George will receive a study excellence award of $10,000 per year at the end of the next 3 years. Calculate the present value today of George's cash flows if the effective annual interest rate is 10%.

Question

BlackTom AI · Accepted Answer

First, identify the cash flows and the discount rate: George receives $10,000 at the end of years 1, 2, and 3, and the annual effective interest rate is 10% (i = 0.10). We are asked for the present value today of these three payments.

Option-by-option analysis:

Option A: $27,355.37
- This value corresponds to the present value of an annuity due (payments at the beginning of each year) rather than an ordinary annuity. If you take the ordinary annuity PV formula and multiply by (1 + i), you get a higher amount: PV_ordinary × (1.1) = 24,868.52 × 1.10 ≈ 27,355.37. Since the problem specifies payments at the end of each year, this option overstates the true PV for an ordinary annuity and is not correct in this context.

Option B: $27,355.37
- This option matches the PV of an annuity due, not an ordinary annuity. It comes from applying the end-of-year payment timing assumption incorrectly by multiplying the ordinary PV by (1 + i). Because the payments occur at year-end, this is not the correct interpretation for the given problem.

Option C: $33,100.00
- This number is larger than the total undiscounted payments of $30,000, so it cannot be the present value of these end-of-year payments discounted at 10%. It might reflect a misapplication such as dividing by a too-small discount factor or summing with an incorrect formula, but it does not align with the standard PV calculation for three end-of-year payments.

Option D: $24,868.52
- This is the correct present value when applying the ordinary annuity PV formula: PV = PMT × [1 − (1 + i)^(−n)] / i, with PMT = 10,000, i = 0.10, n = 3. Compute (1 + i)^(−n) = (1.1)^(−3) ≈ 0.7513148. Then [1 − 0.7513148] / 0.10 = 0.2486852 / 0.10 = 2.486852. Multiply by 10,000 to get 24,868.52. This aligns exactly with three end-of-year payments discounted at 10%.

In summary, the crucial distinctions lie in the timing of payments (end-of-year vs beginning-of-year) and in applying the correct annuity formula. The given problem specifies end-of-year payments, so the correct present value is $24,868.52, while the other options reflect common misapplications or timing assumptions.

George will receive a study excellence award of $10,000 per year at the end of the next 3 years. Calculate the present value today of George's cash flows if the effective annual interest rate is 10%.单项选择题

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