Question

A loan of $2100 is to be repaid with quarterly payments for 9 years at 9.8% interest compounded quarterly. Calculate the quarterly payment.

Answer 1

$\bf ~~~~~~~~~~~~ \textit{Amortized Loan Value} \\\\ pymt=P\left[ \cfrac{\frac{r}{n}}{1-\left( 1+ \frac{r}{n}\right)^{-nt}} \right]$

$\bf ~~~~~~ \begin{cases} P= \begin{array}{llll} \textit{original amount deposited}\\ \end{array}\dotfill & \begin{array}{llll} 2100 \end{array}\\ pymt=\textit{periodic payments}\dotfill \\ r=rate\to 9.8\%\to \frac{9.8}{100}\dotfill &0.098\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &9 \end{cases}$

$\bf pymt=2100\left[ \cfrac{\frac{0.098}{4}}{1-\left( 1+ \frac{0.098}{4}\right)^{-4\cdot 9}} \right] \implies pymt=2100\left[ \cfrac{0.0245}{1-\left( 1.0245\right)^{-36}} \right] \\\\\\ pymt\approx 2100\left[ \cfrac{0.0245}{0.5816215} \right]\implies pymt\approx 2100(0.0421236) \\\\\\ pymt\approx 88.45956$