The formula for calculating the periodic payment (PMT) for an annuity can be expressed as:
\[ PMT = \frac{PV \times r \times (1 + r)^{nt}}{(1 + r)^{nt} - 1} \]
- \( PV \) is the present value or the target amount (in this case, $25,000),
- \( r \) is the periodic interest rate (annual rate divided by the number of compounding periods per year),
- \( n \) is the total number of compounding periods (number of years multiplied by the compounding periods per year), and
- \( t \) is the number of years.
Given the scenario:
- Annual interest rate: 5%
- Quarterly payments: 4 times a year
- Number of years: 20
Let's calculate it:
\[ r = \frac{0.05}{4} \] (quarterly compounding)
\[ n = 4 \times 20 \]
\[ PMT = \frac{25000 \times \frac{0.05}{4} \times \left(1 + \frac{0.05}{4}\right)^{4 \times 20}}{\left(1 + \frac{0.05}{4}\right)^{4 \times 20} - 1} \]
Now, you can compute this value to find the periodic payment (PMT).
Calculating the periodic payment (PMT) using the provided formula:
\[ PMT = \frac{25000 \times \frac{0.05}{4} \times \left(1 + \frac{0.05}{4}\right)^{4 \times 20}}{\left(1 + \frac{0.05}{4}\right)^{4 \times 20} - 1} \]
This results in approximately $285.71. Therefore, the periodic payment (PMT) necessary to accumulate $25,000 in an annuity account under the given conditions is approximately $285.71 per quarter.