Question

Use the savings plan formula to answer the following questions.

At age 3535, you start saving for retirement. If your investment plan pays an APR\mathrm{APR} of 6%6 \% and you want to have $2\$ 2 million when you retire in 3030 years, how much should you deposit monthly?

Solution

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Answered 1 year ago
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To solve for the regular payments PMT\textcolor{#19804f}{\bold{PMT}} needed to achieve the financial goal of $2,000,000\textcolor{#19804f}{\$2,000,000}, we derive an equation from the savings plan formula.

A=PMT×[(1+APRn)(nY)1](APRn)\text{A} = \text{PMT} \times \dfrac{\left[\left(1 + \dfrac{\text{APR}}{n} \right)^{(nY)} -1\right] }{\left( \dfrac{\text{APR}}{n} \right)}

where A\bold{A} is the accumulated balance after a certain number of years Y\bold{Y}, PMT\bold{PMT} is the deposited amount on a regular basis, APR\bold{APR} is the annual percentage rate in decimal form, and n\bold{n} is the number of payment periods per year.


First, we multiply both sides by (APRn)\textcolor{#19804f}{\left( \dfrac{\text{APR}}{n} \right)}.

A×(APRn)=PMT×[(1+APRn)(nY)1](APRn)×(APRn)A×(APRn)=PMT×[(1+APRn)(nY)1]\begin{aligned} \text{A} \times\textcolor{#19804f}{\left( \dfrac{\text{APR}}{n} \right)}&= \text{PMT} \times \dfrac{\left[\left(1 + \dfrac{\text{APR}}{n} \right)^{(nY)} -1\right] }{\cancel{\left( \dfrac{\text{APR}}{n} \right)}} \times \cancel{\textcolor{#19804f}{\left( \dfrac{\text{APR}}{n} \right)}} \\ \\ \text{A} \times \left( \dfrac{\text{APR}}{n} \right) &= \text{PMT} \times \left[\left(1 + \dfrac{\text{APR}}{n} \right)^{(nY)} -1\right] \end{aligned}


Then, divide both sides by [(1+APRn)(nY)1]\textcolor{#19804f}{\left[\left(1 + \dfrac{\text{APR}}{n} \right)^{(nY)} -1\right] }.

A×(APRn)[(1+APRn)(nY)1]=PMT×[(1+APRn)(nY)1][(1+APRn)(nY)1]\begin{aligned} \dfrac{\text{A} \times \left( \dfrac{\text{APR}}{n} \right)}{\textcolor{#19804f}{\left[\left(1 + \dfrac{\text{APR}}{n} \right)^{(nY)} -1\right] }} &= \dfrac{\text{PMT} \times \cancel{ \left[\left(1 + \dfrac{\text{APR}}{n} \right)^{(nY)} -1\right]}}{\cancel{\textcolor{#19804f}{\left[\left(1 + \dfrac{\text{APR}}{n} \right)^{(-nY)} -1\right] }}} \end{aligned}

Lastly, interchange the left and right sides to get the equation for PMT\text{PMT}.

PMT=A×(APRn)[(1+APRn)(nY)1]PMT =\dfrac{\text{A} \times \left( \dfrac{\text{APR}}{n} \right)}{\left[\left(1 + \dfrac{\text{APR}}{n} \right)^{(nY)} -1\right] }

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