You expect to have the given amount in an account with the given terms. Find how much you can withdraw periodically in order to make the account last the specified amount of time. Round your answer to the nearest cent.



Account balance:

$450,000

Interest rate:

3.6%

Frequency

quarterly

Time:

21 years

To calculate the periodic withdrawal amount from the account, we can use the formula for the **Present Value of an Annuity** (PVA). Since the interest is compounded quarterly, we'll adjust the interest rate and number of periods accordingly.

### Formula:
\[
PVA = W \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
Where:
- $PVA$ = Present Value of Annuity (the initial account balance)
- $W$ = periodic withdrawal amount (what we're solving for)
- $r$ = interest rate per period
- $n$ = total number of periods

### Given:
- Present Value $PVA = 450,000$
- Annual interest rate = 3.6% (so quarterly rate $r = \frac{3.6\%}{4} = 0.009$)
- Time = 21 years (so number of quarters $n = 21 \times 4 = 84$)

### Now we rearrange the formula to solve for $W$ (the withdrawal amount):
\[
W = \frac{PVA \times r}{1 - (1 + r)^{-n}}
\]

Substituting the given values:

\[
W = \frac{450,000 \times 0.009}{1 - (1 + 0.009)^{-84}}
\]

Let's calculate this.The periodic withdrawal amount that can be made from the account is approximately **$7,657.86** per quarter, in order to deplete the account in 21 years.

Would you like further details on the calculations or any clarification?

Here are 5 related questions to expand on this topic:
1. How would the withdrawal amount change if the interest rate were 5% instead of 3.6%?
2. How would monthly compounding affect the periodic withdrawal amount?
3. What happens if you want the account to last 30 years instead of 21?
4. How would a higher initial balance impact the periodic withdrawals?
5. Can this formula be used for varying withdrawal frequencies like monthly or annually?

**Tip:** When planning withdrawals from an account, it's important to factor in both the interest rate and the frequency of compounding to ensure the funds last as long as intended.

You expect to have the given amount in an account with the given terms. Find how much you can withdraw periodically in order to make the account last the specified amount of time. Round your answer to the nearest cent.

Account balance: $450,000
Interest rate: 3.6%
Frequency: quarterly
Time: 21 years

Learn how to calculate the periodic withdrawal amount from an account with a $450,000 balance, a 3.6% annual interest rate compounded quarterly, and a time period of 21 years. This problem involves Present Value of Annuity and compound interest formulas, and provides a step-by-step solution. Suitable for advanced high school or college 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