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Let's analyze and solve this investment problem step-by-step.

### Given:
1. **Monthly Deposit**: $650 at the end of every month.
2. **Initial Interest Rate**: 4.56% compounded monthly.
3. **Investment Duration for Regular Deposits**: 9 years.
4. **Change in Interest Rate After 9 Years**: The interest rate changes to 6.87% compounded quarterly.
5. **Additional Growth Period without Deposits**: 4 years (from year 9 to year 13).

### Solution Approach:
We will break down the problem into two parts corresponding to parts (a) and (b).

---

### Part (a): Calculate the Accumulated Balance at the End of 9 Years

For part (a), we have a future value of an annuity problem, where deposits are made monthly, and the interest compounds monthly.

#### Formula for Future Value of an Annuity:
\[
FV = P \times \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}}
\]
where:
- $ P = 650 $: monthly deposit.
- $ r = 0.0456 $: annual interest rate (4.56%).
- $ n = 12 $: compounding periods per year (monthly).
- $ t = 9 $: total years of deposit.

#### Calculation:
Let's calculate the future value after 9 years with the given values.

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### Part (b): Calculate the Accumulated Balance at the End of 13 Years

For part (b), after 9 years, Erik stops making monthly deposits, and the accumulated balance is now subject to compound interest at a rate of 6.87% compounded quarterly for the next 4 years.

#### Formula for Compound Interest:
\[
A = P \times \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $ P $: the balance from part (a) after 9 years.
- $ r = 0.0687 $: new annual interest rate (6.87%).
- $ n = 4 $: compounding periods per year (quarterly).
- $ t = 4 $: additional years of growth without deposits.

---

I’ll calculate each part now.### Results:
1. **Part (a)**: The accumulated balance in the investment fund at the end of 9 years is approximately **$86,595.16**.
2. **Part (b)**: After allowing the balance to grow for an additional 4 years at the new interest rate of 6.87% compounded quarterly, the accumulated balance at the end of 13 years is approximately **$113,716.76**.

Would you like further details on the calculations, or do you have any questions?

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Here are five additional questions related to investment and compound interest:

1. What is the impact of increasing the deposit frequency on the future value of an investment?
2. How does a change in compounding frequency affect the final balance of an investment?
3. What if the interest rate changes multiple times over the investment period?
4. How can inflation influence the real value of the accumulated balance over time?
5. How would the final amount differ if deposits were made at the beginning of each period instead of at the end?

**Tip**: Always note the compounding frequency in investment problems, as it has a significant effect on the final balance.

Erik invested $650 at the end of every month in an investment fund that was earning interest at a rate of 4.56% compounded monthly. She stopped making regular deposits at the end of 9 years when the interest rate changed to 6.87% compounded quarterly. However, she let the money grow in this investment fund for the next 4 years. 

a. Calculate the accumulated balance in her investment fund at the end of 9 years.
b. Calculate the accumulated balance in her investment fund at the end of 13 years.

Learn how to calculate the accumulated balance of an investment with monthly deposits of $650, a 4.56% interest rate compounded monthly for 9 years, and then a 6.87% rate compounded quarterly for an additional 4 years. Detailed solution with formulas for future value of an annuity and compound interest.

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