If you want to have \$1,000,000 saved at retirement in 30 years and you can earn an annual interest rate of 6\% compounded monthly, how much must you deposit at the end of each month? (Assume payments are made at the end of each period.)

• A. Approximately \$2,778 per month
• B. Approximately \$1,074 per month
• C. Approximately \$1,667 per month
• D. Approximately \$555 per month

Answer: (B)

Verified step by step guidance

1. Step 1: Identify the type of problem. This is a future value of an ordinary annuity problem, where you want to determine the monthly payment required to reach a specific future value.
2. Step 2: Use the formula for the future value of an ordinary annuity: FV = P * [(1 + r/n)^(n*t) - 1] / (r/n), where FV is the future value, P is the monthly payment, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.
3. Step 3: Substitute the known values into the formula: FV = \$1,000,000, r = 0.06 (6%), n = 12 (monthly compounding), and t = 30 years. Rearrange the formula to solve for P (the monthly payment).
4. Step 4: Calculate the effective interest rate per period (r/n) and the total number of periods (n*t). For this problem, r/n = 0.06/12 = 0.005, and n*t = 12 * 30 = 360.
5. Step 5: Solve for P by isolating it in the formula: P = FV / [(1 + r/n)^(n*t) - 1] / (r/n). Plug in the values and simplify step by step to find the monthly payment required.