John won a lottery. After taxes, he was able to take home his winnings worth $450000. He decides to deposit 15% of this in a separate savings account for retirement. If the savings account has a nominal interest rate of 8%, compounded monthly, how much will be in the account if he retires in 38 years?

To determine how much John will have in his savings account when he retires in 38 years, we can use the **Future Value of a Lump Sum** formula, since he's making a one-time deposit and letting it grow with compound interest.

The formula for the future value of a lump sum with compound interest is:

\[
FV = PV \times (1 + r)^n
\]

Where:
- $ PV $ is the present value or the initial deposit.
- $ r $ is the monthly interest rate (annual nominal rate divided by 12).
- $ n $ is the total number of compounding periods (number of years multiplied by 12).
- $ FV $ is the future value of the investment.

### Step 1: Calculate the initial deposit
John deposits 15% of his winnings into the account:
\[
PV = 0.15 \times 450000 = 67500 \text{ dollars}
\]

### Step 2: Determine the monthly interest rate
The nominal annual interest rate is 8%, so the monthly interest rate $ r $ is:
\[
r = \frac{8\%}{12} = \frac{0.08}{12} = 0.0066667
\]

### Step 3: Determine the total number of compounding periods
John will save for 38 years:
\[
n = 38 \times 12 = 456 \text{ months}
\]

### Step 4: Plug values into the formula
\[
FV = 67500 \times (1 + 0.0066667)^{456}
\]

Let’s calculate this.John will have approximately **$1,396,939.54** in his savings account when he retires after 38 years.

Would you like more details or have any questions?

### Related Questions:
1. What if John decides to save 20% of his lottery winnings instead of 15%?
2. How would the future value change if the interest rate was 6% instead of 8%?
3. What if John made an additional annual contribution to the account? How much would he have then?
4. How does the compounding frequency affect the future value of his account?
5. What if John decides to withdraw some of the money after 20 years? How much would remain?
6. How would inflation impact the real value of his savings in 38 years?
7. What if John deposited the entire $450,000 instead of just 15%? How much would it grow to?
8. What are the tax implications on the interest earned over these 38 years?

**Tip:** Even a small increase in the initial deposit can have a significant impact over the long term due to the power of compound interest.

Learn how to calculate the future value of savings with compound interest using a real-life scenario. John deposits 15% of his lottery winnings into a savings account with an 8% annual interest rate compounded monthly. After 38 years, find out how much he will have saved. Explore variations with different initial deposits, interest rates, and contribution frequencies.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation