An annuity investment is formed when $500,000 is invested at a rate of interest of 3.1% per annum, compounded monthly.An additional payment of $1500 is made into the account each month. The value of the annuity after 5 years is closest to:

Question

BlackTom AI · Accepted Answer

We start by identifying the two components that contribute to the future value: the initial lump-sum of 500,000 growing at the monthly compounded rate, and the series of monthly payments of 1,500.

First, compute the monthly interest rate: i = 3.1% per year compounded monthly -> i = 0.031 / 12 ≈ 0.0025833. The number of months is n = 5 years × 12 = 60.

1) Future value of the initial investment:
FV_initial = 500,000 × (1 + i)^n = 500,000 × (1.0025833)^{60}.
Using the approximation (1 + i)^n ≈ e^{n·ln(1 + i)} with ln(1.0025833) ≈ 0.002579, we get (1 + i)^60 ≈ e^{0.15474} ≈ 1.1673. Therefore, FV_initial ≈ 500,000 × 1.1673 ≈ 583,650.

2) Future value of the monthly payments (an ordinary annuity, payments at end of each month):
FV_payments = 1,500 × [((1 + i)^n − 1) / i].
We have (1 + i)^n − 1 ≈ 1.1673 − 1 = 0.1673. Divide by i: 0.1673 / 0.0025833 ≈ 64.8. Then FV_payments ≈ 1,500 × 64.8 ≈ 97,200.

3) Total future value after 5 years:
FV_total ≈ FV_initial + FV_payments ≈ 583,650 + 97,200 ≈ 680,850.

Now compare to the given options:
- Option A: $680,927 — very close to the calculated value; within rounding error from the approximations above.
- Option B: $3,376,192 — far too large; this would require much larger growth or deposits, not supported by the data.
- Option C: $486,498 — significantly smaller; would imply much less accumulation than computed, likely underestimating the annuity component.
- Option D: $514,031 — also too small; closer to the initial lump-sum value but does not reflect the substantial accumulation from the monthly payments.

Thus, the correct ballpark is near $680k, with Option A being the closest estimate.

MUF0141 Fund. Mathematics Unit 1 - Semester 2, 2025

An annuity investment is formed when $500,000 is invested at a rate of interest of 3.1% per annum, compounded monthly.An additional payment of $1500 is made into the account each month. The value of the annuity after 5 years is closest to:

查看解析

登录即可查看完整答案

类似问题

Which type of annuity has payments that occur at the beginning of the period?

Which of the following best describes the difference between an annuity due and an ordinary annuity?

Which of the following statements is correct?

(4 marks, difficulty level: Easy) Suppose that you invest $28,000 in an account paying 8% interest. You plan to withdraw $2300 at the end of each year for 20 years. How much money will be left in the account after 20 years?

An ordinary annuity is best defined as:

An ordinary annuity is best defined as:

更多留学生实用工具

考试浏览器助手

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