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 it is a product of the frequency of the insured event on the average amount of damage along the entire totality of the insurance claims that 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 surcharge 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
Notes
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Synonyms:Watch what is a "net premium" in other dictionaries:
SUBS., Number of synonyms: 1 Net rate (1) Dictionary of Synonyms ASIS. V.N. Trishin. 2013 ... Synonym dictionary
Net prize - element gross bets; Designed to form resources for payment for payment insurance compensation. Overhead costs of the Insurer for doing business in the calculation of net rates are not accepted. In the actuarial calculation of the net, the bet is accepted ... ... Large accounting dictionary
See Tariffs Insurance Dictionary Business Terms. Academician. 2001 ... Business Terms Dictionary
p-Stern Mixed Life Insurance
Net prize is calculated by the formula:
Full life insurance, 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 insurance of life with the payment of insurance benefits at 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 by. 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).
Gross Prize, or Insurance Contributionis the amount of insurance payments under the insurance contract paid by the Fear of the Insurer (insurance organization) for a certain period of operation 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. This period by duration may not coincide with total Insurance. The structure of the gross premium reflects the economic mechanism of insurance.
It can be allocated in it. two elements — net Prize, intended for insurance payments under the terms of the insurance contract, and loaddesigned to cover the costs of doing business and make profits from insurance operations (Fig. 1). Note that a net premium designed for a unit of the sum insured, equal, usually 100 rubles, is called net bet.
Fig. 1. Gross premium structure
The ratio of net premium and load depending on the type and volume of insurance, as well as on the level of costs of doing business can be different.
Currently, this ratio changes in the direction of increasing the share of loads up to 15-20%, as is accepted in world practice. This trend is mainly due to the increase in the structural element of the load - the commission of the commission, which indicates the strengthening of the importance of working through the insurance (agent, broker), and to a large extent correspond to global practice.
In general net premium may include the following structural elements — risky fee, risk (warranty) over-lack and accumulative (savings) fee (Fig. 2).
Fig. 2. Possible Structure of Net Prize
Risky fee is designed To cover the risk for all types of insurance, i.e. it is used for insurance payments in the influence of the insured event. In the structure of the net premium, it is always present.
Risk (warranty or stabilization) surcharge It is intended to compensate for the possible exceeding the actual payments over the calculated, challenged in the form of a risky contribution. In the structure of the net premium, this surcharge may not be turned on - it all depends on the selected insurer of the management strategy. If his goal is to win the insurance market due to lower prices compared to 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 It is intended for the accumulation of the amount paid under the terms of the long-term contract of life insurance - in case of harvesting the insured to a certain date (at risk of survival). The accumulative contribution must be in-part in order to obtain income. It is a structural element of a non-prize of long-term life insurance contracts, for example, when insuring surviving, mixed life insurance, pensions insurance (in this case, the Russian classification of insurance types is used).
Risk premium size The net premium depends on the insurance sum and the probability of the occurrence of the insured event.
The size of the risk surcharge Depends on the adopted probability of exceeding actual payments over the calculated. The smaller the given probability exceeds the actual payments above the calculated, the higher the size of the risk surcharge. The ratio between the risk contribution and the risky over the lawls for different species Insurance may be different.
Accumulative contribution size Depends on the adopted rule of de-gentle turnover (simple or complex percent), the size of the insurance (accumulated) amount paid at the risk of recovery, the assessment of the rate of income and the term of the contract (period of accumulation). For the accumulative type of insurance, the ratio of risk and accumulative contributions is determined by the contracts of the contract.
The inclusion of risk and accumulative contributions to the structure of a net premium is determined by the type of insurance - risky wear Prak is fully included in all types of insurance, as it provides for the risk coating, and the accumulative - only in long-term up-talk life insurance.
So, with short-term insurance against accident and bole medical insurance Or insurance in case of death, with the insurance of property and responsibility (risky countries) in 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 into a risky surcharge.
When insuring pensions ( long-term view Life insurance) The structure of a net premium includes a cumulative contribution, which is designed for payments to the insured at risk of recovery up to a certain date, for example, before the date of the next payment. Note that for long-term life insurance agreements, which is envisaged by one-time as a risk coating (risk of death and, perhaps the risk of an accident) and the accumulation of funds in case of survival, on the example, for contracts mixed insurance Life, the need for inclusion in the net-premium of the risk surcharge disappears - the role of Ri-Kova (warranty) surcharge performs a storage contribution.
In tab. 1 presented options for possible gross premium structures for different species Insurance.
Table 1
Options for the structure of gross premiums for various types of insurance
Net-prize elements - risky contribution, risk premium and accumulative contribution - are sources of formation of special insurance funds - insurance reserves intended for payments under the terms of the insurance contract.
As already noted, the load is part of the gross premium, designed to cover the costs of doing business and for profit from insurance operations (Fig. 3).
Fig. 3. Load structure
First Structural Load Element — business costs - refers to the cost of insurance services, the second element is a planned profit of an insurance organization from insurance operations.
The cost of doing business is divided into traditionalwho have a place for any kind of business, and specificCharacteristic precisely for the insurance business.
Specific types of costs from Commission remuneration to agents and brokers for in average activities in the dissemination of insurance products, expenses for warning (preventive) measures, costs related, for example, with the initial examination (at the conclusion of the Agreement), as well as expertise associated with on the onset of the insured event, etc.
Experience economically developed countries It shows that the share of races to conduct warning events may be 4-6% of the gross premium, and the share of commission remuneration may reach up to 20% of the gross premium.
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 data over the past period and is a product of the frequency of the insured event on the average damage at the entire population of the insurance cases 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 amount is a private amount from dividing the total amount of 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 the money collected is enough for the payment of damage in the future in all cases, a risk (insurance) premium is added to the net native award.
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
Net prize
So, it is shown that a net premium that provides break-evenness of insurance should be higher than the risk award, calculated on the basis of the equivalence principle of the obligations of the Parties. The difference between them is called risky allowance, and the attitude of this difference to the risk premium is a relative risk resistance. Consider the procedure for the formation of a net premium in contracts with distributed damage.
In the insurance it is customary to operate special monetary sum - Unit of the sum insured (E.S.S.), depending on the currency of the country, for example, 1 E.S. \u003d 100 rubles.
Consider an example. An individual claim takes three meanings: 0; one; 4 E.S.S. With probabilities 0.9965, 0.0030, 0.0005, respectively. Find a net premium.
Average value and dispersion of an individual claim:
Then the conditions for ensuring a 95% reliability (probability of survival) using normal approximation: using the risk award and considering the number of contracts; Find a net premium:
Then relative allowance is equal to:
So, the risky premium is 0.0050; Risk surcharge is 0.0017; Net prize is 0.0067; The gross premium (at) will be: 0.0067 / 0.88 \u003d 0.76, this will exceed a risk award by 1.5 times.
Analysis of a homogeneous insurance portfolio using normal approximation
We will continue to consider the above task (about risk allowance).
Recall: it is necessary to explore the process:
The collected net premium provides the ability to fulfill their obligations to pay compensation if the number of insurance cases does not exceed 110. For reliability, 96% (if that) it is necessary to be able to pay cases up to the 117th inclusive. It should be noted that the 117th case will either happen, or not, so it is necessary to round 116.6 to the nearest as much longer. It is necessary to ensure the possibility of paying the sum insured in 117 cases. The actual probability of ruin will be:
Reliability is slightly higher than the insangency requires.
If the market has established an average relative risk surcharge of 10%, it will arbitrarily increase it to 16.6% (or up to 17%), the insurer cannot due to competition. Therefore, it is forced to increase its reliability or invest its funds (ie, capital) - to create an initial reserve, or resort to reinsurance.
Consider the first opportunity. So, the insurer does not receive funds for paying 7 insurance cases, i.e. He needed capital in the amount of 7 insurance compensation. For example, if the insurance amount is 500, then the capital, in which the specified reliability is guaranteed is not
We analyze the second opportunity. Suppose that cases from 111st to the 117th inclusive are transmitted to reinsurance. This means that if the number of cases exceeds 117, then the reinsurer pays these cases, and all of the following reimburses the cedent. Therefore, we will use the local Laplace theorem (since the amount of payments is fixed) and find the probabilities:
For example,
Thus obtained probabilities: 0.0021; 0.0019; 0.0016; 0.0014; 0.0012; 0.0010; 0.0008. The probability will have to search for the Laplace integral theorem:
Then the mathematical expectation of the reinsurer payments is:
This is the risk premium in the reinsurance contract.
If there is a relative surcharge at the reinsurer, you can find a net premium in this contract. For example, then: (about 2/3 of one insurance amount.) Consequently, the cedent has an alternative: either to keep the reserve in 7 insurance amounts, or permanently pay the reinsurer 2/3 of one sum insured. If the cedent can invest its temporarily free funds under a percentage, greater than 0.654 / 7.0 \u003d 9.4%, then reinsurance can be paid at the expense of profits.
If the insurer has no funds for the reserve (or he considers it appropriate to let his funds in turnover) is a reinsurance agreement. I distribute the zones of responsibility.
The insurer pays compensation at the expense of the collected net premiums. With justice is divided between the insurer and the reinsurer. The first pays a fixed number of refunds:, and the second is everything else :. Finally, the risk is not provided, this is the entrepreneurial risk of the insurer. (The insurer believes that more than 117 cases may not occur in his portfolio. Therefore, it does not take measures in case of this situation. It does not create a reserve and does not contribute to the reinsurance contract by the reinsurer of compensation in the 118th insurance case. If the 118th insured event occurs, the reinsurer will pay only 7 cases, the technical ruin of the cedentis).
Note that the left limits of the reinsurer responsibility can be shifted. It is necessary to pay for reinsurance, the insurer does not have their own funds, so he tries to pay their customers money. (In principle, the insurer always uses customer money to solve emerging problems. Here it is in mind the lump sum summarice native awards collected this year).
He collected contributions to the amount:, and the average expected payments are made up, so the expected profit (before reinsurance) will be 5,000. The insurer shares the expected profit with the reinsurer to increase its reliability. But this means that the assembled funds are not enough to pay compensation, at least the 110th case.
All risk x can be divided into three parts: y - the risk of insurer, Z is the risk of reinsurer, W is an unsecured risk. Obviously, x \u003d y + z + w, then m (x) \u003d m (y) + m (z) + m (w). When calculating dispersions, you should consider covariance. To analyze the dispersion (and the process as a whole), you need to choose approximation. Since it is impossible to apply the law of Poisson, but a normal approximation is allowed.
However, it is necessary to be prepared for the emergence of inaccuracies caused by the change in the law of the distribution. For example, the loss of "tailings" of the normal distribution, the inability to adopt negative values, errors when replacing the discrete distribution by continuous, difference in the results when using the local Laplace Theorem and the Laplace integral theorem, etc. (By the way, if the damage is fixed, i.e. the total damage in the portfolio of Cathedral of the Insurance Cases, the local theorem is preferable!). Finally, there are computational errors.
This circumstance illustrates the complexity of actuarial tasks. The training course shows only a fundamental approach. In civilized insurance market In the conditions of tough competition wins the one who considers more precisely (!).
So, it is necessary to find m (x), m (y), m (z) (and possibly m (w)).
For normal distribution law density
condition is satisfied:
then it is clear that with the narrowing of the interval of the integration to (0, n) the integral of the positive function will decrease, so the mathematical waiting for the entire risk X will be somewhat less than
For further we need to be different
Denote this integral through
So, it was found that
To calculate the type j integral, we will replace the variables traditional when working with normal distribution:
then: therefore:
So, you only need to calculate and use the properties of the exhibitance and the functions of the Laplace.
1. In practice:
and with a large portfolio
So, the risk of the insurer after reinsurance amounted to:
hence,
insurance Company Damage Contract
In practice, you must specify who reimburses the 110th case, so
The risk of reinsurer is quite small, which is explained relatively large. Interestingly, the total risk of insurer and the reinsurer is equal to this due to the failure of 100% reliability. The difference 4.06 must be an unsecured risk.
Let us summarize: the discrepancy is explained by the factors given at the beginning of the section. It should be noted that the insurer can expect an increase in its expected profit to refunds (7370). And for reinsurance will have to pay all E.S.S. (391 conventional units), which is quite acceptable! The difference is credited to the reserve, which will allow in the future to do without reinsurance (or increase reliability, or reduce the allowance, thereby increasing its competitiveness).
