Payments of $2000 and $1000 were originally scheduled to be paid 1 year and 5 years, respectively, from today. They are to be replaced by a $1500 payment due 4 years from today and another payment due 2 years from today. If money can earn 7% interest compounded semiannually, what is the amount of the unknown payment at the end of year 2 so that the replacement stream is economically equivalent to the original stream?

Introduction / Context:
This question involves restructuring a cash flow by replacing two payments at different future dates with another set of two payments at new dates, under a given interest rate. To maintain economic equivalence, both payment streams must have the same value when measured at a common focal date using the applicable compound interest rate. This concept is widely used in loan refinancing, project evaluation, and financial planning.

Given Data / Assumptions:

• Original payments: 2000 dollars at year 1 and 1000 dollars at year 5
• Replacement payments: 1500 dollars at year 4 and an unknown payment X at year 2
• Nominal annual interest rate r = 7% per annum
• Compounding frequency is semiannual, so m = 2
• We assume the same interest rate applies consistently

Concept / Approach:
To find the unknown payment X, we equate the present values of the original and replacement cash flows at a convenient focal date. A natural choice is today (time 0), but we must convert all future cash flows to present value using the periodic semiannual rate. With nominal 7% compounded semiannually, the periodic rate per half year is 0.07 / 2. The number of half year periods for a payment at time t years is 2 * t. Once we compute the present value of each stream, we set them equal and solve for X.

Step-by-Step Solution:
Nominal rate r = 7% = 0.07, m = 2, periodic rate j = r / m = 0.07 / 2 = 0.035 Present value of original payments: Payment 1 at year 1: 2000 / (1 + 0.035)^(2 * 1) = 2000 / (1.035)^2 Payment 2 at year 5: 1000 / (1 + 0.035)^(2 * 5) = 1000 / (1.035)^10 PV_original = 2000 / (1.035)^2 + 1000 / (1.035)^10 Present value of replacement payments: Payment at year 2: X / (1.035)^(2 * 2) = X / (1.035)^4 Payment at year 4: 1500 / (1.035)^(2 * 4) = 1500 / (1.035)^8 PV_replacement = X / (1.035)^4 + 1500 / (1.035)^8 Set PV_original = PV_replacement and solve for X to get X ≈ 1648

Verification / Alternative check:
Using a calculator, we obtain numerical values for the discount factors and compute both present values. Substituting X = 1648 yields nearly identical present values for the original and replacement cash flows, confirming economic equivalence at the given interest rate. Small rounding differences are acceptable in practice, and 1648 is the closest of the given options to the exact solution, which is approximately 1648.8.

Why Other Options Are Wrong:
Values such as 1348, 1548, and 1748 produce present values for the replacement stream that are either too low or too high compared to the present value of the original stream. Only 1648 balances the equation closely when discounted at the semiannual rate corresponding to 7% nominal interest.

Common Pitfalls:
A common error is to treat the nominal rate as if it were an effective annual rate and ignore the semiannual compounding, which leads to wrong discount factors. Another mistake is to work with years directly instead of converting to half year periods, or to sum the payments first and then discount as a lump sum, which does not account for their different timing. Careful identification of the number of compounding periods for each payment is crucial.

Final Answer:
The amount of the unknown payment at the end of year 2 must be approximately $1648 for the replacement stream to be economically equivalent to the original stream at 7 percent interest compounded semiannually.