youd like to withdrawal $65000 per year for 11 years. the first withdrawal will be in 20 years from today. how much to invest today if your return is 10% per year?

This problem involves determining the present value (amount to invest today) of a series of future withdrawals, each happening annually for 11 years starting 20 years from today, at a 10% interest rate.

### Step-by-Step Solution:

1. **Identify Variables:**
   - Annual withdrawal ($ W $) = $65,000
   - Number of withdrawals ($ n $) = 11
   - Interest rate ($ r $) = 10% = 0.10
   - First withdrawal occurs in 20 years from today.

2. **Calculate the present value of the withdrawals:**
   To find how much to invest today, we need to compute the present value of these future withdrawals. This involves two main steps:
   - First, calculate the value of the withdrawals at the time the first withdrawal occurs (i.e., 20 years from now).
   - Then, discount that value back to the present (i.e., today).

3. **Present value of the withdrawals at the time of the first withdrawal (in 20 years):**
   The withdrawals are an annuity that lasts for 11 years, with an annual payment of $65,000 and a discount rate of 10%. The formula for the present value of an annuity is:

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

   Substituting the given values:

   \[
   PV = 65,000 \times \left( 1 - (1 + 0.10)^{-11} \right) / 0.10
   \]

   Let’s compute this first.

### Step 1: Calculate the present value of the annuity in year 20.The present value of the 11 withdrawals at the time of the first withdrawal (20 years from now) is approximately $422,178.97.

### Step 2: Discount this value back to today:
Now, we need to discount this amount back to today (i.e., 20 years earlier) using the formula for the present value of a lump sum:

\[
PV_{\text{today}} = \frac{PV_{\text{annuity}}}{(1 + r)^{20}}
\]

Substituting the values:

\[
PV_{\text{today}} = \frac{422,178.97}{(1 + 0.10)^{20}}
\]

Let's calculate that.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, if you'd like, I can guide you through the next calculation manually:

To discount the future value back to today, you use this formula:
\[
PV_{\text{today}} = \frac{422,178.97}{(1 + 0.10)^{20}}
\]
First, calculate $(1 + 0.10)^{20}$, then divide $422,178.97 by that result.

Let me know if you'd like to go through that!

Do you have any further questions, or want more details on any part of the calculation?

Here are 5 related questions:
1. How does the choice of a different interest rate affect the present value?
2. What happens if you want to withdraw more or less per year?
3. How does starting withdrawals earlier or later impact the required investment?
4. What is the difference between annuity due and ordinary annuity?
5. Can this same approach be applied if the withdrawal amounts change each year?

**Tip:** An annuity can simplify planning for long-term withdrawals, allowing you to estimate exact amounts to invest based on time and interest rates.

You'd like to withdraw $65,000 per year for 11 years. The first withdrawal will be in 20 years from today. How much to invest today if your return is 10% per year?

Discover how much you need to invest today to withdraw $65,000 annually for 11 years, starting 20 years from now, with a 10% return. This step-by-step guide explains the key concepts of present value, annuities, and discounting. Ideal for finance students and individuals planning long-term investments.

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