Loans
APR as the monthly IRR times twelve
The net cash you receive equals the present value of the payments when discounted at the monthly rate i. There is no closed form; the rate is found by trial or a financial calculator.
The Formula
P - F = sum_{k=1..n} pmt / (1 + i)^k ; APR = 12 x i
The net cash you receive equals the present value of the payments when discounted at the monthly rate i. There is no closed form; the rate is found by trial or a financial calculator.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Principal borrowed | currency | Amount financed before fees. |
| F | Upfront fee | currency | Fee deducted at closing. |
| i | Monthly IRR | rate | Monthly rate solving the equation. |
| n | Number of payments | count | Total monthly payments. |
What Each Symbol Means
Pcurrency
Principal borrowed
Amount financed before fees.
Fcurrency
Upfront fee
Fee deducted at closing.
irate
Monthly IRR
Monthly rate solving the equation.
ncount
Number of payments
Total monthly payments.
See It Worked With Real Numbers
A formula on its own does not tell you which number goes where. The how to calculate apr page applies this equation to a concrete case, shows every intermediate step, and names the mistakes that produce a wrong answer even when the algebra is right.
Where This Formula Appears
This equation is the working core of How To Calculate APR, filed under Finance. Every number the guide produces comes from substituting into this one line.