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

For an ordinary annuity the present value formula yields the value one period before the first payment, so if the first cash flow occurs at the end of year 10 the PV from the ordinary-annuity formula is located at the end of year 9.

1) Relevant knowledge points and concepts
- Ordinary annuity: payments occur at the end of each period. The standard PV formula gives the value one period before the first payment.
- PV of an ordinary annuity: PV_{t-1} = PMT * [1 - (1+i)^{-N}] / i, which is the lump-sum value measured at time t-1 when the first payment is at time t.
- Single cash flow discounting: PV_0 = FV / (1+i)^n to move a value n periods back to time 0.

2) Complete reasoning step by step
- First payment is at end of year 10 (ten years from today). Treat the annuity as an ordinary annuity with first payment at t = 10.
- Use the ordinary-annuity PV formula to compute the value one period before the first payment, i.e., at t = 9 (end of year 9).
- Then discount that lump-sum value from t = 9 back to today (t = 0) using PV_0 = PV_9 / (1+i)^9.

3) Calculations (symbolic)
- PV_9 = PMT * [1 - (1+i)^{-N}] / i
- PV_0 = PV_9 / (1+i)^9 = {PMT * [1 - (1+i)^{-N}] / i} / (1+i)^9

4) Why the correct answer is correct
- The ordinary annuity formula always locates the PV one period before the first payment. With the first payment at end of year 10, that “one period before” timepoint is the end of year 9.

5) Why other options are incorrect
- a (end of year 8): two periods before the first payment, incorrect.
- b (end of year 10): that is the time of the first payment, not where the ordinary-annuity PV formula measures the value.
- d (start of year 9): mixes beginning/ending conventions; the PV formula assumes end-of-period timing.

6) Conclusion
This reasoning confirms the correct choice is c. the end of year 9.

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