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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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Most popular questions from this chapter

You decide to work part-time at a local veterinary hospital. The job pays \(\$ 9.50\) per hour and you work 20 hours per week. Your employer withholds \(10 \%\) of your gross pay for federal taxes, \(5.65 \%\) for FICA taxes, and \(5 \%\) for state taxes. a. What is your weekly gross pay? b. How much is withheld per week for federal taxes? c. How much is withheld per week for FICA taxes? d. How much is withheld per week for state taxes? e. What is your weekly net pay? f. What percentage of your gross pay is withheld for taxes? Round to the nearest tenth of a percent

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The price of a condominium is \(\$ 180,000\). The bank requires a \(5 \%\) down payment and one point at the time of closing. The cost of the condominium is financed with a 30 -year fixed-rate mortgage at \(8 \%\). a. Find the required down payment. b. Find the amount of the mortgage. c. How much must be paid for the one point at closing? d. Find the monthly payment (excluding escrowed taxes and insurance). e. Find the total cost of interest over 30 years.

Describe how to find percent decrease and give an example.

Exercises 19 and 20 refer to the stock tables for Goodyear (the tire d. How many shares of this company's stock were traded company) and Dow Chemical given below. In each exercise, use yesterday? the stock table to answer the following questions. Where necessary, e. What were the high and low prices for a share yesterday? round dollar amounts to the nearest cent. f. What was the price at which a share last traded when the stock a. What were the high and low prices for a share for the past exchange closed yesterday? b. If you owned 700 shares of this stock last year, what dividend g. What was the change in price for a share of stock from the did you receive? h. Compute the company's annual earnings per share using c. What is the annual return for the dividends alone? How does Annual earnings per share this compare to a bank offering a \(3 \%\) interest rate? $$ =\frac{\text { Yesterday's closing price per share }}{P E \text { ratio }} . $$ $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { 52-Week High } & \text { 52-Week Low } & \text { Stock } & \text { SYM } & \text { Div } & \text { Yld \% } & \text { PE } & \text { Vol 100s } & \text { Hi } & \text { Lo } & \text { Close } & \text { Net Chg } \\ \hline 73.25 & 45.44 & \text { Goodyear } & \text { GT } & 1.20 & 2.2 & 17 & 5915 & 56.38 & 54.38 & 55.50 & +1.25 \\ \hline \end{array} $$
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