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Chapter 8: Problem 33

How much should you deposit at the end of each month into an IRA that pays \(6.5 \%\) compounded monthly to have \(\$ 2\) million when you retire in 45 years? How much of the \(\$ 2\) million comes from interest?

Short Answer

Expert verified
First, we use the formula for monthly deposit to calculate the monthly deposit needed for reaching $2 million in 45 years. Then, the total amount invested is calculated by multiplying this monthly deposit by the number of months which is 45 years * 12 months. The interest earned is the difference between the future value ($2 million) and the total amount invested.

Step by step solution

01

Identify the Given Variables

The exercise provides the following information: \n Future Value (FV) = $2 million, \n Annual interest rate (r) = 6.5% or 0.065 annually, \n Number of periods (n) = 45 years, \n Compound frequency (m) = 12 times a year (monthly).
02

Formula for Monthly Deposit

The formula that allows to calculate the monthly deposit into the IRA is: \n \( PMT = \frac{{FV * (r/m)}}{{(1 + r/m)^{n*m} - 1}} \n where PMT is the monthly deposit.
03

Calculate Monthly Deposit

Insert the given parameters into the formula to calculate the monthly deposit needed: \n \( PMT = \frac{{2000000* (0.065/12)}}{{(1 + 0.065/12)^{(45*12)} - 1}} \)
04

Total Amount Invested

The total amount invested over 45 years would be derived by multiplying the monthly deposit by number of months. \n \( Total ~Invested = PMT * n * m \)
05

Calculate the Interest Earned

Finally, calculate the interest earned by subtracting the total amount invested from the future value (2 million). \n \( Interest ~Earned = FV - Total ~Invested \)

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Most popular questions from this chapter

Suppose that you have \(\$ 10,000\) in a rather risky investment recommended by your financial advisor. During the first year, your investment decreases by \(30 \%\) of its original value. During the second year, your investment increases by \(40 \%\) of its first-year value. Your advisor tells you that there must have been a \(10 \%\) overall increase of your original \(\$ 10,000\) investment. Is your financial advisor using percentages properly? If not, what is your actual percent gain or loss of your original \(\$ 10,000\) investment?

In Exercises 11-18, a. Determine the periodic deposit. Round up to the nearest dollar. b. How much of the financial goal comes from deposits and how much comes from interest?$$ \begin{array}{|l|l|l|l|} \hline \$ ? \text { at the end of each month } & 4.5 \% \text { compounded monthly } & 10 \text { years } & \$ 200,000 \\ \hline \end{array} $$

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Suppose that you are thinking about buying a car and have narrowed down your choices to two options: The new-car option: The new car costs \(\$ 28,000\) and can be financed with a four-year loan at \(6.12 \%\). The used-car option: A three-year old model of the same car costs \(\$ 16,000\) and can be financed with a four-year loan at \(6.86 \%\). What is the difference in monthly payments between financing the new car and financing the used car?

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?
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