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I need help restructuring this formula to solve for the payment PMT rather than the Total:

$$ \text{Total} = \text{Compound interest for principal} + \text{Future value of a series} , $$

$$ \text{Total} = P \left( 1+\frac{r}{n} \right)^{nt} + \text{PMT} * \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1) }{\frac{r}{n}} . $$

Where:

  • $A$ = the future value of the investment/loan, including interest
  • $P$ = the principal investment amount (the initial deposit or loan amount)
  • $\text{PMT}$ = the monthly payment
  • $r$ = the annual interest rate (decimal)
  • $n$ = the number of times that interest is compounded per unit t
  • $t$ = the time (months, years, etc) the money is invested or borrowed for
Amaan M
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jeffbartelli
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1 Answers1

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If you know the total, you can just do a bit of algebraic rearranging:

$$ \text{Total} = P \left( 1+\frac{r}{n} \right)^{nt} + \text{PMT} * \frac{\left( 1 + \frac{r}{n} \right)^{nt}-1}{\frac{r}{n}} $$

$$ \Rightarrow \text{Total} - P\left( 1+\frac{r}{n} \right)^{nt} = \text{PMT} * \frac{\left( 1 + \frac{r}{n} \right)^{nt}-1}{\frac{r}{n}} $$

$$ \Rightarrow \frac{r}{n} * \left( \text{Total} - P\left( 1+\frac{r}{n} \right)^{nt} \right) = \text{PMT} * \left(\left( 1 + \frac{r}{n} \right)^{nt}-1\right) $$

$$ \Rightarrow \text{PMT} = \frac{r}{n} * \frac{\text{Total} - P\left( 1+\frac{r}{n} \right)^{nt}}{\left( 1 + \frac{r}{n} \right)^{nt}-1} $$

Amaan M
  • 2,790