Overview - Maple Help

Home : Support : Online Help : Mathematics : Finance : Date Arithmetic : Finance/DayCounters

Finance Package Commands For Day Counting

Overview

The Financial Modeling package supports most standard day count conventions used in the industry, which include Actual/Actual, Actual/360 and 30/360 conventions.

Day counting convention defines the way in which interest accrues over time. Generally, we know the interest earned over some reference period, (for example, the time between coupon payments), and we are interested in calculating the interest earned over some other period.

The day counting convention is usually expressed as $\mathbit{X}\mathbf{/}\mathbit{Y}$, where $\mathbit{X}$ defines the way in which the number of days between the two dates is calculated, and $\mathbit{Y}$ defines the way in which the total number of days in the reference period is measured. The interest earned between the two dates is

$\frac{\mathbit{Number_of_days_between_two_dates}}{\mathbit{Number_of_days_in_reference_period}}\mathbf{×}{\mathbit{Interest}}_{}\mathbit{_earned_in_reference_period}$

Three day counting conventions commonly used in the United States are

 • Actual/Actual
 • Actual/360
 • 30/360

 - return the number of days between two dates according to a given convention - return the interval between two dates as a fraction of a year according to a given convention

Day Count Conventions

Actual/Actual Conventions

The actual/actual interest accrual convention is recommended for euro-denominated bonds. There are at least three different interpretations of actual/actual. These three interpretations are identified as:

 • Actual/Actual (ISDA)
 • Actual/Actual (ISMA)
 • Actual/Actual (AFB)

The difference between the ISDA, ISMA and AFB methods can be reduced to a consideration of the denominator to be used when calculating accrued interest. In all three cases, the numerator will be equal to the actual number of days from (and including) the last coupon payment date or period end date, to (but excluding) the current value date or period end date.

Under the Actual/Actual (ISDA) approach, the denominator varies depending on whether a portion of the relevant calculation period falls within a leap year. For the portion of the calculation period falling within a leap year, the denominator is 366, for the other portion the denominator is 365. The ISDA convention is also known as Actual/Actual (Historical), Actual/Actual, Act/Act, and according to ISDA also Actual/365, Act/365, and A/365.

Under the Actual/Actual (ISMA) approach, the denominator is the actual number of days in the coupon period multiplied by the number of coupon periods in the year. The ISMA and US Treasury convention is also known as Actual/Actual (Bond).

Under the Actual/Actual (AFB) approach, the denominator is either 365 if the calculation period does not contain February 29th, or 366 if the calculation period includes February 29th. The AFB convention is also known as actual/actual (Euro).

Consider some examples:

 > $\mathrm{with}\left(\mathrm{Finance}\right):$

First you will use a day counter that follows the ISDA convention.

 > $\mathrm{DayCount}\left("Jan-01-2006","July-01-2006",\mathrm{ISDA}\right)$
 ${181}$ (2.1.1)

The numerator is equal to the actual number of days from (and including) the last coupon payment date or period end date, to (but excluding) the current value date or period end date. Therefore, the number of days from January 1st, 2006 to July 1st, 2006 can be calculated by adding the number of days in January, February, March, April, May, and June together:

 > $31+28+31+30+31+30$
 ${181}$ (2.1.2)
 > $\mathrm{YearFraction}\left("Jan-01-2006","July-01-2006",\mathrm{ISDA}\right)$
 ${0.4958904110}$ (2.1.3)

The denominator for ISDA is 365 since the year of 2006 is not a leap year:

 > $\frac{\mathrm{DayCount}\left("Jan-01-2006","July-01-2006",\mathrm{ISDA}\right)}{365}$
 $\frac{{181}}{{365}}$ (2.1.4)
 > $\mathrm{evalf}\left(\right)$
 ${0.4958904110}$ (2.1.5)
 > $\mathrm{DayCount}\left("Jan-01-2008","April-20-2008",\mathrm{ISDA}\right)$
 ${110}$ (2.1.6)
 > $\mathrm{YearFraction}\left("Jan-01-2008","April-20-2008",\mathrm{ISDA}\right)$
 ${0.3005464481}$ (2.1.7)
 > $\frac{\mathrm{DayCount}\left("Jan-01-2008","April-20-2008",\mathrm{ISDA}\right)}{366}$
 $\frac{{55}}{{183}}$ (2.1.8)
 > $\mathrm{evalf}\left(\right)$
 ${0.3005464481}$ (2.1.9)
 >
 ${256}$ (2.1.10)
 >
 ${0.6994535519}$ (2.1.11)
 >
 $\frac{{128}}{{183}}$ (2.1.12)
 > $\mathrm{evalf}\left(\right)$
 ${0.6994535519}$ (2.1.13)

In the second example you will use the ISMA convention.

 > $\mathrm{DayCount}\left("Jan-01-2006","July-01-2006",\mathrm{ISMA}\right)$
 ${181}$ (2.1.14)

As you can see the number of days between January 1st, 2006 and July 1st, 2006  is the same according to both conventions. However, the length of the period from January 1st, 2006 to July 1st, 2006 as a fraction of the year is different.

 > $\mathrm{YearFraction}\left("Jan-01-2006","July-01-2006",\mathrm{ISMA}\right)$
 ${0.5000000000}$ (2.1.15)

The denominator is the actual number of days in the coupon period multiplied by the number of coupon periods in the year.

 > $\mathrm{DayCount}\left("Jan-01-2008","April-20-2008",\mathrm{ISMA}\right)$
 ${110}$ (2.1.16)
 > $\mathrm{YearFraction}\left("Jan-01-2008","April-20-2008",\mathrm{ISMA}\right)$
 ${0.3333333333}$ (2.1.17)
 > $\mathrm{DayCount}\left("Jan-01-2008","April-01-2008",\mathrm{ISMA}\right)$
 ${91}$ (2.1.18)
 > $\mathrm{YearFraction}\left("Jan-01-2008","April-01-2008",\mathrm{ISMA}\right)$
 ${0.2500000000}$ (2.1.19)

Finally, consider the AFB day counting convention.

 > $\mathrm{DayCount}\left("Jan-01-2006","July-01-2006",\mathrm{AFB}\right)$
 ${181}$ (2.1.20)
 > $\mathrm{YearFraction}\left("Jan-01-2006","July-01-2006",\mathrm{AFB}\right)$
 ${0.4958904110}$ (2.1.21)

The denominator is either 365 if the calculation period does not include February 29th, or 366 if the calculation period includes February 29th.

 > $\frac{\mathrm{DayCount}\left("Jan-01-2006","July-01-2006",\mathrm{AFB}\right)}{365}$
 $\frac{{181}}{{365}}$ (2.1.22)
 > $\mathrm{evalf}\left(\right)$
 ${0.4958904110}$ (2.1.23)
 > $\mathrm{DayCount}\left("Jan-01-2008","April-20-2008",\mathrm{AFB}\right)$
 ${110}$ (2.1.24)
 > $\mathrm{YearFraction}\left("Jan-01-2008","April-20-2008",\mathrm{AFB}\right)$
 ${0.3005464481}$ (2.1.25)
 > $\frac{\mathrm{DayCount}\left("Jan-01-2008","April-20-2008",\mathrm{AFB}\right)}{366}$
 $\frac{{55}}{{183}}$ (2.1.26)
 > $\mathrm{evalf}\left(\right)$
 ${0.3005464481}$ (2.1.27)
 >
 ${256}$ (2.1.28)
 >
 ${0.7013698630}$ (2.1.29)
 >
 $\frac{{128}}{{183}}$ (2.1.30)
 > $\mathrm{evalf}\left(\right)$
 ${0.6994535519}$ (2.1.31)
 >
 $\frac{{256}}{{365}}$ (2.1.32)
 > $\mathrm{evalf}\left(\right)$
 ${0.7013698630}$ (2.1.33)