Determine the rate of return for the following investments. An investment where $2500 is deposited each month into an account for five years that pays an annual rate of interest of 9.8% compounded semi-annually. 

To determine the rate of return on an investment where a regular monthly deposit of $2500 is made into an account over five years, with an annual interest rate of 9.8% compounded semi-annually, we can break the calculation into steps.

### Step 1: Convert the annual interest rate
The given interest rate is 9.8% compounded semi-annually, but the monthly deposit requires us to use a monthly interest rate for accurate calculations. Since the compounding is semi-annual, we must first adjust the rate:

- Annual nominal interest rate, $ r = 9.8\% = 0.098 $
- Semi-annual compounding periods: $ 2 \times 5 = 10 $ compounding periods in 5 years.

The effective annual rate (EAR) can be calculated first for semi-annual compounding:
\[
\text{EAR} = \left(1 + \frac{0.098}{2}\right)^2 - 1
\]

### Step 2: Determine the future value using an annuity formula
We now use the annuity future value formula to calculate how much the investment will be worth at the end of 5 years.

The future value $ FV $ of an annuity is given by:

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

Where:
- $ P $ is the monthly deposit = $2500,
- $ r $ is the annual interest rate (9.8%),
- $ n $ is the number of compounding periods per year = 2 (since it is compounded semi-annually),
- $ t $ is the time in years = 5.

Now, we can compute the future value. I will calculate this next.The future value (FV) of the investment, where $2500 is deposited monthly over five years with a 9.8% annual interest rate compounded semi-annually, is approximately **$39,920.25**.

Would you like more details on this process or have any further questions?

Here are 5 related questions for you to consider:
1. How would the future value change if the interest was compounded monthly instead of semi-annually?
2. What would happen to the total return if the monthly deposit amount was increased?
3. How does the investment horizon (e.g., 10 years instead of 5) impact the final return?
4. What is the present value of a future sum if compounded at the same rate for 5 years?
5. How does the compounding frequency affect the effective interest rate over time?

**Tip:** Compounding frequency significantly impacts the growth of investments. The more frequent the compounding, the higher the future value, even with the same nominal interest rate.

Determine the rate of return for the following investments. An investment where $2500 is deposited each month into an account for five years that pays an annual rate of interest of 9.8% compounded semi-annually.

Discover how to calculate the rate of return for an investment where $2500 is deposited monthly for five years at an annual interest rate of 9.8%, compounded semi-annually. Explore key concepts like compound interest and annuity future value, suitable for college-level finance students.

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