## Risk premium

The premium that is calculated on the basis of the principle of equivalence and ensures its observance is called * "risk premium." * It corresponds to the cost of that individual risk that the insurer takes over contract of insurance.

In order to derive formulas for calculating the risk premium on the basis of the equivalence principle, an estimate is made of the expected value of the parties' obligations. Equating them, the equation is obtained, where the amount of premium in the obligations of the insured is unknown. As a result of its solution, the required design dependencies are obtained in general form. Let's illustrate this with a small example.

** Example. Equivalent cost of risk **

In the previous example, the risk of the car owner before his participation in the insurance fund was as follows: with a certain probability * q * he could lose a car worth s; and with probability (1 - q) he did not lose anything, i.e. his losses were equal to 0.

The car owner has given this risk to the insurance fund. The expected cost of such a risk by definition of mathematical expectation is equal to the sum of products of possible values of a random variable by the probability of their occurrence, i.e.

For transferring the risk, the owner of the car paid a certain amount (insurance premium) to the insurance fund. This premium was to be paid unconditionally upon joining the fund, so the car owner's expenses in any case amounted to P. There was an exchange of accidental risk of a catastrophic loss (car theft) to a fixed (nonrandom) loss equal to the insurance premium.

This exchange should be based on an equivalent basis. The principle of equivalence, which underlies the transfer of risk in insurance, implies the equality of the expected values of the obligations of the parties. This means that the expected value of premiums under the contract should be equal to the expected cost of payments. In our case, the premium is paid unconditionally, its expected value is equal to the actual value, so you can write down

If we substitute numerical values from the previous example, we get the already familiar result:

The premium value calculated in this way is a fair insurance payment, equivalent to the risk that the participant transfers to the insurance fund.

In the example considered, the insurance period was only one year, and the equivalence principle was applied in a simplified formulation that did not take into account the change in the value of money in time and additional income from investing. In addition, the insurance premium was compulsorily paid at the time of the start of insurance. Therefore, its expected value coincided with the actual value. As a result, the risk premium was equal to the expected cost of payments under the contract:

A similar situation is typical for most types of property and personal insurance, except for life insurance.

## Risk premium

If each risk premium is equivalent to the expected loss under the contract, then their amount will be equal to the expected aggregate value of payments throughout the portfolio. That is, pure fair premiums are sufficient if the actual amount of payments does not exceed the expected. But in practice, there may be deviations from the expected values both in the smaller and in the larger side. Such fluctuations are a consequence of the operation of special risks that arise as a result of the merger of contracts into one portfolio.

First of all, for any insurance portfolio there is always a * chance of accident. * Even if there are known objectively existing theoretical probabilities of occurrence of insurance events and distribution of payments, to predict specific values that random are impossible (therefore they are called random). Even for huge aggregates, the law of large numbers does not guarantee the equality of the actual mean values expected. It only allows us to estimate the interval at which this value falls within a given probability. The width of this interval depends on the number of risks in the portfolio. The more of them, the less likely are the large deviations from the expected values. But such a probability, albeit very small, always exists.

When calculating risk premiums are guided by the expected values. But no one knows the true (theoretical) expected values of the number of insured events and losses for either a separate contract or for the portfolio as a whole. In practice, they have to be evaluated on the basis of available statistics. The resulting estimates always differ to a certain extent from the true value. This introduces an additional uncertainty into the activity of the insurance company, which is called * the risk of evaluation. *

The statistical data, on the basis of which random variables are estimated, are collected in past periods. However, in the future, the level of dangers can change. This applies to all areas of human life. Climate change affects the likelihood of natural disasters. Thanks to scientific progress, new types of technology are emerging, the properties of which are still poorly understood. The development of medicine reduces the risk of mortality and prolongs life expectancy. Therefore, tariffs calculated methodically according to absolutely reliable estimates may not be sufficient. It is required to predict the change in the level of "dangers" in time. As a result, there is a so-called * forecast risk. *

All three components of uncertainty inherent in insurance are combined by the common notion of "** technical insurance risk ** & quot ;. Because of its presence in the activities of each insurer, there is always the possibility of adverse deviation of the actual amount of losses on the portfolio from its expected value. To compensate for possible deviations, the insurance company may use its own or borrowed funds or pre-formed special reserves. One of the main sources of coverage for this risk is * Risk * (

*)*

**Guarantee**

**premium.** For a random value, the probability of taking a value greater than or less than the mathematical expectation is 50%. Therefore, the risk premium, which is guided by the expected values, will be sufficient only in half the cases. The addition of its risk premium increases the likelihood of a breakeven operation of the company to some level specified by the insurer, which is called the "* security guarantee." *

Its practical value ranges from 95 to 99.99%, but it can never reach 100%. This can be explained as follows. If the insurer wants to ensure absolutely surely the excess of premiums over payments, it must form an insurance fund in the amount of the aggregate insurance amount. In this case, the premium for each contract will be equal to the sum insured. Of course, such conditions are unacceptable for policyholders. Therefore, companies are forced to accept a security guarantee of less than 100%, although close enough to it. Thus, even an introduction to premium risk premium can guarantee a break-even operation with only a certain, albeit a very large, probability.

** Example. Risk premium **

In the previous examples, we considered a portfolio of car insurance in case of theft, subject to the law of large numbers. The expected loss values are used to estimate each individual risk. This ensures the equivalence of the obligations of the parties. But how can you be sure that these awards will be enough for all payments?

The portfolio consists of * n * = 1000 contracts. The amount of loss in the insured event is fixed and corresponds to the value of the car

*500 000 rub. Therefore, the total amount of payments on the portfolio is determined by one random value - the number of thefts. For each contract, the insured event may occur with a probability of*

**s =***= 0.03. If all contracts are independent, then the number of hijackings in such a portfolio is a discrete random variable distributed according to binomial law.*

**q**The risk premium under the contract was determined in the amount

0.03 x 500 000 = 15 000 rubles.

** Fig. 5.1. ** Illustration for calculating a risk premium

Their amount is enough for payments if the number of hijackings does not exceed the expected value, which is equal to

* n * x

*1000 x 0,03 = 30.*

**p =** The probability of this event (* m * = 30), determined by binomial law, is 55%. In the remaining 45% of cases, the number of hijackings will be higher than expected, and risk premiums will not be enough. Of course, this level of security is unacceptable for practical work. A risk premium is required, which will ensure compensation for possible adverse deviations in actual payments from the expected ones, and thereby increase the likelihood of break-even operation up to a given level of 98-99.99% security guarantee.

For the portfolio of risks, the number of hijackings that is not exceeded with a 98% probability is 42. With a security guarantee of 99.99%, this value increases to 52 (Figure 5.1)!

If the security level is set at 98%, the size of the fund should be sufficient to pay for 42 thefts. Risk premiums guarantee payment only for the expected 30 insured events. The risk premium should enable the insurance company, if necessary, to make 12 more payments.

The amount of risk premium per contract is

12 x 500 000/1000 = 6000 rubles.

Then the insurance premium, which will provide a probability of failure of 98%, should be equal to

15 000 + 6000 = 21 000 rubles. instead of 15 000 rubles. clean premium.

With respect to a risk premium, the premium is

6000/15000 = 0.4 = 40%.

This is a significant amount. If the insurer succeeded in combining not 1000 risks in one portfolio but 5000, then at the same level of security the 98% surcharge will amount to 2500 rubles. on the contract, i.е. only about 17% of the risk premium. For a portfolio of 10,000 contracts, its value will decrease to 12%.

This effect is manifested as a result of the action of the law of large numbers. The larger the set of random variables, the less probable the deviations of the mean values from the expected values. For an insurance company, increasing the portfolio means the opportunity to make a smaller bonus premium and thereby reduce the price of insurance. This gives a serious competitive advantage to large insurance companies.

The task of calculating a risk premium is quite complex. To solve it, it is necessary, first, to determine the required total amount of risk premiums and, secondly, to establish "fair" the principle of dividing this total amount between all contracts.

In the example considered, the total amount of risk premiums was determined on the basis of the assumption about the binomial distribution of the cumulative loss on the portfolio. Then it was divided equally among all the participants, because under the conditions all insured risks were the same. In practice, the portfolio combines different degrees of "danger" contracts, and the cumulative loss can have a more complex distribution. Therefore, there are different approaches to dividing the premium: the principle of mathematical expectation (ie, proportionally risk premium), the standard deviation principle, the dispersion principle, etc. Their combinations are also possible.

The amount of the risk premium depends on the given level of the security guarantee and the spread of the cumulative loss in relation to the expected value. The latter, in turn, is determined by the number of contracts in the portfolio and the variance (standard deviation) of the risks that make up this portfolio.

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