Day count conventions

Day count conventions specify how to count the number of days between two dates and how to calculate the size of an interest period when the number of days is a fraction of a normal period.

Day count conventions differ in assumptions on the number of days in a year as well as the number of days in a month. The notation used is d / y where:

the numerator d is the assumed number of days in a month
the denominator y is the assumed number of days in a year

Day count conventions can be divided in three groups:

the first group contains conventions that are based on the actual number of days between two dates, with or without adjustments for leap days
the second group contains conventions that are based on 30-day months and 360-day years, with or without adjustments when a date falls on the 31th or when it is the last day of February
a third group is based on the actuarial method used in calculations to determine the APR of a loan

Although the rules seem simple, many different interpretations have lead to a large number of slightly different conventions. To add to the confusion, many different names have been used for the same convention.

The following table lists all the abbreviations and their corresponding day count conventions we encountered. ParanzaSoft accepts no liability for any inaccuracies or errors it might contain.

Group 1: day count conventions based in the actual number of days

Convention	Alternative names	Rules
act/act	actual/actual actual/actual (Historical) actual/actual (ISDA)	This method splits up the actual number of days falling in leap years and in non-leap years. The year fraction is the sum of the actual number of days falling in leap years divided by 366 and the actual number of days falling in non-leap years divided by 365. ISDA = International Swaps and Derivatives Association
act/act (Euro)	actual/actual (Euro) actual/actual (AFB) actual/365 (Actual)	This method first calculates the number of full years counting backwards from the second date. For any resulting stub periods, the numerator is the actual number of days in the period, the denominator being 365 or 366 depending on whether February 29th falls in the stub period. AFB = Association Française de Banques
act/365 F	actual/365 fixed English	The numerator is the actual number of days between the two dates. The denominator is always 365 days.
act/365 L	actual/365 leap actual/365L	The actual number of days is used as the numerator. If the second date falls in a leap year, the denominator is 366, otherwise it is 365.
NL/365	actual/365 no leap NL365	The actual number of days minus the number of leap days is used as the numerator. The denominator is always 365 days.
actual/360	actual/360 French	The actual number of days between two dates is used as the numerator. The denominator is always 360 days.

Group 2: based on 30 day months and 360 day years

The year fraction is calulated based on a 360 day year with 30-day months, after applying the following rules:

Convention	Alternative names	Rules
30/360	30/360 Bond Basis	If the first date falls on the 31st, it is changed to the 30th. If the second date falls on the 31th, it is changed to the 30th, but only if the first date falls on the 30th or the 31st.
30E/360	30/360 ISMA 30/360 European 30S/360 Special German Eurobond Basis Special German	If the first date falls on the 31st, it is changed to the 30th. If the second date falls on the 31th, it is changed to the 30th.
30E+/360		If the first date falls on the 31st, it is changed to the 30th. If the second date falls on the 31th, it is changed to the 1st and the month is increased by one.
30/360 ISDA	30/360 U.S. Municipal	If the first date falls on the 31st, it is changed to the 30th. If after the previous test the fist day is the 30th and the second date falls on the 31st, it is changed to the 30th. ISDA = International Swaps and Derivatives Association
30/360 US (NASD)	30/360 Bond Basis 360/360	If the first date falls on the 31st, it is changed to the 30th. If the second date falls on the 31st and the first date is earlier than the 30th, then the second date is changed to the 1st of the next month, otherwise it is changed to the 30th.
30E/360 ISDA	30/360 German German	If the first date falls on the 31st or if it's the last day of February, it is changed to the 30th. If the second date falls on the 31st or if it's the last day of February, it is changed to the 30th. Try: Jan 29th, 2007 to Feb 29th, 2008
30/360 US	30U/360 30US/360	If both the first date and the second date are the last day of February, the second date is changed to the 30th. If the first date is the last day of February, it is changed to the 30th. If after the previous tests the second date is the 31st and the first date is the 30th or the 31st, the second date is changed to the 30th. If after the previous tests the first date is the 31st, it is changed to the 30th. Try: Jan 29th, 2007 to Jan 31th, 2008
30/360 SIA		If the first date and the second date are the last day of February, the second date is changed to the 30th. If the first date falls on the 31st or if it's the last day of February, it is changed to the 30th. If after the preceding test the first day is the 30th and the second day is the 31st then the second day is changed to the 30th. SIA = Securities Industry Association SIA merged on 1 Oct 2006 with the BMA to form the SIFMA = Securities Industry and Financial Markets Association Try: Jan 29th, 2007 to Jan 31th, 2008
30/360 BMA	30/360 PSA	If the first date falls on the 31st or if it's the last day of February, it is changed to the 30th. If after the preceding test the first day is 30 and the second day is 31 then the second day is changed to the 30th. BMA = Bond Market Association, formerly PSA = Public Securites Association BMA merged on 1 Oct 2006 with the SIA to form the SIFMA = Securities Industry and Financial Markets Association Try: Feb 28th, 2007 to Feb 29th, 2008
30/360 IT	30/360 Italian	If the first date falls on the 31st or if it is February 28th or 29th, then it is changed to the 30th. If the second date falls on the 31st or if it is February 28th or 29th, then it is changed to the 30th. Try: Feb 1st, 2007 to Feb 28th, 2008

Group 3: based on unit periods (actuarial method)

The year fraction is calculated according to the unit-period definitions for determining the annual percentage rate of a loan as described in Regulation Z, appendix J to Part 226 of theTruth in Lending Act.

Below are some of the relevant definitions (retrieved on January 4, 2010), for the full text, please see FDIC's online version.

Topic	Rules
Definitions of time intervals	A period is the interval of time between advances or between payments and includes the interval of time between the date the finance charge begins to be earned and the date of the first advance thereafter or the date of the first payment thereafter, as applicable. A common period is any period that occurs more than once in a transaction. A standard interval of time is a day, week, semimonth, month, or a multiple of a week or a month up to, but not exceeding, 1 year. All months shall be considered equal. Full months shall be measured from any point in time on a given date of a given month to the same point in time on the same date of another month. If a series of payments (or advances) is scheduled for the last day of each month, months shall be measured from the last day of the given month to the last day of another month. If payments (or advances) are scheduled for the 29th or 30th of each month, the last day of February shall be used when applicable.
Number of unit-periods between 2 given dates	The number of days between 2 dates shall be the number of 24- hour intervals between any point in time on the first date to the same point in time on the second date. If the unit-period is a month, the number of full unit-periods between 2 dates shall be the number of months measured back from the later date. The remaining fraction of a unit-period shall be the number of days measured forward from the earlier date to the beginning of the first full unit-period, divided by 30. If the unit-period is a month, there are 12 unit-periods per year. If the unit-period is a semimonth or a multiple of a month not exceeding 11 months, the number of days between 2 dates shall be 30 times the number of full months measured back from the later date, plus the number of remaining days. The number of full unit-periods and the remaining fraction of a unit-period shall be determined by dividing such number of days by 15 in the case of a semimonthly unit-period or by the appropriate multiple of 30 in the case of a multimonthly unit-period. If the unit-period is a semimonth, the number of unit-periods per year shall be 24. If the number of unit- periods is a multiple of a month, the number of unit-periods per year shall be 12 divided by the number of months per unit-period. If the unit-period is a day, a week, or a multiple of a week, the number of full unit-periods and the remaining fractions of a unit-period shall be determined by dividing the number of days between the 2 given dates by the number of days per unit-period. If the unit-period is a day, the number of unit-periods per year shall be 365. If the unit-period is a week or a multiple of a week, the number of unit-periods per year shall be 52 divided by the number of weeks per unit-period. If the unit-period is a year, the number of full unit-periods between two dates shall be the number of full years (each equal to 12 months) measured back from the later date. The remaining fraction of a unit-period shall be (A) The remaining number of months divided by 12 if the remaining interval is equal to a whole number of months, or (B) The remaining number of days divided by 365 if the remaining interval is not equal to a whole number of months. In a single advance, single payment transaction in which the term is less than a year and is equal to a whole number of months, the number of unit-periods in the term shall be 1, and the number of unit-periods per year shall be 12 divided by the number of months in the term or 365 divided by the number of days in the term. In a single advance, single payment transaction in which the term is less than a year and is not equal to a whole number of months, the number of unit-periods in the term shall be 1, and the number of unit-periods per year shall be 365 divided by the number of days in the term.
Notes	The actuarial method is very good at handling end of month situations. Please try out the Date Series calculation to compare it with the other day count conventions. It is important to note that when converting to a year fraction, the proper denominator should be used for odd days (fractions of a full period). To comply with the above definitions when converting to year fractions the denominator for odd days should be 365 for day based intervals, 360 for month based intervals, and 364 for week based intervals (a 365 day year has 52 full periods and one 1/7 odd day fraction). It is not uncommon though to have month based intervals with odd day fractions that are calculated with 365 as the denominator, since this produces year fractions that are very close to those calculated using actual based day counts.

How does FinKit use these day count methods?

FinKit uses a wide variety of day count conventions to calculate time and yields:

Most of the calculations (ranging from the Single Payment calculation up to the Capitalized Cost calculation) assume even length periods: this corresponds with a 30/360 day count for annual to semimonthly payments and an act/act day count for weekly, biweekly and daily payments.

For bonds, the time between coupon dates is calculated using the 30/360 day count and the accrued interest for fractions of a coupon period is calculated using the act/act day count.

For zero-coupon bonds, the time between the settlement date and the redemption date is calculated using the act/act day count.

In the Time between Dates calculation, exact time is calculated using the actual number of days, while approximate time is calculated using the 30/360 day count.

The Term Date calculation uses the actual number of days to add or subtract days from a given date.

The Date Series calculation lets you set up a series of dates and view the number of days for each interval, using different methods (act/act, 30/360, 30E/360 and acuarial).

The Year Fraction calculation lets you calculate the time between two dates as a fraction of a year for the day count conventions in all groups.

The Interest between Dates calculation uses theday count conventions from the first and second group to calculate simple and compound interest.

Related topics

Interest between Dates