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

Present value of an ordinary annuity is used because the $10,000 payments occur at the end of each of the next 3 years. The PV equals the sum of each payment discounted at the effective annual rate.

1) Relevant knowledge points and concepts
- Present value (PV) discounts future cash flows using PV = CF / (1 + i)^t.
- For an ordinary annuity of n equal payments, PV = PMT * [1 - (1 + i)^(-n)] / i.
- Here PMT = $10,000, i = 0.10, n = 3.

2) Complete reasoning process step by step
Use the annuity formula for speed and verify by discounting each cash flow:
PV = 10000 * [1 - (1 + 0.10)^(-3)] / 0.10.

3) Calculations with intermediate steps
Compute powers and reciprocals:
(1 + 0.10)^2 = 1.21
(1 + 0.10)^3 = 1.331
(1 + 0.10)^(-3) = 1 / 1.331 = 0.7513148009

Annuity factor = [1 - 0.7513148009] / 0.10 = 0.2486851991 / 0.10 = 2.486851991
PV = 10000 * 2.486851991 = 24,868.51991 ≈ $24,868.52

Verification by discounting each payment:
PV1 = 10000 / 1.10 = 9,090.90909
PV2 = 10000 / 1.21 = 8,264.46281
PV3 = 10000 / 1.331 = 7,513.14801
Sum = 9,090.90909 + 8,264.46281 + 7,513.14801 = 24,868.51991 ≈ $24,868.52

4) Why the correct answer is correct
Both the annuity formula and the term-by-term discounting yield the same PV, $24,868.52, which is the present value today of three $10,000 year-end payments at 10% interest.

5) Why the other options are incorrect
- $36,410.00 is larger than the undiscounted total ($30,000), so it cannot be a present value at a positive discount rate.
- $27,355.37 and $33,100.00 do not match the correct discounted sum obtained by proper discounting; they likely arise from incorrect rates or arithmetic errors.

6) Conclusion
The present value is $24,868.52, which matches the provided correct answer.

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%.Single choice

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