Net premium. Practice Plan
Net premium- part of the insurance premium intended directly to cover the damage. Net premium is the main integral part gross premiums..
The net premium consists of the net net risk premium and the risk (insurance) premium.
Net net premium
Determining the net risk premium has traditionally been the domain of actuarial calculations and insurance mathematics. The net net premium is calculated on the basis of historical damage data and is the product of the frequency of occurrence insured event on the the average size loss for the entire set of insured events that occurred in the past.
Net risk premium = Damage frequency x Average damage
The frequency of damage is defined as the quotient of the number of damage cases in the observed set divided by the number of observation units included in this set.
The average damage is the quotient of the division total amount loss for the observed period by the number of cases of damage for the same period.
Risk (insurance) premium
The risk premium is intended to increase the reliability of insurance coverage.
When identifying the pattern of occurrence of damage as a result of random events in the past and determining, on the basis of this past experience, unprofitability in the future, errors of two types are inevitable:
- Misdiagnosis resulting from 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.
- Forecast error, which consists in the fact that in the future there will not be a complete coincidence with the circumstances of the previous period, on the basis of which the net risk premium was determined. This may be due to the influence of unaccounted for or changed factors. It has been proven that even with very good information about damages, future damage exceeds its value in half the cases.
In order to guarantee reliable insurance protection, i.e. to increase the probability that the collected money will be enough to pay damages in the future in all cases, a risk (insurance) premium is added to the net net premium.
The value of the risk premium cannot be less than the value of the standard deviation of the loss ratio of the sum insured.
Using the net insurance rate to determine the net premium
The expected value of the net premium can be defined as the product of the sum insured and the net rate. The net rate is a percentage that reflects the probability of loss, calculated on the basis of the ratio of damage to the total sum insured of the insured objects.
The amount of the net premium is determined by the formula:
Net premium = Sum insured x Net rate/100
Part insurance premium, used to cover insurance payments for a specific type of insurance for a certain period of time, is called a net premium. Its value is directly dependent on the development of risk. The parameter may correspond to the risk premium at systematic development dangers.
What is it used for and what does it affect?
Part of the insurance premium is intended for compensation payments, the purpose of which is to cover the damage. Its value is determined by the parameter of the net premium, which is a constituent element of the gross premium. The formation of the net premium is based on risk and insurance premium. The determination of the risk value is made using actuarial calculations, taking into account the section of insurance mathematics. To identify the value, information about the damage caused for the past period is used.
The parameter corresponds to the product of the frequency of occurrence of an insured event for the selected period and the average amount of damage caused. When determining it, the calculation includes the losses caused to the insured as a result of circumstances classified as an insured event for the entire allocated time period to be analyzed. The frequency of damage is calculated by the private number of the total amount of damage in their observed set and the number of observed units included in it. The average amount of damage is determined by the quotient of its total amount and the number of cases of damage. All parameters are taken into account for the allocated time interval, interpreted as being observed.
What elements does it consist of?
The insurance premium determines average value net premium, which causes positive and negative deviations of the parameter. To compensate for it, a guarantee premium is included in the amount of the risk premium, which is used to stabilize the indicator. Its structure is formed in accordance with the type of insurance and its subject. In property and personal insurance product It is made up of different constituent elements. Net premium property insurance determined by the risk premium and the stabilization allowance. For personal insurance an actuarial calculation is characteristic, which takes into account the risk premium and the funded contribution. In some situations, a guarantee surcharge is taken into account.
How are settlement operations performed?
The net premium is relevant for insurance operations, the subject of which is property, health and life of a person. It corresponds to the difference between the total amount of the insurance premium and the agency or brokerage fee. The premium is necessary to ensure insurance protection against possible damage. It does not include that part of it, the expenditure of which is intended to cover other expenses. In life insurance, the parameter is interpreted as the difference between the initial insurance premium and the amount of dividends paid to the insured if they were used by the beneficiary to pay premium payments under the life insurance policy. The expected value of the net premium is determined by the formula:
NP \u003d SS x NS / 100, where:
- NP- net premium;
- SS- sum insured;
- NS- net rate.
The net bet is represented by a percentage that reflects the probability of loss. The parameter is calculated by the ratio of the damage caused to the total sum insured of the insured objects. The amount of damage is determined by the quotient of the total amount of damage and the number of recorded similar cases. All values are taken into account for the observed period.
To increase the reliability of protection by the decision of the insurer, the risk premium may be taken into account in the calculation. It cannot be less than the standard deviation of the loss ratio applied to the sum insured. Accounting for the value in the calculations increases the likelihood that the collected money in terms of the insurance premium will be enough to make compensation payments for the damage suffered by the insured as a result of the occurrence of an insured event. The calculation of the new premium value will look like the sum of the basic net premium and the insurance premium.
The risk premium parameter must be taken into account in calculations when identifying certain regularities in the fact of damage caused to the insured object as a result of random events in the past period. On the basis of statistical information, it is possible to predict unprofitability in advance. Analyzing the parameters, it is necessary to avoid diagnostic errors, which consist in the processing of information not in full, as well as forecast inaccuracies, expressed in the impossibility of repeating the past in the future period. In order to avoid inaccuracies and overstatement of the payment amount, the average statistical value of the value is used, determined by several time episodes.
Conclusion
Thus, the expected value of the net premium can be determined knowing the size of the sum insured and the value of the net rate. In this case, the net rate is a percentage that reflects the probability of loss (damage). This probability is calculated on the basis of the ratio of damage to the total sum insured of the insured objects. The net net premium is determined in the course of actuarial calculations, which require knowledge of statistical data for past periods, including the frequency of occurrence of insured events and the average damage from them.
The concept and structure of the gross premium
Definition 1
Gross premium is the amount determined by the terms of the insurance contract Money, which the insured is obliged to pay to the insurance company for a certain period of time.
In the structure of the gross premium, a net premium and a load are distinguished.
The net premium is necessary to fulfill the obligations of the insurance company under insurance contracts. May consist of the following elements:
- risk premium intended to cover damage in the event of an insured event;
- risk premium required to compensate for increased damage in the event of a possible increase in the likelihood of a risk event;
- a savings deposit used only in life insurance and intended to accumulate a certain amount of money during the term of the contract with subsequent payment.
The risk premium is always included in the net premium and is intended to form the insurance reserve fund, while the risk premium is taken into account when calculating the net premium at the discretion of the insurance company and goes to the formation of the reserve fund.
The burden included in the structure of the gross premium is the cost of the insurance company to carry out its activities and its profit.
Costs include traditional costs that are typical for any enterprise ( wage, rent, travel expenses, utility bills, etc.) and specific costs that apply only to the insurance industry (payment commissions insurance agents and brokers, implementation of preventive measures, examinations to determine the amount of damage, etc.).
Remark 1
Depending on the type of insurance, as well as the costs of the insurance company for its activities, the ratio of net premium and load may be different. Most often, in the total gross premium, 70-80% is the net premium, the rest is the load.
In the general case, the gross rate $Tb$ is equal to:
$Tb \u003d Tn / (100 - N) $ 100, where:
$Tn$ – net rate,
$H$ - load, defined as a percentage of the gross rate.
If the load is defined in rubles, then the gross rate is:
$Tb = Tn + H$
When calculating the gross premium, the most important thing is to determine the optimal size of the net premium, because. subsequent solvency depends on it and financial stability insurer. Therefore, its calculation is given increased attention.
Calculation of the net rate for risky types of insurance
Definition 2
The net rate is an indicator equal to the net premium calculated per unit (usually 100 rubles) of the sum insured.
The methodology for calculating the net rate for risky types of insurance implies the availability of a sufficient amount of statistical data necessary for accurate calculations, the conclusion is predicted a large number contracts (for the same period), as well as the absence of events that may entail payments for several insured events at once.
In accordance with the methodology, the formula for calculating the net rate $Тн$ is as follows:
$Tn = To + Tr$, where:
$To$ is the risk premium (part) of the net rate,
$Tr$ is the risk premium.
The risk premium is calculated as follows:
$To = Q Sb ⁄ S 100$, where:
$Q$ is the probability with which the occurrence of an insured event is possible,
$Sb$ is the average amount of insurance payment,
$S$ is the average sum insured.
$Q = M ⁄ N$, where:
$M$ is the number of insured events that have occurred,
$N$ - the number of contracts concluded for a certain period of time.
The average insurance payment is equal to the ratio of the amount of payments under all contracts to the number of contracts:
$Sb = (∑Sbi) ⁄ M$
The average sum insured is equal to the ratio of the total amount of sums insured for all contracts to the number of these contracts:
$S = (∑Si) ⁄ N$
The risk premium $Tr$ is equal to:
$Тр = Тo α(γ) √ ((1 – Q + (Rb ⁄ Sb)^2) / (N Q))$, where:
$Rb$ is the standard deviation of the average insurance payment,
$α(γ)$ is a coefficient that depends on the probability γ chosen by the insurance company that the premiums will be enough to cover the damage. The value is taken from the table:
Figure 1. Coefficient values. Author24 - online exchange of student papers
Calculation of net life insurance rate
The main factors affecting the size of the net rate for life insurance include:
- age and gender of the insured person;
- the duration of the contract and the procedure for paying fees;
- predicted profitability of funds received by the insurance reserve fund life insurance, if invested.
The calculation of the net rate is based on the data of the tables on the mortality of the population certain age and average life expectancy.
To begin with, the necessary indicators are calculated
The probability of death in a given year of life $Qx$ is calculated by the formula:
$Qx = Bx ⁄ Lx$, where:
$Bx$ is the number of people who die in the period from $x$ to $x + 1$ years,
$Lx$ is the total number of people who survived to x years;
The probability with which a person will live to a given age, $Px$ is equal to:
$Px = L(x+1) ⁄ Lx$, or:
Taking into account the fact that contracts for this type of insurance have a long period of validity, and the funds received from the insured can be used by the insurance company to invest in order to receive additional income, to adjust the final net rate, use the multiplier $V^n$ equal to:
$V_n = 1 ⁄ (1+i)_n$, where:
$i$ is the rate of return on investment,
$n$ is the number of years for which the funds are invested.
As a result, the size of the net premium $(Ex)_n$ for survival will be equal to:
$(Ex)_n = (L(x+n) V_n) / Lx S$, where:
$L(x+n)$ - the number of people who survived until the end of the period for which the contract was concluded,
$n$ - the period for which the contract was concluded,
$S$ – the sum insured.
The net bet on the possibility of death $(Az)_n$ is equal to:
$(Az)_n = (Bx ∙ V + B(x+1) ∙ V_2 + ⋯ +B (x+n-1) ∙ V_n) / Lx ∙ 100$, where:
$Bx, B(x+1)…B(x+n-1)$ is the number of people dying between $x$ years and $x+1$, calculated for each year of the contract period.
When concluding a combined insurance contract for both survival and the possibility of death, the net rate will be equal to:
$Tn = (Ex)_n + (Az)_n$
This method of calculating the net rate is applicable provided that the entire amount of the insurance payment is paid immediately for the entire insurance period. If the policyholder wishes to divide the premium amount into several parts equal to the number of years of insurance, then the amount of the annual payment $P^x$ will be equal to:
$Р_x = (Ed)_x / α_x$, where:
$(Ed)_x$ – the amount of the calculated one-time payment,
$α_x$ – installment factor, which is the cost of payments in the amount of one monetary unit. In fact, this indicator is close in magnitude to the value of the number of years for which the contract is concluded, but it turns out to be slightly lower than it. As a result, the amount of annual payments exceeds the value equal to the simple division of the lump sum by the number of years of insurance. In this case, the insurer compensates for the losses that it incurs from the inability to invest the entire amount at once and receive income from this.
Net premium
Thus, it is shown that the net premium ensuring the break-even of insurance should be higher than the risk premium calculated on the basis of the principle of equivalence of the obligations of the parties. The difference between them is called the risk premium, and the ratio of this difference to the risk premium is called the relative risk premium. Let us consider the procedure for forming a net premium in contracts with distributed damage.
In insurance, it is customary to operate with a special sum of money- unit of sum insured (u.c.c.), depending on the currency of the country, for example, 1 u.s.c. = 100 rubles.
Consider an example. Individual claim takes three values: 0; one; 4 e.s.s. with probabilities 0.9965, 0.0030, 0.0005 respectively. Find net premium.
Mean and variance of an individual claim:
Then the conditions for ensuring 95% reliability (probability of survival) using normal approximation we get: using the risk premium and taking into account the number of contracts; find the net premium:
Then the relative premium is:
So, the risk premium is 0.0050; the risk premium is 0.0017; net premium is 0.0067; gross premium (at) will be: 0.0067/0.88=0.76, which will exceed the risk premium by 1.5 times.
Analysis of a Homogeneous Insurance Portfolio Using Normal Approximation
We continue to consider the above problem (on the risk premium).
Recall: we need to investigate the process:
Collected net premiums provide an opportunity to fulfill their obligations to pay compensation if the number of insured events does not exceed 110. For a reliability of 96% (if that), it is necessary to be able to pay for events up to and including the 117th. Note that the 117th occurrence either happens or it doesn't, so 116.6 needs to be rounded up to the nearest higher integer. It is necessary to ensure the possibility of paying the sum insured in 117 cases. The actual probability of ruin in this case will be:
Reliability is somewhat higher than required by Insurance.
If an average relative risk premium of 10% has been established on the market, then the insurer cannot arbitrarily increase it to 16.6% (or up to 17%) due to competition. Therefore, in order to increase its reliability, it is forced either to invest its funds (ie, capital) - to create an initial reserve, or to resort to reinsurance.
Let's consider the first possibility. So, the insurer does not have enough funds to pay out 7 insured events, i.e. he needs capital in the amount of 7 insurance claims. For example, if the sum insured is 500, then the capital at which the given reliability is guaranteed is equal to and not
Let's analyze the second possibility. Let's assume that cases from the 111th to the 117th inclusive are transferred for reinsurance. This means that if the number of cases exceeds 117, then the reinsurer pays for the indicated cases, and the assignor reimburses all the following ones. Therefore, we will use the local Laplace theorem (since the size of payments is fixed) and find the probabilities:
For instance,
So the probabilities are obtained: 0.0021; 0.0019; 0.0016; 0.0014; 0.0012; 0.0010; 0.0008. The probability will have to be searched for by the Laplace integral theorem:
Then expected value the reinsurer's payout is:
This is the risk premium in the reinsurance contract.
If the relative premium from the reinsurer is known, then you can find the net premium in this contract. For example, then: (About 2/3 of one sum insured.) Therefore, the assignor has an alternative: either to keep a reserve of 7 sums insured, or to pay irrevocably 2/3 of one sum insured to the reinsurer. If the assignor can invest his temporarily free funds at an interest rate greater than 0.654 / 7.0 = 9.4%, then reinsurance can be paid at the expense of profit.
If the insurer does not have his own funds for the reserve (or he considers it expedient to put his funds into circulation), a reinsurance contract is concluded. Let's allocate areas of responsibility.
When the insurer pays compensation at the expense of collected net premiums. When liability is shared between the insurer and the reinsurer. The first pays a fixed number of refunds: , and the second - everything else: . Finally, when the risk is not secured, this constitutes the entrepreneurial risk of the insurer. (The insurer believes that more than 117 cases cannot occur in his portfolio. Therefore, he does not take measures in case of this situation. He does not create a reserve and does not enter into the reinsurance contract the condition for the reinsurer to pay compensation in the 118th insured event. If 118- th insured event, the reinsurer will pay only 7 cases, the technical ruin of the assignor occurs).
Note that the left limit of the reinsurer's liability may be shifted. You have to pay for reinsurance, the insurer does not have its own funds, so he tries to pay with the money of his clients. (In principle, the insurer always uses clients' money to solve emerging problems. Here we mean the one-time total net premium collected this year).
He collected contributions in the amount of: , and the average expected payments are, so the expected profit (before reinsurance) will be 5000. The insurer shares the expected profit with the reinsurer to increase its reliability. But this means that the collected funds are not enough to pay for the reimbursement of at least the 110th case.
All risk X can be divided into three parts: Y - insurer's risk, Z - reinsurer's risk, W - unsecured risk. Obviously, X=Y+Z+W , then M(X)=M(Y)+M(Z)+M(W). Covariance should be taken into account when calculating variances. To analyze the variance (and the process as a whole), one must choose an approximation. Since, it is impossible to apply Poisson's law, but normal approximation is admissible.
However, one must be prepared for the appearance of inaccuracies caused by a change in the distribution law. For example, the loss of the “tails” of the normal distribution, the inability to take negative values, errors when replacing a discrete distribution with continuous ones, the difference in results when using the local Laplace theorem and the Laplace integral theorem, etc. (By the way, if the loss is fixed, i.e. the total loss in the portfolio is a multiple of the number of insured events, then the local theorem is preferable!). Finally, there are also computational errors.
This circumstance illustrates the complexity of actuarial problems. V training course only a basic approach is shown. On a civilized insurance market in conditions of fierce competition, the one who calculates more accurately wins (!).
So, we need to find M(X), M(Y), M(Z) (and possibly M(W)).
For the normal distribution law, the density
condition is met:
then it is clear that when the integration interval is narrowed to (0,n), the integral of the positive function will decrease, so the mathematical expectation of the entire risk X will be slightly less than
For what follows, we will need for different
Let us denote this integral by
So, it has been established that
To calculate the integral of type J, we make a change of variables, which is traditional when working with a normal distribution:
then: hence:
So, it is only necessary to calculate and use the properties of the exponent and the Laplace function.
1. in practice:
and with a large portfolio
So, the risk of the insurer after reinsurance was:
hence,
insurance company contract damage
In practice, it is necessary to indicate who reimburses the 110th case, therefore
The risk of the reinsurer is quite small, which is explained by the relatively large. Interestingly, the total risk of the insurer and the reinsurer is equal to This is due to the rejection of 100% reliability. The difference of 4.06 should constitute an unsecured risk.
To summarize: The discrepancy is explained by the factors cited at the beginning of the section. Note that the insurer may expect to increase its expected earnings before claims (7370). And for reinsurance, you will have to pay only u.s.c. (391 conventional units), which is quite acceptable! The difference is credited to the reserve, which will allow in the future to do without reinsurance (either to increase reliability, or to reduce the premium, thereby increasing its competitiveness).
- Net premium - part of the insurance premium intended directly to cover the damage. The net premium is the main component of the gross premium.
The net premium consists of the net net risk premium and the risk (insurance) premium.
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