You want to be able to withdraw $45,000 each year for 30 years. Your account earns 5% interest. How much do you need in your account at the beginning?

This problem involves determining the present value of an annuity. The goal is to find the amount of money you need to deposit initially in order to make withdrawals of $45,000 each year for 30 years, assuming the account earns an interest rate of 5%.

### Step 1: Identify the formula
The present value of an annuity formula is:
\[
PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}
\]

where:
- $ PV $ = Present Value (the amount you need initially)
- $ PMT $ = Annual payment ($45,000)
- $ r $ = Interest rate per period (5% or 0.05)
- $ n $ = Number of periods (30 years)

### Step 2: Substitute the values into the formula
\[
PV = 45,000 \times \frac{1 - (1 + 0.05)^{-30}}{0.05}
\]

### Step 3: Calculate the present value
Let's break down the calculations:
1. Calculate $(1 + 0.05)^{-30}$
2. Subtract that result from 1
3. Divide by 0.05
4. Multiply by $45,000

\[
PV = 45,000 \times \frac{1 - (1.05)^{-30}}{0.05}
\]
\[
PV = 45,000 \times \frac{1 - 0.2314}{0.05}
\]
\[
PV = 45,000 \times \frac{0.7686}{0.05}
\]
\[
PV = 45,000 \times 15.372
\]
\[
PV \approx 691,740
\]

### Answer
You need approximately **$691,740** in your account at the beginning.

Would you like more details on any step or have any questions about this solution?

Here are some related questions you might find interesting:
1. What if the interest rate changes to 6%? How much would you need initially?
2. How does the formula for present value of an annuity differ from the future value formula?
3. What happens to the initial amount if you want to withdraw money for a different number of years?
4. How would you calculate the amount left in the account after 15 years?
5. What if the withdrawals are monthly instead of yearly?

**Tip:** Understanding the present value concept helps in evaluating how much a future cash flow is worth today.

Calculate how much you need to deposit initially to withdraw $45,000 each year for 30 years at a 5% interest rate. This problem involves the present value of an annuity formula, ideal for advanced high school or college-level students. Learn step-by-step calculations and related financial concepts.

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