Answer
$PMT = \frac{P~(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}]}$
Work Step by Step
This is the formula we use to calculate the value of an annuity:
$A = \frac{PMT~[(1+\frac{r}{n})^{nt}-1]}{\frac{r}{n}}$
Note that $PMT$ is the monthly payment.
This is the formula we use when we make calculations with compound interest:
$A = P~(1+\frac{r}{n})^{nt}$
Note that $P$ is the amount of the loan.
We can equate $A$ in both equations and solve for $PMT$
$\frac{PMT~[(1+\frac{r}{n})^{nt}-1]}{\frac{r}{n}} = P~(1+\frac{r}{n})^{nt}$
$PMT~[(1+\frac{r}{n})^{nt}-1] = P~(1+\frac{r}{n})^{nt}~(\frac{r}{n})$
$PMT = \frac{P~(1+\frac{r}{n})^{nt}~(\frac{r}{n})}{[(1+\frac{r}{n})^{nt}-1]}$
$PMT = \frac{P~(\frac{r}{n})}{\frac{[(1+\frac{r}{n})^{nt}-1]}{(1+\frac{r}{n})^{nt}}}$
$PMT = \frac{P~(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}]}$
