## Pension scheme with return of paid contributions in case of death at the pre-retirement age

In this case, along with the fund intended for the implementation of pension payments, it is necessary to create a corresponding fund for the return of contributions paid in the event of the death of the insured person before the retirement age. The calculations were carried out on the assumption that there was no waiting period: * x + k = p, * where * p * is the retirement age.

In case the return of the contributions paid up to the moment of death is carried out * at the end of the insurance year, * the expected current value of this fund at the time of signing the insurance contract is:

(9.25)

where - the amount of annual contributions paid;

- the expected current cost of standard insurance * in case of death * with an increasing sum insured, which is defined by the formula:

(9.26)

* Standard Increasing Life Insurance * means an annual increase in the insured amount per unit, compared to the single insurance amount of the first year of insurance.

The amount of the net contribution for a single annual pension is determined from the condition of equality of current costs of contributions and payments:

(9.27)

The left side of equation (9.27) characterizes the contributions, the right side - payments. In this agreement, the insurer is provided with payment of installments with installments in * (p * -. Y) years, therefore on the left side of the equation there is a value of an urgent annuity with a period * (p - x) * years . Obligations of the insurer are formed from two terms: the first takes into account the life insurance (annuity, deferred to * (p - x) * years), the second - death insurance with an increasing insured amount with payment at the end of the year.

Solving the equation for , we get:

(9.28) In practice, the insurance contract usually provides for payment * immediately after the fact of death * of the insured. Therefore, in calculating the current value of insurance payments in actuarial mathematics, the factors are:

a) - if payments are made immediately after death;

b) - if payments occur within the year of death.

To solve the task, it is also necessary to determine the expected number of deaths during the year. Interpolation of mortality tables in the interval * (η, n + * 1) can be carried out under the assumption of one of three hypotheses:

* Linear interpolation. * If the lifetime * T * x * - * any (not necessarily an integer) number in the range ,

then it can be represented in the form

the whole part, - the fractional part of the age, at 0 & lt; * t < 1. *

According to the hypothesis

which leads * to a uniform distribution * of the moments of death within each year-old age interval. Under this assumption, *, p x is a linear function. *

* Demonstration interpolation. * Survival function * s (x) * on the segment approximates an exponential function

This interpolation corresponds to the assumption of * the constant mortality rate * between the two nodes.

* The Balducci condition * (hyperbolicity) implies interpolation by linear functions

In the event that the contributions are refunded directly * after the insured's death *, the expected current value of this fund is equal to the value determined by the formula (9.25) multiplied by :

** EXAMPLE 9.4 **

Determine the amount of the annual contribution of pension insurance for a man of 55 years, subject to the return of contributions in the event of death at the dopentsyiim age (retirement at 60 years).

* Solution *

If the insurance contributions are paid back * at the end of the insurance year, * the net contribution for a single annual pension will be calculated using the formula

In our example, we use the data of the switching tables of P8,119 applications. We get:

If the insurance contributions are paid back * immediately after death, * the net contribution for a single annual pension will be calculated using the formula

where * i * is the nominal annual interest rate.

Taking into account that * i = 5%, * we get * Р ~ * = 1,98094.

* Response *: the amount of the annual fee is 1,97864 (refund of contributions at the end of the insurance year); 1,98094 (refund of contributions immediately after death).

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