An annuity is set up that will pay $1 500 per year for ten years. What is the present value (PV) of this annuity given that the discount rate is 6%? (round to the nearest $1)

Question

BlackTom AI · Accepted Answer

Present value of an ordinary annuity is given by PV = PMT × [1 − (1 + r)^−n] / r, so use PMT = $1,500, r = 0.06, n = 10.

1) Relevant knowledge points and concepts
- An ordinary annuity pays at the end of each period; its present value discounts each payment back to today.
- Formula: PV = PMT * (1 − (1+r)^−n) / r.
- Compute (1+r)^n, then its reciprocal (1+r)^−n, then the bracketed factor, then multiply by PMT.

2) Step-by-step reasoning and computations
- Compute (1 + r)^n = 1.06^10.
  Using a standard exponential evaluation: 1.06^10 ≈ 1.790847697.
- Compute (1 + r)^−n = 1 / 1.06^10 ≈ 1 / 1.790847697 ≈ 0.558395.
- Compute bracket: 1 − (1+r)^−n = 1 − 0.558395 = 0.441605.
- Divide by r: 0.441605 / 0.06 = 7.360083.
- Multiply by PMT: PV = 1,500 × 7.360083 = 11,040.1245.

3) Intermediate steps shown
- 1.06^10 ≈ 1.790847697
- (1.06)^−10 ≈ 0.558395
- PV factor = (1 − 0.558395) / 0.06 = 7.360083
- PV = 1,500 × 7.360083 = 11,040.1245 → rounded to nearest $1 gives $11,040.

4) Why the correct answer is correct
- The computed PV directly follows the ordinary annuity formula and uses precise discounting of ten yearly $1,500 payments at 6%. The arithmetic yields $11,040 (rounded), matching option c.

5) Why the other choices are incorrect
- a. $19,771 is much larger than the correct PV; it could reflect an incorrect use of a lower discount rate or treating payments as perpetual. 
- b. $8,441 is too small; it would correspond to using a much higher discount rate or too few periods.
- d. $14,721 is also too large; it might come from miscomputing the annuity factor (for example, using future value instead of present value).

6) Conclusion
- The correct present value is $11,040, which matches option c.

An annuity is set up that will pay $1 500 per year for ten years. What is the present value (PV) of this annuity given that the discount rate is 6%? (round to the nearest $1)单项选择题

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