You are graduating from college at the end of this semester and have decided to invest ​$3 comma 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 30 years. If you earn 5 percent compounded annually on your investment of ​$3 comma 000 at the end of each​ year, how much will you have when you retire in 30 ​years? How much will you have if you wait 10 years before beginning to save and only make 20 payments into your retirement​ account?
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Part 1
Click on the table icon to view the FVIFA​ table: LOADING....
When you retire in 30 ​years, you will have ​$

enter your response here.  ​(Round to the nearest​ dollar.)
Part 2
If you wait 10 years before beginning to save and only make 20 payments into your retirement​ account, when you retire you will have ​$

enter your response here.  ​(Round to the nearest​ dollar.)

This problem involves calculating the future value of an annuity, which can be done using the future value of an ordinary annuity formula:

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

Where:
- $FV$ is the future value of the annuity.
- $P$ is the annual payment (in this case, $3,000).
- $r$ is the interest rate per period (5% or 0.05).
- $n$ is the number of payments.

### **Part 1: Investing for 30 years**
In this case:
- $P = 3,000$
- $r = 0.05$
- $n = 30$

Now we will calculate the future value:

\[
FV_{30} = 3,000 \times \left(\frac{(1 + 0.05)^{30} - 1}{0.05}\right)
\]

### **Part 2: Delaying for 10 years (20 payments)**
If you wait 10 years to start investing, you will only make 20 payments over the 30-year period. Thus, the same formula applies, but $n = 20$.

\[
FV_{20} = 3,000 \times \left(\frac{(1 + 0.05)^{20} - 1}{0.05}\right)
\]

Let me calculate these values for you.It looks like there was an issue with the calculation environment. Let me walk you through the steps so you can perform these calculations manually or using a calculator.

### **Part 1: Investing for 30 years**

Use the future value of an annuity formula:
\[
FV_{30} = 3,000 \times \left(\frac{(1 + 0.05)^{30} - 1}{0.05}\right)
\]

Breaking it down:
1. Calculate $1 + 0.05 = 1.05$.
2. Raise $1.05$ to the power of 30: $1.05^{30} \approx 4.3219$.
3. Subtract 1: $4.3219 - 1 = 3.3219$.
4. Divide by 0.05: $\frac{3.3219}{0.05} = 66.4379$.
5. Multiply by 3,000: $3,000 \times 66.4379 = 199,313.71$.

So, the future value after 30 years is approximately **$199,314**.

### **Part 2: Delaying for 10 years (20 payments)**

Similarly, for 20 payments:
\[
FV_{20} = 3,000 \times \left(\frac{(1 + 0.05)^{20} - 1}{0.05}\right)
\]

Steps:
1. $1 + 0.05 = 1.05$.
2. Raise $1.05$ to the power of 20: $1.05^{20} \approx 2.6533$.
3. Subtract 1: $2.6533 - 1 = 1.6533$.
4. Divide by 0.05: $\frac{1.6533}{0.05} = 33.066$.
5. Multiply by 3,000: $3,000 \times 33.066 = 99,198.58$.

So, if you wait 10 years, you will have approximately **$99,199**.

### Summary:
- After 30 years, you'll have **$199,314**.
- If you delay 10 years, you'll have **$99,199**.

Would you like further clarification or have any other questions?

Here are five questions to deepen your understanding:
1. What factors affect the growth of an investment in a Roth IRA?
2. How does delaying investment impact the overall returns?
3. Can you use a different interest rate for retirement projections?
4. How would monthly payments instead of yearly change the final outcome?
5. What are the tax advantages of a Roth IRA versus other accounts?

**Tip**: Starting early with investments allows compounding to work over a longer period, significantly boosting your retirement savings.

You are graduating from college at the end of this semester and have decided to invest ​$3,000 at the end of each year into a Roth IRA for the next 30 years. If you earn 5 percent compounded annually on your investment of $3,000 at the end of each year, how much will you have when you retire in 30 years? How much will you have if you wait 10 years before beginning to save and only make 20 payments into your retirement account?

Learn how to calculate the future value of a Roth IRA with $3,000 annual contributions compounded at 5% for 30 years. Discover how delaying payments for 10 years and making 20 contributions impacts the final amount. Suitable for college-level finance students.

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