Find the periodic withdrawals PMT for the given annuity account. HINT [See Quick Example 4.] (Assume end-of-period withdrawals and compounding at the same intervals as withdrawals. Round your answer to the nearest cent.)
$55,000 at 4%, paid out quarterly for 20 years
PMT = $ 

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

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

Where:
- $PV$ is the present value of the annuity (the amount available to withdraw, i.e., $55,000),
- $PMT$ is the periodic withdrawal (which we are solving for),
- $r$ is the interest rate per period (quarterly in this case),
- $n$ is the number of compounding periods per year,
- $t$ is the number of years.

### Step 1: Define the known values
- $PV = 55,000$
- Annual interest rate = 4% → quarterly interest rate $r = \frac{4\%}{4} = 0.01$ (since the interest is compounded quarterly)
- $n = 4$ (quarters per year)
- $t = 20$ years

### Step 2: Use the formula
Now we solve for $PMT$:

\[
55,000 = PMT \times \left( \frac{1 - (1 + 0.01)^{-4 \times 20}}{0.01} \right)
\]

\[
55,000 = PMT \times \left( \frac{1 - (1.01)^{-80}}{0.01} \right)
\]

### Step 3: Calculate the factor
- First, calculate $(1.01)^{-80} \approx 0.44936$,
- Then, $1 - 0.44936 = 0.55064$,
- Now, divide by 0.01: $\frac{0.55064}{0.01} = 55.064$.

### Step 4: Solve for PMT
\[
55,000 = PMT \times 55.064
\]

\[
PMT = \frac{55,000}{55.064} \approx 998.84
\]

### Final Answer:
The periodic withdrawal (PMT) is approximately **$998.84**.

Would you like further details on any part of the process? 

Here are 5 related questions to deepen your understanding:
1. How does compounding frequency affect the periodic withdrawal?
2. What happens if the interest rate changes over time?
3. How would the PMT change if the withdrawal was monthly instead of quarterly?
4. Can this formula be used for withdrawals at the beginning of each period?
5. How do we handle inflation in real-life annuities?

**Tip:** Increasing the number of compounding periods (e.g., monthly vs. quarterly) results in a lower interest rate per period but increases the total number of periods, which can slightly alter the total withdrawals.

Find the periodic withdrawals PMT for the given annuity account. HINT [See Quick Example 4.] (Assume end-of-period withdrawals and compounding at the same intervals as withdrawals. Round your answer to the nearest cent.) $55,000 at 4%, paid out quarterly for 20 years. PMT = $

Learn how to calculate the periodic withdrawals (PMT) for an annuity account with $55,000 at 4%, paid out quarterly for 20 years. This step-by-step guide covers key concepts like the present value of an annuity and compound interest, suitable for advanced high school and college students.

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