# Estimation of the reserve in supplementary pension insurance...

## Estimating the reserve in additional pension insurance

In accordance with the actuarial principles, in general, the insurance reserve is defined as the difference between the present value of payments that the insurer is required to pay and the present value of future contributions of the insured.

## Provision for a one-time contribution

Provided that the net premium is paid at a time, at the time of the contract, the reserve (at the time (x + T) ) is equal to the value of the annuity pension for the same moment. In this case, the total contract term can be divided into two intervals.

In the first - before the beginning of the pension payments (G k) - the reserve is equal to the cost of the annuity annulled for (k - T) . With a lifetime pension, the reserve is determined as follows:

(9.39)

In the second period, a pension is paid.

a) At the time pension payments begin, (T = k) the reserve is equal to the value of the immediate annuity:

(9.40)

b) At the moment after the beginning of pension payments (D & gt; k) we find:

(9.41)

Similarly, we obtain the reserve formula for urgent pensions (the repayment period is n years).

c) For the moment preceding the payment of pensions (T we find:

(9.42)

d) At the moment of commencement of payments of pensions (T = k):

(9.43)

e) At the moment after the beginning of the pension payments (T & gt; k) the remaining period of pension payments is equal to k + n - T. Hence

(9.44)

As mentioned earlier, in practice, contributions are rarely paid at a time, more often - in installments. In this case, the insurer in the reserve must keep the amount in accordance with the volume of the obligations of the insured (contributions paid).

EXAMPLE 9.6

Using the data in the table of switching functions (Appendix 12), determine the amount of the reserve at the time of the commencement of pension payments but the supplementary pension insurance contract for a man of 30 years, provided that the pension is paid from 50 years for life.

Solution

At the time of pension payment, the reserve is equal to the cost of an immediate annuity. Since the pension is paid from 50 years, we have k = 20.

Answer: The amount of the reserve at the moment of commencement of pension payments will be 13.86 (ESS).

The total period of validity of the insurance policy in this case can be divided into several stages. The scheme of the contract is analyzed, under which the pension is paid starting from the age (x + k), the installment of premiums takes t years (i & lt; k) and is carried out at the beginning of the year. Then, in the first stage (x, x + t) net premiums are received and accelerated accumulation of funds occurs. On the second (x + t, x + k) - the amount of savings increases only due to the accretion of interest. On the third (x + k, w), if the pension is lifelong (or (x + q> k, x + k + n), if the pension is urgent), the funds go to pay pensions. Interest is regularly accumulated on the balance of savings. If the contributions are calculated correctly, then at the age of w (or x + k + n) the accumulations should be fully used (the balance undermines competitiveness, and the lack - stability). >

Let's analyze lifetime pensions. The reserve is assessed on the 7th year after the beginning of insurance at the client's age (x + 7). If T & lt; t, i.e. an estimate is made of the period of premium payments, then by definition:

This formula takes into account that at the time of valuation (the valuation is made at the beginning of the year) the premium for this year has already been paid.

## Provision for installment payments

(9.45)

Therefore, the difference in the value of annuities at the beginning of the insurance contract is equal to the first premium contribution, i.e. P x.

For the cases when T & gt; t, we obtain formulas similar to the expressions given above for the estimation of reserves with a one-time premium contribution, namely:

- if t & lt; T & lt; k, then the size of the reserve is determined by the formula (9.39), because the semantic content of the time interval is identical - the contributions have been made in full, and payments have not yet begun;

- for the last period when T & gt; k, the reserve is calculated using the formula (9.41), as the reserve is valued at the time when the insured has fulfilled its obligations, and the insurer has already begun payments.

For lower pensions in this situation, we get:

a) for T & lt; t

(9.46)

b) for t

(9.47)

c) for T & gt; k the amount of the reserve is determined by the same formula (9.44) as for urgent pensions provided a lump-sum contribution. In this case, the analytical meaning of the moment of reserve estimation is absolutely identical - the insurer has received contributions in full and has already begun to make payments in accordance with the terms of the contract.

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