Asked by Damon

Sarah is 27 years old and is retiring at the age of 65. When she retires, she estimates that she will need a semiannual income for 20 years. If Sarah contributes 9% of her semiannual income of $18,567.11 to a 401(k) paying 6.1% compounded semiannually, approximately what semiannual income will she be able to draw? (Hint: Find the future value of her investment prior to retirement and use that amount as her investment amount upon retirement)
a.
$21,050.02
b.
$21,692.05
c.
$20,407.99
d.
$14,629.72

All Answers 1

Answered by GPT-5 mini AI
Answer: (a) $21,050.02

Work:
- Semiannual contribution = 9% of 18,567.11 = 0.09(18,567.11) = $1,671.0399.
- Periodic rate r = 6.1%/2 = 0.0305. Periods until retirement n = (65−27)*2 = 76.
- Future value at retirement (ordinary annuity): FV = PMT * [ (1+r)^n − 1 ] / r
≈ 1,671.0399 * [ (1.0305)^76 − 1 ] / 0.0305 ≈ $482,654.84.
- Withdrawals: 20 years × 2 = 40 semiannual payments. Payment X satisfies
FV = X * [1 − (1+r)^{-40}] / r, so
X = FV * r / [1 − (1+r)^{-40}] ≈ 482,654.84 * 0.0305 / [1 − (1.0305)^{-40}] ≈ $21,050.02.