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Question

# Suppose you have a student loan of $\ 30,000$ with an $\mathrm{APR}$ of $9 \%$ for $20$ years.a. What are your required monthly payments?b. Suppose you would like to pay the loan off in $10$ years instead of $20$. What monthly payments will you need to make?c. Compare the total amounts you'll pay over the loan term if you pay the loan off in $20$ years versus $10$ years.

Solution

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Answered 1 year ago
Answered 1 year ago
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$\bold{a.}$ The monthly payments can be calculated using the formula

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

where $\text{PMT}$ is the regular payment amount, $\text{APR}$ is the annual percentage rate in decimal form, $P$ is the amount borrowed, $n$ is the number of payment periods per year, and $Y$ is the loan term in years.

We compute for the monthly payments $PMT$ for a loan with the following conditions:

• $P = \30,000$
• $APR = 9\%$
• $Y = 20$

Substituting these values, we have

\begin{aligned} PMT &=\dfrac{\text{\30,000} \times \left( \dfrac{\text{0.09}}{12} \right)}{\left[1-\left(1 + \dfrac{\text{0.09}}{12} \right)^{(-12 \times 20)}\right] } \\ \\ &= \dfrac{\225}{0.8335872} \\ \\ &= \textcolor{#4257b2}{\269.92} \end{aligned}

The monthly payments would be $\textcolor{#4257b2}{\269.92}$ for the student loan.

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