## Commutation functions: principles of construction and use in actuarial calculations

Switching functions were invented in the XVIII century. and gained a high level of popularity. Switching functions greatly simplify the calculation of the values of many actuarial indicators. However, with the development of information and computer technology this advantage is not so relevant and significant. Expected values (one-time net premiums) can be derived in a deterministic model, closely related to commutation functions. At the same time, the transition to probabilistic models allows a deeper understanding of the essential aspects of insurance. However, at present, switching functions are still used in actuarial calculations.

Switching functions are divided into two groups. The commutation functions of the first group are based on the numbers of surviving, the basis of the functions of the second group - the number of deceased. The commutation functions of both groups are calculated from the data of the mortality tables. To calculate tariffs and reserves, individual pensions insurance requires switching functions of only the first group. In the calculations for verifying the degree of balancing of production pension funds (group pension insurance), the switching numbers of both types are used.

Net premiums in life insurance are calculated in the same way as in risk insurance, based on the equality of the obligations of the insured and the insurer. First of all, two factors will influence the amount of the net premium.

The first factor is connected with the possibility of extracting profits from investing capital of an insurance company. This capital is formed mainly through insurance premiums, the company's own funds and retained earnings. In actuarial calculations, the factor of possible extraction of income from investing funds is accounted for in the form of the so-called * technical (expected) interest rate. * The actual value depends on fluctuations in the financial market, inflation and other factors.

Another important factor affecting the value of calculated net premiums is the statistical evaluation of demographic events on the basis of mortality tables. These two factors in reality always work together and in actuarial calculations are accounted for by switching numbers.

An obligatory element of the switching function is the * discount factor * (reflecting the change in the price of money), which is calculated as follows:

(8.1)

where * i * - the annual rate of compound interest (technical, embedded in the calculations, the rate of return); * n * is the period for which discounting is performed.

The main in the first group are commutation numbers and . They are defined as follows:

(8.2)

(8.3)

where * w * is the age limit to which the mortality table is compiled.

From the above formulas it follows:

Sometimes, in order to simplify the calculation, it is necessary to know the sum of the commutation numbers * D * x for some age range: from * x * to * x * + * t. * Then the function * N * x * is used: *

The most important representatives of the switching functions of the second group are the functions . The value of * C * x is the discounted number of deaths from * x * to * x + * 1:

(8.4)

where * d * x is the number of deaths from * x * to .r + 1, determined by the mortality table; v - discount factor at the accepted interest rate.

Therefore

And for * x = w. *

The sum of the numbers C, is denoted as * M * x *: *

(8.5)

The value of * M * x is calculated recurrently, starting at the age limit:

In the insurance literature, the following transformation is often used, which makes it possible to obtain * M * x through * N * x.

The commutation function * R * x is the sum of the numbers * M * x *: *

(8.6)

Switching functions are difficult to directly interpret meaningfully. They should be perceived as purely technical, auxiliary values. However, if we assume that each of the * 1 * X people (surviving to the age * x * years) is paid 1 ruble, then * D * r can be considered as < i> the modern value of the sum of /, paid through * x * years. In turn, * N * x is the current value of the payment sequence/v * l * x + {*, ..., l * w rubles.

For the insurance of the combined life of * two persons *, there is a need for additional switching functions:

(8.7)

In turn,

(8.8)

Since products of commutation numbers have a large dimension, they are usually multiplied by 10 3.

By analogy with the function * N * x, we find:

(8.9)

For payments made * t * once a year, we get:

(8.10)

Switching functions are built based on the specified mortality table and the set interest rate. Switching functions and their tables played a big role before the emergence of powerful computers, since they significantly reduced the amount of computing work. And the tables of these functions were given for a limited range of interest rates. Now their role has significantly decreased, because their direct calculation on the computer takes much less time than searching the table.

In addition, Western actuaries have convincingly shown that the use of switching functions leads to somewhat higher (up to 1%) tariffs, compared to exact methods based on direct calculation of tariffs. The overestimation of tariffs may lead to a decrease in the competitiveness of the insurance company. Since the problem of reliability and stability for them is solved, this circumstance is considered as a serious drawback. Today it is not so significant in the Russian conditions. Since if you require this accuracy of calculations, you need to require the same accuracy in everything else: in data, methods, etc. To date, the main problem for Russian insurers is the lack of reliable data for a long period of time.

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