Identify the APR type: nominal APR or effective APR
Gather inputs: periodic payment, principal (loan amount), number of payments, payment frequency, and any fees/discounts added to or subtracted from principal
Choose the cash-flow sign convention: inflows to borrower positive, outflows negative (or vice versa, consistently)
If payments are level (ordinary annuity), model the loan as an equation in the periodic interest rate (r):
(PV = sum_{k=1}^{n} dfrac{PMT}{(1+r)^k})
If there are fees/discounts: use (PV = text{net amount received})
Solve for the periodic rate (r) (typically requires numerical methods if APR is not directly given)
Convert periodic rate to nominal APR:
( text{APR}_{text{nominal}} = r times m)
where (m) = number of payment periods per year
Convert periodic rate to effective annual rate (if required):
( text{APR}_{text{effective}} = (1+r)^m – 1)
If the APR is given as a function of time-varying rates or irregular payments, compute via IRR of the full cash-flow schedule and convert to annualized form using the same periodic-to-annual conversion above
For credit cards (common convention), use the periodic rate implied by the balance growth and then annualize:
(r = dfrac{text{interest charged}}{text{balance}times text{days or periods}})
then apply nominal/effective conversions with (m) based on the statement period frequency
Use a financial calculator or software to compute the rate from the payment equation (APR/IRR input)
Report the result with the correct convention (nominal vs effective) and the payment frequency used