How this calculation works
Subtract entered withheld fees from the stated principal to obtain net proceeds. Solve the monthly annuity equation using a bounded monotone search, then display nominal and effective annual summaries separately. The present-value residual shows whether the solved rate actually reproduces the proceeds rather than returning an unchecked starting guess.
Net proceeds = payment × [1−(1+r)^−n]/r; nominal annual comparison = 12r; effective annual = (1+r)^12−1A worked example
Receiving 12,000 and repaying twelve monthly payments of 1,000 implies zero interest. If a fee is withheld but those payments stay the same, the net proceeds fall and the implied rate increases. An effective annual equivalent compounds the monthly rate instead of simply multiplying it by twelve.
Good to know
A regular cash-flow rate, not a legally certified APR or a complete jurisdiction-specific credit disclosure. Equal monthly payments start one month after disbursement; no irregular dates, balloon, variable rate, insurance or later fees. The bounded search supports monthly rates from −99% to 100%; unsupported roots produce an error.
Reference: CFPB: compare loan amounts, payments and borrowing costs (opens in a new tab) ↗
A payment amount alone does not identify the borrowing cost
The same scheduled payments can correspond to different costs if the borrower receives different net proceeds. A withheld fee reduces the money available on day one without reducing those later payments. It belongs in the cash-flow equation when you want to compare that scenario.
The effective annual rate calculator converts a stated nominal rate and compounding frequency. Here the rate is unknown and must first be solved from cash amounts. A legally defined APR may impose additional fee-inclusion and timing rules, so the numerical result is deliberately labeled as an implied comparison rate.
Use the residual and the timing assumptions together
A numerical solver should show that the answer satisfies the equation. The residual is modeled payment present value minus net proceeds. A small residual confirms that equation within numerical precision; it does not prove that your entered cash flows match the actual contract.
The zero-interest case is supported rather than rejected. A payment stream smaller than net proceeds can mathematically imply a negative rate, for example under a subsidy assumption. Such a result is a property of the entered cash flows, not confirmation that a negative-rate product exists. The loan payment budget calculator solves a different unknown using a rate you already know.
A few questions, answered
Is the nominal annual number a certified legal APR?
No. It is a regular monthly cash-flow comparison rate; disclosure rules can differ.
What should the pricing residual look like?
It should be near zero within numerical precision if the solved rate reproduces the entered proceeds.