(4 marks, difficulty level: Easy) Ellen just turned 35 years old, and she has decided it is time to plan seriously for her retirement. On each birthday, beginning in one year and ending when she turns 60, she will save $13,000 in an RRSP account. Although $13,000 is the most she can save in the first year, she expects her salary to increase each year so that she will be able to increase her savings by 6% per year. If the account earns 15% per year, how much will Ellen have saved at age 60?

Question

(4 marks, difficulty level: Easy) Ellen just turned 35 years old, and she has decided it is time to plan seriously for her retirement. On each birthday, beginning in one year and ending when she turns 60, she will save $13,000 in an RRSP account. Although $13,000 is the most she can save in the first year, she expects her salary to increase each year so that she will be able to increase her savings by 6% per year. If the account earns 15% per year, how much will Ellen have saved at age 60?

BlackTom AI · Accepted Answer

We need to restate the problem in our own words and then evaluate each answer choice.

Question restated: Ellen will make a sequence of annual savings in an RRSP starting one year from now, with the first year’s contribution of 13,000 and each subsequent year increasing by 6%. She will contribute each birthday from age 36 up to and including age 60 (a total of 25 contributions). The account earns 15% per year. What is the accumulated value of all contributions at age 60 (i.e., immediately after the 60th birthday contribution, with no further growth after that)?

Option by option analysis:

Option A: $3,293,576
- This figure is notably lower than the expected magnitude of a 25-year series of growing deposits at a high rate (15% return). If we approximate the future value using the method below, a value in the low 3 millions would require either many years of much smaller growth or a much smaller interest/annuity combination. The calculation indicates the geometric growth effect and the 15% return are not being captured fully here, so this option likely undervalues the accumulation.

Option B: $4,135,023
- This matches the scale we expect when summing a 25-term growing annuity with initial payment 13,000, growing at 6% per year, and accumulating at 15% annual interest to the final date. A compact way to compute is to factor the final-year accumulation and use the ratio of growth rates:
  - Let the first payment be 13,000 at t=0 and grow by 6% each year: 13,000*(1.06)^t for t=0..24.
  - Let the accumulation factor be (1.15)^{24-t} for the payment made at year t to reach year 60.
  - This yields S = 13,000*(1.15^{24}) * sum_{t=0}^{24} (1.06/1.15)^t.
  - With r = 1.06/1.15 ≈ 0.9217, the sum is (1 - r^{25})/(1 - r) ≈ (1 - 0.130)/(1 - 0.9217) ≈ 0.870/0.0783 ≈ 11.12, and 1.15^{24} ≈ 28.66. Therefore S ≈ 13,000 * 28.66 * 11.12 ≈ 4,140,000, which is consistent with the provided 4,135,023 when computed precisely. This option aligns with the standard formula for a growing annuity with a high interest rate and a modest growth in payments.

Option C: $3,784,454
- This value is below the estimate we just calculated (around 4.14 million). It suggests either a smaller final growth factor, a smaller number of contributions, or a misapplication of the growing annuity formula. Given the 25 deposits and the strong 15% return, this result underestimates the compounding impact and the effect of the 6% growth in payments, making it an unlikely correct value.

Option D: $3,714,071
- Similar to option C, this figure is also below the expected ballpark for a 25-year sequence of growing deposits at 15% interest. It might correspond to an arithmetic misstep, such as treating the deposits as level (not growing) or incorrectly accounting for the timing of the contributions. The growth of payments and the high rate both push the final amount higher than this option.

Summary of reasoning:
- The problem uses a 25-term growing annuity: payments start at 13,000 and grow by 6% each year, with each payment accumulating at 15% to age 60.
- The correct approach is to factor the final-year accumulation and sum a geometric series with ratio 1.06/1.15, yielding a result around 4.135 million.
- Options A, C, and D are notably lower than the computed magnitude, indicating they undervalue either the growth in payments, the compounding effect, or both, whereas option B aligns with the correct mathematical outcome for the scenario described.

FINE 2000 A, B & C Midterm Quiz-All Sections - F2025- Requires Respondus LockDown Browser

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