A new start-up company promises to pay an investor each quarter for the next two years. The company will pay $21,100.00 per quarter for the first four quarters, and then $25,775.00 per quarter for the following four quarters. If the investor wants a 11.44% APR return with quarterly compounding, what is the value of the investment opportunity today?

To calculate the present value of the investment opportunity today, we need to discount each of the company's future payments back to the present using the given interest rate of 11.44% APR compounded quarterly. This involves using the formula for the present value of an annuity with different cash flows in two periods.

### Given:
- Payments for the first 4 quarters: \$21,100 each.
- Payments for the next 4 quarters: \$25,775 each.
- APR (Annual Percentage Rate) = 11.44%, compounded quarterly.
- Total time: 8 quarters (2 years).
  
#### Step 1: Calculate the quarterly interest rate:
The quarterly interest rate is obtained by dividing the APR by 4 (since the compounding is quarterly).

\[
r = \frac{11.44\%}{4} = 2.86\% = 0.0286
\]

#### Step 2: Discount each set of cash flows:

1. **For the first 4 quarters** (cash flow = \$21,100):
   Each payment is discounted back to the present value using the formula:

   \[
   PV = \frac{C}{(1 + r)^n}
   \]
   where:
   - $C$ is the cash flow for the period.
   - $r$ is the quarterly interest rate.
   - $n$ is the number of quarters from the present.

   Summing the present values of the first four quarters:

   \[
   PV_{\text{first 4 quarters}} = \frac{21,100}{(1+0.0286)^1} + \frac{21,100}{(1+0.0286)^2} + \frac{21,100}{(1+0.0286)^3} + \frac{21,100}{(1+0.0286)^4}
   \]

2. **For the next 4 quarters** (cash flow = \$25,775):
   Since these payments occur after the first 4 quarters, they need to be discounted further. So the discount factors are higher for these payments:

   \[
   PV_{\text{next 4 quarters}} = \frac{25,775}{(1+0.0286)^5} + \frac{25,775}{(1+0.0286)^6} + \frac{25,775}{(1+0.0286)^7} + \frac{25,775}{(1+0.0286)^8}
   \]

#### Step 3: Add the two present value sums to get the total present value of the investment.

Let me calculate that for you.The present value of the investment opportunity today is approximately \$164,570.12.

Would you like further details or have any questions about the calculation? 

Here are some related questions:
1. How does quarterly compounding differ from annual compounding?
2. What happens to the present value if the interest rate increases?
3. How would the present value change if the payments were spread over 3 years instead of 2?
4. Can you calculate the future value of the investment after 2 years at 11.44% APR?
5. What if the payments varied each quarter; how would you handle that?

**Tip:** When dealing with multiple cash flows, always discount each one individually to account for its timing.

Learn how to calculate the present value of an investment with quarterly cash flows. This example covers two cash flow periods: $21,100 for the first 4 quarters and $25,775 for the next 4, with a return rate of 11.44% APR. Suitable for finance and business students.

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