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.

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:单项选择题

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