# Day Count Conventions and Accrual Factors

## Overview

### Description of Supported Accrual Methods

An accrual method or day count convention is used to calculate an accrual factor, which represents the fraction of a year a given period accounts for.  There are two components that make up an accrual factor.  The first component uses a day count convention to determine how many days fall in the accrual period, which will be the numerator in the calculation of the accrual factor.  The second component is a day count convention to determine the number of days that make up a full period, which will be the denominator in the calculation of the accrual factor.

Accrual Method List

 Accrual Method Description Actual/365 (fixed) The number of accrued days is equal to the actual number of days between the effective date and the terminating date.  The accrual factor is the number of accrued days divided by 365. Actual/360 The number of accrued days is equal to the actual number of days between the effective date and the terminating date.  The accrual factor is the number of accrued days divided by 360. Actual/365 (actual) The number of accrued days is equal to the actual number of days between the effective date and the terminating date.  Calculation of the accrual factor assumes the year basis to be 365 days for non-leap years and 366 for leap years.  If a short stub period (< 1 year) contains a leap day, the number of days is divided by 366; otherwise, the number of days is divided by 365. 30/360 (ISDA) (same as U.S. Muni – 30/360) The number of accrued days is calculated on the basis of a year of 360 days with 12 30-day months, subject to the following rules: 1.       If the first date of the accrual period falls on the 31st of the month, the date will be changed to the 30th. 2.       If the first date of the accrual period falls on the 30th of the month after applying (1) above, and the last date of the accrual period falls on the 31st of the month, the last date will be changed to the 30th. The accrual factor is calculated as the number of accrued days divided by 360. 30E/360 (30/360 ISMA) The number of accrued days is calculated on the basis of a year of 360 days with 12 30-day months, subject to the following rules: 1.       If either the first date or last date of the accrual period falls on the 31st of a month, that date will be changed to the 30th.  2.       If the last day of the accrual period falls on the last day of February, the month of February will not be extended to a 30-day month.  Rather, the actual number of days in February will be used. The accrual factor is calculated as the number of accrued days divided by 360. 30E+/360 The number of accrued days is calculated on the basis of a year of 360 days with 12 30-day months, subject to the following rules: 1.       If the first date of the accrual period falls on the 31st of a month, it will be changed to the 30th of that month. 2.       If the last date of the accrual period falls on the 31st of a month, it will be changed to the 1st of the next month. The accrual factor is calculated as the number of accrued days divided by 360. Actual/Actual (ISMA-99) This accrual method is primarily related to bonds.  In the context of accrual factors, the time in years is calculated as follows: if the period is less than one year the accrual factor is equal to the actual number of days between the effective date (d_e) and the terminating date (d_t) divided by the number of days in the period from (d_t – 1 year) to d_t (either 365 or 366).  If the period is greater than one year, the accrual factor is equal to the number of whole years plus the accrual of a stub period calculated as above.  In the context of bonds, there are two ISMA-99 methods: Normal and Ultimo.  The methods differ only in the assumption made regarding coupon dates.  The ISMA-99 Normal method assumes that regular coupons fall on the same day of the month (non end-of-month), and the ISMA-99 Ultimo method assumes that regular coupons fall on the last day of the month (end-of-month).  The ISMA-99 methods make a distinction between regular and irregular interest periods.  Regular interest periods are always an exact multiple of a number of months long, whereas irregular interest periods require that notional interest periods be generated.  The accrual factor for a period is the number of accrued days falling in that period divided by the actual number of days in the period.  The overall accrual factor is then the sum of the individual interest period accrual factors, multiplied by the year fraction of a regular interest period.  For more details, see Reference . Actual/Actual (ISDA) The number of accrued days is equal to the actual number of days between the effective date and the terminating date.  The accrual factor is the sum of the accrued days falling in a non-leap year divided by 365 and the accrued days falling in a leap year divided by 366. 30/360 (Old) This method is old and should not be used.  This method used to be labeled 30/360 (ISDA). 30E/360 (Old) This method is old and should not be used.  This method used to be labeled 30E/360 (ISDA). 30/360 (SIA) The number of accrued days is calculated on the basis of a year of 360 days with 12 30-day months, subject to the following rules: 1.       If the last date of the accrual period is the last day of February and the first date of the period is the last day of February, then the last date of the period will be changed to the 30th.  2.       If the first date of the accrual period falls on the 31st of a month or is the last day of February, that date will be changed to the 30th of the month. 3.       If the first date of the accrual period falls on the 30th of a month after applying (2) above, and the last date of the period falls on the 31st of a month, the last date will be changed to the 30th of the month. The accrual factor is calculated as the number of accrued days divided by 360.  Note that these rules assume that the security follows the end-of-month rule. (see the Date Generation Functions FINCAD Math Reference document.) If the security does not follow the end-of-month rule, then 30/360 (ISDA) should be used. 30/360 (BMA) The number of accrued days is calculated on the basis of a year of 360 days with 12 30-day months, subject to the following rules: 1.       If the first date of the accrual period falls on the 31st of a month or is the last day of February, the date will be changed to the 30th. 2.       If the first date of the accrual period falls on the 30th of the month after applying 1) above, and the last date of the accrual period falls on the 31st of the month, the last date will be changed to the 30th. The accrual factor is calculated as the number of accrued days divided by 360.  Note that prior to 1997, the BMA was known as the PSA, and this method was referred to as 30/360 (PSA). 30/360 (German) The number of accrued days is calculated on the basis of a year of 360 days with 12 30-day months, subject to the following rules: 1.       If either the first date or last date of the accrual period falls on the 31st of a month, that date will be changed to the 30th. 2.       If either the first date or last date of the accrual period is the last day of February, that date will be changed to the 30th. The accrual factor is calculated as the number of accrued days divided by 360. Bus/252 The number of accrued days is calculated as the number of market days in the accrual period.  The accrual factor is calculated as the number of accrued (market) days divided by 252. Actual/365L The number of accrued days is calculated as the actual number of days between the effective date and the terminating date.  This number is divided by 366 if the terminating date falls in a leap year and 365 otherwise. NL365 The number of days is calculated as the actual number of days between the effective date and the terminating date without including any occurrences of the leap day, February 29th.  This number is divided by 365.

The main differences between the various 30/360 methods is the treatment of dates landing on the 31st of a month, or the end of February.  The ISMA, ISDA, and 30E+/360 methods make adjustments for dates landing on the 31st of a month, but not for dates landing on the last day of February.  The SIA, BMA, and German methods make adjustments for dates landing on the 31st of a month, as well as for dates landing on the last day of February.

### Accrual Switches

There are three main accrual switches used in FINCAD:

sw_376

 Switch Value Accrual Method 1 actual/365 (fixed) (eom) 2 actual/360 (eom) 3 actual/365 (actual) (eom) 4 30/360 (ISDA) (eom) 5 30E/360 (30/360 ISMA) (eom) 6 30E+/360 (eom) 7 actual/actual (ISMA-99 Ultimo) (eom) 8 actual/actual (ISDA) (eom) 9 30/360 (old) (eom) 10 30E/360 (old) (eom) 11 actual/365 (fixed) 12 actual/360 13 actual/365 (actual) 14 30/360 (ISDA) 15 30E/360 (30/360 ISMA) 16 30E+/360 17 actual/actual (ISMA-99) 18 actual/actual (ISDA) 19 30/360 (old) 20 30E/360 (old)

This switch is used by bond functions that are from versions prior to Version 9.  It is a dual purpose switch that is used to define the accrual method, as well as the end-of-month rule.

sw_1102

 Switch Value Accrual Method 1 actual/actual (ISMA-99) 2 actual/actual (ISDA) 3 actual/365 (actual) 4 actual/365 (fixed) 5 actual/360 6 30E/360 (30/360 ISMA) 7 30E+/360 8 30/360 (SIA) 9 30/360 (ISDA) 10 30/360 (BMA) 11 30/360 (German) 12 bus/252 13 actual/365L 14 NL365

This switch is used by bond functions that are from Version 9 or later.  It includes some accrual methods that sw_376 does not, and is strictly an accrual method switch.  Bond functions using this switch will have an additional switch indicating the end-of-month rule.

sw_331

 Switch Value Accrual Method 1 actual/365 (fixed) 2 actual/360 3 actual/365 (actual) 4 30/360 (ISDA) 5 30E/360 (30/360 ISMA) 6 30E+/360 7 actual/actual (ISMA-99) 8 actual/actual (ISDA) 9 30/360 (old) 10 30E/360 (old) 11 30/360 (SIA) 12 30/360 (BMA) 13 30/360 (German) 14 actual/365L 15 NL365

This switch is widely used in many areas, such as swaps, interest rate derivatives, credit derivatives, and others.

The supported accrual methods described above are used throughout FINCAD functions.  Note that not all methods are supported by all functions that use accrual methods.  Specifically, the Bus/252 accrual method is only supported by functions new to Version 9 or later.  In addition, there are FINCAD functions for calculating the number of accrued days, the accrual factor, or the accrued interest using a specific method.  These functions are:

aaAccrual_days(d_e, d_t, acc)

Calculates the number of days between two dates according to an accrual method.

aaAccrual_factor(d_e, d_t, acc)

Calculates the accrual factor for a time period according to an accrual method.

aaAccrual_factor2(d_e, d_t, d_cycle, freq, acc, eom, hl)

Calculates the accrual factor that applies between two dates using the appropriate accrual method (day count basis).

aaAccrued(princ_m, cpn, freq, d_n_cf, d_v, d_prev_cf, acc)

Calculates accrued interest according to an accrual method.

aaAccrual_days_act252(d_e, d_t, hl)

Calculates the number of accrual days between two dates using the Bus/252 accrual method.  Days counted are market days only.

aaAccrual_factor_act252(d_e, d_t, hl)

Calculates the accrual factor for a time period using the Bus/252 accrual method.

aaAccrued_act252(princ_m, cpn, freq, d_v, d_prev_cf, hl)

Calculates accrued interest using the Bus/252 accrual method.

## Description of Inputs

 Input Argument Description d_e effective date of the accruing period d_t termination date of the accruing period d_cycle date from which coupon periods are generated only from act/act ISMA-99 Acc accrual method Eom end-of-month rule (used by actual/actual (ISMA-99)) Hl holiday list (for act252 and actual/actual (ISMA-99) functions only) princ_m principal at maturity Cpn Coupon Freq coupon frequency (used by actual/actual (ISMA-99)) d_n_cf next cash flow date d_v value (settlement) date d_prev_cf previous cash flow date Acc accrual method

## Description of Outputs

aaAccrual_days and aaAccrual_days_act252

 Output Description Days number of accrual days for given accrual method

aaAccrual_factor, aaAccrual_factor_act252 and aaAccrual_factor2

 Output Description acc_fac accrual factor for given accrual method

aaAccrued and aaAccrued_act252

 Output Description acc_int accrued interest

## Examples

### Example 1

Suppose d_e (effective date of the accruing period) = 25-Oct-1996 and d_t (termination date of the accruing period) = 31-Dec-1996.  Using aaAccrual_days and aaAccrual_factor with various accrual methods, we get:

Results

 Day Count Number of Accrual Days Accrual Factor Act/365 (fixed) 67 0.183561644 Act/360 67 0.186111111 Actual/365 (actual) 67 0.183561644 30/360 (ISDA) 66 0.183333333 30E/360 65 0.180555556 30E+/360 66 0.183333333 Actual/Actual (ISMA-99) 67 0.183060109 Actual/Actual (ISDA) 67 0.183060109 30/360 (Old) 66 0.183333333 30E/360 (Old) 66 0.183333333 30/360 (SIA) 66 0.183333333 30/360 (PSA) 66 0.183333333 30/360 (German) 65 0.180555556 Actual/365L 67 0.183060109 NL365 67 0.183561644

### Example 2

Suppose d_e (effective date of the accruing period)  = 27-Jan-1998 and d_t (termination date of the accruing period) = 1-Feb-1999.  Using aaAccrual_days and aaAccrual_factor with various accrual methods, we get:

Results

 Day Count Number of Accrual Days Accrual Factor Act/365 (fixed) 370 1.01369863 Act/360 370 1.027777778 Actual/365 (actual) 370 1.01369863 30/360 (ISDA) 364 1.011111111 30E/360 364 1.011111111 30E+/360 364 1.011111111 Actual/Actual (ISMA-99) 370 1.01369863 Actual/Actual (ISDA) 370 1.01369863 30/360 (Old) 365 1.013888889 30E/360 (Old) 364 1.011111111 30/360 (SIA) 364 1.011111111 30/360 (PSA) 364 1.011111111 30/360 (German) 364 1.011111111 Actual/365L 370 1.01369863 NL365 370 1.01369863

### Example 3

Suppose the accrual days and accrual factor using the Bus/252 method are required for various effective and terminating dates given the following table of holidays:

Holiday List

 1-Jan-2005 21-Feb-2005 14-Apr-2005 23-May-2005 4-Jul-2005 4-Sep-2005 9-Oct-2005 23-Nov-2005 25-Dec-2005 26-Dec-2005

Using aaAccrual_days_act252 and aaAccrual_factor_act252, we get:

Results

 Effective Date Terminating Date Number of Accrual Days Accrual Factor 1-Jan-2005 1-Jan-2006 254 1.007936508 22-Mar-2005 22-Apr-2005 22 0.087301587 10-Jun-2005 23-Oct-2005 95 0.376984127 30-Aug-2005 23-Oct-2005 39 0.154761905 28-Sep-2005 25-Dec-2005 62 0.246031746

## References

          Christie, David, (2003), Accrued Interest & Yield Calculations and Determination of Holiday Calendars, SWX Swiss Exchange,

          Mayle, Jan. (1993), Standard Securities Calculation Methods: Fixed Income Securities Formulas, Volume 1, Third Edition, New York, Securities Industry Association.

          Annex to the 2000 ISDA Definitions, (2000), New York, International Swap Dealers Association, Inc.

          BMA Uniform Practices, Second Edition, (1999), New York, The Bond Market Association.

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