## Periodic Awards

The insurance premium can be paid by the insured not only in full at the time of the conclusion of the contract, but also paid periodically during the term of the insurance agreement (installment plan).

** Periodic premium ** - premium paid by the insured in installments, within the time limits and amounts stipulated in the insurance contract. Periods for the payment of the periodic premium depend on the term of the insurance contract. Contributions can be paid monthly, quarterly, annually, etc.

When concluding an insurance contract, the moment of payment of the insurance amount is not known in advance. The process of payment of insurance premiums, as a rule, is extended for the entire period of the contract. Therefore, if the contract is concluded for a relatively long period, then it is necessary to take into account the change in the price of money in time, since the time factor in the long run plays no less a role than the amount of money.

The need to take into account the time factor is determined by the * principle of the inequality of money, * relating to different points in time. Inequality is determined by the fact that theoretically any amount of money can be invested and generate revenue. Received income, in turn, can be reinvested, etc. Therefore, today's money in this sense is more valuable than future money, and future receipts are less valuable than modern ones. More details of the basics of financial mathematics we consider in Ch. 6.

The consequence of the principle of "money inequality" is the modification of the * principle of equivalence * of the obligations of the insurer and the policyholder, which takes the following form: * the current prices of the risks of the insurer and the insured are equal. *

The general algorithm for finding periodic premiums is:

1) determine the mathematical expectation of the current price of the insurance indemnity paid - a one-time risk premium;

2) calculate the current cost of insurance protection, taking into account the risk premium - a one-time net premium;

3) take into account the current cost of the load and receive a modern price of gross premiums;

4) using the mathematical apparatus of rent and taking into account the risk of underpayment of contributions is the size of the necessary periodic contributions.

In financial computing, the time factor is accounted for by calculating simple or compound interest. Simple interest is usually applied in financial transactions with a term of up to one year. In long-term financial transactions, compound interest is used, when interest is not paid periodically, but is added to the amount of debt. In actuarial calculations, compound interest is usually used (for more details, see Chapter 6). The formula * accretion by compound interest * has the form:

where * C * is the initial amount; * C (n) * - the amount accumulated for * n * sums; * i - * Effective interest rate for one period (usually annual); * n * - interest accrual period.

The calculation of the original sum * C * for a given accumulated sum * C (n) * at the end of the insurance period is a task inverse to the interest accrual, it is called * discounting. *

The value of * C, * found by discounting is called * the current value * (current value) of the sum * C (n). * The value * C * is equivalent to the sum * C (n *) in the sense that after a certain period of time and with a specified interest rate, the sum * C * becomes, as a result of the build-up, * C (n) . *

The formula * discounting on compound interest * looks like:

where - the discount factor.

Consider the algorithm for determining periodic contributions based on the renting vehicle. Suppose that the size of the one-time gross premium of the BP was calculated. The value of the annual interest rate * is known * It is assumed that contributions will be paid * t * once a year. The price of the first contribution made by the insured will be equal to . The current price of the second periodic payment will be less by 1/v times (according to the discount operation) (where the discount factor v is calculated through the effective interest rate of the gap 1 */t. * The current price of the third installment will be 1/v2 times less etc. The current price of the last installment will be less by 1/vm 1 times Thus, proceeding from the principle of the equality of the present value of the premium and the amount of future payments, it is possible to compile the * balance equation, which must be solved with respect to the unknown quantity BPperio - the size of the periodic premiums:*

When determining the size of the periodic premium, the considered method on the basis of the classic renting vehicle does not take into account the risk of non-receipt of the next installment by the insurer. In the classical apparatus of rent, the payment flows are determined by time and size. In insurance, if the insurer has not received a one-time premium, he does not have full confidence that he will receive all the agreed amount of contributions. If the insured event occurred earlier than the next installment, the client is released from all further contributions (unless otherwise stipulated in the insurance contract), and the insurance company must fulfill its obligations in full. To take into account this random factor, the classical rent machine is modified. If the current price of liabilities takes into account only the change in the price of money, then the actuarial price also takes into account the likelihood of shortage of part of the contributions by the insurer.

If the probability of occurrence of the insured event is equal to * p * and this probability is distributed evenly (does not depend on the time of the contract), then the insured event in each period may occur with probability * p*

The first installment of the insurance company will receive a probability of 1, since without it the insurance contract will not take effect. Up to the second installment, one period will pass for which the event will occur with probability * p/t * or will not happen with probability 1 - * p/t. * In the latter case, the insurance company will receive a second installment probability 1 - * p/m, *, etc. Thus, based on the principle of the equality of the present value of the premium and the amount of future payments taking into account the risk of the insurer, the equation for finding the periodic premium BPperiod takes the form: * p/t. *

(2.26)

If the law of probability distribution of the occurrence of the insured event was not uniform, but some other character, it would be necessary to calculate the probability of receiving each next payment according to the given distribution law.

Thus, the size of the periodic premium depends on the following factors:

• the size of a one-time premium;

• the discount rate and the type of interest accretion;

• probability of occurrence of an insured event;

• the law of probability distribution of the occurrence of an insured event;

• long-term insurance contract and periodicity of payments.

According to the general rules of insurance, periodic contributions must be paid at the beginning of each period, since without the first installment an insurance contract is not concluded, and the earlier money is deposited, the longer they "work", i.e. bring more income, therefore, the nominally introduced amount decreases. This decrease will be the greater, the higher the interest rate * i. *

Thus, the actuarial price of periodic premiums is more expensive due to the risk that the insurer will not receive a part of the contributions. The lump sum of the premium is always less than the amount of periodic premiums. However, with long-term insurance (for example, pension insurance), the insurer can provide an additional discount, despite the payment by installments. The fact is that the premiums received by the insurer can be invested, due to which additional income can be obtained. And with a long term of insurance, starting from a certain moment of time * t, * the interest rate starts to influence more than

* Fig. 2.10. * ** Comparison in time of nominal contributions and the accrued amount with interest **

the amount of probability of occurrence of the insured event, which affects the risk of non-receipt of contributions (Figure 2.10). This effect underlies the discounts provided to the customer when the insurance period is extended.

** EXAMPLE 2.8 **

In a one-year contract, a one-time risk premium is 30 cu. Find a periodic quarterly risk premium if the probability of occurrence of an insurance event is evenly distributed, and within a year the case may occur with a probability of 0.05. The annual interest rate is 15% with a quarterly interest charge, and the contributions that are not paid by the time of the insured event are not deducted from the refund.

* Solution *

Probabilities of the insured event * p * = 0.05 for one year with a uniform distribution law correspond to the quarterly probability of the insured event, equal to:

* p * kv * = * 0.05/4 = 0.0125.

Thus, only the first installment will receive the company with a probability of 1 (without a first installment the contract is not concluded). Up to the second installment, one quarter will pass, for which an insured event can occur with a probability of 0.0125 or not occur (then the company will receive a second installment) with a probability of * p = * 1 - 0.0125 = 0.9875. Reasoning similarly, we find that the probability of the company receiving each next installment is reduced by 0.0125, and the third installment will receive an insurer with a probability of 0.975, and the fourth installment will be 0.9625.

Consider the second factor - the flow of four payments at the beginning of each quarter, equivalent to a given one-time premium, with a certain interest rate * g. *

*g*= 3.75%, then for a quarterly interest calculation it is necessary to consider for each contribution a discount factor equal to V = 1/(1 + "), raised to the power corresponding to the period in the quarters that has passed since the present moment. The current price of the second periodic contribution will, according to the discounting operation, be less by 1/v times. The current price of the third installment will be less by 1/v2 times, etc.

Thus, proceeding from the principle of the equality of the present value of the premium and the amount of future payments:

Since the insurer must retain the premium providing the principle of equivalence of the parties, we make a balance equation that takes into account all the above factors (2.26):

How do we express the quarterly award:

Substituting the values, we get the value of the quarterly premium:

The total summary risk premium for the year will be:

The difference between the total periodic and lump sum premium will be:

Thus, by paying installments in installments, the customer will pay 2.24 cu per year. (or by 7.5%) more than a one-time payment.

* Response : * a quarterly risk premium with a 15% annual interest rate for a quarterly interest calculation is 8,066 cu, while the annual risk premium is 32,264 cu. , which is 7.5% more than a one-time contribution.

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