You are promised $5,000 at a fixed positive interest rate. Compare two scenarios: (1) You receive $5,000 in 3 years, and (2) You receive $5,000 in 8 years. Which statement must be true?[Fill in the blank]

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Present value equals the future payment divided by (1 + r)^t, so PV = FV / (1 + r)^t. With a fixed positive interest rate r and times t measured in years, a payment received later has a smaller present value. 1) Relevant knowledge points and concepts - Time value of money: one dollar today is worth more than one dollar in the future when r > 0. - Present value formula: PV = FV / (1 + r)^t. - For fixed FV and r > 0, PV is a strictly decreasing function of t. 2) Step-by-step reasoning Let FV = $5,000. For Scenario 1 (t = 3): PV3 = 5000 / (1 + r)^3. For Scenario 2 (t = 8): PV8 = 5000 / (1 + r)^8. Compare PV3 and PV8. 3) Calculations and intermediate steps PV8 / PV3 = [5000 / (1 + r)^8] ÷ [5000 / (1 + r)^3] = 1 / (1 + r)^(8-3) = 1 / (1 + r)^5. Since r > 0, (1 + r)^5 > 1, so PV8 / PV3 < 1, which implies PV8 < PV3. Example: if r = 5%, (1.05)^3 = 1.157625 → PV3 ≈ 5000/1.157625 ≈ $4,317. (1.05)^8 = 1.477455 → PV8 ≈ 5000/1.477455 ≈ $3,384. 4) Why the correct answer is correct Because discounting by (1 + r)^t reduces the value as t increases when r > 0, the payment received in 8 years has a lower present value than the identical nominal payment received in 3 years. 5) Why the other options are incorrect - (a) Claims the future value measured today is identical: incoherent and false; “measured today” means present value, which differs by timing. - (c) Asserts PV is the same because the dollar amount is identical: ignores time value of money; timing matters when r > 0. - (d) Says PV is higher in Scenario 2: contradicts the discounting effect; receiving later lowers PV. 6) Conclusion Therefore the correct statement is b. The present value is lower in Scenario 2 because the money is received later. This matches the provided answer.

You are promised $5,000 at a fixed positive interest rate. Compare two scenarios: (1) You receive $5,000 in 3 years, and (2) You receive $5,000 in 8 years. Which statement must be true?[Fill in the blank]单项选择题

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