How much do you  need   in your  account today  if you expect to  make quarterly withdrawals of $10,900 for 11 years and also make a special withdrawal of $22,400 in 11 years. The expected return for the account is 3.17 percent per quarter and the first regular withdrawal will be made today.

$______ Round your answer to the nearest dollar

This is a problem involving the calculation of the present value of an annuity (the quarterly withdrawals) and a lump-sum amount (the special withdrawal at the end of 11 years), given that the first withdrawal is made today. Here's how to solve it:

### Step-by-step approach:

1. **First regular withdrawal made today**: Since the first withdrawal is made today, we need to adjust the annuity formula for the immediate withdrawal.

2. **Quarterly withdrawals**: You are making quarterly withdrawals of $10,900 for 11 years. This results in a total of $ 11 \times 4 = 44 $ withdrawals.

3. **Interest rate per quarter**: The interest rate given is 3.17% per quarter, or $ r = 0.0317 $.

4. **Special withdrawal**: A special withdrawal of $22,400 is made at the end of 11 years, i.e., after 44 quarters.

5. **Present value of an immediate annuity**: The formula for the present value (PV) of an annuity due (where the first payment is made immediately) is:
   \[
   PV_{\text{annuity}} = C \times \left[1 + \frac{1 - (1 + r)^{-n}}{r} \right]
   \]
   Where:
   - $ C = 10,900 $ (the withdrawal amount),
   - $ r = 0.0317 $ (the quarterly interest rate),
   - $ n = 44 $ (the number of withdrawals).

6. **Present value of the special withdrawal**: The present value of the special withdrawal of $22,400 in 11 years is:
   \[
   PV_{\text{special}} = \frac{22,400}{(1 + r)^{44}}
   \]
   This discounts the $22,400 back to today's value using the same quarterly interest rate.

Now, let's compute the values.The amount you need in your account today to make the quarterly withdrawals of $10,900 for 11 years, with a special withdrawal of $22,400 in 11 years, and an expected return of 3.17% per quarter is **$273,323**.

Would you like more details, or have any further questions?

Here are 5 related questions to explore:
1. How would the required initial amount change if the interest rate was 4% per quarter?
2. What is the future value of the account after all withdrawals, if no special withdrawal was made?
3. How much would you need today if the special withdrawal were $30,000 instead of $22,400?
4. How does the timing of the first withdrawal (immediate vs. delayed) affect the initial amount required?
5. What if you increase the withdrawal amount by 5% each year?

**Tip**: Always remember to consider the impact of interest rates on both annuities and lump-sum payments when calculating present values.

How much do you need in your account today if you expect to make quarterly withdrawals of $10,900 for 11 years and also make a special withdrawal of $22,400 in 11 years? The expected return for the account is 3.17 percent per quarter and the first regular withdrawal will be made today.

Learn how to determine the amount needed today for quarterly withdrawals of $10,900 over 11 years, with a special withdrawal of $22,400 at the end, using a 3.17% quarterly return rate. This solution covers present value calculations for annuities and lump sums, ideal for students in Grades 11-12 or college-level courses.

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