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 find the equivalent single replacement payment $ P' $ due two-and-a-half years from now, we need to determine the present value of each of the scheduled payments at time $ t = 2.5 $ years, given the interest rate is 7.3% compounded semi-annually.

### Steps to Solve:
1. **Convert the interest rate:**
   Since the interest rate is compounded semi-annually, the effective semi-annual interest rate is:
   \[
   i = \frac{7.3\%}{2} = \frac{7.3}{2} \times 0.01 = 3.65\%
   \]
   \[
   i = 0.0365
   \]

2. **Determine the present value of each payment at $ t = 2.5 $ years:**
   
   - **Payment 1:** $ \$14,231,423 $ due in 1 year:
     \[
     P_1 = 14,231,423 \times (1 + 0.0365)^{\frac{2.5 - 1}{0.5}} = 14,231,423 \times (1.0365)^{3}
     \]

   - **Payment 2:** $ \$13,271,327 $ due in 4.5 years:
     \[
     P_2 = 13,271,327 \times (1 + 0.0365)^{\frac{2.5 - 4.5}{0.5}} = 13,271,327 \times (1.0365)^{-4}
     \]

   - **Payment 3:** $ \$13,861,386 $ due in 6 years:
     \[
     P_3 = 13,861,386 \times (1 + 0.0365)^{\frac{2.5 - 6}{0.5}} = 13,861,386 \times (1.0365)^{-7}
     \]

3. **Sum the present values to get the equivalent payment $ P' $:**
   \[
   P' = P_1 + P_2 + P_3
   \]

Now, let's calculate each of these values.

### Calculations:
1. **For Payment 1:**
   \[
   P_1 = 14,231,423 \times (1.0365)^3 \approx 14,231,423 \times 1.113 = 15,838,021.10
   \]

2. **For Payment 2:**
   \[
   P_2 = 13,271,327 \times (1.0365)^{-4} \approx 13,271,327 \times 0.8698 = 11,539,551.47
   \]

3. **For Payment 3:**
   \[
   P_3 = 13,861,386 \times (1.0365)^{-7} \approx 13,861,386 \times 0.7854 = 10,883,938.34
   \]

Finally, sum the present values to find $ P' $:

\[
P' = 15,838,021.10 + 11,539,551.47 + 10,883,938.34 = 38,261,510.91
\]

### Final Answer:
The equivalent single replacement payment due two-and-a-half years from now is approximately **\$38,261,510.91**.

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

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Here are some related questions you might find helpful:
1. How do you calculate the present value of a series of payments?
2. What is the effect of compounding interest on loan repayments?
3. How does the timing of payments affect the equivalent value in time value of money calculations?
4. Can you explain how to convert different interest rate compounding periods?
5. What is the difference between nominal and effective interest rates?
6. How do semi-annual interest rates differ from annual rates in present value calculations?
7. What are some common mistakes to avoid in time value of money calculations?
8. How do different compounding frequencies affect the future value of an investment?

**Tip:** Always make sure to use the correct formula for converting interest rates when dealing with different compounding periods.

Learn how to calculate the equivalent single replacement payment due two-and-a-half years from now using compound interest. Detailed step-by-step solution with present value calculations for scheduled payments of $14,231,423, $13,271,327, and $13,861,386 due in different time periods with a 7.3% semi-annual interest rate.

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