Assume you are 25 and started a job. You will work until 60 and live until 90. You can earn 6% EAR per year on your investments during your lifetime. Assume you are saving 10% of your salary of $8,000 per month, hoping to have at least $1,000,000 saved up by retirement.

How much you will have saved up by retirement at age 60? $1,098,883.18

Will you have met your goal of saving at least $1,000,000? Yes, you will have actually saved even more

Question

Assume you are 25 and started a job. You will work until 60 and live until 90. You can earn 6% EAR per year on your investments during your lifetime. Assume you are saving 10% of your salary of $8,000 per month, hoping to have at least $1,000,000 saved up by retirement.

How much you will have saved up by retirement at age 60? $1,098,883.18

Will you have met your goal of saving at least $1,000,000? Yes, you will have actually saved even more

BlackTom AI · Accepted Answer

We start by identifying the key components of the problem and the two chosen options.

Option 1: "$1,098,883.18"
- The person saves 10% of a monthly salary of $8,000, which equals $800 per month.
- This monthly saving is invested at an EAR of 6% per year. To handle monthly contributions with an annual rate, we can approximate using an equivalent monthly rate i where (1+i)^12 = 1.06. Solving gives i ≈ 0.004867 (about 0.4867% per month).
- The investment horizon is from age 25 to 60, which is 35 years or 420 months.
- The future value of a regular monthly deposit P over n periods with monthly rate i is FV = P * [((1+i)^n - 1) / i]. Plugging in P = 800, i ≈ 0.004867, n = 420, we get FV ≈ 800 * [(1.004867^420 - 1) / 0.004867].
- Compute (1.004867^420) ≈ 1.06^35 ≈ about 7.68. Then (7.68 - 1) ≈ 6.68. Divide by i: 6.68 / 0.004867 ≈ 1,370–1,373. Multiply by 800 gives roughly $1,096,000 to $1,098,000 depending on precise rounding. The provided figure $1,098,883.18 is within this rounded range and aligns with the standard formula for the future value of a monthly ordinary annuity at the given rate.
- Therefore, the computed amount of about $1.099 million is consistent with the inputs and method, supporting the value $1,098,883.18 as plausible.

Option 2: "Yes, you will have actually saved even more"
- If the ending balance is around $1.098 million, that is indeed greater than the $1,000,000 goal. So the statement that you will have saved even more than the goal is true in this calculation.
- However, to assess correctness, we should note that the conclusion rests on the same assumptions used to compute the FV: a 6% EAR, monthly contributions of $800, 35 years, and the same compounding approach. If any of these inputs were different (for example, a lower return, changes in savings rate, or a shorter/longer horizon), the conclusion could change. Given the numbers as stated, the result exceeding $1,000,000 makes the claim accurate.
- In summary, under the stated assumptions, the projective FV is about $1.099 million, which indeed surpasses the $1,000,000 goal, so the assertion that you will have saved even more is correct.

Notes on potential sources of confusion or error:
- Some solutions use a monthly effective rate derived from the annual rate; small differences in rounding of i can lead to slightly different FV figures, but they should stay near the million-dollar mark and typically support the conclusion that the goal is met.
- The problem uses a single consistent set of assumptions (salary, savings rate, return, time horizon). If any assumption changes, recalc would be required.

COMM_V 298 201-207 2024W2 Class 6: Valuing Loans Practice Quiz

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