Scheduled payments of 

​$14231423​, 

​$13271327​, 

and 

​$13861386 

are due in 

oneone 

yearyear​, 

four dash and dash a dash halffour-and-a-half 

​years, and 

sixsix 

years respectively. What is the equivalent single replacement payment 

two dash and dash a dash halftwo-and-a-half 

years from now if interest is 

7.3 %7.3% 

compounded semi dash annually question mark semi-annually?

To solve this problem, let's break it down into steps.

### 1. **Understanding the Problem:**
You have three scheduled payments:
- $ \$14,231,423 $ due in 1 year.
- $ \$13,271,327 $ due in 4.5 years.
- $ \$13,861,386 $ due in 6 years.

You want to find the equivalent single payment due in 2.5 years, given an interest rate of 7.3% compounded semi-annually.

### 2. **Interest Rate Conversion:**
The nominal interest rate is 7.3% per annum, compounded semi-annually. The semi-annual interest rate is:
\[
i = \frac{7.3\%}{2} = 3.65\%
\]
\[
i = 0.0365
\]

### 3. **Calculate the Present Value of Each Payment at 2.5 Years:**
To find the present value of each payment at 2.5 years, we discount each payment back to 2.5 years using the formula:
\[
PV = \text{Future Value} \times (1 + i)^{-n}
\]
Where $ n $ is the number of semi-annual periods between the time of the payment and 2.5 years.

- **Payment 1:** $ \$14,231,423 $ due in 1 year ($ t = 1 $ year)
  - $ n = \frac{2.5 - 1}{0.5} = 3 $ semi-annual periods
  - Present Value:
    \[
    PV_1 = 14,231,423 \times (1.0365)^{-3}
    \]

- **Payment 2:** $ \$13,271,327 $ due in 4.5 years ($ t = 4.5 $ years)
  - $ n = \frac{4.5 - 2.5}{0.5} = 4 $ semi-annual periods
  - Present Value:
    \[
    PV_2 = 13,271,327 \times (1.0365)^{-4}
    \]

- **Payment 3:** $ \$13,861,386 $ due in 6 years ($ t = 6 $ years)
  - $ n = \frac{6 - 2.5}{0.5} = 7 $ semi-annual periods
  - Present Value:
    \[
    PV_3 = 13,861,386 \times (1.0365)^{-7}
    \]

### 4. **Sum the Present Values to Find the Equivalent Payment at 2.5 Years:**
The equivalent single payment at 2.5 years ($ P' $) is the sum of the present values calculated above:
\[
P' = PV_1 + PV_2 + PV_3
\]

### 5. **Calculations:**

Now, let's calculate the present values for each payment.

- **Calculate $ PV_1 $:**
  \[
  PV_1 = 14,231,423 \times (1.0365)^{-3} = 14,231,423 \times 0.904 = 12,867,266.82
  \]

- **Calculate $ PV_2 $:**
  \[
  PV_2 = 13,271,327 \times (1.0365)^{-4} = 13,271,327 \times 0.8698 = 11,539,551.47
  \]

- **Calculate $ PV_3 $:**
  \[
  PV_3 = 13,861,386 \times (1.0365)^{-7} = 13,861,386 \times 0.7854 = 10,883,938.34
  \]

Finally, sum these present values to get $ P' $:
\[
P' = 12,867,266.82 + 11,539,551.47 + 10,883,938.34 = 35,290,756.63
\]

### **Final Answer:**
The equivalent single replacement payment due two-and-a-half years from now is approximately **\$35,290,756.63**.

Would you like more details or have any questions about this solution?

---

Here are some related questions you might find helpful:
1. How do you determine the equivalent value of multiple cash flows at a specific time?
2. What effect does the timing of payments have on their present value?
3. How do different compounding intervals affect the present value calculation?
4. Can you explain how to calculate future values given semi-annual compounding?
5. How does changing the interest rate impact the equivalent single payment?
6. What methods can be used to validate the accuracy of time value of money calculations?
7. How does the concept of discounting apply to financial decision-making?
8. What are the implications of using different discount rates in present value calculations?

**Tip:** When dealing with semi-annual compounding, always adjust both the interest rate and the number of periods accordingly to get accurate results.

Learn how to calculate the equivalent single replacement payment due in 2.5 years using compound interest. This problem involves determining present values of future payments with semi-annual compounding at an interest rate of 7.3%. Suitable for advanced college-level students and professionals.

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