How much must be invested today to make four annual withdrawals of $20,000 each for tuition payments if you can earn 8% compounded annually on your investment and the first withdrawal will take place in one year?

To determine how much must be invested today to make four annual withdrawals of $20,000, earning 8% compounded annually, we can treat this as a present value of an ordinary annuity problem. The formula for the present value (PV) of an ordinary annuity is:

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

Where:
- $ C $ = the annual withdrawal ($20,000)
- $ r $ = the annual interest rate (8% or 0.08)
- $ n $ = the number of years (4)

### Step 1: Input the values into the formula
\[
PV = 20,000 \times \left(1 - (1 + 0.08)^{-4}\right) / 0.08
\]

### Step 2: Solve for $ PV $
First, calculate $ (1 + 0.08)^{-4} $:

\[
(1 + 0.08)^{-4} = 1.08^{-4}
\]

Now we will compute the rest of the expression.

Let me calculate that.The amount that must be invested today to make four annual withdrawals of $20,000, with an 8% compounded annually, is approximately **$66,242.54**.

Would you like further details or have any questions?

Here are 5 related questions to explore this concept further:
1. How would the required investment change if the interest rate were 6% instead of 8%?
2. What if the number of withdrawals were 5 instead of 4?
3. How would the result differ if withdrawals occurred at the beginning of each year instead of the end?
4. How does the concept of the present value of annuities apply to retirement planning?
5. What is the impact of more frequent compounding (e.g., semi-annual or quarterly) on the investment required?

**Tip**: When planning for future withdrawals, understanding the present value of annuities helps determine how much you need to invest today!

Find out how much you need to invest today to make four annual $20,000 withdrawals with an 8% annual interest rate. This problem explores the present value of annuities, compound interest, and the time value of money. Suitable for advanced high school or undergraduate finance students.

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