What is a net premium and its structure. Analysis of a homogeneous insurance portfolio using normal approximation. P-Stern Mixed Life Insurance

p-Stern Mixed Life Insurance

Net prize is calculated by the formula:

Full insurance Life delayed on t. years

At this type of insurance, the net premium is calculated by the formula:

p-summer temporary life insurance, delayed on t. years

Full life insurance with a continuously increasing benefit

19. Calculation of net premiums with full life insurance with payment insurance benefit in the end last year Life.

Calculation of net premiums for major discrete

Fights insurance

Based on the definition of discrete types of insurance, and the concept of actuarial cost can be obtained by the following formulas for calculating net premiums:

1. Full life insurance with the payment of insurance benefits at the end of the last year of life.

Net prize is calculated as

it is a discrete analysis of a continuous simplifying function.

20. Calculation of net premiums with P-year-old temporary and mixed life insurance with

payment of insurance benefits at the end of the last year of life.

p-summer temporary life insurance with benefits at the end of the year of death

3. p-summer mixed life insurance with payment benefits at the end of the year of death

4. Full life insurance with the payment of insurance benefits at the end of the last year of life, delayed for tons

5. Complete life insurance with annually ascending benefits and benefits at the end of the last year of life

Describing, we can write in the form

Here is a discrete simplifying function.

21. The relationship between continuous and discrete life insurance.

Discrete life insurance insurance amount paid at the end of the year of death. Calculations can be carried out directly on the life expectancy tables.

Calculating a net premium with a discrete insurance of life, it is possible to calculate the net premium with the appropriate types of continuous insurance. In order to associate continuous and discrete types of insurance, it is necessary to make certain assumptions about the lifetime distribution law for fractional ages.

It is usually assumed that this law is uniform. It is known that in this case random variables and independent, and has a uniform distribution to. Then we can get the following formulas binding a net premium for relevant continuous and discrete types of insurance:

The above formulas make it possible to calculate one-time net premium for continuous types of insurance through the characteristics, which are sufficiently calculated according to the data given in general life expectancy tables.

22. . Analysis of a summary claim in a long-term life insurance model.

Let at the time of time insurance Company He concluded life insurance agreements. Denote by - premiums, and through - the amount of insurance benefits paid to the Treaty in a random point of time. Place the values \u200b\u200bin order of increasing :. Then at the time of time, the company's capital can be calculated as

and the company will not go broke if the condition of the type is satisfied:

where is the current payment of the payment of the Insurance Treaty. The probability of inconvenient will be calculated by the formula:

which is similar to the corresponding formula for short-term life insurance. That is, the calculation of the probability of inconvenient with long-term insurance is made in the same way as with short-term insurance with loss values.

Then the fee for insurance will look:

where is a non-premium to the Agreement, but the corresponding insurance premium, which is calculated similar to the short-term life insurance.

In the simplest case, when the insurance premium is divided in proportion to mathematical expectations, we get:

With more complex long-term insurance models, it is not always possible to express:

a) the probability of incommodation in the form of a simple formula of the form (32);

b) Net premium and insurance premiums in the form (34).

Net prize - Part of the insurance premium intended directly to cover damage. Net prize is the main part of Gross premium ..

The net premium consists of a pure net risk premium and risk (insurance) surcharge.

Net Net Prize

The definition of a net risk premium traditionally refers to the field of actuarial calculations and insurance mathematics. Pure net premium is calculated on the basis of damages for damages last period and is a product of the offensive frequency insurance case The average amount of damage along the entire totality of the insurance claims occurred in the past.

Pure risk premium \u003d damage frequency x average damage

Frequency of damage is defined as a particular damage from dividing the number of cases in the observed set, the number of observation units included in it.

The average damage is a private from division. total amount Damage for the observed period by the number of damage cases during the same period.

Risk (insurance) allowance

Risk premium is designed to increase the reliability of insurance protection.

When identifying the patterns of damage as a result of random events in the past and determination on the basis of this past experience of unprofitability in the future, the errors of two species are inevitable:

• A diagnosis error that appears as a result of incomplete information. This is due to the fact that the statistical sample is limited and does not meet the requirements of the law of large numbers.
• The error of the forecast, which is that in the future there will be no complete coincidence with the circumstances of the previous period, on the basis of which the net risk premium was determined. This may be a consequence of the influence of unrecorded or changed factors. It has been proven that even with very good information about damages, the future damage exceeds its value in half cases.

In order to guarantee reliable insurance protection, i.e. increase the likelihood that money collected Enough for the payment of damage in the future in all cases, a risk (insurance) premium is added to the net native premium.

The magnitude of the risk surcharge cannot be less than the value of the standard deviation of the sumprintment of the sum insured.

Using the net raising of the insurance factor to determine the net premium

The expected amount of net premium can be defined as a work of the sum insured on the net rate. The net rate is a percentage that reflects the likelihood of a loss, calculated on the basis of the ratio of the aggregate insurance sum of the insured objects.

Net prize size is determined by the formula:

Net Prize \u003d Insurance Sum X Net-Bet / 100

Concept and structure of gross premium

Definition 1.

Gross premium - this is a certain amount of insurance contract moneywhich is obliged to pay the insured company for a certain period of time.

In the structure of gross premiums allocate a net premium and load.

Net prize is necessary to fulfill the obligations of the insurance company under insurance contracts. May consist of the following elements:

• risk premium designed to cover damage when an insured event occurs;
• risk premium required for compensation of increased damage in case of a possible increase in the likelihood of the occurrence of a risky event;
• the savings contribution used only in life insurance and intended for accumulating a certain amount of funds during the term of the contract with the subsequent payment.

The risk premium is always present in the net premium and is intended for the formation of the insurance reserve fund, and the risky surcharge is taken into account when calculating the net premium at the discretion of the insurance company and goes to the formation of a reserve fund.

The load, which is part of the gross premium structure, is the cost of an insurance company to carry out its activities and its profits.

Costs include traditional costs characteristic of any enterprise ( wage, rental, travel expensesCommunal payments, etc.) and specific costs that are applicable only to the insurance industry (payment of commission remuneration to insurance agents and brokers, carrying out preventive measures, conducting examinations in order to determine the amount of damage, etc.).

Note 1.

Depending on the type of insurance, as well as the cost of an insurance company to carry out its activities, the ratio of net premium and the load may be different. Most often, in the total magnitude of the gross premium, 70-80% is a net premium, the rest is the load.

In general, the $ TB $ gross rate is:

$ TB \u003d TN / (100 - H) $ 100, where:

$ Tn $ - net bid,

$ H $ is a load defined as a percentage of gross betting.

If the load is defined in rubles, the gross rate is equal to:

$ TB \u003d tn + H $

When calculating the gross premium most important It has the determination of the optimal size of the net premium, because The subsequent solvency and financial sustainability of the insurer depends on this. Therefore, its calculation is paid to increased attention.

Calculation of net raising of risk insurance

Definition 2.

Net rate is an indicator, equal value Net premium designed for one unit (usually 100 rubles) Insurance amount.

The method of calculating a net-rack of risk insurance implies the presence of a sufficient amount of statistical data required for the implementation of accurate calculations, the conclusion of a large number of contracts (for the same time) is predicted, and the absence of events that can entail payments immediately Several Insurance Cases.

In accordance with the methodology of the formula for calculating the net $ TN $, it looks like:

$ Tn \u003d then + tr $, where:

$ Then $ - a risky premium (part) of a net rate,

$ Three $ is a risky surcharge.

The risky premium is calculated as follows:

$ TO \u003d Q SB / S $ 100, where:

$ Q $ - the probability with which the occurrence of the insurance event is possible,

$ SB $ - medium size insurance payment,

$ S $ - the average size of the sum insured.

$ Q \u003d M / N $, where:

$ M $ - the number of insurance events that occurred,

$ N $ - the number of contracts prisoners for a certain period of time.

Medium size of insurance payments equal to relation Amounts of payments for all contracts for the number of contracts:

$ SB \u003d (ΣSBI) / M $

The average size of the sum insured is equal to the ratio of the total magnitude of the insurance amounts on all contracts for the number of these treaties:

$ S \u003d (ΣSI) / N $

Risk surcharge $ tru $ equal:

$ Tr \u003d that α (Γ) √ ((1 - Q + (Rb / Sb) ^ 2) / (n q)) $, where:

$ RB $ - the standard deviation of the average insurance payment,

$ α (γ) $ is a coefficient that depends on the selected insurance company of the probability of Γ that the contributions are enough to cover the damage. The value is taken from the table:

Figure 1. The values \u200b\u200bof the coefficients. Author24 - Student Internet Exchange

Calculation of Net Life Insurance Rate

The main factors affecting the size of a net rate during life insurance can be attributed:

• age and sex of the insured person;
• the term of the contract and the procedure for payment of contributions;
• the predicted yield of funds enrolled in the insurance reserve life insurance fund, in case of investing.

The net rate calculation is based on data tables about the mortality of the population of a certain age and the average life expectancy.

For the beginning, the necessary indicators are calculated

The probability of the occurrence of death at the given year of life is $ qx $ is calculated by the formula:

$ Qx \u003d bx / lx $, where:

$ Bx $ - the number of people who dies between $ x $ to $ x + 1 $

$ Lx $ is the total number of people who lived up to x years;

The probability with which a person lives to a predetermined age, $ px $ is:

$ Px \u003d L (x + 1) / LX $, or:

Taking into account the fact that agreements on this kind Insurance have a long period of action, and funds coming from the insured can be used by an insurance company for investing in order to obtain additional income, a multiplier $ V ^ n $ is used to adjust the total net rate:

$ V_n \u003d 1 / (1 + i) _n $, where:

$ I $ - the rate of return on investment,

$ n $ is the number of years to which funds are invested.

As a result, the amount of net premium $ (ex) _n $ will be equal to the survival:

$ (Ex) _n \u003d (L (x + n) v_n) / LX S $, where:

$ L (x + n) $ - the number of people who lived before the deadline for which the contract was concluded

$ n $ - the term for which the contract is concluded

$ S $ - the amount of the sum insured.

Net rate on the possibility of death $ (AZ) _n $ is:

$ (AZ) _n \u003d (BX ∙ V + B (x + 1) ∙ v_2 + ⋯ + b (x + n - 1) ∙ v_n) / lx ∙ $ 100, where:

$ BX, B (X + 1) ... B (x + N - 1) $ is the number of people dying in a period of C $ x $ ten $ x + 1 $, calculated for each year of the contract.

When concluding a contract of combination insurance and to survive, and the ability to death net will be equal to:

$ Tn \u003d (ex) _n + (az) _n $

This method of calculating net rates is applicable provided that the whole amount insurance payment Imagined immediately for the news of the insurance period. If the policyholder wants to divide the amount of the contribution to several parts, equal to the number of years of insurance, then the size of the annual payment $ p ^ x $ will be:

$ P_x \u003d (ED) _x / α_x $, where:

$ (ED) _x $ - the size of the calculated one-time payment,

$ α_h $ - installment coefficient, which is the cost of payments in the amount of one monetary unit. In fact, this indicator of greatness is close to the value of the number of years to which the contract is concluded, but it turns out to be slightly lower. As a result, the magnitude of annual payments exceeds the value equal to the simple division of a one-time contribution to the number of years of insurance. In this case, the insurer reimburses the loss, which he carries from the inability to invest the entire amount immediately and get income from this.

Part insurance feeused to cover insurance payments on a specific type of insurance for a certain time interval is called a net premium. Its value is in direct dependence on risk development. The parameter may correspond to a risk premium with a systematic development of hazards.

What is used and what does it affect?

Part of the insurance premium is intended for compensatory payments, the purpose of which is to cover damage. Its value is determined by the parameter of the net premium, which is the component of the element of the gross premium. The formation of a non-premium is based on risk and insurance premium. The definition of risk value is made with the help of actuarial calculations that take into account the section of insurance mathematics. To identify the value, information about the damage caused over the past period is used.

The parameter corresponds to the product of the incidence of an insured event for the dedicated period and the average amount of damage. When it is determined, the calculation includes the losses caused as a result of the circumstances attributed to the category of the insured event for the entire selected time period to be analyzed. The frequency of damage is calculated by the private number of the total amount of damage in the observed set and the number of the observed units included in it. The average amount of damage is determined by the private total amount and the number of damage cases. All parameters are taken into account for the selected time interval, interpreted as observed.

What elements consist of?

The insurance premium determines middle value Net premium, which causes positive and negative parameter deviations. For its compensation in the risk premium size, a warranty surcharge applied to stabilize the indicator. Its structure is formed in accordance with the type of insurance and its subject. In property and personal insurance ProductIt consists of different components of the elements. Net prize property insurance Determined by the risk premium and stabilization premium. For personal insurance, an actuarial calculation is characterized, which takes into account the risky premium and the accumulative fee. Some situations take into account the guaranteeing allowance.

How are the calculated operations?

Net prize is relevant for insurance operations, the subject of which is the property, health and life of a person. It corresponds to the difference in the total sum of the insurance premium and agency or brokerage remuneration. The premium is necessary to provide insurance protection against possible damage. It does not include that part of which is intended to cover other expenses. In life insurance, the parameter is interpreted by the difference between the initial insurance premium and the amount of dividend paid by the insured in the event that they were used by the beneficiary to pay for premium payments on the life insurance policy. The expected net premium is determined by the formula:

Np \u003d ss x ns / 100, where:

• NP- Net prize;
• SS- Insurance amount;
• Ns.- Net rate.

The net bid is represented by the percentage displaying the likelihood of a loss. The parameter is calculated by the ratio of damage to the aggregate insurance sum of the insured objects. The amount of damage is determined by a private total amount of damage and the number of fixed such cases. All values \u200b\u200bare recorded for the observed period.

To increase the reliability of protection by the decision of the insurer, the permanent allowance may be taken into account. It cannot be less than the standard deviation of the unprofitability parameter applied to the insurance amount. Accounting in the calculations increases the likelihood that the money collected in the view of the insurance premium will be sufficient for the work of compensatory payments for the damage incurred by the insured as a result of the occurrence of the insured event. The calculation of the new value of the award will look like the amount of the basic net premium and the insurance premium.

The risk premium parameter must be taken into account in the calculations in the detection of certain patterns of the fact of damage caused to the insured object as a result of random events in the last period. On the basis of statistical information, you can predict the unprofitability in advance. Analyzing the parameters should not allow diagnostic errors in the processing of information not in full, as well as the inaccessibility of the forecast expressed in the impossibility of repetition of the past in the future. To avoid unreliability and overestimation of the amount of payment, the average value of the value determined by several temporary episodes is applied.

Conclusion

Thus, the expected net premium can be determined, knowing the size of the sum insured and the value of the net. In this case, the net rate is a percentage reflecting the likelihood of a loss (damage). This probability is calculated based on the ratio of the aggregate insurance sum of the insured objects. The net prize is determined in the course of actuarial calculations, for which knowledge of statistical data is needed over past periods, including the frequency of occurrence of insurance cases and the average damage from them.

Gross premium, or insurance premium, is the amount of insurance payments paid by the insurer to the insurer (insurance organization) for a certain period from the entire sum insured. The gross premium depends on the amount of the sum insured, the degree of risk and the period for which this insurance premium is made. Such a period of duration may not coincide with total Insurance. The structure of the gross premium reflects the economic mechanism of insurance.

It can select two elements in it: Net prize, intended for insurance payments under the terms of the insurance contract, and loadintended to cover the costs of doing business and profit from insurance operations (Fig. 10.1). Note that a net premium designed for a unit of an insured amount equal to 100 rubles, has a name "Net-rate."

Fig. 10.1. Brutte-prize structure

The ratio of net premium and load depending on the type and volume of insurance, as well as the level of costs of doing business may vary. Currently, this ratio changes in the direction of increasing the share of the load to 15-20%, as is customary in world practice. This trend is determined mainly by increasing the structural element of the load - Commission remuneration What speaks to enhance the value of the intermediary's work in insurance (agent, broker), and to a large extent corresponds to world practice.

In general, a net premium may include the following structural elements: a risky contribution, risk (warranty, or stabilization) premium and accumulative (savings) fee (Fig. 10.2).

Fig. 10.2. Possible Structure of Net Prize

Risky fee Designed to cover risk on all, i.e. It is used for insurance payments when an insured event occurs. In the structure of the net premium it is always present.

Risk (warranty, or stabilization) allowance Designed to compensate for possible exceeding actual payments over the calculated, challenged in the form of a risky contribution. In the structure of a net premium, this surcharge may not be included - it all depends on the management strategy chosen by the insurer. If his goal is to conquer the insurance market at the expense of prices lower than that of other insurers, this element (risk surcharge) is not included in the structure of the net premium. If the insurer wishes to strengthen his financial sustainabilityThis element is included in the net premium.

Cumulative (savings) fee Designed for the accumulation of the amount paid under the conditions of the long-term contract of life insurance - in the event of a survival of the insured to a certain date (at risk of survival). The accumulative fee must be investing in order to obtain income. It is a structural element of a non-prize of long-term life insurance contracts, for example, when insuring live, mixed life insurance, pensions insurance (in this case The Russian classification of types of insurance is used).

The size of the risk contribution to the net premium depends on the sum insured and the probability of the occurrence of the insured event. The size of the risk surcharge depends on the adopted probability of exceeding the actual payments over the calculated one. The smaller the given probability of exceeding the actual payments above the calculated, the higher the size of the risk surcharge. The ratio between the risky contribution and risk surcharge for different species Insurance may be unequal.

The size of the accumulative contribution depends on the decision taken monetary turnover (simple or complex percent), the size of the insurance (accumulated) amount paid at risk of recovery promised to the insured rate of income and the term of the contract (accumulation period). For the accumulative type of insurance, the ratio of risk and accumulative contributions is determined by the terms of the contract.

The inclusion of risk and accumulative contributions to the structure of a net premium is determined by the type of insurance: the risk premium condition is practically incorporated into all types of insurance, as it provides for the coating of risk, and the condition of cumulative - only in long-term life insurance agreements. So, with short-term insurance against accident and illness, medical insurance Or insurance in case of death, property insurance and responsibility (risky insurance) The structure of a net premium necessarily includes a risky contribution, and depending on the company's chosen management strategy may include or not enter a risky surcharge.

When insuring pensions ( long-term view Life insurance) The structure of a net premium includes a cumulative fee, which is intended for payments to the insured person at risk of survival to a certain date, for example, before the date of the next payment. Note that for long-term life insurance contracts, in which it is envisaged simultaneously as a risk coating (the risk of death and may be, the risk of accidents) and the accumulation of funds in case of survival. Thus, for contracts of mixed life insurance, the need for inclusion in the net-premium of the risk surcharge disappears, since the role of the risk (warranty) allowance performs a storage contribution.

In tab. 10.1 are presented possible options Gross premium structures for different types of insurance.

Net-prize elements: risky contribution, risky surcharge and accumulative contribution - serve as sources of formation of special insurance funds - insurance reserves intended for payments under the terms of the insurance contract.

Table 10.1. Options for the structures of gross premiums for different types of insurance

Note: "+" means the obligation to include in the structure gross Prizia; "±" means that this element may be included or not included in the gross premium structure.

As already noted, the load is part of the gross award, intended to cover the costs of doing business and produce profits from insurance operations (Fig. 10.3).

First Structural Load Element - business costs - refers to the cost of insurance services, the second element - Profit from insurance operations - This is the planned profit of the insurance organization from such operations.

The cost of doing business is divided into Traditionalwho are typical for any type of business, and Specificcarried out precisely in the insurance business. Specific costs include commission remuneration to agents and brokers for intermediary activities in the dissemination of insurance products, the costs of fee carrying preventive (preventive) activities, costs associated, for example, with a curial examination (at the conclusion of the contract), as well as expertise associated with the onset of insurance case, etc.

Fig. 10.3. Load structure

Experience economically developed countries It shows that the share of expenses for warning measures can be 4-6% of gross premiums, and the share of commission remuneration can reach up to 20% of the gross premium.