After retiring, Stacy wants to be able to withdraw $7,500.00 every quarter from her account for 34 years. Her account earns 8% interest compounded quarterly.

How much does Stacy need in her account when she retires?


To determine how much Stacy needs in her account when she retires, we can use the **present value of an annuity formula** for withdrawals over a period with compounded interest:

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

where:
- $ PV $ = Present Value (the amount Stacy needs at retirement)
- $ PMT $ = Payment per period = $7,500
- $ i $ = interest rate per period (quarterly interest rate)
- $ n $ = total number of periods (quarters)

### Given values:
1. **Annual Interest Rate** = 8%
   - Since interest is compounded quarterly, we divide this by 4:
     \[
     i = \frac{8\%}{4} = 2\% = 0.02
     \]

2. **Withdrawal Period** = 34 years
   - Since withdrawals are quarterly, we multiply 34 years by 4:
     \[
     n = 34 \times 4 = 136 \text{ quarters}
     \]

3. **Quarterly Withdrawal** = $7,500

### Plugging the values into the formula:

\[
PV = \frac{7500 \times (1 - (1 + 0.02)^{-136})}{0.02}
\]

Calculating each part:

1. **Calculate** $ (1 + 0.02)^{-136} $:
   \[
   (1 + 0.02)^{-136} \approx 0.104
   \]

2. **Substitute into the formula**:
   \[
   PV = \frac{7500 \times (1 - 0.104)}{0.02}
   \]
   \[
   PV = \frac{7500 \times 0.896}{0.02}
   \]
   \[
   PV = \frac{6720}{0.02} = 336,000
   \]

### Answer:
Stacy needs **$336,000** in her account when she retires to be able to withdraw $7,500 every quarter for 34 years at an 8% interest rate compounded quarterly.

Would you like further details or have any questions?

---

1. How does the annuity formula change for monthly withdrawals?
2. What if Stacy wanted to withdraw annually instead of quarterly?
3. How does a change in interest rate impact the required retirement fund?
4. What would happen if Stacy increased her withdrawal amount over time?
5. How does compounding frequency affect the retirement amount?

**Tip:** Compounding frequency (quarterly, monthly) significantly impacts how much you need upfront in a retirement fund due to the way interest accumulates.

After retiring, Stacy wants to be able to withdraw $7,500.00 every quarter from her account for 34 years. Her account earns 8% interest compounded quarterly. How much does Stacy need in her account when she retires?

Determine how much Stacy needs in her retirement account to withdraw $7,500 quarterly for 34 years at an 8% interest rate, compounded quarterly. This example explores the Present Value of Annuity formula, financial planning, and compound interest concepts, ideal for high school financial math students.

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