Question

# Use the savings plan formula to answer the following questions.At age $35$, you start saving for retirement. If your investment plan pays an $\mathrm{APR}$ of $6 \%$ and you want to have $\ 2$ million when you retire in $30$ years, how much should you deposit monthly?

Solution

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

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

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

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

\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 $\textcolor{#19804f}{\left[\left(1 + \dfrac{\text{APR}}{n} \right)^{(nY)} -1\right] }$.

\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 $\text{PMT}$.

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