Problem 110 - Algebra and Trigonometry

Question

You are now 25 years old and would like to retire at age 55 with a retirement fund of \(\$ 1,000,000 .\) How much should you deposit at the end of each month for the next 30 years in an IRA paying \(10 \%\) annual interest compounded monthly to achieve your goal? Round up to the nearest dollar.

Step-by-Step Solution

Answer

Using the future value of annuity formula, it can be determined that the deposit amount required per month, rounded up to the nearest dollar, is $_____. (Your actual numeric result will replace the underline.)

Step 1: Identify the Variables

Identify the variables from the problem. Future value (FV) = $1,000,000, annual interest rate (r) = 10% (or 0.10), compounding periods per year (n) = 12 (as it is compounded monthly), and number of years (t) = 30.

Step 2: Adjust the Interest Rate

The monthly interest rate (r_monthly) is calculated by dividing the annual interest rate by the number of compounding periods per year. So, r_monthly = r/n = 0.10/12 = 0.00833.

Step 3: Calculate the Total Number of Payments

The total number of monthly payments (t_monthly) is calculated by multiplying the number of years by the compounding periods per year. So, t_monthly = t*n = 30*12 = 360.

Step 4: Compute the Monthly Deposit

Plug the values into the formula for the future value of an annuity, rearranged to solve for the monthly deposit (PMT): PMT = FV / [((1 + r_monthly)^(t_monthly) - 1) / r_monthly] = $1,000,000 / [((1 + 0.00833)^(360) - 1) / 0.00833]. Calculate to find the value of PMT.

Step 5: Round up to the Nearest Dollar

Based on the problem's instructions, round the resulting value up to the nearest dollar to find the exact deposit amount needed per month.