You plan to retire in 8 years with $X. You plan to withdraw $147,200 per year for 17 years. The expected return is 17.92 percent per year and the first regular withdrawal is expected in 8 years. What is X? Round to the nearest dollar for your answer.

To solve for $X$, we need to compute the present value of the withdrawals given the return rate of 17.92% and then discount that amount back to the present time. This is essentially solving for the present value of an annuity problem, adjusted for the delay before the withdrawals begin.

### Step-by-Step Breakdown:

1. **Withdrawal Details:**
   - Annual withdrawal: $147,200
   - Number of withdrawals: 17 years
   - First withdrawal starts in 8 years.

2. **Interest/Return Rate:**
   - Annual return rate: 17.92% or 0.1792.

3. **Calculate Present Value of Withdrawals (at Year 8):**
   The present value of an annuity formula is used to calculate the value of the withdrawals in year 8 (just before the withdrawals begin):

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

   Where:
   - $ C = 147,200 $ (annual withdrawal)
   - $ r = 0.1792 $ (annual return rate)
   - $ n = 17 $ (number of withdrawals)

4. **Discounting the Present Value to Today (Year 0):**
   Once we have the present value of the withdrawals in year 8, we discount it back to today using the following formula:

   \[
   X = \frac{PV}{(1 + r)^t}
   \]

   Where:
   - $ PV $ is the present value of the annuity from step 3.
   - $ r = 0.1792 $ (annual return rate)
   - $ t = 8 $ (years before the withdrawals start)

Let's calculate this step by step.The amount you need to have at retirement, $X$, is approximately **$206,389**.

Would you like more details on any part of the calculation?

Here are 5 related questions to expand your understanding:
1. How would the required $X$ change if the withdrawal amount was increased?
2. What happens to the result if the return rate is lower than 17.92%?
3. Can this method be adapted for varying withdrawal amounts?
4. How does the time until the first withdrawal impacts the required retirement amount?
5. What is the impact of inflation on the real value of the withdrawals?

**Tip:** Always reassess your retirement needs if market conditions change drastically, especially if your expected return rate drops.

Learn how to calculate the required retirement savings (X) if you plan to retire in 8 years, withdraw $147,200 per year for 17 years, and expect a return rate of 17.92%. This step-by-step guide covers the present value of annuities and discounting cash flows, suitable for advanced high school or undergraduate finance students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation