# Deterministic fixed rents, deferred for years - Actuarial calculations

## Deterministic fixed rents, deferred for years

Rent payments discussed above began on the first interval (0, 1) (at the beginning of it, ie at the moment of 0 = 0, for anticipatory rent and at the end, ie at the moment & pound ;, = 1, for delayed rent). For practical use, the so-called deferred annuities are also important. To determine them, consider consecutive unit time intervals (0,1), (1,2), ..., (off- 1, from), ..., (from + n - 1, from + and). As before, under the moment t 0 = 0 we mean the present moment.

A series of n payments, each a value of 1, made at the end of the intervals (from, from + 1), ..., (from + η - 1, from + i), i.e. at moments from +1, ..., from + n is called (Figure 6.5). Its value at the moment is 0 = 0 is denoted by aul. To calculate this value, we give each of the n payments at the moments from + 1, ..., > m + n to the initial instant of time, and then add the values ​​obtained: (6.29)

Similarly, a series of n payments, each a value of 1, made at the beginning of the intervals (from, from + 1), ..., (m + n - 1, from + and), i.e. at moments from, ..., m + n - 1, is called deferred annuity due. Its value at the moment is 0 = 0 is denoted by m dt. To calculate this value, we give each of the n payments at the moments from, ..., m + n - 1 k the present moment of time, and then add the obtained values: (6.30) Fig. 6.5. The reduced (current) value of the deferred annuity (postnumbero) period

Equations (6.29) and (6.30) can be transformed in a certain way so as to express deferred rents through ordinary. First, taking the common factor ν 'in the right-hand sides of (6.29) and (6.30) for the bracket, we obtain: (6.31) (6.32)

These formulas are easy to interpret and easy to understand. They show that in order to estimate at the moment 0 a series of deferred payments on the intervals (te, te + 1), ..., (te + η - 1 , you can first evaluate this series at the time te (since this is the starting point of the first interval of the series, we are talking about the usual rent or , and then bring the result back to those units of time to the left, i.e., to time 0.

In addition, supplementing the given sums (6.29) and (6.30) with the initial summands up to sums of the form (6.19) and (6.20), we obtain: (6.33)

Similarly for pre-emerge rent: (6.34)

For delayed rents special designations for savings by the end of the payment period are not needed, since from the point of view of the last period of time, deferred rent does not differ from the corresponding ordinary one. In other words, let's say that we decide to designate by the value of delayed delayed rent at the end of the last payment period (i.e., at time t + n ). Assuming n single payments at the moments m + 1, ..., m + n the moment m + n, we would have

EXAMPLE 6.8

A new participant in a non-state pension fund that has just entered into a pension agreement will acquire the right to receive a non-state pension after 7 years. He would like to:

• make a one-time pension contribution at the time of the conclusion of the pension agreement;

• Receive a pension once a year for 12 years;

• The size of the pension should take into account inflation and be equal to 5000 USD. in current prices.

It is assumed that over the next 6 years, inflation will be almost zero, and then will be about 1.2% per year.

The Fund places pension reserves and provides the fund's participants with an investment income of i = 6.3% per annum.

Determine the size of a one-time pension contribution.

Solution

Taking into account inflation, the nominal size of the n retirement benefit (n = 1, ..., 12) will be 5000 • 1.012 \$, and this payment will be made will be at the moment 6 + n (we accept the moment of the conclusion of the contract as the initial, and one year - as a unit of time). The current cost of this series of 12 payments is equal to

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