Evelyn is planning on making an investment that will generate cash flows of $1,000 per quarter for 3 years with the first payment occurring immediately. Similar risk investments are offering a return of 10% p.a. compounded quarterly. How much should Evelyn be prepared to pay for this investment?

Question

BlackTom AI · Accepted Answer

Begin by identifying the cash flow structure and the required rate. The investment pays 1,000 every quarter for 3 years, with the first payment occurring immediately. That means we have an annuity due with n = 12 payments and a periodic (quarterly) rate i = 10% per year compounded quarterly, so i = 0.10/4 = 0.025 per quarter.

Option A: $10,514.21. To evaluate this, compute the present value of an annuity due: PV = Payment × [(1 − (1 + i)^(−n)) / i] × (1 + i).
Plugging in numbers: (1 + i)^(−n) = 1.025^(−12) ≈ 0.7430, so (1 − that) ≈ 0.2570. Then [(1 − (1 + i)^(−n)) / i] ≈ 0.2570 / 0.025 ≈ 10.28. Multiply by the payment: 10.28 × 1,000 ≈ 10,280. Then adjust for the annuity due by multiplying by (1 + i) ≈ 1.025: 10,280 × 1.025 ≈ 10,537. This is very close to $10,514.21, suggesting this option is plausible given rounding differences in intermediate steps.

Option B: $7,382.60. This value is far smaller than the calculated PV for a 12-payment annuity due at 2.5% per period; it would correspond to either far fewer payments, a much lower rate, or both. Since we have 12 payments of 1,000 at a modest rate, this option underestimates the value by a wide margin.

Option C: $9,274.98. This is still substantially below the computed PV for a 12-payment annuity due at 2.5% per quarter. It would imply a noticeably smaller PV factor or fewer payments; given 12 payments of 1,000, it remains too low.

Option D: $11,593.06. This amount exceeds the PV we calculated using the standard annuity-due formula with i = 2.5% and n = 12. The discrepancy could arise if someone accidentally used an incorrect factor or miscounted the payments, but with the given structure, this is too large.

Option E: $12,573.15. This is even larger than Option D and far above the correct PV for the specified cash flows and rate. It would require an overestimation of the PV factor or an assumption of more than 12 payments, which conflicts with the problem’s data.

In summary, the calculation steps align with the general method for valuing an annuity due with quarterly payments at a periodic rate of 2.5% and n = 12. The result around 10,536, which is very close to $10,514.21 when accounting for rounding of the intermediate powers and factors, makes Option A the best fit among the provided choices. The other options are either clearly too small given 12 payments of 1,000 at this rate, or too large for the specified cash-flow pattern.

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