Find the periodic payments necessary to accumulate the given...

Find the periodic payments necessary to accumulate the given amount in an annuity account. (Assume end-of-period deposits and compounding at the same intervals as deposits.) [HINT: See Quick Example 2.]
$\$ 20,000$ in a fund paying $3 \%$ per year, with monthly payments for 10 years

Key Concepts

Future Value of an Annuity

This concept refers to the total amount accumulated after making a series of equal payments into an interest-bearing account over a specific period. The future value of an annuity takes into account both the individual deposits and the interest earned on these deposits over time, acknowledging that each payment has a different duration to accrue interest. It is central to problems where one wants to know how much will be available at the end of the period given regular contributions.

Periodic Compounding and Interest Rate Conversion

Since interest accrues at specific intervals (e.g., monthly, quarterly), the annual nominal interest rate must be appropriately converted to match the deposit frequency. This ensures that the compounding effect is accurately captured for each period. The periodic interest rate is obtained by dividing the annual rate by the number of compounding intervals per year, which is crucial when aligning the payment schedule with the rate at which interest is credited.

Determining the Number of Compounding Periods

In problems where deposits are made regularly over several years, it is necessary to compute the total number of periods by multiplying the number of years by the number of payments per year. This total period count is used in the formula for the future value of an annuity, ensuring that the growth of each deposit, due to compounding, is correctly summed over the entire duration of the investment.

Solving for the Periodic Payment

To determine the amount of each periodic payment required to achieve a specified future value, one rearranges the future value of an annuity formula. This formula, FV = Payment × ((1 + i)^n - 1)/i, where 'i' is the periodic interest rate and 'n' is the total number of periods, is algebraically manipulated to solve for the Payment. Understanding this step is vital as it connects the theory of time value of money with practical financial planning and decision-making.

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