A lottery winner will receive $8 million at the end of each of the next twelve years. What is the future value (FV) of her winnings, given that the interest rate is 2.6% per year?

Question

BlackTom AI · Accepted Answer

Future value of an ordinary annuity is PMT * [((1 + r)^n − 1) / r], so use PMT = $8,000,000, r = 0.026, n = 12.

1) Relevant concepts:
- An ordinary annuity pays at period ends; its FV formula is FV = PMT * ((1+r)^n − 1)/r.
- Compute the accumulation factor (1+r)^n, subtract 1, divide by r, then multiply by the payment.

2) Step-by-step calculation:
- Compute (1 + r)^n = (1.026)^12.
  Using repeated multiplication:
  1.026^2 = 1.052676
  1.026^3 = 1.0800456
  1.026^4 = 1.1081267856
  1.026^5 = 1.136938102436
  1.026^6 = 1.166498493099336
  1.026^7 = 1.1968274539199187
  1.026^8 = 1.2279449677218366
  1.026^9 = 1.2598715368826036
  1.026^10 = 1.2926281968415513
  1.026^11 = 1.3262365299594316
  1.026^12 = 1.3607186797383768

- Compute ((1+r)^n − 1) = 1.3607186797383768 − 1 = 0.3607186797383768.

- Divide by r: 0.3607186797383768 / 0.026 = 13.87302614301449.

- Multiply by PMT: FV = $8,000,000 * 13.87302614301449 = $110,984,209.14 ≈ $110.9 million.

3) Intermediate steps shown above for the exponentiation, subtraction, division, and final multiplication.

4) Why this answer is correct:
- The winner receives end-of-year payments (ordinary annuity), so the ordinary-annuity FV formula is appropriate. Plugging the given r, n, and PMT yields about $110.984 million, which rounds to $110.9 million.

5) Why the other choices are incorrect:
- $165.2M and $188.8M are substantially larger—these would result from using a much larger interest rate, more periods, or treating payments as invested earlier (annuity due with extra compounding) repeatedly; they do not match the given r and n.
- $118.0M is higher than the correct computed FV and likely comes from rounding errors or miscomputing (for example using a slightly higher r or mis-evaluating (1+r)^n).

6) Conclusion:
The calculated future value is approximately $110.98 million, so the correct choice is $110.9 million, matching the provided answer.

A lottery winner will receive $8 million at the end of each of the next twelve years. What is the future value (FV) of her winnings, given that the interest rate is 2.6% per year?单项选择题

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