(4 marks, difficulty level: Easy) Ellen just turned 30 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 65, she will save $10,000 in an RRSP account. Although $10,000 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 4% per year. If the account earns 9% per year, how much will Ellen have saved at age 65?

Question

BlackTom AI · Accepted Answer

We start by restating the problem in our own words to ensure clarity: Ellen will make a sequence of annual savings beginning one year from now (at age 31) and continuing on each birthday until she turns 65, with the first-year deposit capped at 10,000 and subsequent deposits growing by 4% each year. The account earns 9% annually. We want the value of all deposits at age 65.

First, determine how many deposits there are. From age 31 through age 65 inclusive gives 35 deposits (one for each birthday from 31 up to and including 65). The payment in year n (counting from the first deposit at age 31 as n=1) is P_n = 10,000 × 1.04^{n-1}.

Next, express the future value at age 65 of all deposits. Each deposit P_n earns interest at 9% for (N − n) years, where N = 35 is the total number of deposits. Therefore the future value is:
FV = sum_{n=1}^{35} [10,000 × 1.04^{n-1} × 1.09^{35−n}].

A convenient closed form is to factor and recognize a geometric series. Let i = 0.09 (9%) and g = 0.04 (4%). Then FV = 10,000 × (1.09)^{34} × sum_{k=0}^{34} [(1.04/1.09)^k], where k = n−1. Here r = (1.04)/(1.09) ≈ 0.954128.

Compute the geometric sum: sum_{k=0}^{34} r^k = (1 − r^{35})/(1 − r).
- r ≈ 0.954128, so 1 − r ≈ 0.045872.
- r^{35} ≈ exp(35 × ln(0.954128)) ≈ exp(35 × (−0.04696)) ≈ exp(−1.6436) ≈ 0.193.
Thus sum ≈ (1 − 0.193)/0.045872 ≈ 0.807/0.045872 ≈ 17.58.

Now compute (1.09)^{34}. Since ln(1.09) ≈ 0.08618, 34 × ln(1.09) ≈ 2.931, so (1.09)^{34} ≈ e^{2.931} ≈ 18.75.

Putting it together: FV ≈ 10,000 × 18.75 × 17.58 ≈ 10,000 × 329.25 ≈ 3,292,500. This is very close to the choice around 3.29 million.

Now evaluate the given options:
- Option A: $4,135,023. This is notably larger than our calculated FV; it would require either more deposits, larger growth, or more years of compounding than specified. The 9% return and 4% growth in contributions do not support such a high value here.
- Option B: $3,784,455. This is closer to the ballpark but still higher than the computed estimate; it would imply a higher effective accumulation factor than produced by the combination i = 9% and g = 4% over 35 deposits.
- Option C: $3,293,576. This aligns with the computed approximation (roughly 3.293 million) using the closed-form geometric-sum approach and matches the magnitude from the calculation.
- Option D: $3,714,071. This is another overestimate relative to the derived value and would require a larger contribution path or interest effect than the problem specifies.

In summary, comparing the calculated magnitude to each option shows that Option C is the one that matches the expected future value given a 9% return and 4% annual growth over 35 deposits with the initial 10,000 cap.

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

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