If the current rate of interest is 8%, then the future value (FV) of an annuity that pays $1 000 per year and lasts 20 years is closest to:

Question

BlackTom AI · Accepted Answer

The future value of an ordinary annuity is given by FV = PMT * [ (1 + r)^n − 1 ] / r. Using PMT = $1,000, r = 0.08, n = 20:

1) Relevant concepts
- Ordinary annuity (payments at the end of each period) FV formula above.
- Compound growth factor (1 + r)^n and how it accumulates each payment to the end of period n.

2) Step-by-step reasoning and computation
- Compute the compound factor (1.08)^20.
  Using repeated squaring for accuracy:
  1.08^2 = 1.1664
  1.08^4 = 1.1664^2 ≈ 1.360489
  1.08^8 = 1.360489^2 ≈ 1.850931
  1.08^16 = 1.850931^2 ≈ 3.425933
  1.08^20 = 1.08^16 * 1.08^4 ≈ 3.425933 * 1.360489 ≈ 4.660943
- Subtract 1: (1.08)^20 − 1 ≈ 4.660943 − 1 = 3.660943
- Divide by r: 3.660943 / 0.08 = 45.7617875
- Multiply by PMT: FV = 1000 * 45.7617875 ≈ $45,761.79 ≈ $45,762

3) Intermediate steps shown above (powers, subtraction, division, multiplication) lead to the numeric FV.

4) Why this answer is correct
- The formula precisely accumulates each $1,000 payment to the end of year 20 at 8% interest. The arithmetic yields roughly $45,762, which matches option a.

5) Why other choices are incorrect
- b ($9,818) is approximately the present value of the same 20-year $1,000 annuity at 8% (PV formula), not the future value.
- c ($20,000) is simply the sum of payments (1000 × 20) ignoring interest.
- d ($18,519) does not correspond to the correct FV or PV calculations for the given parameters.

6) Conclusion
The correct future value is approximately $45,762, matching option a.

If the current rate of interest is 8%, then the future value (FV) of an annuity that pays $1 000 per year and lasts 20 years is closest to:单项选择题

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