You are graduating from college at the end of this semester and have decided to invest 

​$5 comma 0005,000 

at the end of each year into a Roth IRA​ (a retirement investment account that grows tax free and is not taxed when it is​ liquidated) for the next 

4040 

years. If you earn 

44 

percent compounded annually on your investment of 

​$5 comma 0005,000 

at the end of each​ year, how much will you have when you retire in 

4040 

​years? How much will you have if you wait 10 years before beginning to save and only make 

3030 

payments into your retirement​ account?  




We will approach this problem using the **future value of an ordinary annuity** formula, which is commonly used for investments made at regular intervals (e.g., at the end of each year). The formula is:

\[
FV = P \times \frac{{(1 + r)^n - 1}}{r}
\]

Where:
- $ FV $ is the future value of the investment.
- $ P $ is the annual payment (in this case, \$5,000).
- $ r $ is the annual interest rate (4%, or 0.04 as a decimal).
- $ n $ is the number of payments.

### Case 1: Starting immediately and making payments for 40 years
- $ P = 5000 $
- $ r = 0.04 $
- $ n = 40 $

Let's calculate the future value using these values.

### Case 2: Waiting 10 years before starting and making payments for 30 years
In this scenario, the number of payments is reduced to 30 years. So:
- $ P = 5000 $
- $ r = 0.04 $
- $ n = 30 $

We will calculate the future value for both cases. Let's compute the results.### Results:
1. If you start investing \$5,000 annually for 40 years, earning 4% interest compounded annually, you will have approximately **\$475,127.58** at the end of the 40 years.
   
2. If you wait 10 years and only invest for 30 years, you will accumulate approximately **\$280,424.69**.

This shows that starting earlier significantly boosts your retirement savings due to the power of compound interest.

Would you like further details or clarification?

Here are 5 related questions:
1. How does the interest rate affect future investment values?
2. What is the difference between compounding annually and monthly?
3. How would your savings change if the annual contribution were increased?
4. What happens if the interest rate is higher, say 6% instead of 4%?
5. How much would you need to save annually to reach \$1,000,000 in 40 years?

**Tip**: The earlier you start saving, the more time your money has to grow due to compound interest!

You are graduating from college at the end of this semester and have decided to invest $5,000 at the end of each year into a Roth IRA for the next 40 years. If you earn 4 percent compounded annually on your investment of $5,000 at the end of each year, how much will you have when you retire in 40 years? How much will you have if you wait 10 years before beginning to save and only make 30 payments into your retirement account?

This problem explains how to calculate the future value of a Roth IRA investment of $5,000 annually over 40 years with a 4% annual return. Learn how compound interest and time affect your retirement savings. Also, discover how waiting 10 years to start saving impacts your total retirement fund.

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