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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

In Exercises 1-10, a. Find the value of each annuity. Round to the nearest dollar b. Find the interest. $$ \begin{array}{|l|l|l|} \hline \begin{array}{l} \$ 4000 \text { at the end of } \\ \text { each year } \end{array} & \begin{array}{l} 5.5 \% \text { compounded } \\ \text { annually } \end{array} & 40 \text { years } \\ \hline \end{array} $$

Suppose that at age 25 , you decide to save for retirement by depositing \(\$ 50\) at the end of each month in an IRA that pays \(5.5 \%\) compounded monthly. a. How much will you have from the IRA when you retire at age 65 ? b. Find the interest.

Describe two advantages of using credit cards.

Solve for \(P\) : $$ A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} . $$ What does the resulting formula describe?

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