The rules for computing an Actuarial APR are specified in Appendix J to Regulation Z, along with numerous samples. These rules include a special calendar method, commonly called the "Federal Calendar", for counting time between cash flows, as well as how loans are to amortize. We could calculate payments for a given loan using Actuarial APR as the interest rate, following the calendar and amortization rules that define the method. Even so, these payments would still differ from those computed by the standard interest methods in use today. You see lenders do not generally use the Actuarial method in computing loan payments. It is for this reason that we cannot expect the disclosed actuarial APR to match the interest rate, for the vast majority of loans.
United States (U.S.) Rule method
Regulation Z also allows the lender to disclose an APR computed according to the U.S. Rule. This rule forbids the capitalization of interest during the amortization of a loan, but does not specify the calendar method to be used in counting time between cash flows. Many lenders employ the U.S. Rule method to calculate their interest accrual, but different lenders can use different calendar methods for counting time. If a lender discloses a U.S. Rule APR and uses the same calendar method to compute interest, the APR and interest rate should be the same, providing there are no contributions to finance charge other than interest.
So, to summarize, for the majority of lenders that disclose an APR computed by the actuarial method, it will be extremely rare, in fact only coincidental, for the APR to match the interest rate. But those lenders that accrue interest by the U.S. Rule method, and who choose to compute the APR by the same method, will have agreement between interest rate and APR for those loans where the entire finance charge is due to interest.