You are planning to retire 40 years from now. If your retirement account pays an annual rate of 6% compounded monthly and you start making a monthly contribution of $400 a month today, how much will you have when you retire in 40 years?

Question

BlackTom AI · Accepted Answer

Approaching this retirement savings problem, I’ll lay out the key parameters and compute the future value step by step.
First, identify the interest rate per period: the annual rate is 6%, compounded monthly, so the monthly rate i = 0.06 / 12 = 0.005 (0.5%).
Next, determine the total number of contributions: 40 years of saving with monthly contributions gives n = 40 × 12 = 480 months.
Crucially, the problem states you start contributing today, which means the deposits occur at the beginning of each month. This makes the series an annuity due, not an ordinary annuity. Therefore, after calculating the regular future value of the ordinary annuity, we multiply by (1 + i) to account for the extra compounding on the first payment.
Now apply the annuity-due future value formula: FV = P × [ ((1 + i)^n − 1) / i ] × (1 + i), where P is the monthly payment.
Plug in the numbers: P = 400, i = 0.005, n = 480.
Compute (1 + i)^n: (1.005)^480 ≈ e^(480 × ln(1.005)). Since ln(1.005) ≈ 0.0049875, we get 480 × 0.0049875 ≈ 2.3922, and e^2.3922 ≈ 10.95. So (1.005)^480 ≈ 10.95.
Then ((1 + i)^n − 1) / i ≈ (10.95 − 1) / 0.005 = 9.95 / 0.005 ≈ 1990.
Multiply by (1 + i) to reflect the annuity-due timing: 1990 × 1.005 ≈ 1999.95.
Finally, multiply by the monthly payment: FV ≈ 400 × 1999.95 ≈ 799,980.
With more precise calculation, this value aligns with about $800,579, which corresponds to the option stated in the data.
Note on the missing answer options: since the provided data includes an answer of $800,579 but does not list any alternative choices to compare, I cannot assess which distractors would be correct or incorrect. If other options were available, I would evaluate each one against the same formula and reasoning to explain why some are off due to timing (annuity due vs ordinary annuity), rounding, or misapplication of the rate.
In summary, the core reasoning uses the monthly rate 0.005, 480 periods, and the annuity-due adjustment (1 + i) to reflect starting contributions today, leading to a value near $800k.

You are planning to retire 40 years from now. If your retirement account pays an annual rate of 6% compounded monthly and you start making a monthly contribution of $400 a month today, how much will you have when you retire in 40 years?单项选择题

类似问题

You want to buy a new sports car from Muscle Motors for $54,000. The contract is in the form of a 36-monthannuity due at a 8.25 percent APR. Required: What will your monthly payment be?

Theresa adds $1,500 to her savings account on the first day of each year. Marcus adds $1,500 to his savings account on the last day of each year. They both earn 6.5 percent annual interest. What is the difference in their savings account balances at the end of 35 years?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单