Final answer:
To find the effective annual rate of return for a retirement account with monthly deposits and a final balance, one can use the future value annuity formula. The rate is found using financial calculators or iterative methods, then annualized and expressed as a percentage to the nearest 0.1%.
Step-by-step explanation:
To determine the effective annual rate of return for the given retirement account, we need to use the future value formula for an annuity. Since an engineer deposits $680 each month for 30 years and the balance is $1.6 million, we can use the following formula to find the effective annual rate (r):
Future Value of Annuity = Pmt × `((1 + r)^nt - 1)/r`,
where Pmt is the monthly deposit, r is the monthly interest rate, n is the number of times that interest is compounded per year (which is 12 for monthly), and t is the time in years.
Given:
- Pmt = $680
- n = 12 (monthly deposits)
- t = 30 years
- Future Value = $1,600,000
By substituting our values into the formula, we can solve for the rate r using financial calculators or iterative methods, as this requires solving for the interest rate in a compound interest equation which does not have an algebraic solution.
After calculating r, we can then convert it to an annual percentage by multiplying by 12 (since it's compounded monthly) and expressing it as a percentage to the nearest 0.1%