Find the periodic withdrawals PMT for the given annuity account. (Assume end-of-period withdrawals and compounding at the same intervals as withdrawals. Round your answer to the nearest cent.)
$200,000 at 7%, paid out monthly for 19 years

To find the periodic withdrawals (PMT) for an annuity account, we can use the formula for the present value of an annuity:

\[
PV = \dfrac{PMT \times \left(1 - (1 + r)^{-n}\right)}{r}
\]

Where:
- $ PV $ is the present value (the initial amount in the account, $200,000).
- $ PMT $ is the periodic withdrawal amount we need to find.
- $ r $ is the interest rate per period (monthly in this case).
- $ n $ is the total number of periods (months in this case).

### Given Information:
- $ PV = 200,000 $
- Annual interest rate = 7%, so the monthly interest rate is $ r = \dfrac{7\%}{12} = 0.005833 $ (7% divided by 12 months).
- The time period is 19 years, so the total number of periods $ n = 19 \times 12 = 228 $ months.

Now, we rearrange the formula to solve for $ PMT $:

\[
PMT = \dfrac{PV \times r}{1 - (1 + r)^{-n}}
\]

Substitute the values:

\[
PMT = \dfrac{200,000 \times 0.005833}{1 - (1 + 0.005833)^{-228}}
\]

Now, let's calculate this step by step.

\[
1 + r = 1 + 0.005833 = 1.005833
\]

\[
(1.005833)^{-228} \approx 0.25025
\]

\[
1 - 0.25025 = 0.74975
\]

\[
PMT = \dfrac{200,000 \times 0.005833}{0.74975} \approx \dfrac{1,166.6}{0.74975} \approx 1,556.29
\]

So, the periodic monthly withdrawal (PMT) is approximately **$1,556.29**.

### Would you like a detailed breakdown of any step? Feel free to ask!

Here are some related questions:
1. What happens if the interest rate increases or decreases?
2. How do periodic withdrawals change if the number of years is shortened or extended?
3. How does the compounding frequency affect the periodic withdrawal amount?
4. What is the total amount withdrawn over the 19-year period?
5. What if the withdrawals were at the beginning of the period instead of the end?

**Tip:** If you're uncertain about the future, building a flexible annuity with varying withdrawal options can help adapt to changing needs.

Find the periodic withdrawals PMT for the given annuity account. (Assume end-of-period withdrawals and compounding at the same intervals as withdrawals. Round your answer to the nearest cent.) $200,000 at 7%, paid out monthly for 19 years.

Learn how to calculate the periodic monthly withdrawals for an annuity account with an initial amount of $200,000 at 7% interest, paid out monthly over 19 years. This detailed step-by-step solution covers the annuity formula, time value of money, and compound interest. Suitable for advanced high school and college students.

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