Final answer:
Marissa's retirement account balance after 40 years would be $223,012.04, calculated by using the compound interest formula for the initial deposit and the future value of an annuity formula for the monthly contributions.
Step-by-step explanation:
To calculate Marissa’s retirement account balance after 40 years with an initial deposit of $1,000 and monthly contributions of $100 at an annual interest rate of 2%, we must use the future value of an annuity formula for the monthly deposits and combine that with the future value of a single lump sum for the initial deposit.
First, we calculate the future value of the initial $1,000 deposit using the compound interest formula:
- Future Value = Present Value * (1 + interest rate)^number of periods
- Future Value = $1,000 * (1 + 0.02)^40
- Future Value = $1,000 * (1.02)^40
- Future Value = $1,000 * 2.20804
- Future Value = $2,208.04
Next, calculate the future value of annuity for the monthly contributions:
- Future Value of Annuity = Contribution * [(1 + r)^nt - 1] / r
- Here, r = monthly interest rate, n = number of times the interest is compounded per year, and t = number of years.
- Monthly interest rate (r) = 2% annual interest / 12 months = 0.00166667
- Compounding frequency (n) = 12
- Time in years (t) = 40
- Future Value of Annuity = $100 * [(1 + 0.00166667)^(12*40) - 1] / 0.00166667
- Future Value of Annuity = $100 * [(1.00166667)^480 - 1] / 0.00166667
- Future Value of Annuity = $100 * 2.20804 = $220,804
Add the future value of the initial deposit to the future value of the annuity to get Marissa’s total retirement savings:
- Total Retirement Savings = Future Value of initial deposit + Future Value of Annuity
- Total Retirement Savings = $2,208.04 + $220,804
- Total Retirement Savings = $223,012.04
Therefore, after 40 years, Marissa’s retirement account balance would be $223,012.04.