Both methods count backwards in unit periods from the end date to the start date.
Months are considered as being equal to 1/12th of a year, regardless of the number of days in a month.
If there is not a whole number of unit periods between the end and start date, there will be a so called stub period for the remainder.
From here on things are different...
Unit periods
The EU method uses only three unit periods: week, month and year.
The US method defines standard intervals as a day, week, semimonth, month, or a multiple of a week or month up to, but not exceeding one year.
The most commonly used unit periods (or standard intervals) are weekly, biweekly, four-weekly, semimonthly, monthly, bimonthly, quarterly, semiannually and annually.
Note: According to the US definition, you could also have other common intervals such as three-weekly, five-weekly, four-monthly, five-monthly, and so on. The software only uses the most commonly used unit periods.
Stub periods
When there is a stub period, odd days are handled differently in each method.
The EU method counts the actual number of days in the stub period and divides it by 365 or 366 depending on whether there is a leap day in the year before the start of the regular periods.
The fraction for a single day would be either 1/365 or 1/366.
In the US method the days fraction is based on the assumed number of days in a unit period and the number of unit periods per year.
For month based unit periods the fraction for a single day used is 1/30 for monthly, 1/60 for bimonthly, 1/90 for quarterly, 1/180 for semiannually and 1/360 for annually.
Multiplied by the frequency this will always result in 1/360, as multiplying the number of days in a month based period by its frequency (30 x 12, 60 x 6, 90 x 4, 180 x 2 or 360 x 1) always results in a 360 denominator.
Note: The stub period itself is subdivided in whole months and odd days, not the actual number of days.
Example: Using semiannual periods as the unit period, the interval between November 30th, 2024 to May 1st, 2025 is calculated as 1 day + 5 months of 30 days or 151 days, not 1 + 31 + 31 + 28 + 31 + 30 days or 152 days.
A special provision is made for single advance - single payment transactions with a term that is less than one year.
If the stub period can not be expressed as a number of whole months, the denominator is 365 otherwise the denominator is 360.
Note: Appendix J also provides for the whole months case to use a denominator of 365 divided by the number of days in the term. The software does not calculate for that option as it is not clear whether it means using actual days or standard month days.
For semimonthly or half month unit periods the fraction for a single day will be 1/15.
Multiplied by the frequency this will also result in 1/360, as multiplying 15 by 24 results in 360.
For week based unit periods the fraction for a single day used is 1/7 for weekly, 1/14 for biweekly and 1/28 for four-weekly.
When multiplied by the frequency this will result in 1/364, as multiplying the number of days in a period by its frequency (7 x 52, 14 x 26 or 28 x 13) always results in 364.
Calculating the NPV
Effective annual rate vs nominal annual rate
Both methods calculate their APR by solving an equation in which the net present value (NPV) of all advances is equal to the NPV of all payments.
Where they differ is in how they actually calculate the net present value of each data item.
The EU method solves for the annual rate directly, and so the result is the effective annual rate.
The US method solves for the periodic rate for the used unit period, and multiplies this periodic rate by the number of unit periods per year, thus resulting in the nominal annual rate.
If you import one of the non-year based examples for either the EU or US method, you’ll notice that there is a significant difference between the APR that each method produces.
As the U.S. method produces nominal annual rates that correspond to the payment frequencies, you may still need to convert them to the corresponding annual rate to be able to compare them effectively.
Just using the nominal annual rate as it is calculated in the US method is not sufficient to compare offers with different unit periods.
For your convenience, APR Calculator also calculates the effective annnual rate based on the US unit period and the calculated periodic rate, it is displayed beneath the APR result.
EXAMPLE
For US example 1.1 Monthly payments (regular periods) the US APR is 9.69 %, which is the solved periodic rate of 0.80729233 % multiplied by 12 to obtain 9.687508 %, which is rounded to 9.69 %.
The effective annual rate as is calculated as:
(1 + 0.80729233 %) ^ 12 - 1 = 10.127465 %
Suppose you obtain the same 9.69 % US APR for a loan with semiannual payments. The periodic rate is 9.69 % / 2 or 4.85 %, and the effective annual rate becomes:
(1 + 4.85 %) ^ 2 - 1 or 9.92474 %.
Even though both loans have the same nominal annual rate, their effective annual rate is quite different!
Compound interest vs compound interest + simple interest
The two methods also use different methods to calculate interest over stub periods.
The EU method uses compound interest on both the periods and the stub periods. It adds the period fraction to the stub fraction to obtain a year fraction and uses that to calculate interest.
The US method on the other hand calculates interest over the periods fraction as compound interest and interest over the stub fraction as simple interest.