The present value of an annuity with the first payment starts 10 years from today can be calculated in two steps: (1) using the PV of an ordinary annuity formula calculate the present value of the annuity at _____ (2) then discount back the answer found in part 1 to time zero by calculating the present value of this amount using single cash flow PV formula PV=FV/(1+i)^n

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The question describes valuing an annuity whose first payment begins 10 years from today and states a two-step process: (1) compute the PV of an ordinary annuity at a certain time point, and (2) discount that amount back to time zero using the single cash flow PV formula.

Option a: 'the end of year 10' — If you compute the PV at the end of year 10 for an ordinary annuity that starts 10 years from today, you would be aligning the cash flows with a different time anchor. The ordinary annuity PV formula assumes payments occur at the end of each period, so evaluating at the end of year 10 would imply the first payment is at the end of year 11, which contradicts the given starting point (first payment occurs at year 10). This mismatch makes this choice incorrect.

Option b: 'the end of year 9' — Here, by evaluating the PV of the annuity at the end of year 9, you place the time anchor one period earlier than the first payment. Since the first payment occurs at the end of year 10, valuing the sequence as an ordinary annuity one period before the first cash flow matches the standard PV-at-a-time-point-of-just-before-first-payment approach. After obtaining the PV at the end of year 9, you would then discount it back 9 years to time zero using PV = FV/(1+i)^n, which is consistent with the two-step process described.

Option c: 'the end of year 8' — Valuing at the end of year 8 would place the time anchor two periods before the first payment. While you could still discount back to time zero, this would misalign the period count with the structure of a standard ordinary annuity, which typically uses the end-of-period timing for PV calculations. This choice introduces an extra discount step mismatch and is not the standard approach described.

Option d: 'the start of year 9' — The start of year 9 is not aligned with the end-of-period timing assumed by the ordinary annuity PV formula. PV formulas for ordinary annuities assume payments occur at the end of each period, so evaluating at the start of year 9 would require a different treatment (e.g., annuity-due conventions). This would not correspond to the stated method, making this option incorrect.

In summary, the method calls for first locating the PV of an ordinary annuity at a point that aligns with the period immediately before the first payment (so that discounting back to time zero uses the correct number of periods). Among the choices, the end of year 9 is the point that best fits this interpretation, ensuring the timing of cash flows and the subsequent discounting steps are consistent with the described approach.

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