(4 marks, difficulty level: Easy) Your grandmother bought an annuity from Manulife Financial for $400,000 when she retired. In exchange for the $400,000, Manulife will pay her $60,000 per year until she dies. The interest rate is 6%. How long must she live after the day she retired to come out ahead (that is, to get more in value than what she paid in)?

Question

(4 marks, difficulty level: Easy) Your grandmother bought an annuity from Manulife Financial for $400,000 when she retired. In exchange for the $400,000​, Manulife will pay her $60,000 per year until she dies. The interest rate is 6%. How long must she live after the day she retired to come out ahead​ (that is, to get more in value than what she paid​ in)?

BlackTom AI · Accepted Answer

The problem asks how long she must live after retirement for the total value of the payments to exceed the amount paid for the annuity, given a 6% annual return and payments of 60,000 per year. First, set up the present value of an ordinary annuity (payments at the end of each year): PV = 60,000 × a-angle-n, where a-angle-n = (1 − (1.06)^(−n)) / 0.06. We want PV > 400,000, so solve 60,000 × (1 − (1.06)^(−n)) / 0.06 > 400,000. Dividing both sides by 60,000 gives (1 − (1.06)^(−n)) / 0.06 > 6.666..., which leads to 1 − (1.06)^(−n) > 0.4, i.e., (1.06)^(−n) < 0.6. Taking natural logs, −n·ln(1.06) < ln(0.6). Since ln(1.06) > 0, this gives n > −ln(0.6)/ln(1.06). Numerically, −ln(0.6) ≈ 0.5108 and ln(1.06) ≈ 0.0583, so n > 0.5108 / 0.0583 ≈ 8.76. Therefore, she would need to survive a little over 8.7 years for the value of the payments to surpass the initial 400,000, so in whole years she must live at least 9 years to come out ahead. Now evaluate the provided options one by one: - 11 years: This is longer than required; if she lives 11 years, the cumulative payments clearly exceed the purchase price, but the question is about the threshold when the value first surpasses 400,000. Since 8.76 years is the break-even point, 11 years would be well beyond break-even, which is compatible but not the earliest threshold. - 10 years: This is slightly above the break-even point (8.76 years), so living 10 years would indeed yield more value than paid, but it is not the smallest integer year above break-even. - 9 years: This is the smallest integer year above 8.76, so after 9 full years the total value of payments crosses the 400,000 threshold, meeting the condition to come out ahead. - 8 years: This is below the break-even point of approximately 8.76 years, so at 8 years the payments would not yet have accumulated enough value to exceed the 400,000 purchase price. In summary, the key idea is solving for the break-even point with the annuity present value formula and then mapping that to whole-year outcomes to identify the earliest year that surpasses the initial cost. The option corresponding to 9 years aligns with that break-even calculation and represents the threshold just after which the payment stream has accrued enough value.

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