Originally, payments of $2000 and $1000 were scheduled to be made one year and five years from today, respectively. They are to be replaced by a $1500 payment due four years from today and another payment due two years from today. If money earns 7% interest compounded semiannually, what is the unknown payment that makes the replacement stream economically equivalent to the original stream?

Introduction / Context:
This is a cash flow equivalence problem under compound interest with semiannual compounding. Two original payments at different future dates are to be replaced by two new payments at other dates. To preserve economic equivalence, the present value of both streams must be the same when discounted at the given interest rate.

Given Data / Assumptions:

• Original payments: $2000 at t = 1 year, $1000 at t = 5 years
• Replacement payments: $1500 at t = 4 years, unknown amount X at t = 2 years
• Interest rate: 7% per annum, compounded semiannually
• Semiannual rate i = 7% / 2
• We compare present values at t = 0

Concept / Approach:
For nominal rate r compounded semiannually, the period rate is:
i = r / 2and each year has 2 periods. The present value of a future amount A at time t years is:
PV = A / (1 + i)^(2 * t)Economic equivalence requires that:
PV(original stream) = PV(replacement stream)We solve this equation for X.

Step-by-Step Solution:
Step 1: Compute semiannual rate.i = 0.07 / 2 = 0.035Step 2: Present value of original payments.PV1 = 2000 / (1.035)^(2 * 1) + 1000 / (1.035)^(2 * 5)PV1 ≈ 2000 / (1.035)^2 + 1000 / (1.035)^10PV1 ≈ 2575.94 (approximate)Step 3: Present value of replacement payments.PV2 = 1500 / (1.035)^(2 * 4) + X / (1.035)^(2 * 2)PV2 = 1500 / (1.035)^8 + X / (1.035)^4Step 4: Set PV1 = PV2 and solve for X.2575.94 = 1500 / (1.035)^8 + X / (1.035)^4Rearranging and multiplying by (1.035)^4 gives X ≈ 1648.79 ≈ $1648

Verification / Alternative check:
Substitute X = 1648 back into PV2 and recompute with more precise calculator values. PV2 will match PV1 up to rounding error, confirming the correctness of the unknown payment.

Why Other Options Are Wrong:
$1548 and $1448: These are smaller than required and would reduce the present value of the replacement stream below the original.$1748: Too large; it would make the replacement stream more valuable than the original.$1348: Much too low to be equivalent at the given interest rate.

Common Pitfalls:
Some learners mix up the number of compounding periods (using years instead of half-years), or compare future values instead of present values. Others incorrectly move all payments in the wrong direction in time or forget that both streams must be evaluated at the same reference date. Drawing a time line and carefully applying the present value formula for each payment help avoid mistakes.

Final Answer:
The unknown replacement payment due two years from today must be approximately $1648 to be economically equivalent to the original payment stream.