Net premium, load - insurance. Analysis of a homogeneous insurance portfolio using normal approximation Practical lesson plan

Net premium

So, it has been shown that the net premium that ensures the insurance break-even 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 amount of money - a unit of the insured amount (s), depending on the currency of the country, for example, 1 s. = RUB 100

Let's look at an example. An individual claim takes three values: 0; one; 4 e.w. with probabilities 0.9965, 0.0030, 0.0005, respectively. Find a net premium.

Average and variance of an individual claim:

Then the conditions for ensuring 95% reliability (probability of survival) using the normal approximation will be obtained: using the risk premium and taking into account the number of contracts; find a net premium:

Then the relative markup is equal to:

So, the risk premium is 0.0050; the risk premium is 0.0017; the net premium is 0.0067; the gross premium (if) will be: 0.0067 / 0.88 = 0.76, this will exceed the risk premium by 1.5 times.

Analysis of a homogeneous insurance portfolio using normal approximation

Let's continue to consider the above problem (on the risk premium).

Let's remind: it is necessary to investigate the process:

The collected net premiums provide an opportunity to fulfill their obligations to pay indemnities if the number of insured events does not exceed 110. For 96% reliability (if that), it is necessary to be able to pay for cases up to 117 inclusive. Note that the 117th case will either happen or not, so you need to round 116.6 to the nearest higher integer. It is necessary to ensure the possibility of payment of the sum insured in 117 cases. The real probability of ruin in this case will be:

The reliability is slightly higher than that required by Insurance Supervision.

If the market has established an average relative risk premium of 10%, then the insurer cannot arbitrarily increase it to 16.6% (or up to 17%) due to competition. Therefore, in order to increase his reliability, he is forced either to invest his funds (i.e., 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 specified reliability is guaranteed is equal to and not

We analyze the second possibility. Suppose that cases from 111th to 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. 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 sought by the integral Laplace theorem:

Then the mathematical expectation of the reinsurer's payments is:

This is the risk premium in the reinsurance contract.

If you know the relative premium from the reinsurer, then you can find the net premium in this contract. For example, then: (About 2/3 of one insured sum.) Consequently, the assignor has an alternative: either to keep a reserve of 7 insured sums, or irretrievably pay the reinsurer 2/3 of one insured sum. If the assignor can invest his temporarily surplus funds at an interest rate greater than 0.654 / 7.0 = 9.4%, then reinsurance can be paid for at the expense of profits.

If the insurer does not have its own funds for the reserve (or it considers it expedient to put its funds into circulation), a reinsurance contract is concluded. We will distribute areas of responsibility.

When the insurer pays compensation from the collected net premiums. In this case, the responsibility is divided between the insurer and the reinsurer. The first pays out a fixed number of refunds: and the second pays everything else:. Finally, if the risk is not secured, this constitutes the business risk of the insurer. (The insurer believes that more than 117 cases cannot occur in its portfolio. Therefore, it does not take measures in case of this situation. It 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, there is a technical ruin of the assignor).

Note that the left border of the reinsurer's liability can be shifted. You have to pay for reinsurance, the insurer does not have its own funds, so it tries to pay off with the money of its clients. (In principle, the insurer always uses clients' money to solve problems that arise. This refers to the lump sum net premium collected this year).

He has collected contributions in the amount of:, and the average expected payments are, therefore the expected profit (before reinsurance) will be 5000. The insurer shares the expected profit with the reinsurer to improve its reliability. But this means that the funds raised are not enough to pay for the reimbursement of at least the 110th case.

All risk X can be broken down 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). When calculating variances, the covariance should be taken into account. To analyze variance (and the process as a whole), an approximation must be chosen. Since, then Poisson's law cannot be applied, but normal approximation is acceptable.

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 impossibility of accepting negative values, errors in replacing the 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 damage is fixed, i.e. the total damage in the portfolio is a multiple of the number of insured events, then the local theorem is preferable!). Finally, there are computational errors.

This circumstance illustrates the complexity of actuarial tasks. The training course demonstrates only a principled approach. In a civilized insurance market in conditions of tough competition, the one who thinks more accurately (!) Wins.

So, we need to find M (X), M (Y), M (Z) (and possibly M (W)).

For a normal distribution law, the density

the condition is met:

then it is clear that as the interval of integration narrows 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,

We denote this integral by

So, it has been established that

To calculate an integral of type J, we make a change of variables, which is traditional when working with a normal distribution:

then: hence:

So, you only need 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 reinsurer's risk is quite small, which is explained by its 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 be the unsecured risk.

To summarize: The discrepancy is explained by the factors given at the beginning of the section. Note that the insurer can expect to increase its expected profit before claims (7370). And for reinsurance you will have to pay only e.s.s. (391 conventional units), which is quite acceptable! The difference is credited to the reserve, which will make it possible to do without reinsurance in the future (or to increase reliability, or to reduce the premium, thereby increasing its competitiveness).

• The net premium is the portion of the insurance premium earmarked 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.

Related concepts

Interest income - income received by the owner of funds from providing them for a while to other economic entities. Represents compensation paid for the use of funds. Usually expressed in the form of an annual percentage rate.

Cash flow statement - a statement of the company on the sources of funds and their use in the reporting period, directly or indirectly reflecting the company's cash receipts classified by main sources and its cash payments classified by main areas of use during the period. The report provides an overall picture of operating results, short-term liquidity, long-term creditworthiness and makes it easier to conduct financial analysis of the company.

Interest rate risk is the risk (possibility) of financial losses (losses) due to unfavorable changes in interest rates. Interest rate risk may be caused by the mismatch of the terms of demand (repayment) of claims and liabilities, as well as the unequal degree of change in interest rates on claims and liabilities.

A risk measure is a function that allows you to obtain an estimate of the financial risk for a certain portfolio of assets in quantitative terms (most often in monetary terms). The risk measure is used to determine the amount of capital reserve required to meet the requirements of the regulator.

Mezzanine Loan is a relatively large loan, usually unsecured (i.e. provided without collateral) or having a deeply subordinated collateral structure (for example, a lien on property of the third priority, but without recourse in relation to borrower). The loan usually has a maturity of more than five years if the principal is repaid at the end of the loan term. As part of a standard offer, the loan is accompanied by a tear-off certificate (coupon) giving the right to ...

Costs - the amount of resources (to simplify, measured in monetary terms) used in the course of economic activity over a certain period of time. Or in simple terms: cost is the cost estimate of resources.

Mortgage insurance is insurance against the risk of losses from creditors that may arise in the event of defaults by mortgage borrowers and the subsequent sale of the pledged property (lack of funds from the sale of the pledged property and the inability to recover the balance of funds from the borrower).

Accident insurance is a type of personal insurance. It is intended to compensate for damage caused by the loss of health or death of the insured.

The (zero-coupon) yield curve or term structure of interest rates is the dependence (dependence curve) of the yield of homogeneous financial instruments on their maturities (duration). The basic yield curve is plotted for government securities (G-curve, G-Curve) of various maturities (in Russia - for OFZ). You can also build your own profitability curve for a specific organization based on the cost of attracted resources, depending ...

Income tax is a direct tax levied on the profits of an organization (enterprise, bank, insurance company, etc.). Profit for the purposes of this tax is usually defined as income from the company's activities minus the amount of established deductions and discounts (however, it is never less than 12.5%). The deductions include ...

Net national product (NPP) is the total volume of goods and services that a country has produced and consumed in all sectors of its national economy over a certain period of time.

The total cost of the loan (PSK) is the borrower's payments under the loan agreement, the amounts and terms of payment of which are known at the time of its conclusion, including taking into account payments in favor of third parties determined by the agreement, if the borrower's obligation for such payments arises from the terms of the agreement. The total cost of the loan is calculated as an annual percentage.

The weighted average cost of capital (WACC) is the average interest rate across all funding sources for a company. The calculation takes into account the share of each source of funding in the total cost.

Coinsurance is the joint insurance of the same object by several insurers. This method of providing insurance coverage is used, as a rule, when insuring large objects, when one insurance company is not able to take on large risks.

Financial mathematics is a branch of applied mathematics that deals with mathematical problems related to financial calculations. In financial mathematics, any financial instrument is considered from the point of view of some (possibly random) cash flow generated by this instrument.

Tail risk or residual risk is the risk that the price of an asset or portfolio of assets will change by more than three standard deviations from the current price. At the same time, most asset managers control only the risk of loss, that is, the risk of a price decline by more than three standard deviations below the current price.

Bank liquidity - the ability of the bank to ensure the timely and full performance of its obligations.

Insurance risk is an event, the occurrence of which is not defined in time and space, independent of the will of a person, dangerous and, as a result, creates an incentive for insurance; this is the risk that can be assessed in terms of the likelihood of an insured event and the amount of possible damage.

The minimum acceptable rate of return (generally accepted abbreviation - MARR) is the minimum rate of return for a project that a manager or company is ready to accept before starting a project, taking into account its risk and opportunity costs other projects in business and engineering. A synonym, seen in many contexts, is the attractive minimum rate of return.

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Internal rate of return (IRR) is the percentage rate at which the net present value (net present value - NPV) is equal to 0. NPV is calculated based on the cash flow discounted to date ...

Property insurance is a type of insurance in which the property interest associated with the possession, use and disposal of property acts as an object of insurance. It is carried out mainly in the form of voluntary insurance, with the exception of insurance of state property leased. The policyholders are any enterprises and organizations of various organizational and legal forms, as well as individuals.

Economic profit is the profit remaining with the enterprise after deducting all costs, including the opportunity cost of distributing capital to the owner. In the case of a negative value of economic profit, the option of leaving the enterprise from the market is considered.

Casco (from Spanish casco helmet or Dutch casco body) - insurance of means of transport (cars, ships, planes, wagons) against damage, theft or theft. It does not include insurance of transported property (cargo, English cargo), liability to third parties, etc. Various types of franchises are actively used in casco insurance, often the insurance rules provide for the possibility of abandon.

Double insurance (multiple insurance) - simultaneous insurance of the same property interest, the same object and risk from different insurers, in which the total liability limit of insurers (the total insured amount for all contracts) exceeds the insured value ...

Pricing for research and development work (R&D) - setting prices for exploratory, theoretical and experimental work carried out with the aim of creating new technology.

Gross premium, or insurance premium, is the amount of insurance payments paid by the policyholder to the insurer (insurance organization) for a certain period from the entire insured amount. The gross premium depends on the amount of the insured amount, the degree of risk and the period for which this insurance premium is made. This period may not be the same as the general insurance period. The structure of the gross premium reflects the economic mechanism of insurance.

Two elements can be distinguished in it: net premium, intended for insurance payments under the terms of the insurance contract, and load, designed to cover the costs of doing business and making a profit from insurance operations (Figure 10.1). Note that the net premium calculated per unit of the insured amount, which is, as a rule, equal to 100 rubles, has the name "Net rate".

Rice. 10.1. Gross premium structure

The ratio of the net premium to the load, depending on the type and volume of insurance, as well as the level of costs of doing business, may differ. Currently, this ratio is changing towards an increase in the share of the load up to 15-20%, as is customary in world practice. This trend is mainly due to an increase in the structural element of the load - commission, which speaks of the growing importance of the work of an intermediary in insurance (agent, broker), and to a large extent corresponds to world practice.

In the general case, the net premium can include the following structural elements: risk contribution, risk (guarantee, or stabilization) premium and accumulative (savings) contribution (Fig. 10.2).

Rice. 10.2. Possible net premium structure

Risk contribution designed to cover the risk for all, i.e. it is used for insurance payments in the event of an insured event. It is always present in the structure of the net premium.

Risk (guarantee, or stabilization) premium is intended to compensate for the possible excess of actual payments over the calculated ones, taken into account in the form of a risk contribution. This premium may not be included in the structure of the net premium - it all depends on the management strategy chosen by the insurer. If its goal is to conquer the insurance market at the expense of prices lower than those of other insurers, then this element (risk premium) is not included in the structure of the net premium. If the insurer wishes to strengthen its financial stability, this element is included in the net premium.

Cumulative (savings) contribution is intended to accumulate the amount paid under the terms of a long-term life insurance contract - if the insured person survives until a certain date (at the risk of survival). The funded contribution must be invested in order to generate income. It is a structural element of the net premium of long-term life insurance contracts, for example, for survival insurance, mixed life insurance, pension insurance (in this case, the Russian classification of types of insurance is used).

The amount of the risk contribution in the net premium depends on the insured amount and the likelihood of an insured event. The amount of the risk premium depends on the accepted probability of the excess of actual payments over the calculated ones. The less the specified probability of excess of the actual payments over the calculated ones, the higher the amount of the risk premium. The ratio between the risk premium and the risk premium for different types of insurance may not be the same.

The amount of the accumulative contribution depends on the accepted rule of monetary circulation (simple or compound interest), the amount of the insurance (accumulated) amount paid for the risk of survival, the income rate promised to the policyholder and the duration 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 in the structure of the net premium is determined by the type of insurance: the condition of the risk premium is practically included in all types of insurance, since it provides for coverage of risk, and the condition for accumulative contributions is only included in long-term life insurance contracts. So, in the case of short-term insurance against accident and illness, medical insurance or death insurance, property and liability insurance (risk types of insurance), the structure of the net premium necessarily includes a risk contribution, and depending on the chosen company management strategy, it may or may not include a risk premium.

When insuring pensions (long-term type of life insurance), the structure of the net premium includes a funded contribution, which is intended for payments to the insured person at the risk of surviving until a certain date, for example, until the date of the next payment. Note that for long-term life insurance contracts, which provide for both the coverage of risk (risk of death and, perhaps, the risk of an accident), and the accumulation of funds in case of survival. So, for contracts of mixed life insurance, there is no need to include the risk premium in the net premium, since the role of the risk (guarantee) premium is performed by the accumulative contribution.

Table 10.1 presents possible options for the structure of the gross premium for different types of insurance.

The elements of the net premium: risk premium, risk premium and accumulative premium - serve as sources for the formation of special insurance funds - insurance reserves intended for payments under the terms of an insurance contract.

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

Note: “+” means that it is obligatory to include in the structure of the gross premium; “±” means that this element may or may not be included in the gross premium structure.

As already noted, the load is a part of the gross premium intended to cover the costs of doing business and making a profit from insurance operations (Figure 10.3).

The first structural element of the load is 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 by traditional that are typical for any type of business, and specific carried out precisely in the insurance business. Specific types of costs include commissions to agents and brokers for intermediary activities in the distribution of insurance products, expenses for carrying out preventive (preventive) measures, costs associated, for example, with conducting an initial examination (upon concluding a contract), as well as an examination related to the occurrence of insurance case, etc.

Rice. 10.3. Load structure

The experience of economically developed countries shows that the share of expenses on preventive measures can be 4-6% of the gross premium, and the share of commissions can be up to 20% of the gross premium.

The portion of the premium used to cover the insurance premiums for a particular type of insurance over a certain period of time is called the net premium. Its value is directly dependent on the development of risk. The parameter can correspond to the risk premium with the systematic development of hazards.

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 damage. Its value is determined by the parameter of the net premium, which is a constituent element of the gross premium. The net premium is formed on the basis of risk and insurance premium. Determination of the amount at risk is carried out 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 the insured event for the selected period and the average value of the damage caused. When determining it, the calculation includes the losses caused to the insured, received as a result of circumstances classified as an insured event for the entire allocated time period subject to analysis. The frequency of damage is calculated by the quotient of the total amount of damage in the 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 damage cases. All parameters are taken into account for a specified time period, interpreted as observed.

What elements does it consist of?

The insurance premium determines the average value of the net premium, which determines the 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 consists of different constituent elements. The net property insurance premium is determined by the risk premium and the stabilization premium. Personal insurance is characterized by an actuarial calculation, which takes into account the risk premium and the accumulation premium. In some situations, a guaranteeing premium is taken into account.

How are settlement transactions carried out?

The net premium is relevant for insurance operations, the subject of which is property, health and human life. It corresponds to the difference between the total amount of the insurance premium and the agency or brokerage fee. The premium is necessary to provide insurance coverage 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 policyholder 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 = SS x NS / 100, where:

• NP- net premium;
• SS- sum insured;
• NS- net rate.

The net rate is represented by a percentage representing the likelihood of a loss. The parameter is calculated by the ratio of the damage caused to the total insured amount 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 can be taken into account in the calculation. It cannot be less than the standard deviation of the loss ratio applied to the insured amount. Taking the value into account in the calculations increases the likelihood that the collected money in the view of the insurance premium will be sufficient to make compensation payments for the damage incurred by the policyholder as a result of the insured event. The calculation of the new premium value will look like the sum of the base net premium and the insurance premium.

The risk markup parameter must be taken into account in the calculations when identifying certain patterns of the fact of damage caused to the insured object as a result of random events in the past period. Based on statistical information, it is possible to predict unprofitability in advance. Analyzing the parameters, one should avoid diagnostic errors, which consist in the processing of information not in full, as well as inaccuracies of the forecast, 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 applied, determined over several time episodes.

Conclusion

Thus, the expected value of the net premium can be determined by knowing the amount of the insured amount and the value of the net rate. In this case, the net rate is a percentage reflecting the likelihood of loss (damage). This probability is calculated based on the ratio of damage to the total insured amount 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.