Assume you are currently planning for retirement. You want to deposit $X amount of money today (year 0), so that when you retire in 35 years from now, you can make annual withdrawals for the subsequent 21 years, with your first withdrawal being at time 36. You want your first withdrawal (in year 36) to be $55,000 and then increase this payment by 2% each year thereafter to compensate for inflation. If your interest rate is 5% APR, how much is $X?

To solve this problem, we need to calculate the present value $ X $ of the deposit required today to fund a series of withdrawals that start 36 years from now and grow with inflation over the following 21 years.

Here's the step-by-step breakdown:

### Step 1: Define the problem
- You want to deposit $ X $ at time 0.
- Withdrawals start in year 36 and continue for 21 years (from year 36 to year 56).
- The first withdrawal is $55,000 at year 36, and each subsequent withdrawal increases by 2% annually.
- The annual interest rate is 5% APR.

### Step 2: Calculate the present value of withdrawals at year 35
Withdrawals form a growing annuity, which starts at year 36. To find its present value at year 35, we use the formula for the present value of a growing annuity:

\[
PV_{35} = C \times \frac{1 - \left(\frac{1+g}{1+r}\right)^n}{r - g}
\]

Where:
- $ C = 55,000 $ (first payment)
- $ r = 0.05 $ (interest rate)
- $ g = 0.02 $ (growth rate)
- $ n = 21 $ (number of withdrawals)

Substituting the values:

\[
PV_{35} = 55,000 \times \frac{1 - \left(\frac{1.02}{1.05}\right)^{21}}{0.05 - 0.02}
\]

### Step 3: Discount the present value to today (year 0)
To find $ X $, the present value at year 0, we discount $ PV_{35} $ back to the present using the formula:

\[
X = PV_{35} \times (1 + r)^{-35}
\]

Substituting the values:

\[
X = PV_{35} \times (1.05)^{-35}
\]

Let's calculate these values step-by-step.The present value of the withdrawals at year 35 is approximately \$835,932.42. To find how much you need to deposit today, the required amount $ X $ is approximately:

\[
X \approx 151,546.43
\]

So, you need to deposit about **\$151,546.43** today to meet your retirement withdrawal goals.

### Follow-up Questions:
1. How does the required deposit change if the interest rate changes?
2. What if the withdrawals grow at a different rate, say 3% instead of 2%?
3. How would the result change if withdrawals started earlier, such as year 30?
4. What would happen if the annual withdrawal amount was fixed instead of growing?
5. How does the number of years of withdrawals affect the required deposit?

**Tip:** Always consider adjusting your investment strategy based on expected inflation and interest rate changes to ensure your retirement plan remains robust!

Learn how to calculate the present value required for retirement planning. This problem involves determining how much to deposit today to fund withdrawals that increase by 2% annually over 21 years, starting 36 years from now, with a 5% interest rate. Ideal for college-level finance students.

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