A day count convention (aka, day count fraction, day count basis, day count method) determines how interest is calculated on instruments such as bonds, notes, swaps, repurchase agreements, and various forms of loans. They have three primary applications (collected on this site under Coupon Interest):

• Calculation of the accrued interest (aka, "interest bought/sold") on buys and sells of notes and bonds. This also applies to loan-type instruments such as commercial loans and repos. See Trade Interest Bought/Sold.
• Calculation of interest income/expense (end-of-day accrual). See Interest Income and Expense.
• Calculation of interest (coupon) payments. See Coupon Payments.

This page is independent of, but related to, the other pages on this site associated with the Architecture for Transaction Management. Those pages provide guidance on transaction modeling, maintaining positions and balances, performing a variety of P&L accruals, and establishing a knowledge base. Site Info has a Roadmap, Learning Guide, and Site Search for all topics.

Introduction

If you have any questions, comments, or corrections on this material, including information on other normative references and current uses, we welcome you to e-mail us at DCC@eclipsesoftware.biz.

Sources

There is no central agency responsible for defining, standardizing, and documenting day count conventions. There are many day count conventions, but there is no standard terminology or notation. Not only are multiple terms used for the same convention, but the same term may be used for different conventions. The definitions of certain agencies are also vague at best.

We base this discussion on three of the agencies that have the most widespread application. They are also normative. Their acronyms appear frequently in references. It is useful to be aware of their history, as the acronyms have changed over time (e.g., the AIBD Rule 251 became ISMA Rule 251, and is now ICMA Rule 251).

Org	Description
ICMA	The International Capital Market Association (ICMA) is a self-regulatory organization (SRO) and trade association representing financial institutions active in the international capital markets. It was created in 2005 from the merger of the International Securities Market Association (ISMA) and the International Primary Market Association (IPMA). ISMA was known as the Association of International Bond Dealers (AIBD) until 1991.
ISDA	The International Swaps and Derivatives Association, Inc. (ISDA) represents participants in the privately negotiated derivatives industry. It was chartered in 1985.
SIFMA	The Securities Industry and Financial Management Association (SIFMA) is a US-based financial industry trade association. It was created from the 2006 merger of The Securities Industry Association (SIA) and The Bond Market Association (TBMA or BMA). TBMA was known as the Public Securities Association (PSA) until 1997.

Determining the Applicable Convention

Before trying to understand what is involved in a given day count convention, it is first necessary to determine the convention to be used for the situation at hand. This is not always trivial.

• For commercial loans and medium-term notes, the convention is typically specified in the transaction documents, either directly or by reference to another source.
• Derivatives transactions (e.g., swaps, FRAs) usually take the conventions from the trade organization, e.g., the [ISDA].
• Bonds and notes can include the convention in their prospectus or take it from the market in which they trade. For instance, "30/360" can refer to just about any of the 30/360 Conventions, depending on the market. The situation is similar with the term "Actual/Actual".

Historical Background

For any investment that accrues interest, it is obviously important to be able to determine the value exactly on a day-by-day basis. Because of conflicting needs in terms of ease of computation, consistency, and other factors, a number of different conventions arose.

The markets at that time were not as large or interconnected as they are today. This was also long before computers were invented. The result was a large number of conventions.

There is a movement towards convergence regarding the conventions, as well as for other market practices. For examples, see [ISDA_EMUMKT] for a discussion or convergence associated with the euro and Recommendations on the Principal Characteristics and Calculation Conventions of Listed Bonds and Bills for the viewpoint of the OMX Nordic Exchange in Iceland.

As a result, the number of conventions in use is decreasing, and the conventions used by particular investments are changing. One needs to be careful when encountering statements of the sort "convention XYZ is used in the ABC markets", because the convention may well have changed recently.

For more on these topics see Interest Figuration Routines and Day Count Conventions. Those pages also address the handling of interest income/expense accruals, whereas this discussion is focused on interest bought/sold and coupon payments.

Day Count Convention Quick Reference

The following table summarizes the conventions. See The Calculations for more extensive descriptions.

The table below gives the values of N (numerator) and Den (denominator) for each convention. It uses the following notation (see also Terms and Definitions):

• Fact (factor) is N / Den, except for A.01.
• Bulleted entries denote conditional rules to be performed.
  • "&" means both of the adjacent conditions must be true for the rule to apply.
  • "^" indicates end of month. For example, "M1^" means the last day of M1. "Feb^" means the last day of February (i.e., the 28th or 29th).
  • ":" denotes the end of the condition. If the condition is true, perform the action to the right.
  • "→" means replace the value on the left with the value to the right.

Code	Short Form	N	Den
3.01a	30U/360	EOM & D1 = Feb^ & D2 = Feb^: D2 → 30. EOM & D1 = Feb^: D1 → 30. D2 = 31 & D1 in [30..31]: D2 → 30. D1 = 31: D1 → 30.	360
3.01b	30U/360	D2 = 31 & D1 in [30..31]: D2 → 30. D1 = 31: D1 → 30.	360
3.02	30E/360	D1 = 31: D1 → 30. D2 = 31: D2 → 30.	360
3.03	30E/360 ISDA	D1 = M1^: D1 → 30. D2 = M2^: D2 → 30. Unless Date2 = MatDt & M2 = February.	360
3.04	30E+/360	D1 = 31: D1 → 30. D2 = 31: M2.D2 → (M2+1).1.	360
A.01	Act/Act (ISDA)	Fact = (Nnl / 365) + (Nly / 366)
A.02	Act/Act (ICMA)	JulianDays(Date1, Date2)	Fr * JulianDays(Date1, Date3)
A.03	Act/365 (Fixed)	JulianDays(Date1, Date2)	365
A.04	Act/360	JulianDays(Date1, Date2)	360
A.05	Act/365L	JulianDays(Date1, Date2)	Fr = 1: February 29 in Date1[ex] to Date3[in]: 366. Else: 365. Else: Date3 in a leap year: 366. Else: 365.

Day Count Convention Cross-Reference

The following table lists the conventions we describe and links them to the primary references.

The first column below is a code we use for precision in discussing conventions; it has no meaning in any other context. The second column is a convenient (but non-standardized) short form. The remaining columns are the sources.

Code	Short Form	ICMA	ISDA_2006	ISDA_2000	SIFMA	SIA	SWX AI
3.01a	30U/360				30/360		US
3.01b	30U/360		4.16(f)	4.16(e)		30/360
3.02	30E/360	251.1(ii), 251.2	4.16(g)				Special German
3.03	30E/360 ISDA		4.16(h)	4.16(f)			German
3.04	30E+/360
A.01	Act/Act (ISDA)		4.16(b)	4.16(b)
A.02	Act/Act (ICMA)	251.1(iii)	4.16(c)		Actual/Actual	Actual/Actual	ISMA-99
A.03	Act/365 (Fixed)		4.16(d)	4.16(c)	Actual/365	Actual/365	English
A.04	Act/360	251.1(i)(a) (not sterling)	4.16(e)	4.16(d)	Actual/360	Actual/360	French
A.05	Act/365L	251.1(i)(b) (Euro-sterling)					ISMA-Year

The conventions are organized to reflect the two major categories in use:

Cat.	Description.
3	The 30/360 set of conventions.
A	The Actual set of conventions.

The conventions which differ only by a final letter (e.g., 3.01a and 3.01b) are variants. They are essentially identical, differing only in a special condition. They may denote the same convention, but we have not been able to verify this. We use the same Short Form in the interim.

The Short Form represents a common usage, but should not be relied on without understanding the context. Many terms are also used within a particular market context, and thus differ between markets (e.g., "30/360", "Actual/Actual", and "Money Market Basis").

ICMA Rule 251.1(i) describes two conventions, which we have notated as "(a)" and "(b)", but those letters are not used by the ICMA.

In general we are concerned only with conventions which have a current normative source, though we include certain other conventions that are frequently encountered.

Normative Changes

Two agencies have changed the definition and/or terminology of their conventions over the years:

Agency	Changes
SIFMA	[SIFMA_SSCM] adds a clarification for the 30U/360 method when the EOM convention is in use and Date1 and Date2 are the end of February. Because of the special handling of end-of-month dates under all the 30/360 methods, 30/360 securities generally avoid having coupon dates (and thus Date1, which is the source of the difficulty) at the end of the month. If you review the bonds using 30U/360 on [SIFMA_BMP], you will be hard-pressed to find a bond with such a coupon date. Except for such securities, the old and new methods are identical.
ISDA	The 2006 version of the conventions ([ISDA_4.16_2006]) adds a convention (A.02) to those documented in the 2000 Annex ([ISDA_4.16_2000]) and changes the convention referred to as "Eurobond Basis".

The difficulty this introduces is that many secondary sources have picked up this terminology. They usually give their interpretation but not their source, which makes it difficult to determine if they mean to exclude the revised definition or not. The frequent references to superseded organizations also draw this into question.

Terms and Definitions

Inputs

BDC

The business day convention. This determines how to handle situations where the scheduled interest payment date falls on a non-business date (in this context, a business date is one on which cash can be moved, i.e., a day the banks are open). There are two components to it.

• The day on which payment will be made. For example, [IADB_IM] lists three typical options:
  • Following business day.
  • Preceding business day.
  • Following business day, unless that day is in the next calendar month. In that case use the preceding business day.

  See [ACT_DCC] for a more thorough discussion, covering eight different business day conventions.

• Whether the interest payment amount is adjusted for the changed length of the coupon period.

We are not generally concerned with this topic because bonds and notes typically do not adjust the interest payment amount, so the scheduled date for Date3 can be used as is. Bonds and notes also generally use the "Following business day" convention.

CR

Annual coupon rate.

DCC

The day count convention.

Den

The denominator for conventions where the Fact is N / (some number of days). It is often referred to as "number of days in the year", though this is a bit or a misnomer.

DiY

The number of days considered to be in a year, used as Den in many conventions. Again, a bit of a misnomer.

EOM

Indicates that the coupon payments occur on the last day of the month. Non-EOM securities always pay on the same day of the month, e.g., February 15, May 15, August 15, and November 15. EOM securities often pay on different days, e.g., February 28 (or 29), May 31, August 31, and November 30.

Fact

Factor to multiply the annual coupon rate (CR) by to get the accrued interest (AI). It is often N / Den.

Fr

Coupon frequency. 1 = annual coupons, 2 = semi-annual, 4 = quarterly, etc.

MatDt

The maturity date of the security.

N

Number of "days" between Y1.M1.D1 and Y2.M2.D2. We say "days" because this number can vary depending on the day count convention (DCC). This is often referred to as the number of days in the accrual period.
The two following terms are used by A.01:

Nnl

Number of days between Date1 and Date2 that are not in a leap year.

Nly

Number of days between Date1 and Date2 that are in a leap year.

Y1.M1.D1

Starting date of the interval, aka Date1. For a security with periodic payments, this is the coupon payment date immediately preceding Y2.M2.D2.

Y2.M2.D2

Ending date of the interval, aka Date2. For accrued interest on a trade, this will be settlement date. For end-of-day income/expense accounting, it is the day through which valuation is determined (often the business date following the valuation date).

Y3.M3.D3

Coupon payment date immediately succeeding Y2.M2.D2, aka Date3.

Outputs

AI

Amount of interest calculated based on the dates and the day count convention. This is for one unit. To determine the actual cash amount in any situation, it must be multiplied by the quantity.

AI = CR * Fact.

Functions

JulianDays (StartDt, EndDt)

This function returns the number of days between StartDt and EndDt. It can be obtained by subtracting the Julian value of StartDt from the Julian value of EndDt. For example, JulianDays(2-Nov-2007, 15-Nov-2007) returns 13.

Determining Coupon Payment Dates

All day count conventions require values for Date1 and Date2, and some also require Date3. Date2 is normally provided as part of the trade information (or accrue-through date). In many IT environments, the security ID is also provided, but not Date1 or Date3. It is up to the programmer to derive them.

The normal procedure is to start by looking the security ID up in the security master database. Date1 and Date3 are not normally there, either, but you will often find the following fields:

• First coupon date.
• Maturity date (MatDt).
• Coupon payment frequency (Fr).
• EOM convention.

If you have first coupon date, use it as your base and work forward using the frequency. If you only have maturity date, use it and work backward (this works the majority of times, but not in the cases with an odd last coupon).

If the EOM convention is in use, use the last day of each month as the day portion of the coupon payment dates. Otherwise, use the actual day of the base date.

Apply the frequency to determine the coupon payment dates that bracket the settlement date.

Example

The Calculations

This section presents the formal definitions for regular coupon periods. For irregular coupon periods, see Irregular Coupon Periods.

The primary purpose of a day count convention is to assist in the determination of accrued interest (AI). The basic relation is:

AI = CR * Fact

With the exception of A.01:

Fact = N / Den.

Den is often interpreted as DiY, but that can be misleading.

Another way to think of the calculation (except for A.01) is as a three step process:

• Determine the value of one day's interest.
• Determine the number of days.
• Multiply the two.

Under this view, the basic relation becomes:

AI = N * (CR * Den)

This lessens the inclination to interpret Den as DiY.

30/360 Conventions

This category of conventions uses 30 days per month (though this is only approximately true) and a Den of 360, often referred to as DiY.

All of the 30/360 conventions compute accrued interest as follows:

AI	=	CR * Fact
Fact	=	N / Den
N	=	360 * (Y2 - Y1) + 30 * (M2 - M1) + (D2 - D1)
Den	=	360

In calculating N, Date1 and Date2 may be adjusted depending upon the convention. When the adjustments are given, it is to be understood that if a date is changed in one rule, the changed value is used in subsequent rules.

Actual Conventions

This category of conventions uses the actual number of calendar days in the accrual period. The differences come in the value of Den.

N is simply the number of days between Date1 and Date2. For example, the number of days between 2-Nov-2007 and 15-Nov-2007 is 13. Just subtract. This is notated as JulianDays(Date1, Date2).

This convention is referred to as:

Term	Sources
ISMA-Year	[SWX_AI]
Actual/365L	[SWX_AI]
Actual/Actual AFB/FBF Master Agreement	[EBF_MA]
AFB Actual/Actual (Euro)	[ISDA_EMUMKT]

Irregular Coupon Periods

Coupon payments normally follow a regular pattern based on the first coupon date, frequency (Fr), and EOM convention. It is possible, however, that the first and/or last coupon period may be long or short.

We won't address this topic in detail. The basic idea is to embed the irregular period in one of more "quasi-coupon periods" (also called "notional periods") that are determined in the regular manner (see Determining Coupon Payment Dates). The interest for each period is calculated and then summed to get the appropriate amount.

For details see [SIFMA_SSCM] (pp. 30 - 33), [ISDA_EMUMKT] (section 4), and [SWX_AI] (Appendix A.2).

Geometry

It is helpful to have a visual understanding of the various conventions. The figures below are based on a security with a 7.2% annual coupon paying four times a year, with coupon payments on March 1, June 1, September 1, and December 1. (This is the example bond used in Coupon Interest, with $10,000 par.)

The figures show one or more coupon periods (always starting with the June 1 payment), plus the two days before and after. The vertical scale shows a $180 quarterly coupon payment. The data points are the trade interest bought/sold for the given settlement dates. They are zero on each coupon payment date.

Figure 1 – 30U/360 Single Period

This graph shows one period for the 30U/360 convention.

This illustrates the common characteristics of all the 30/360 methods.

30/360 Convention Characteristics

Characteristics	Comments
Constant slope	The interest goes up in a series of lines with the same slope (i.e., daily interest increment), separated by breaks. In this example there are two such lines, to the left and right of July 31. The slope is constant not only within the period, but across periods as well. All 30/360 conventions have the same slope.
Breaks	There can be breaks from the constant slope at month end. For this convention, there is a break on July 31, where the interest amount stays unchanged for a day. There can also be a jump, for the end of February. The location and size of the breaks is what differentiates the 30/360 conventions.
Constant monthly amounts	This is achieved by the breaks.
Constant coupon amounts	Derives from having constant monthly amounts.
Constant yearly amounts	Derives from having constant monthly amounts.

The table has the common characteristics of all the 30/360 conventions. Where they differ is where the month-end breaks (jumps and flats) occur, as given in the calculations above.

Figure 2 – 30U/360 Multiple Periods

This graph shows three coupon periods for the 30U/360 convention (one period was shown above).

The first period is the same as in the earlier figure. This kind of saw-tooth pattern is common for all conventions, not just 30/360.

For this convention we see the flat areas at the end of July, October, December, and January. The other months with 31 days (May and August) have $180 in interest on the last day of those months, which are followed by a coupon payment date.

There is a gap at the end of February, with the following day being a coupon payment date.

Figure 3 – Act/360 and Act/Act (ICMA)

This graph shows two Actual conventions, Act/Act (ICMA) and Act/360, over a single coupon period.

These are the common characteristics of these methods.

Actual Convention Characteristics

Characteristics	Comments
No breaks	There are none of the breaks (jumps or flat areas) that characterize the 30/360 conventions.
Variable monthly amounts	The amount for each month is not constant. They more closely reflect the actual number of days in the month.

Though the Actual conventions don't have breaks, nor do they necessarily have constant slopes, either within or between coupon periods.

Slope for the Actual Conventions

Convention(s)	Slope behavior
Act/Act (ISDA)	There are two slopes, depending upon whether the day (not period) is in a leap year or not. If a coupon period spans year-end, it will have a kink at that point if one of the years is a leap year.
Act/Act (ICMA)	The slope is constant within each coupon period, but can vary between periods (depending upon the number of days in the period). In the figure above the final day of the coupon period never reaches (or exceeds) the normal coupon amount. It is done this way in order that the coupon payments are constant, and thus the annual amounts are also constant.
Act/365 (Fixed)	The slope is constant for all periods. This means that the coupon payments (as well as annual amounts) aren't constant. It is similar to Act/360.
Act/360	The slope is constant for all periods. This means that the coupon payments (and annual amounts) aren't constant. In the figure above you see how it goes above $180 for the coupon period.
Act/365L	Each coupon period has one of two slopes, depending upon whether Date3 is in a leap year or not.

Figure 4 – 30U/360 and Act/Act (ICMA)

This figure compares the 30U/360 and Act/Act (ICMA) conventions. Both were shown in the earlier figures.

We note the following:

• Both conventions have the same coupon payment amount, $180.
• Because the slopes are different, 30U/360 has to have a break at the end of July and another at the end of August so that you meet the fixed coupon payment amount.

References

We have mentioned that the references and definitions of day count conventions can be difficult to straighten out. For this reason we have partitioned the references.

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