Calculate the future value.
Present Value	Interest Rate	Payments	Timing of Payment	Years	Future Value
$0.00	8.5% monthly	$400.00 monthly	End	24	???
Determine the annuity type.
Ordinary Simple Annuity
Ordinary General Annuity
Simple Annuity Due
General Annuity Due
Identify the following pieces of information to be used to calculate the future value of the annuity.
Periodic Payment:
Number of Payments per Year:
Total Number of Payments:
Annual Interest Rate:
Number of Compoundings per Year:
Determine the future value of the annuity.

Let's break down this problem and solve it step by step.

### Annuity Type
First, let's determine the type of annuity:

1. **Ordinary Annuity**: Payments are made at the end of each period.
2. **Annuity Due**: Payments are made at the beginning of each period.
3. **Simple Annuity**: The compounding period and payment interval are the same.
4. **General Annuity**: The compounding period and payment interval are different.

Given:
- Payments are made **monthly**.
- Compounding is done **monthly** (since interest is given monthly).
- Payments are made at the **end** of each period.

**Type of Annuity**: **Ordinary Simple Annuity** (because payments are made at the end of each period, and the compounding period and payment interval are the same).

### Information Identification
- **Periodic Payment (PMT)**: $400.00
- **Number of Payments per Year**: 12 (monthly payments)
- **Total Number of Payments**: $12 \times 2 = 24$ (for 2 years)
- **Annual Interest Rate**: 8.5%
- **Number of Compoundings per Year**: 12 (since it's compounded monthly)

### Formula for Future Value of an Ordinary Annuity
The formula for the future value (FV) of an ordinary annuity is:

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

Where:
- $PMT$ is the periodic payment
- $i$ is the periodic interest rate
- $n$ is the total number of payments

### Calculation
1. **Periodic Interest Rate (i)**: 
   \[
   i = \frac{8.5\%}{12} = \frac{0.085}{12} \approx 0.007083
   \]

2. **Total Number of Payments (n)**: 
   \[
   n = 24
   \]

3. **Future Value (FV)**:
   \[
   FV = 400 \times \left(\frac{(1 + 0.007083)^{24} - 1}{0.007083}\right)
   \]

Let's compute this.The future value of the annuity is approximately **$10,424.17**.

### Summary of Information:
- **Periodic Payment (PMT)**: $400.00
- **Number of Payments per Year**: 12
- **Total Number of Payments**: 24
- **Annual Interest Rate**: 8.5%
- **Number of Compoundings per Year**: 12
- **Future Value (FV)**: $10,424.17

### Do you have any questions, or would you like a deeper explanation?

### Follow-up Questions:
1. What would be the future value if the payments were made at the beginning of each period?
2. How does increasing the interest rate affect the future value?
3. What if the payments were made quarterly instead of monthly?
4. How would a different compounding frequency (e.g., quarterly) impact the future value?
5. What would be the impact if the annuity was extended to 3 years instead of 2?
6. Can we calculate the present value if the future value and interest rate are known?
7. How does inflation affect the real value of the future sum?
8. What would the future value be if the interest rate were compounded daily?

### Tip:
Always double-check the timing of payments and the compounding frequency, as they can significantly affect the future value of an annuity.

Learn how to calculate the future value of an ordinary simple annuity with monthly payments. This problem involves understanding annuities, compound interest, and applying the future value formula for an ordinary annuity. Suitable for students in Grades 10-12.

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