## Features of calculating premiums in life insurance

In life insurance, uncertainty is associated with the random nature of the duration of human life. Therefore, insurers must have information that will allow them to assess the risk of death or survival before a certain period for people of different ages and sex. The main source of this type of data is the * mortality tables *, which are compiled by state statistical agencies with a certain periodicity on the basis of information obtained from the population census. In addition, in some countries, insurers, long time life insurance and having a large amount of data on their clients, create their own mortality tables, which more accurately characterize the death rate among the insured. It is believed that the first summary mathematical tables of mortality were made by the English astronomer Edmund Halley (1656-1742).

The life insurance contract distinguishes four aspects from other insurance contracts:

- as a rule, it is a long-term contract that has a validity period of 5-15 years (urgent) or throughout the lifetime of the insured (lifelong);

- life insurance contract is a contract of insurance of the amount, according to which the insured amount is agreed in advance, because it is not correct to estimate the cost of human life and, accordingly, the harm done to it;

- for life insurance contracts there is no excessive insurance and, accordingly, restrictions on payments. For all agreements concluded by the client, payment of insurance amounts is made upon the occurrence of an insured event. The only restriction for the assignment of the insurance amount is the ability of the customer to pay the corresponding insurance premiums;

- Under the life insurance contract, the insurer can usually estimate the value of the insured event (the insured amount signed in the contract), as well as the probability of occurrence of the insured event, i.e. probability for the client to live or die at a certain age, obtained from the tables of mortality of the population.

* Tariff rate - * is the price of the unit of consequences of the onset of insurance risk and other expenses, an adequate monetary expression of the insurer's obligations under the concluded insurance contract. Tariff rates are determined by actuarial calculations. The aggregate of tariff rates forms an insurance tariff.

The basic definitions and structure of the insurance tariff in life insurance are similar to those that we examined in Ch. 2 for non-life insurance.

* Insurance rate * is the contribution rate from a single insured amount or insurance object. With the help of the tariff rate, the amount of the insurance premium that the policyholder must pay at the conclusion of the insurance contract is determined. For this purpose, the value of the tariff rate is multiplied by the insured amount specified in the contract.

Tariff rate is the basis for determining the share of each insurer's participation in the formation of a monetary fund. Due to this fund, insurance payments must be made, other expenses of the insurer covered and profit earned. Therefore, the main task that is put in calculating the tariff rate is connected with the determination of the probable amount of payments for insured events and other expenses of the insurer per unit of the insured amount or one object of insurance.

If the tariff rates are calculated correctly, then the insurer, through the received insurance premiums, can fully fulfill its obligations, cover its insurance costs and make a profit. The overestimation of tariffs in comparison with the probability of the existing risk does not contribute to the conclusion of insurance contracts with potential insurers, reduces the competitive capabilities of the insurer in the insurance market.

Lowering the tariff rate may result in the insurer not having enough funds to effect insurance payments, and as a result, the damage incurred by insurants or other insurance participants will be compensated for. The latter situation negatively affects the financial position of the insurer and causes distrust of insurance from insurers. Therefore, the insurance supervisory authority establishes control over the reasonableness of the applied rate of the tariff rate and can take strict sanctions for reducing rates by insurers without sufficient reason.

The tariff rate at which the policyholder pays the insurance premium is called the * gross rate. * The gross rate is greater than the net rate by the amount of the load, which compensates for all costs of the business, and, as a rule, provides the insurer normative profit. In pension insurance, the costs of running a business are the largest, in comparison with other types of personal insurance, as they cover not only one-off, but also some periodic expenses related to the receipt of premiums paid in installments and with pensions.

The gross rate (as discussed in Chapter 2) is the insurer's rate, which can be conditionally represented in the form

where * f * is the load expressed in fractions of the gross rate.

Then

* Net-rate * is intended for the formation of a monetary fund from which insurance payments are made.

* The load * is used to cover the costs of the insurer for carrying out insurance operations. Such expenses include the payment of labor of the employees of the insurance organization, the costs of making insurance documents (applications, policies, acts, etc.), advertising, business expenses (rent of premises, payment for utilities, etc.), etc. In The load can also include the profit that the insurer provides for from insurance activities. In the structure of the gross rate, the main is the net rate, which accounts for 60-95%, depending on the type of insurance, and the load is respectively 5-40%.

The basic principle, from which the net rate is determined, is the * principle of equivalence of obligations * of the insured and the insurer. In general, this principle is realized by equating the net premium received by the insurer (the policyholder's obligation), the generalized amount of pensions paid (the insurer's obligation).

In life insurance * there is no risk premium (PH) *, as there is insurance for survival (before the due time, payments are not made). In pension insurance, this element (PH) appears, because it is not known in advance how many times you will have to pay a pension (how many years the client will survive after reaching a certain age). Usually PH is taken in proportion to the * risk premium (PH), * so it's enough to calculate the RP. Then the net premium is determined, then the gross premium. Moreover, the calculation of a net premium does not have an independent meaning, so you can immediately "fix" coefficient and the RP to find a gross premium.

The share of the load in the gross rate is determined, as a rule, but the accounting data for the previous reporting periods. However, since the main component of the load is a commission, its magnitude will be significantly influenced by * the way the contract is concluded. * Therefore, if other sales opportunities are provided for the contract being developed, this fact must be taken into account when choosing the size of the load.

At the time of calculating net rates, the insurer can not say for sure what percentage it will be able to invest insurance reserves, so the calculation of tariff rates applies the * planned * rate of return. In some countries, * the minimum guaranteed interest rate that the insurer must provide is established by state insurance supervision bodies.*

Peculiarities of calculation of life insurance tariff rates are that the formation of a reserve of contributions and calculation of tariff rates are made using actuarial methods, based on mortality tables and rates of return on investment of temporarily spare life insurance reserves.

In life insurance, tariff rates are determined by different age categories and depending on the sex of the insured. Insurance companies usually offer policyholders the choice of several possible insurance terms and options for paying premiums. Therefore, the rates will be determined separately for each term of insurance and the order of payment. The tariffs calculated in this way are reduced to tables, which are then used by the insurer in practical work when concluding insurance contracts.

The amount of the * net rate * of the life insurance premium is calculated according to the following factors:

• probability of occurrence of an insured event;

• the age and gender of the insured at the time of entry into force of the agreement;

• terms of the contract: the type of contract; amount and term of payment of insurance coverage; term and period of payment of insurance premiums; term of the insurance contract; of the planned rate of return.

Based on the * net rate *, insurance premiums (premiums) are calculated that can be:

- * one-off. * The policyholder pays the insurance fee immediately for the entire insurance period forward. Its amount is determined by the time the insurance contract is concluded;

- * annual. * In personal insurance, there are urgent and life-long annual insurance premiums. Urgent - these are those that are paid within a certain period of time (for how many years the contract is concluded), life is paid annually, while the insured is alive;

- * Deferred * - for example, monthly or quarterly payments.

One-time premiums can be replaced by annual ones taking into account the economic possibilities of the insured. In turn, the annual fee can be replaced by a monthly, quarterly, semi-annual. Calculations, for example, of monthly contributions are made on a lump-sum basis, rather than on an annual basis. In practice, monthly installments are more often used. Contributions are paid at the beginning of the period.

According to the theory of actuarial calculations, the nominal amount of installments (for example, monthly) of insurance contributions is always greater than the annual contribution (due to loss of profit by the insurer under these contracts). This is the * modern * price commitment. In addition, the insurer takes into account the * risk of shortage of part of the contributions * due to the occurrence of the insured event. There is an * actuarial * cost.

In the formulas, this circumstance is taken into account with the help of the corresponding installment factors (annuities). The calculation is based on a scheme with payment of compensation at the end of the year of death of the insured person. Transition to other points of payment of insurance coverage is carried out by multiplying the main net premium by the appropriate correction factor.

The principle of transition from annual contributions to installments (for example, quarterly, * т * = 4, or monthly, * т = * 12) relies on the assumption of the distribution of mortality during the year under study life. If we start from the * uniform * distribution, then we can use the general approach.

The probability of paying the next installment is:

The change in the price of money is accounted for by the coefficients:

therefore the installment factor is equal to the sum of pairwise products:

It is necessary to divide the annual contribution by this coefficient.

If the researcher intends to use another assumption (constant intensity or Balducci condition), it is necessary to modify the probability formula accordingly. Note that the installment factor is always less than * m, * since it contains exactly * t * items, of which only one equals 1, and all the others are less than 1. Therefore, the symbol * a/m * means only the fact of payment of contributions * t * once a year, and * not the operation of dividing * annual contribution into * t * equal parts.

More significant is the fact that if the term of the agreement is equal to several years, then it is not entirely correct to perform the transition from year to quarter * separately for each year. * It should be noted that each year corresponds to its value q x, which increases with increasing * x. * Therefore, the built-in separate the installment rates will become * unequal *, which is unacceptable in insurance practice. In addition, contributions (for death insurance) will increase with the age of the insured. Thus, a pensioner will not be able to pay for a policy when the need for insurance coverage is maximum! To ensure that contributions are the same, you need to determine the quarterly contribution in the same way as the annual fee, i.e. based on the * one-time fee *, using the corresponding installment factor.

Analyzing not one year in isolation, but the whole term of the contract, the actuary ensures the equivalence of the parties' obligations for the whole term, and not every year separately. First there is the probability of surviving until the expiration of the contract:

Then the probability of the opposite event is determined, * q * x, which plays the role of * q * v finally, taking into account the new number of periods: * N = tm * and the rate of return * i * is determined by: (1 + r) -1/ ', and the installment factor is calculated.

An alternative is to refuse a uniform change in the probability of survival until the next payment is made. For large * t * the assumption of a uniform distribution of mortality is not entirely correct. The hypothesis of constant intensity is more preferable. Then for each year its own step is defined: * qjm * - the change in the probability of getting another installment. But in practice - it is not important.

In the study of life insurance for death, the actuary is guided by the expected duration of the life to come. And here there is a risk caused by the fact that the client will die too soon, before paying all the contributions, and the insurer will have to fulfill its obligations in full. This determines the practice of including a wait-and-see period in the contract, the modification of the contract so that the client pays all the contributions (or most of them) in the initial period of the contract, etc.

In the contract for a clean survival of the opposite effect due to the increase of * q * x does not arise, because in this policy the insurer's risk is absent (the client can claim payment only if he survives to the specified age or a term, which means that he will necessarily pay all contributions). Therefore, here a risk premium is generally needed. More precisely, it is designed to compensate for negative (for the insurer) changes in the behavior of the survival curve and the interest rate. But the installment factor in this contract is determined on the basis of a one-time contribution.

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