Question

Between the two investment plans listed below, which will have the greatest future value and by what amount? Round all answers to the nearest cent.
A 2-column table with 3 rows. Column 1 is labeled 401 (k) with entries An employee contribution of 9 percent on an annual salary of 45,624 dollars and Employer matches 3 percent of employee contribution, compounded annually at 1.2 percent, annual contributions for 30 years. Column 2 is labeled Roth I R A with entries monthly deposit of 352 dollars and 45 cents, compounded monthly at 1.2 percent, monthly contributions for 30 years.

Answer 1

Assumptions: the 401(k) contributions are made once per year at year-end, the Roth IRA deposits are made monthly at month-end, the quoted 1.2% is an annual nominal rate (compounded annually for the 401(k) and monthly for the Roth), and the employer "matches 3 percent" is interpreted as 3% of salary (a common interpretation).

1) 401(k)
- Employee annual contribution = 9% of 45,624 = 0.09(45,624) = $4,106.16
- Employer annual match = 3% of 45,624 = 0.03(45,624) = $1,368.72
- Total annual contribution C = 4,106.16 + 1,368.72 = $5,474.88
- Annual rate r = 0.012, n = 30
- FV = C * [ (1+r)^n − 1 ] / r
= 5,474.88 * [ (1.012)^30 − 1 ] / 0.012
≈ 5,474.88 * 35.890917
≈ $196,498.49

2) Roth IRA (monthly)
- Monthly deposit P = $352.45
- Monthly rate i = 0.012/12 = 0.001, months N = 360
- FV = P * [ (1+i)^N − 1 ] / i
= 352.45 * [ (1.001)^360 − 1 ] / 0.001
≈ 352.45 * 433.019
≈ $152,617.90

Comparison
- 401(k) FV ≈ $196,498.49
- Roth IRA FV ≈ $152,617.90
- Difference = 196,498.49 − 152,617.90 = $43,880.59

Answer: The 401(k) will have the greater future value by $43,880.59 (rounded to the nearest cent).