I'm trying to create three new formulas from a compound interest formula: I'm trying to respectively isolate M, r, and P. The original formula comes from https://www.thecalculatorsite.com/articles/finance/compound-interest-formula.php#regular-contributions, and it's the standard formula for finding future compound interest with an initial investment.
$$ A = P\left(1 + \frac{r}{n}\right)^ {n t} + M\biggl(\frac{\left(1 + \frac{r}{n}\right)^ {n t} – 1}{\frac{r}{n}}\biggr)$$
These are what the variables stand for:
- A = the future value of the investment/loan, including interest
- P = the principal investment amount
- M = the monthly payment
- r = the annual interest rate
- n = the number of times that interest is compounded per unit t
- t = the number of years the money is invested or borrowed for
I contacted the site owner and he wasn't able to help with this, since he's not on the mathematical side of things, he just builds calculators on the back of formulas that already exist. I've also been trying to achieve this in Wolfram Alphra with no luck:
solve P * (1 + r / n) ^ (n * t) + PMT * (((1 + r / n) ^ (n * t) - 1) / (r * n)) = A for r
Any help here would be appreciated! Thanks. It's been too many years since my college math.
Best Answer
There is no problem isolating $P$ or $M$, the equation is linear in these ($a=bP+c\to P=\dfrac{a-c}b$).
On the other hand, there is no analytical solution for $r$ and you have to resort to a numerical approximation approach. Newton's iterations will work fine.
As $(1+\frac rn)^{nt}\approx 1+rt$, a starting approximation is otained by solving
$$ A = P(1+rt) + Mnr$$ for $r$.