quartely payment

Question

A Loan of R65 000 with an interest rate of 16% per annum compounded quartely is to be amortised by equal quartely payments over 3 years

Question : how do I calculate the size of the quartely payment?

Will I = 0,016/4 and will n =36 months (3years)?

Answer 1

For this question you can use the formula $A=P(1+\frac rn)^{nt}$. Where $P$ is your loan amount, $r$ is the rate, $n$ is the number of times compounded per year, and $t$ is the number of years.

In your case you would use $65000(1+\frac{.16}{4})^{4(3)}$.

I hope this helps.

Answer 2

$\displaystyle{% r \equiv 3\times\left(16/1200\right).\quad x_{0} \equiv 65,000.\quad n \equiv 12.\quad x \equiv \mbox{quarterly payment}\ =\, ?}$

\begin{align} &\\[5mm] x_{1} &= x_{0}\left(1 + r\right) - x \\[1mm] x_{2} &= x_{1}\left(1 + r\right) - x = x_{0}\left(1 + r\right)^{2} - x\left(1 + r\right) - x \\[1mm] x_{3} &= x_{2}\left(1 + r\right) - x = x_{0}\left(1 + r\right)^{3} - x\left(1 + r\right)^{2} - x\left(1 + r\right) - x \\[1mm]\vdots &= \vdots \\ x_{n} &= x_{0}\left(1 + r\right)^{n} - x\sum_{k = 0}^{n - 1}\left(1 + r\right)^{k} = x_{0}\left(1 + r\right)^{n} - x\,{\left(1 + r\right)^{n} - 1 \over r} \end{align}

$x_{n} = 0\quad\Longrightarrow$

$$ \begin{array}{|c|}\hline\\ \color{#ff0000}{ {\large% x = {r \over 1 - \left(1 + r\right)^{-n}}\ x_{0}}} \\ \\\hline \end{array} $$