5. In recent years, with the emergence of non-state pension funds in our country, a comprehensive program for increasing the future pension of citizens has been established. In particular, a number of such funds offer their clients the opportunity to create an individual pension plan. According to this plan, the client can voluntarily and independently choose the amount and frequency of contributions to their individual account. The client's return on such a deposit is determined by the fund's rate of return and does not depend on the amount and frequency of contributions. The corresponding interest is credited annually to the balance of the client's individual pension plan account until the depositor reaches retirement age. Upon reaching retirement age, the accumulated amount on the account at that time is fixed, evenly distributed, and paid out to the client monthly as a supplement to the state pension over a period established in the contract. The conditions for early termination of such a deposit are determined by the fund and are recorded in the contract.

Suppose that your 39-year-old acquaintance, who will turn 40 next month, is considering whether to enter into an individual pension plan agreement with one of these funds starting next month. For simplicity, assume that the fund's return is stable and will be 9% annually for the entire possible term of the agreement. Your acquaintance has calculated that until the onset of the retirement period (age 60), they will be able to contribute 7,000 rubles monthly to their individual pension account in the fund. Determine the monthly supplement to the state pension that your acquaintance will be able to receive for 15 years after reaching retirement age as a result of participating in the individual pension plan program.

You may need to use additional information about the comprehensive program for increasing future pensions provided by one of these funds to answer the question.
# Solution:

Before retirement, over 20 years, the client will deposit $7000 * 12 = 84000$ rubles into their account each year.

After the first year of savings, the amount on their pension account will be $84000 + 84000 * 0.09 = 91560$ rubles.

After 2 years: $91560 + 84000 + (91560 + 84000) * 0.09 = 191360$ rubles, and so on.

Or:

$(\ldots(((7000 * 12)(1+0.09) + (7000 * 12))(1+0.09) + (7000 * 12))(1+0.09) + \cdots) \ldots)$

Thus, after 20 years, the amount on their pension account will be 4684220.554 rubles. In this case, the monthly supplement over 15 years will be $4684220.554 / 15 / 12 \approx 26023.45$ rubles.

## Criteria:

15 points - fully correct and justified solution.

10 points - errors were made in accounting for capitalization.

5 points - incorrect formulas were used to calculate the final amount in the account, provided that the savings procedure is correctly described.

0 points - the savings procedure is incorrectly described.