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Chapter 8: Problem 34

How much should you deposit at the end of each month into an IRA that pays \(8.5 \%\) compounded monthly to have \(\$ 4\) million when you retire in 45 years? How much of the \(\$ 4\) million comes from interest?

Short Answer

Expert verified
The monthly deposit should be about \$946.86. Of the \$4 million, \$2,890,080 comes from interest.

Step by step solution

01

Understand the Variables

Here, the interest rate \( r = 8.5\% \) annually, compounded monthly, i.e., \( r = \frac{8.5\%}{12} \) monthly. Time \( t = 45 \) years, i.e., \( t = 45 * 12 \) months. The future value \( FV = \$4 \) million.
02

Calculate Monthly Rate and Period

Next, convert the annual interest rate into a monthly interest rate and the time period into month. Let, \( i = \frac{r}{100 * 12} \) and \( n = 12 * t \). Thus, \( i = \frac{8.5}{100 * 12} = 0.00708333 \) and \( n = 12 * 45 = 540 \) months.
03

Calculate Monthly Deposit

Now, calculate the amount to be deposited each month \( PMT \) using the formula for the Future Value of a series of payments i.e., \( PMT = \frac{FV * i}{((1+i)^n - 1)} \). So, \( PMT = \frac{4000000 * 0.00708333}{((1+0.00708333)^{540} - 1)} = \$946.86 \)
04

Calculate Amount of Interest

Finally, the amount that comes from interest is the difference between the future value and the sum of all deposits made over time. Therefore, the total savings are \( PMT * n \) and the amount from interest \( MI \) is \( FV - PMT * n \). Therefore, \( MI = 4000000 - 946.86 * 540 = \$2890080 \)

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