actuarial calculations

The central place in the insurance contract is the cost of insurance services.

actuarial calculations is a process during which the cost and cost of insurance services are determined.

insurance actuaries- citizens of the Russian Federation who have a qualification certificate and carry out, on the basis of an employment or civil law contract with an insurer, the activity of calculating insurance tariffs, insurance reserves, evaluating its investment projects using actuarial calculations.

With the help of actuarial calculations, the tariff rate is calculated - the price of insurance risk and other expenses of the insurer. The set of tariff rates is called tariff.

Tariff rates are of two types: gross rate, net rate.

Gross rate- the rate at which the insurance contract is concluded. Net rate- the price of insurance risk, this rate is used to create a fund for payments to the insured.

Gross rate = net rate + load. The load includes:

A. Expenses for organizing and conducting insurance business, including:

- organizational expenses - when establishing an insurance company;
– acquisition costs – for attracting policyholders through insurance agents;
- collection costs - for servicing cash circulation;
– liquidation expenses – for liquidation of damage;
- administrative expenses.

B. Deductions to reserve funds.

B. Profit of the insurance company.

Where T n – tariff net rate; P(A)- the probability of an insured event, P(A) = Kv / Kd; TO- coefficient, K= Sv / Ss; Kd - the number of payments for a particular period; Kd - the number of concluded contracts; Св - average payment under one contract; Сс is the average sum insured per contract.

T n is an indicator of unprofitability per 100 rubles. sum insured.

Where T h - gross tariff rate; H is the load, expressed as a percentage of the gross rate.

The gross rate is used to compensate for damages, cover the expenses of the insurance company, form insurance funds and make a profit. Profit is distributed according to the usual scheme: first taxes are paid (profit tax, property tax, etc.), net profit is used to create a reserve fund, a company development fund, pay dividends, encourage employees of the company and other purposes.

Classification of insurance. Insurance systems

There are many types of insurance, so you need to classification. It can be carried out according to various criteria.

According to the goals of insurance activity There are two areas - commercial and non-commercial insurance. Non-commercial insurance includes social insurance, compulsory health insurance, etc. Commercial insurance includes primary or direct insurance, coinsurance, reinsurance.

By levels of protection for workers there is a distinction between insurance against industry risk, enterprise risk and individual risk (at the expense of citizens' own funds).

By insurance industry distinguish between personal insurance (life insurance and accident insurance), property insurance (insurance of various material assets, property rights and capital; insurance against possible loss of income and unforeseen expenses) and civil liability insurance (liability insurance for causing harm and liability insurance under the contract) . Liability insurance for causing harm includes the most important types of insurance; civil liability of vehicle owners, enterprises - sources of increased danger, employer's liability to hired workers and employees, professional liability of doctors, pharmacists, accountants, builders, etc., liability for environmental pollution. Contractual liability insurance covers liability for any contractual relationship between the parties: under contracts for supply, transportation, contracting, etc.

By the amount of insurance liability Distinguish between compulsory and voluntary insurance. Compulsory insurance includes insurance of passengers, police officers, foreign intelligence, military personnel, employees of tax inspections and tax police, astronauts, etc.

By class of insurance distinguish between fire, transport, engineering, etc.

According to the form of organization of the insurance business insurance is divided into group and individual.

By orientation of insurance interests A distinction is made between family-oriented insurance and business risk insurance. Entrepreneurial risk insurance includes insurance in case of non-receipt of profit, decrease in profitability, formation of losses; in case of non-payment on the invoices of the supplier of products; lost profits on failed transactions; from equipment downtime, etc.

Table 1 shows the dynamics of development and the ratio of certain types of insurance in the Russian market. 18.1.

Table 18.1

Dynamics of volumes of insurance premiums and insurance payments of subjects of insurance business of the Russian Federation for 2005–2009, billion rubles.

For those considered in Table. 18.1 For 5 years, insurance premiums have increased by 2 times, insurance payments - by 2.7 times. The structure of insurance premiums and insurance payments has also changed. If in 2005–2007 insurance premiums of voluntary insurance prevailed, then in 2008-2009. there is a predominance of compulsory insurance premiums. In the structure of insurance payments, the picture is different - payments in the compulsory insurance industry prevail.

Changes in the structure of insurance premiums and insurance payments of voluntary and compulsory insurance show the data in Table. 18.2.

The share of compulsory insurance in the total volume of insurance premiums increased (from 41 to 57%) due to a corresponding reduction in the share of voluntary insurance. The change in the structure of insurance payments shows the same trend - an increase in the share of compulsory insurance (from 60 to 69%).

Table 18.2

Dynamics of the structure of insurance premiums and insurance payments of voluntary and compulsory insurance for 2005–2009, % to the end

Indicators
	insurance	insurance	insurance	insurance	insurance

Voluntary insurance: a) life insurance
b) personal insurance
c) property insurance
d) liability insurance
e) risk insurance

Compulsory insurance: a) liability insurance of motor vehicle owners
b) compulsory health insurance
c) other types of insurance

In the volume of insurance premiums for voluntary insurance, the share of personal (by 6%) and property insurance (by 10%) significantly decreased. In the volume of insurance payments on voluntary insurance, the share of personal insurance also decreased, but the share of property insurance increased, insurance of business and financial risks appeared.

In the volume of insurance premiums for compulsory insurance, the share of compulsory medical insurance increased significantly (from 29 to 48%), while the share of OSAGO decreased (from 11 to 9%). The same trend is observed in the volume of insurance payments for compulsory insurance.

Pay attention to the ratio of insurance premiums and insurance payments. If for compulsory insurance there is an excess of payments over premiums, then for voluntary insurance, on the contrary, the share of insurance premiums is much higher than the share of insurance payments.

One of the important indicators characterizing the level of development of insurance in the country is the ratio between insurance premiums and GDP. In developed countries, its value usually ranges from 8 to 12%. In the Russian Federation in 2009, this ratio was less than 3%. This indicates a low level of insurance development.

Evidence of the insufficient level of development of insurance in Russia is a sharp reduction in the number of insurance companies. From July 1, 1996 to December 1997, their net reduction was almost 500, or 18%. As of January 1, 1998, 2,334 insurance companies were listed in the State Register, and 1,864 as of January 1, 1999. In 1998, licenses were revoked from 496 insurance companies, and licenses from 116 insurance companies were suspended. As of June 30, 2010, the number of insurers was 666, including 660 insurance companies and 6 mutual lending companies.

Insurance systems

There are five insurance systems.

1. Insurance for the actual value of the property.
2. Proportional liability insurance. This is incomplete, partial insurance of the object. In this case, the amount of insurance compensation is reduced in proportion to the share of the sum insured in the actual value of the object.

where Q - insurance compensation; T - the actual amount of damage; S– insurance amount under the contract; W- market assessment of the object of insurance.

Suppose a car worth $6,000 was insured for $3,000 and the actual damage was $2,000. The insurance indemnity will be 50% of the amount of damage, i.e. $1000

3. First risk insurance. In this case, the insurance indemnity is paid in the amount of damage, but within the sum insured. Damage in excess of the sum insured is not paid at all. If the loss in the previous example was $5,000, then the insured received only $3,000.
4. Fractional system. The insurance contract establishes two sums insured: the shown value and the actual value. For the shown value, the policyholder receives compensation expressed in % or in natural fractions. The liability of the insurer is limited to the size of the fractional part. If the value shown is equal to the actual value, then the insurance of the fractional part turns into insurance under the first risk system. If the value shown is less than the actual value, then the insurance of the fractional part turns into insurance under the proportional liability system.
5. Replacement cost insurance. In this case, the insurance indemnity is equal to the price of a new property of the corresponding type. Of course, insurance premiums will also be higher than with other insurance systems.

Commonly used in insurance contracts franchise, which represents the personal participation of the insured in covering the damage. The deductible is set in rubles or as a percentage of the sum insured or damage. The franchise is beneficial to both the insured and the insurer. The insured receives discounts from the insurance rate, and the insurer transfers part of the damage to the insured. Franchise is of two types - conditional and unconditional.

Conditional (non-deductible) deductible means that the insurer is released from liability for damage if it does not exceed the deductible percentage. If the damage is greater than the deductible, then the insurer is obliged to compensate the damage in full. If there is a conditional deductible, an entry "free from .... interest" is made in the insurance contract.

Unconditional (deductible) deductible means that the insurance indemnity is always equal to the damage minus the unconditional deductible. If there is an unconditional deductible, an entry "free from the first ... percent" is made in the insurance contract.

Main questions

1. The essence of actuarial calculations in insurance and their classification. Tariff policy.

2. Insurance statistics as a basis for calculating the insurance premium. The main indicators of insurance statistics.

3. Insurance rates. The structure of the tariff rate.

4. Calculation of the insurance rate for risky types of insurance.

5. Calculation of the insurance rate for life insurance.

Basic concepts: actuarial calculations; actuary; tariff policy; insurance rate; net rate; gross rate; business expenses; indicators of insurance statistics; insurance case; probability of an insured event; risk premium, mortality table; rate of return; switching numbers; rent; annuity.

7.1. The essence of actuarial calculations in insurance and their classification.

Tariff policy.

actuarial calculations- the process by which the costs required for insurance are determined. With the help of actuarial calculations, the cost of insurance services is determined. As in any business activity, in insurance, the insurer needs to determine the amount of expenses necessary for insuring a particular object. The form in which the insurance costs of a given object are presented is called insurance (actuarial) costing.

actuary(actnarins) in ancient Rome was the official name given to the person who wrote down the decisions of the Senate and kept notes of the debates daily. The first use of the term "actuary" in relation to business was in 1762, when the Equitable Life Insurance and Survival Society was formed in London. In 1775, the mathematician William Morgan was appointed to this post, who limited the scope of his activity to calculating insurance premium rates and ensuring the reliability of financial transactions. Since then, the title "actuary" has been applied to those who did this financial and mathematical work. The term “actuary” was first used in UK law in 1819. In the modern sense, an “actuary” is a person who has certain qualifications to assess the risks and probabilities in the field of finance and business activities associated with random events.

Features of the insurance business that affect the conduct of actuarial calculations:

The probabilistic nature of the events under study;

The calculation of the cost of insurance services is carried out in relation to the entire insurance aggregate;

The need for special reserves of the insurer.

The methodological basis of actuarial calculations is the observance of the principle of equivalence, i.e. establishing a balance between payments and insurance payments of the company.

The main tasks of actuarial calculations:

Research and grouping of risks;

Calculation of the mathematical probability of the occurrence of an insured event, determining the frequency and degree of its consequences, both in risk groups and in the whole insurance population;

Mathematical justification of the required amount of expenses for doing business;

Mathematical substantiation of the necessary insurance funds, determination of methods for their formation.

As a task of actuarial calculations, one can also consider the study of the rate of capital investment (interest rate) when the insurer uses insurance reserves as investment resources.

Classification of actuarial calculations

By industry of insurance:

Actuarial calculations for risky types of insurance;

Actuarial calculations for life insurance.

By type of risk:

Risks related to mass types of insurance;

Rare and catastrophic risks.

On a temporary basis:

Scheduled calculations that are made when a new type of insurance is introduced in the absence of reliable risk observations;

Corrective (reporting) calculations are corrected planned calculations after three to four years of accounting and analysis of statistical data.

On a territorial basis:

Federal actuarial calculations intended for the entire territory of the Russian Federation;

Regional actuarial calculations made for individual regions (republics, regions, territories, cities);

Actuarial calculations at the level of a particular insurance organization.

The methodology of actuarial calculations depends on the insurance industry (life insurance and risk types of insurance), as well as on the availability of statistical data for the calculation.

Under tariff policy refers to the purposeful activity of an insurance organization to develop, establish, clarify and streamline insurance rates. The purpose of the tariff policy is the successful and break-even development of the insurance organization.

Tariff policy principles:

Equivalence of insurance relations. This principle means that net rates should correspond as much as possible to the probability of damage in order to ensure the return of the insurance fund for the tariff period;

Availability of insurance tariffs - tariff rates should not be burdensome for a wide range of insurers, while the effectiveness of insurance as a method of insurance protection increases significantly;

Stability of the size of insurance rates - the invariance of tariff rates for a long time gives insurers confidence in the reliability of the insurer. An increase in tariff rates is permissible only with a steady increase in the unprofitability of the sum insured;

Expansion of the volume of insurance liability - is ensured by a decrease in the loss ratio of the sum insured, and tariff rates become more affordable for the insured;

Self-sufficiency and profitability of insurance operations i.e. insurance rates should be built in such a way that the receipt of insurance payments constantly cover the costs of the insurer and provide him with a certain profit.

7.2. Insurance statistics as a basis for calculating the insurance premium.

Main indicators of insurance statistics

In actuarial calculations, insurance statistics are widely used, which is a systematic study and generalization of the most widespread and typical phenomena in insurance and their change over time. With the help of insurance statistics, insurance organizations receive data to predict the statistical probability of insurance risk, which makes it possible to predict the future amount of damage. At the same time, the greater the number of objects of observation, the more accurate the assessment of the probability of an insured event.

To determine the calculated indicators of insurance statistics, the following initial data are used:

Number of insurance objects;

Number of insured events;

The number of affected objects as a result of insured events;

The amount of collected insurance payments;

The amount of paid insurance compensation;

Sum insured for any object of insurance;

The sum insured attributable to the damaged object of the insurance aggregate.

Estimated insurance statistics indicators present in the table. 7.1.

Table 7.1.

Insurance statistics indicators

Conventions
Number of insured events
Number of insurance objects
Number of objects affected
The total amount of insurance payments
Total sum insured for all insured objects
Total sum insured attributable to damaged objects
Total amount of insurance premiums
Insurance statistics indicators
Name	Calculation procedure
Loss ratio of the sum insured (the indicator is measured in the range from 0 to 1, considered as a measure of the risk premium)
Frequency of insured events (the indicator determines how many insured events occur per one insured object).
The devastation of an insured event or the risk cumulation coefficient (the indicator determines how many insured objects are affected by this or that event, the minimum value is -1)
Degree of inferiority (degree of unprofitability) (the indicator is measured in the range from 0 to 1)
Average sum insured per damaged object
Average sum insured per contract (object) of insurance
Severity of risk
Loss rate,% (the indicator characterizes the financial stability of this type of insurance)
Average security for damaged objects
Severity of damage (the indicator determines to what extent the property is destroyed
Damage frequency (probability) (the indicator expresses the frequency of occurrence of an insured event)

In addition, for the purposes of factor analysis of the loss ratio of the sum insured, the following model can be used:

N)/(N*C*M*S) (7.1)

7.3. Insurance rates. The structure of the insurance rate

Insurance service, like any other product, has its cost or price. The price of the insurance service is expressed in the insurance tariff (contribution, premium).

Insurance rate is a set of rates. In turn, the tariff rate is the price of insurance risk and other expenses of the insurer for organizing insurance; adequate monetary expression of the obligations of the insurer under the concluded insurance contracts. The tariff rate at which an insurance contract is concluded is called the gross rate.

The main purpose of calculating insurance rates is to determine and cover the probable amount of damage attributable to each insured or per unit of the sum insured, therefore, the calculation of the insurance rate is based on such signs of insurance as a closed distribution of losses and the return of insurance payments intended for payments.

The tariff rate (gross rate) as the price of an insurance service has a certain structure (see Fig. 7.1). Separate elements of the structure of the tariff rate should provide funding for all the functions that the insurance company performs. The main elements of the tariff rate are: net premium (net rate) and burden, which includes the costs of doing business; deductions provided for by law and a profit margin.

Rice. 7.1. Gross tariff structure

The main part of the tariff rate - net rate, which directly expresses the price of the insured risk, provides coverage for damage. It is quite clear that at the time of pricing the amount of future damage is unknown, so the amount of damage is determined on the basis of damage data for the past period. Therefore, when determining the net rate for mass risk types of insurance, it is necessary to take into account such factors as the probability of an insured event, the frequency and severity of the risk, the size of the insurance amount of the contract. The expected amount of damage, called the net net premium, acts as the minimum price for risk.

To guarantee insurance protection, the net rate (net net premium) includes a risk or delta premium, which is intended to finance random deviations of real damage from the expected value.

A part of the premium falls on the load in the structure of the tariff rate, approximately from 5% to 30%, depending on the type of insurance.

For different types of insurance, the composition of the load may differ slightly from the one mentioned above. So, for life insurance, only the costs of doing business and profits are included in the burden.

Consider the main components of the load.

The bulk of the load is occupied by the cost of doing business. The costs of doing business can be divided for analysis purposes as follows:

Organizational - expenses associated with the establishment of an insurance company;

Acquisition costs - costs associated with attracting new insurers and with the conclusion of new insurance contracts. The main part of the acquisition costs is occupied by commissions to insurance agents and brokers;

Collection - expenses associated with settlement and cash services. In addition, these costs include the cost of preparing forms, receipts, accounting registers, etc.;

Liquidation costs - expenses associated with the settlement of losses, legal costs, travel expenses to the place of the insured event, payment for expert services, etc.;

Management, which are divided into general expenses and property management expenses. In particular, management costs include labor costs and social security contributions; household and office expenses; transport; connection; rent; entertainment expenses; depreciation, etc. .

Legislative deductions. As a rule, these expenses are associated with the implementation of preventive measures aimed at reducing the risk of an insured event and/or reducing the damage caused by it. The limit of such deductions in the tariff structure is legally set - no more than 15%. Funds of preventive measures in the amount provided for by the structure of the tariff rate are directed to the formation of a reserve of preventive measures. Directions for using the reserve of preventive measures can be as follows: acquisition and operation of fire and security alarms; financing of development and/or acquisition of means of protection against diseases (for example, vaccination); financing the construction of water protection structures, facilities, protection against accidents of technical systems, etc. In addition to the reserve of preventive measures, other deductions provided for by law may be used as the specified load element, for example, deductions to the reserves of compensation payments for OSAGO (2% of the gross rate to the reserve of current compensation payments and 1% of the gross rate to the reserve of guarantees).

The last component of the load is the profit margin (planned profit), i.e. income from insurance activities that the insurer expects to receive. The presence of this element in the structure of the gross rate emphasizes the entrepreneurial nature of insurance activities.

The components of the load can be calculated as per 100 rubles. sum insured, and as a percentage of the gross rate.

The gross rate is calculated according to the formula (7.2).

where Tb-s - tariff gross rate;

Tn-s - tariff net rate;

f - share of the load in the gross rate.

The procedure for determining the tariff net rate will be defined below in the following paragraphs of this topic.

Let us determine how the insurance rate is used to determine the price of an insurance service. Recall that the payment for insurance is the insurance premium (contribution). So, if the insurance rate is set at 2% (2 rubles per 100 rubles of the sum insured), then with an insurance sum of 1000 thousand rubles, the insurance premium will be calculated in the amount of 20 thousand rubles. (2 rubles * 1000 thousand rubles / 100 rubles).

7.4. Calculation of the tariff net rate for risky types of insurance

In this paragraph, we will define the methods for calculating the tariff rate for risky types of insurance, i.e. types of insurance other than life insurance. Note that the methods under consideration will be valid when calculating the tariff rate for mass types of insurance. Mass types of insurance cover a significant number of insurance objects and insured persons, characterized by homogeneity of risks, for which there is a sufficient amount of statistical material that allows calculating the tariff. The random distribution of the amount of loss in mass types of insurance can be described with sufficient accuracy by a normal or logarithmically normal distribution. In addition to mass risks, the risks of man-made and man-made disasters are subject to insurance. In these cases, the calculation of the insurance rate will differ from the methodology typical for mass types by the procedure for calculating the risk premium, which, due to the lack of statistics, will be assessed qualitatively (by experts). In this case, the state of a particular hazardous facility, as well as scenarios of possible accidents, should be taken into account. Among the risks of catastrophes, it is necessary to single out especially rare dangerous events for which there is no statistics. For example, the fall of a meteorite, etc. Due to the fact that the probability of such events and their consequences are not quantified, they are not taken into account when insuring, i.e. such risks are not insured.

The algorithm for calculating the net rate is shown in fig. 7.2.

Determination of the main part of the net rate (To)

Determining the risk premium (Tr)

Determination of the net rate (Tn)

Rice. 7.2. Net rate calculation algorithm

Consider various methods for determining the net rate for mass risk types of insurance:

Method #1. Refers to cases where statistical information is available for the type of insurance under consideration in terms of the probability of an insured event, the average sum insured and the average compensation under one insurance contract (object).

1. Calculation of the main part of the net rate (To) is made according to the formula (7.3).

where q is the probability of an insured event under one insurance contract;

Average insurance indemnity under one insurance contract;

Average sum insured under one insurance contract;

100 - the basic amount of the sum insured. Recall that traditionally the size of the insurance rate is determined in rubles from 100 rubles. sum insured or in % of the sum insured.

Indicators and should be determined using the calculations in Table 7.1. In practice, when defining the relation /, it is recommended to take values not lower than:

0.3 - for insurance against accidents and illnesses and in voluntary health insurance;

0.4 - when insuring ground transport means;

0.5 - for cargo and property insurance (except vehicles);

0.6 - when insuring air and water transport means;

0.7 - for insurance of liability and financial risks.

Let's transform the formula (7.3) and get one more formula for calculating To (7.4):

where Sv is the total amount of insurance payments;

S - total cumulative sum insured for insured objects /

Recall that the Sv / S indicator is called the unprofitability indicator of the sum insured. Often this indicator is determined in rubles per 100 rubles of the sum insured, i.e. Sv/S*100.

2. Calculation of the risk premium (Tr).

The second part of the net rate is the risk or delta premium. The basis for calculating the main part of the net rate is information based on statistical data on the frequency of occurrence of an insured event. At the same time, in different periods, these indicators can deviate, and sometimes quite significantly. To avoid a situation related to the insufficiency of the insurance fund for payments, a risk premium is applied.

Consider the methods for calculating the risk premium:

2.1. Risk premium calculation for every risk is determined by formulas (7.5), (7.6) depending on the availability of data for calculating the dispersion of insurance claims.

where is the dispersion of insurance claims, which is determined by the formula (7.7)

where - the amount of insurance compensation for the i-th case.

A factor that depends on the security assurance, its value is taken from Table 7.2. Safety guarantee - the required probability with which the collected contributions should be enough for insurance payments for all insured events.

Table 7.2

2.2. The risk premium is calculated for several types of risks(formulas (7.8), (7.9), (7.10)).

2.3. In some cases, the size of the risk premium is determined by experts in % of the main part of the net rate.

3. The tariff net rate is calculated for 100 rubles. sum insured or in %.

Tn \u003d T o + T p (7.11)

Method #2. Refers to cases where, for the type of insurance under consideration, there is statistical information on the dynamics of the loss ratio of the sum insured for a number of periods and the dependence of the loss ratio on time is close to linear.

1. Calculation of the main part of the net rate (To).

The main part of the net bet in the following order:

1.1. The loss ratio of the sum insured (Sv/S) is determined for each settlement period (year);

1.2. The predicted level (indicator) of unprofitability is determined from the linear regression equation:

where - equalized indicator of unprofitability of the sum insured;

Linear trend parameters;

Serial number of the corresponding year.

Linear trend parameters can be determined using the least squares method by solving a system of equations (formula (7.13)).

where is the number of years of the calculation period.

2. Calculation of the risk premium (Tr) is made according to the formula (7.14).

where is the standard deviation of the actual values of the loss ratio of the insurance company from its average size for the considered period t;

A factor that depends on the safety assurance, its value is taken from Table 7.3.

Table 7.3

Number of periods (years) of analysis (p)	The probability of not exceeding payments over contributions is a guarantee of security ()
Number of periods (years) of analysis (p)

As can be seen from the values of Table 7.3, with an increase in the calculation period, the accuracy of the tariff is ensured by a lower value of the coefficient and, ultimately, the risk premium (Tр).

3. The tariff net rate is calculated for 100 rubles. sum insured or as a percentage.

Features of actuarial calculations for voluntary medical insurance. Voluntary medical insurance (VHI) in terms of actuarial calculations differs from other risky types of insurance in that as a result of these calculations, not the tariff rate, but the cost of the insurance policy should be obtained. This is due to the peculiarities of VHI as a type of insurance:

Insurance payments under VMI are made not to the Insured, but to the medical institutions that provided the medical service;

There is no such concept as the sum insured in VHI, which, along with the insurance rate, is the basis for determining the cost of insurance services. As an analogue of the sum insured in health insurance, such a concept as "insurance coverage" is used.

When calculating the cost of an insurance policy for VHI, the method of actuarial calculations for risky types of insurance is used.

Information base - indicators of medical statistics. In particular, data on morbidity for certain classes of diseases or types of medical services per 1000 people.

The procedure for calculating the cost of an insurance policy has the following stages:

1. Determination of the indicator of the probability of occurrence of insured events for each type of medical services included in the insurance coverage under this insurance program.

2. Determination of the main part of the net rate (Tosn.) To determine the main part of the net rate, the following formulas are used:

where q is the probability of occurrence of at least one of the considered n events included in the insurance coverage under this insurance program;

S - the size of the base sum insured (100 rubles).

3. Determination of the risk premium (Trisk.). Determined by the formulas above.

4. Determination of the net rate (Tn):

5. Determining the maximum amount of insurance coverage (Sm)

where n is the maximum number of requests for medical care by one insured person during the insurance period;

C - the cost of one treatment, rub.

6. Calculation of the risk ratio (K c.r.). Its use makes sense when the average number of visits for medical care by the insured is less than the maximum. Using this coefficient allows you to reduce the size of the insurance rate.

where S s.r. - average insurance coverage;

where is the average number of requests for medical care by one insured person during the insurance period.

7. Determination of the net cost of an insurance policy for VMI (Mon):

where T n - net rate in%.

8. Determination of the gross cost of the policy for VHI (Pb):

where d is the share of the load in the gross rate.

7.5. Calculation of the insurance rate for life insurance

The information base for calculating insurance rates for life insurance is the mortality table, which is formed on the basis of population census data.

Let's define the content of the information and the order of constructing the mortality table in Table. 7.4.

Table 7.4

Mortality table

Number of people living according to the census

Number of deaths according to the census

mortality

Gr.2 and gr.3 - statistical data.

Group 4 = group 3: group 2, i.e. 116490: 632698 = 0.18412.

The death table shows the number of deaths from year to year at each age from a given number of births.

Gr.5 is an arbitrary number for age 0. The number 100000 is often used. By multiplying this arbitrary number (eg 100000) ha the number in gr. 4 for age 0, we get the number of deaths before reaching one year (gr.6). In our case,

group 6 = 100000 * 0.18412 = 18412.

Gr.5 for the next year is determined by the difference in the value of gr. 5 of the previous year and column 6 of the previous year.

To calculate insurance rates, data common to the population of the region are used, both the population census and statistical information collected directly from the insurance company over a number of years.

When calculating insurance rates for life insurance, a technical percentage is used. The essence of technical interest lies in the fact that it is a form of participation of the insured in the investment income of the insurer. The technical interest is determined using the compound interest formula:

where i - annual income of capital (in insurance terminology - rate of return);

K1, K0 - accumulated and invested capital, respectively.

In insurance, the inverse problem is solved, i.e. it is required to determine what amount must be invested at the present moment in order to receive an amount equal to a unit of capital after a certain time (n). Thus, here it is required to determine the present value of future capital. In this case, the technical percentage (discount factor) will be determined by the formula (7.23):

Let's illustrate the use of technical percentage in calculations.

Let us determine the amount of the insurance payment, which in 2 years will provide an insurance amount of 10,000 rubles. at a rate of return of 9% per annum.

The insurance payment (C) in this case will be determined by:

If the payment is not one-time (one-time), but annual, i.e. in this case it will be produced 2 times, then it can be determined by the formula (7.24):

In our case, Сyear = 10000 * [ 0.09 / (1.09 - 1) ] = 4785 rubles.

Life insurance usually comes in two forms: sum (capital) insurance and annuity (annuity) insurance. The differences are caused by the form of payments. When insuring capital, the payment is made to the insured in the event of an insured event in a lump sum in the amount of the sum insured. Annuity insurance makes periodic payments. Next, consider the calculation of tariff rates for capital life insurance and annuity insurance.

The gross rate (TB) for life insurance is determined in the same way as for risky types of insurance according to the formula (7.2):

Consider procedure for calculating the net rate for life insurance (capital) using the mortality table and the switching number table.

Determination of the net rate (Tn-s) is carried out according to the formula (7.25):

where is a one-time survival rate for the insured age x years with an insurance period of years;

Lump sum in case of death for insured age x years with insurance period years.

This structure of the tariff rate is explained by the presence of two insured events in classical life insurance.

Determining the net rate is possible in two ways: using the mortality table, as well as using the switching number table.

A) Determine the net rate using the mortality table. First let's calculate one-time survival rate. For this, formula (7.26) is used:

where is the sum insured, which is traditionally taken as 100 rubles in the calculations under consideration;

Number of people surviving to age ;

V - discount factor, the size of which depends on the rate of return on life insurance, is determined by the formula (7.27).

Let's consider an example of calculation. We use the following data entered in the mortality table (see Table 7.5).

Table 7.5

	Number of survivors up to age x	The number of deaths at transition from age x by age x+1

For an insured person aged 40 years with an insurance period of 5 years and a rate of return of 3% per annum, the lump sum survival rate will be:

\u003d (86805.0 * 0.86261) / 88565.0 * 100 \u003d 84.55 rubles. from 100 rub. sum insured.

Calculate lump sum in case of death() according to the formula (7.28):

The number of deaths during the transition from age to age.

If the insured is 40 years old and the insurance period is 5 years, the death rate will be:

40А5 \u003d (319 * 0.97087 + 336 * 0.94260 + 352 * 0.91514 + 369 * 0.88849 + 384 * 0.86261) 88565.0 * 100 \u003d 1.82 rubles. from 100 rub. sum insured.

Thus, the tariff net rate (TN-s) in this example will be 86.37 rubles. from 100 rub. sum insured or 86.37%.

In the practice of insurance, lump-sum rates are used quite rarely. Most often, the conditions of insurance provide for the insured to make periodic insurance premiums, say annual. To get annual premiums, you cannot simply divide the lump sum by the corresponding number of years of insurance, because. it is necessary to take into account the loss in income from investing temporarily free funds, as well as the decrease in the number of insured persons due to mortality, therefore, the so-called installment coefficients (7.29) are used.

To obtain an annual tariff rate, its one-time value should be divided by the installment factor.

B) Calculate the net rate using the switching numbers table.

First, we determine the values of commutation numbers. Commutation numbers are a mathematical combination of data from a mortality table and serve to simplify without having a specific economic meaning.

where is the last value of the commutation number table.

In the notation of switching numbers, the formulas for determining the net rates for survival and for the event of death look like this:

Survival Lump Rate (7.30)

Lump sum on death (7.31)

When calculating tariff rates using switching numbers, you can use special formulas (7.32), (7.33) to calculate annual fees:

where is the annual contribution in case of death of the insured age years by years.

where is the annual premium for the life of the insured X years on n years.

Consider procedure for calculating the net rate for life insurance with the condition of payment of annuity.

Annuity formulas are used to determine insurance rates with the condition of annuity payment. Switching numbers are used for calculation. The calculation methodology assumes that annuity insurance is a kind of sequential, repeatable survival insurance:

Let's define different types of annuities for the insured age X with an annual payment of an annuity of 1 rub. in table.

Table 7.6

Formulas for calculating the insurance rate for life insurance with the condition of payment of annuity

annuity

Immediate

life

Deferred

on P years

life

Limited

on t years

immediate

Limited

on t years

deferred for n years

Prenumrando

If payments

produced

m times a year:

If payments

produced

m times a year:

Postnumerando

If payments

produced

m times a year:

If payments

produced

m times a year:

What do insurance rates depend on?

Who determines the cost of OSAGO?

Why do each insurance company have different tariffs for Casco?

How are tariffs calculated for various types of property and liability insurance, life and health insurance?

Answers to all these questions can be given by actuaries - specialists in actuarial calculations.

What do insurance rates depend on?

Each insurance company has licenses for several types of insurance. For each of these types, the insurer develops insurance rules, not against the law even in the smallest nuances.

These rules available to every client, you can read them at any time on the website of the insurance company or contact an employee in the office with a request to give you printed rules.

In addition to other provisions, these rules include tariff rates. When drawing up a contract for each insurance transaction, the company is guided by the provisions and tariffs precisely from the rules of insurance.

That is, it cannot be such that an employee gives you a tariff at random, the tariff is taken from the insurance rules. When applying for a license for a certain type of activity, the company's lawyers prescribe insurance rules, and actuaries calculate the optimal tariff rates.

According to these tariffs, adopted according to actuarial calculations, the company works with clients.

Life does not stand still, political and economic situations are constantly changing, society is changing, insurance rates are also changing, so do not consider it a tautology or a pun, actuarial calculations are always relevant.

What are actuarial calculations in insurance and who makes them

actuarial calculations- these are complex mathematical calculations built on the methods and formulas of the science of statistics and financial and economic analysis of macro- and microeconomic indicators.

Actuarial calculations are based on the demographic situation, the state of the economy and long-term forecasts of its development, the political situation, and an assessment of society's expectations. In actuarial calculations, the theory of probability is widely used.

Actuary is a high-class specialist with an appropriate qualification certificate. Most often this insurance company employee, but also an actuary can work on free bread, providing one-time services to insurer clients.

At the moment, the liability limits for OSAGO are as follows:

for the life and health of the victims - 500,000 rubles per person. That is, if the owner of OSAGO knocked down a pedestrian, then his insurance company will pay this pedestrian no more than 500,000 for treatment;
on property - 400,000 rubles. That is, if the owner of OSAGO broke someone else's car, the insurer will pay the victim no more than 400,000 rubles to repair his car.

These limits are the same throughout the Russian Federation and for all OSAGO policies. But the cost of the policy varies due to the engine size of the car, the number of people allowed to drive, their driving experience, region and other factors.

The essence of this difference in cost is clear to everyone: the younger the driver and the less experience he has, the more likely he is to provoke an accident, and the higher the risk of the insurance company that sells the policy to him.

Therefore, such a policy will cost more than OSAGO for a person with 20 years of experience. Again, in Moscow or St. Petersburg, the traffic is much more intense than in a small village, respectively, and the risk of accidents in the capitals is many times higher, therefore OSAGO is much more expensive here.

How much more expensive will OSAGO cost in all these cases, determine the coefficients. But the basic tariff itself and each coefficient are not taken from the ceiling, they are calculated by actuaries.

Moreover, they expect people to be able to afford to buy an OSAGO policy. without a significant impact on the budget, but at the same time so that insurers can make insurance payments to the same people - their customers.

Let's say citizen N. bought an OSAGO policy for 3,000 rubles. And several hundred more citizens bought various OSAGO policies in the amount of 1,000,000 rubles. A fund has been formed from which the insurer will make payments if citizen N. and other insurers become the culprits of an accident.

Now the question arises, Will this fund be enough to cover all the shoals of insurers? And the answer to this question should be given actuarial calculations.

After all, if the cost of OSAGO is too low, then the insurer simply does not have enough money to pay out. And if for no reason to raise the tariff, then people not be able to buy expensive insurance.

It is also necessary to take into account the fact that cars keep getting more expensive, spare parts for them too, and more and more expensive cars appear on the streets. As a result, insurers do not have enough money in the funds.

For example, the previous limit of 120,000 no longer covered real damage, and the person responsible for the accident had to pay extra for the repair of the victim's car.

Thus, the current the increase in the price of a car citizen is justified and everyone needs it, and the exact amounts are calculated by actuaries, taking into account all aspects of our lives.

Why each insurance company has different tariffs for Casco

The same principle is used to calculate tariffs for hull insurance, property insurance, accident insurance and other types- the probability of an insured event is determined, a variety of factors are taken into account and the optimal tariff for all is calculated.

Pay attention to the Casco rates: the insurance contract lists the risks against which your car is insured, opposite each risk there is a percentage rate, and if you add these rates, you get the total rate under the contract.

For the risk of an accident, the tariff will be much higher than for the risk of “natural phenomena”. The reason is simple: the probability of damage to a car in an accident is ten times higher than due to natural disasters(the same hail can happen at most twice a year).

But it is difficult to calculate specific figures, this is done by actuaries, they take into account traffic police statistics on accidents, police statistics on thefts and thefts, data from weather centers, the demographic situation in the region, the standard of living of the population (the ability to buy expensive cars) and dozens of other factors.

Moreover, actuarial calculations for Casco tariffs are not limited to determining the probability of the occurrence of an event, they also take into account details of the insurance product– franchises, insurance with or without depreciation of parts, and others.

Every year since 2006, the Independent Actuarial Information and Analytical Center, commissioned by the RAMI, conducts an actuarial study on the topic: "Monitoring of current insurance rates for OSAGO for their compliance with the emerging loss ratio for this type of insurance."

The calculation of tariffs is based on the primary and aggregated data of insurance companies, full members of the RAMI. The generalized linear model is used as the main methodological basis for calculations.

According to the results of the study, proposals are formed and sent to the ministries and departments to adjust the current tariffs for OSAGO.

The content of the scientific report, with the results of the study, is as follows:
1. Main results and conclusions
2. Collecting and summarizing insurance statistics on OSAGO
2.1. Research information base
2.2. Aggregate data check system
2.3. Analysis of key indicators of aggregated data
2.3.1. Comparative analysis of the data from the "Reporting Forms for Calculating Insurance Tariffs" and data obtained from other statistical reporting forms of the RSA
2.3.2. Comparative analysis of the data "Reporting forms for calculating insurance rates" for different reporting periods
3. Description of the model used to calculate tariffs
4. Calculation of tariffs for OSAGO based on consolidated Data
4.1. The structure of the data submitted for analysis and the procedure for performing calculations
4.2. Procedure for making calculations
4.3. Loss development analysis based on data from loss development triangles
4.3.1. Estimated average time to settle claims
4.4. Calculation of the frequency of occurrence of an insured event and the average loss for all types of vehicles using the "basic" multifactor model
4.5. Calculation of coefficients of insurance rates
4.5.1. Coefficient of insurance rates depending on the territory of primary use of the vehicle (CT)
4.5.2. Coefficient of insurance rates depending on the engine power of a car and taxi (KM)
4.5.3. Checking the accuracy of approximation by a generalized linear model of real data
4.5.4. Calculation of insurance rates for a pool of insurance companies
4.5.5. Coefficient of insurance rates depending on the age and length of service of the driver admitted to driving a vehicle (FAC) and the availability of information on the number of persons admitted to driving a vehicle (KO)
4.5.6. Coefficient of insurance rates depending on the period of use of the vehicle (CS)
4.6. Calculation of the net premium taking into account the impact of loss reserves, inflation of the average loss, changes in the frequency of an insured event, as well as a risk premium
4.6.1. Inflation accounting for the average loss
4.6.2. Accounting for changes in the frequency of an insured event
4.6.3. Accounting for a risk premium
4.7. Calculation of the gross premium taking into account the influence of the bonus-malus system
4.7.1. Accounting for the impact on the insurance gross premium of the Bonus-Malus system
4.7.2. Adjustment of the basic gross premium, taking into account the peculiarities of the structure of available statistical data
4.7.3. Comparison of the calculated gross premium with the values specified in Government Decree No. 739
4.8. Comparison of proposed OSAGO tariffs with current values
5. Study of the compliance of the territorial coefficients of OSAGO with real data in the context of cities