You’ve decided you’re going to start saving money to eventually put a down payment on a house. You have found an investment account that will pay you 8% interest compounding semi-annually. You determine that you can afford to save $800 every six months, and you decide to save your first $800 today. Approximately how much will you have in your investment account in 5 years? 单项选择题
A
$5,069
B
$6,748
C
$9,605
D
$9,989

Question

You’ve decided you’re going to start saving money to eventually put a down payment on a house. You have found an investment account that will pay you 8% interest compounding semi-annually. You determine that you can afford to save $800 every six months, and you decide to save your first $800 today. Approximately how much will you have in your investment account in 5 years? 单项选择题

A

$5,069

B

$6,748

C

$9,605

D

$9,989

BlackTom AI · Accepted Answer

Because payments are made at the beginning of each semiannual period, this is an annuity-due. Key facts: nominal annual interest = 8% compounded semiannually → periodic rate i = 8%/2 = 4% = 0.04 per period; number of semiannual periods n = 5 years × 2 = 10; payment PMT = $800 at the beginning of each period.

1) Relevant concepts
- Annuity-due future value formula: FV = PMT × [((1+i)^n − 1) / i] × (1 + i).
- If payments were at the end of each period (ordinary annuity), you would not multiply by (1 + i).

2) Step-by-step calculation
- Compute (1 + i)^n = (1.04)^10.
  (1.04)^5 = 1.2166529024, so (1.04)^10 = (1.2166529024)^2 = 1.480244284 (approx).
- Compute ((1+i)^n − 1) = 1.480244284 − 1 = 0.480244284.
- Divide by i: 0.480244284 / 0.04 = 12.0061071.
- Multiply by PMT: 12.0061071 × $800 = $9,604.88568 (this is the future value for an ordinary annuity).
- Convert to annuity-due by multiplying by (1 + i) = 1.04:
  $9,604.88568 × 1.04 = $9,988.08111.

3) Rounding and final amount
- Rounding to the nearest dollar gives $9,989.

4) Why the correct answer is correct
- Because the first payment is made immediately, each payment accumulates one additional period of interest compared with an ordinary annuity; multiplying the ordinary-annuity FV by (1 + i) yields the annuity-due FV of approximately $9,988.08, which rounds to $9,989.

5) Why other options are incorrect
- $9,605 is essentially the ordinary-annuity future value (payments at period end), not the annuity-due (it equals $9,604.89 ≈ $9,605).
- $6,748 and $5,069 are far below the correct accumulation and would correspond to incorrect interest rates, fewer periods, or failing to compound; they do not match the correct annuity-due calculation.

Final confirmation: The correct future value is $9,989, which matches the provided answer.

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