Solving for the Rate: Why r = n((A/P)^(1/(nt)) - 1) Breaks in Real Code

Solving for the Rate: Why r = n((A/P)^(1/(nt)) - 1) Breaks in Real Code You rarely need to quote an interest rate. You need to back one out of a number you already have: an advertised APY, a loan's effective cost, a savings target you must hit. That is a different arithmetic problem, and it is where most implementations quietly go wrong. The forward formula and its inverse The compound amount for principal $P$, nominal annual rate $r$, compounded $n$ times per year for $t$ years is: $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$ Every implementation of this is fine. The interesting direction is the inverse, given $A$, $P$, $n$, $t$: $$r = n\left[\left(\frac{A}{P}\right)^{1/(nt)} - 1\right]$$ Derivation is three steps: divide by $P$, take the $(nt)$-th root, subtract 1, multiply by $n$. In code: function impliedRate(P, A, n, t) { if (P