.

We can offer the following modification of the method of determination of reduction factors with a clear economic meaning.

Suppose the stock market consists of n large brokerage firms that divide the market among themselves. Let us rst for simplicity assume that their accessibility to the shares of all companies are the same and also the same return on investments of financial capital in all shares. Suppose also that the investment policy of each broker — independent, i.e. between brokers is no strong coordination of their actions.

The companies significantly differ in their capitalization: there are very large companies (such as OJSC "Gazprom"), there is a small company.

Then the process of selecting a broker the direction of investment of their funds can be likened to shooting at a difficult target (Fig. 1), square parts which are proportional to the capital of a company (each circle represents one company).

**Fig. 1** Investment target

Naturally, in large portions of the target corresponding to the large cap stocks will "get" a few "shooters" (i.e. brokers), and in small fragments — most likely, only 1 arrow. In other words, the large capital companies will share several brokers a capital small companies will get only 1 broker. Therefore, from the point of view of one of the broker in filling its investment portfolio will involve small companies with 100% capital as well as large companies, but only *with part* of their capital. Thus, when calculating the total capitalization of the brokerage investment portfolio limiting factors appear in the most natural way.

Consider the scheme of the determination of these coefficients from a formal point of view.

We define the probability P of the choice of broker companies:

We define the mathematical expectation of the brokers, applying for shares of i-th company on the condition that one bidder is already in place.

A probability Q_{1} , one more contender will be joined by one will be equal

where C^{1}_{n-1} is the number of combinations of n elements one at a time, which is equal to n — 1.

Similarly, the probability Q_{2} , one more contender will be joined by one will be equal

where C^{2}_{n} is the number of combinations of n elements one at a time, which is equal to

For the case of adding to the existing one still other bidder j, the probability Q_{j} will be equal to

In the end, the average number of b brokers, applying for shares of a company, provided that broker 1 is already available, will be determined as follows:

That is why the 1st broker you will have only part of the capital of the company (shares), defined as

This is the desired weighting factor.

In Fig.2 for various values of n shows the graphs of dependences of the value of the weight coefficient C of P is the relative capitalization of the company. The values of the weighting factor calculated by the ratio (1).

**Fig. 2**

For small values of relative capitalization P the value of C tends to 1; when P 1 approximation to the value of C monotonically decreases and tends to (1/n).

To Refine methodologies need to bear in mind two circumstances.

First, the brokers not all stocks, but only part of them W_{i}, being in free circulation (the ratio of the free float). So logically instead of P in equation (1) is to substitute the parameter P*, equal

Secondly, the shares of different companies have *different* profitability. To account for this circumstance naturally instead of P^{*} to apply the parameter P is^{**}equal to

where C_{i} and C_{0i} , respectively the current and initial (for example, at the time of update of the methodology for calculating RTS index) pictures 1-th action of the i-th company. Large values (C/C_{0})_{i} will be promoting the big incentives to buy the brokers shares the most successful companies.

The value of P^{**}_{i} and you must substitute in equation (1) to calculate the weighting factor. The area of the ellipses in Fig.1 will be proportional to the quantities P^{**}_{i}.

Thus there is a clear and relatively simple scheme of determining the weights and the RTS index.

If there are 7 major brokerages, for example:

- "Troika-Dialog"
- Aton
- "Renaissance Broker"
- "Kit Finance"
- "URALSIB Capital"
- Alpha Bank
- Investment Bank "kit Finance",

the dependence of the weighting factor from the option is P^{**}i will be in the form depicted in figure 3.

**Fig. 3**

**Tags:**
assessment
, financial center
, Exchange
, finances

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