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

Question

BlackTom AI · Accepted Answer

In this problem, we’re dealing with a delayed annuity where the first payment occurs 10 years from today. To price it in two steps, we first find the value as of the moment immediately before the first payment, and then discount that amount back to today.

Option a: the start of year 9. This would place the valuation one period before the typical end-of-period (first payment) timing for an ordinary annuity if the first payment were at the end of year 9. Since the problem states the first payment is 10 years from today, valuing at the start of year 9 does not align with the standard convention for an ordinary annuity valuation point. It would mix up the timing by one year and would not correctly reflect the value just before the first payment.

Option b: the end of year 9. This matches the standard approach: for an ordinary annuity whose first payment occurs at the end of year 10, the present value of the annuity as of the end of year 9 represents the value immediately before the first payment is made. This is the appropriate point to apply the PV of an ordinary annuity formula, since the payments then occur one year apart beginning at year 10.

Option c: the end of year 10. Valuing at the end of year 10 would be the moment of the first payment, not just before it. PV at that moment would include or be affected by the actual first payment, which is inconsistent with using the ordinary annuity formula for a series of payments. After that, one would still need to discount back from year 10 to today, but the standard approach is to value just before the first payment, not at the moment of the first payment.

Option d: the end of year 8. This is two years before the first payment and is not the conventional point to apply the PV of an ordinary annuity formula for a payment stream starting at year 10. Valuing this far back would misplace the timing and complicate the stepwise discounting process unnecessarily.

Pulling these threads together: the conventional two-step method requires locating the present value one period before the first payment (the end of year 9) using the ordinary annuity PV formula, then discounting that amount back to time zero. The other options misalign with the standard timing framework for ordinary annuities with a delayed start.

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