Table of Contents
Introduction
An Annual Percentage Rate (APR) is a metric used to compare different loans. The APR is a percentage telling you how much interest will accrue in a year. If you take out a loan for $100 and $20 of interest accrues in a year, you've got a 20% APR. It's not the same as the interest rate, though—APR also includes charges, fees, and some escrow buckets.
This article will go over what APR is and some basics of how it’s calculated. It will also link you to government resources that provide official information about APR.
What Is APR?
In 1968, Congress passed the Truth in Lending Act, which mandated that lenders disclose information about loans to borrowers. One important aspect of that disclosure was APR, which aims to give consumers a simple, accurate way to compare loan rates between lenders. The original APR calculation (now called the actuarial method) can be found in Appendix J of Regulation Z. However, the government now lets lenders calculate APR with the US Rule Method as well, which diminishes APR's value as a onetoone comparison.
Wait, what's the difference between Regulation Z and TILA?
Let's start with a quick refresher on how the US government works. The national government is divided into three branches: Legislative, Executive, and Judicial. The legislative branch is Congress, and they write laws like TILA. The Executive is the President and all the agencies under their control, like the CFPB and FDIC. These agencies create more specific regulations, like Regulation Z, based on the acts passed by Congress. The Judicial branch is the court system, who resolve disputes about laws and regulations.
In short, Congress passed TILA, and then executive agencies codified it into Regulation Z. (And Appendix J is a document attached to Regulation Z.)
Actuarial Method
LoanPro uses the actuarial method to calculate APR. The rules for computing an Actuarial APR, along with numerous examples, are specified in Appendix J of Regulation Z. 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.
US Rule Method
Regulation Z also allows the lender to disclose an APR computed according to the US Rule. This rule forbids the capitalization of interest during the amortization of a loan—principal can accrue interest, but interest can't accrue more interest. It does not, however, specify the calendar method to be used in counting time between cash flows.
Many lenders employ the US Rule method to calculate their interest accrual, but different lenders can use different calendar methods for counting time. If a lender discloses a US 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.
The US Rule method of calculating APR is not outlined in Appendix J because there isn’t a single calculation that will compute an accurate APR in all scenarios.
Actuarial Method Calculation Basics
The actuarial method calculates APR as the interest rate that will cause the present value of future cash flows (payments) on a loan to equal the loan amount. The interest rate is calculated using an iterative method, because there isn’t an easy way to calculate exactly what it should be. Because of this, LoanPro has to guess at the APR and then raise or lower it in order to make the sum of the present value of future cash flows equal the loan amount.
This causes the calculation of truth in lending (TIL) numbers to take longer than you might expect because sometimes it takes many iterations to arrive at the correct APR.
Our amortization software calculations match that of the FFIEC's APR Computational Tool, as recommended by the FDIC and OCC.
How to Calculate APR Unit Periods and Odd Days for Irregular First Periods
If the difference between the contract date and the first due date is not equal to a standard unit period, then the loan has an irregular first period. If this is the case, you will need to calculate the number of unit periods in the term of the loan and the number of odd days in the first period in order to calculate APR. This will have an impact on the disclosed APR. Below are methods for calculating the unit periods and odd days when the loan has an irregular first period. The methods are organized according to unit period length.
Number of Days
If the unit period is number of days such as: Weekly (7), BiWeekly (14), or any custom frequency (X), then the number of unit periods and odd days is easy to calculate. Simply follow these steps:

Step 1 – Take the number of days between the contract date and the first due date.
For example, if you have a contract date of 12/15/2015 and a first due date of 01/15/2016, the number of days between the contract and first payment dates is 33. 
Step 2 – Divide the number from step 1 by the number of days in a unit period.
In our example, suppose the unit period is BiWeekly (14 days). So we divide 33 by 14 (the number of days in a unit period). In this case we get 33 ÷ 14 = 2 remainder 5. This means there are 2 unit periods and 5 odd days.
Monthly
If the unit period is monthly, you can calculate the number of unit periods and odd days in this way:

Step 1 – Take the day of the month from the first due date and work backwards in full unit periods until you reach or cross the contract date. This will equal the number of unit periods.
As an example, if you have a first due date of 12/07 and a contract date of 11/05, work backwards from 12/07 by whole months. There is one whole month between 11/07 and 12/07, so that is 1 unit period. 
Step 2 – Count the number of additional days to get to the contract date after working backwards in unit periods from the first due date. This will equal the number of odd days.
In our example, there are an additional 2 days to get to 11/05 (contract date) from 11/07 (unit period back from first due date). This means there are 1 unit period and 2 odd days.
If the payment falls on the last day of the month, you can calculate whole unit periods by moving back through the last days of the months. So if the first payment falls on the 31st and there is no 31st of the previous month, just move to the last day of that month.
SemiMonthly
For semimonthly unit periods, calculating unit periods and odd days can be done in this way:

Step 1 – Calculate the number of full months between the first due date and the contract date.
For example, if the contract date is 12/15/2015 and the first due date is 01/17/2016, start on the first due date and go back in onemonth increments until you get as close to the contract date as you can without going past it. The date we get to is 12/17/2015, so 1 full month. 
Step 2 – Find the number of days between the date you went back to and the contract date.
In our case the date we went back to is 12/17/2016 and the contract date is 12/15/2016. The number of days between those dates is 2. 
Step 3 – Multiply the number of full months between the contract date and the first payment date by 30.
We had only 1 full month, so the number will be 30. 
Step 4 – Add together the (number of full months x 30) + (number from step 2) to get the APR Days in first Period.
In our case, that is (1 x 30) + 2 = 32 APR Days. (This CAN be different than the ACTUAL number of days in the calendar between the two dates.) 
Step 5 – Take the APR Days in the first period (step 4) and divide by 15.
In our case this will give 32 ÷ 15 = 2 remainder 2. This means we have two unit periods and two odd days.
Negative APR
LoanPro will calculate negative APR. If the APR is negative, it means you have entered amounts and settings that result in a loan that loses money. This means the original amount lent will not be completely returned. LoanPro will display a warning if the APR is negative.
Resources
 https://www.fdic.gov/regulations/laws/rules/65003550.html
 http://www.consumerfinance.gov/eregulations/1026J/201330108_20150718#1026Ja
 http://www.consumerfinance.gov/eregulations/1026J/201330108_20150718#1026Jb4i
 http://www.gpo.gov/fdsys/pkg/CFR2015title12vol3/pdf/CFR2015title12vol3part226appJ.pdf
 http://www.occ.gov/publications/publicationsbytype/comptrollershandbook/truthinlendinghandbook.pdf