You are scheduled to receive annual payments of $10,100 for each of the next 23 years. Your discount rate is10 percent. What is the difference in the present value if you receive these payments at the beginning ofeach year rather than at the end of each year?

Question

BlackTom AI · Accepted Answer

An annuity due's present value equals the present value of an ordinary (end-of-period) annuity multiplied by (1 + r).

1) Relevant concepts
- PV of an ordinary annuity (payments at end of each period): PV_ordinary = PMT * [1 - (1 + r)^(-n)] / r.
- PV of an annuity due (payments at beginning of each period): PV_due = PV_ordinary * (1 + r).
- The difference = PV_due − PV_ordinary = PV_ordinary * r = PMT * [1 - (1 + r)^(-n)].

2) Step-by-step reasoning
- Given PMT = $10,100, n = 23, r = 0.10.
- Compute the discount factor (1 + r)^(-n) = (1.10)^(-23).

3) Calculations (intermediate steps shown)
- Compute (1.10)^23 ≈ 8.95430243, so (1.10)^(-23) ≈ 1 / 8.95430243 ≈ 0.111704.
- Compute factor [1 - (1 + r)^(-n)] = 1 − 0.111704 = 0.888296.
- Multiply by PMT to get the difference:
  Difference = PMT * [1 − (1 + r)^(-n)] = 10,100 * 0.888296 ≈ 8,971.7896 ≈ $8,972 (rounded to nearest dollar).

4) Why this is correct
- Shifting payments from ends to beginnings increases each payment’s present value. The algebraic derivation PV_due − PV_ordinary = PMT * [1 − (1 + r)^(-n)] matches the numeric computation above, giving the increase in PV when payments are at the beginning.

5) Why other options are incorrect
- $10,100 would be the increase only if n were infinite (difference → PMT), but for finite n the difference is slightly less.
- $11,110, $9,487, and $9,836 do not match the algebraic formula or the numerical evaluation shown; they are inconsistent with PMT * [1 − (1 + r)^(-n)].

6) Conclusion
- The difference in present value when payments are at the beginning rather than at the end is approximately $8,972, matching the provided correct answer.

You are scheduled to receive annual payments of $10,100 for each of the next 23 years. Your discount rate is10 percent. What is the difference in the present value if you receive these payments at the beginning ofeach year rather than at the end of each year?单项选择题

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