### Day count conventions described

Also known as Day Count Fraction (DCF) convention describes how accrued interest is calculated on a variety of financial products like bonds, notes, FRAs, Interest rate swaps etc. While Interest rates are usually expressed on a per annum basis (reference period = 1 year), the periodic payments are generally due over shorter intervals (monthly, quaterly etc.). The Day Count Fraction (DCF), expressed as a number of days in the accrual period divided by the total number of days in the reference (often 360 or 365) period, determines the accrual payment for the period. Different conventions (or rules) determine how number of days are calculated for the accrual and the reference period. The followed convention generally depends on the market type, location and (or) the curriency in which the instrument of interest is denominated. Some of the most commonly followed conventions have been described here.

Accrued interest is calculated using the following formula:

##### Accrued Interest (AI) = Principal amount * Rate (per annum basis) * DCF (1)

A single convention may be referred by different names depending on the market(Money/Bond/Swaps), currency denomination (USD or EUR etc.) and the partes involved. Table 1 lists the most common day count conventions along with some of the alternate names they may be referred to as.

Table 1: Alternate names for day conventions

Convention Alternate Name(s)
Act/Act Actual/Actual, Actual/Actual (ISDA)
Act/365F Actual/365 Fixed, English
Act/360 Actual/360 , French
Act/365A Actual/365 Actual
Act/365L Actual/365 Leap year
NL/365 Actual/365 No leap year , NL365
30/360 ISDA 30/360 U.S. Municipal, Bond basis
30E/360 30/360 ISMA, 30/360 European, 30S/360 Special German, Eurobond Basis
30E+/360 30E+/360
30/360 German 30E/360 ISDA
30/360 US 30U/360,30US/360
##### Quantobjects' Schedules and business calendar library

QO's schedules and business calendar library can be downloaded from here. Other libraries and their respective documentation are available here.

##### Calculating DCFs

Let the dates D1.M1.Y1 (Period start date) and D2.M2.Y2 (Period end date) define the accrual period for interest rate calculations. Table 2 below describes how day count fraction is calculated for various day count conventions. These day conventions are amongst the most commonly used in the financial world today.

Table 2: DCF calculations

Day count method DCF calculation
Act/Act

DCF = Days1 /366 + Days2 / 365

Days1 = Actual number of days in period that fall in a leap year.
Days2 = Actual number of days in period that fall in a normal year.

Act/365F

DCF = Num/Den

Num = Actual number of days within the accrual period
Den = 365

Act/360

DCF = Num/Den

Num = Actual number of days within the accrual period
Den = 360

Act/365A

DCF = Num/Den

Num = Actual number of days within the accrual period
Den = 366 if the Leap day (29th Feb) falls within the accrual period else 365

Act/365L

DCF = Num/Den

Num = Actual number of days within the accrual period
Den = 366 if the accrual period end date (D2.M2.Y2) falls in a leap year else 365

NL/365

DCF = Num/Den

Num: If the Leap day (29th Feb) does not fall within the accrual period
then,
Actual number of days within the accrual period
Otherwise,
Actual number of days within the accrual period -1

Den=365

30/360 ISDA

DCF = Num/Den

Num:
1. If D1 = 31, Set D1 = 30
2. If D1 = 30 after applying 1 and D2 = 31, Set D2 = 30
3. Num = (D2 – D1) + 30 * (M2 – M1) + 360 * (Y2 – Y1)
Den = 360

30E/360

DCF = Num/Den

Num:
1. If D1 = 31, Set D1 = 30
2. If D2 = 31, Set D2 = 30
3. Last day of February not treated specially
4. Num = (D2 – D1) + 30 * (M2 – M1) + 360 * (Y2 – Y1)
Den = 360

30E+/360

DCF = Num/Den Num:
1. If D1 = 31, Set D1 = 30
2. If D2 = 31, Set D2.M2.Y2 to the 1st day of the next month - (D2 = 1, Y2 = Y2 + Integer part of (M2+1)/12, M2= M2 + 1 Mod 12)
3. Num = (D2 – D1) + 30 * (M2 – M1) + 360 * (Y2 – Y1)
Den = 360

30/360 German

DCF = Num/Den
Num:
1. If D1 (and/or D2) = 31, Set D1 (and/or D2) = 30
2. If D1.M1.Y1 (and/or D2.M2.Y2) falls on the last day of the February set use D1 = 30 (and/or D2 = 30)
3. Num = (D2 – D1) + 30 * (M2 – M1) + 360 * (Y2 – Y1)
Den = 360

30/360 US

DCF = Num/Den
Num:
1. If D2.M2.Y2 is the last day of February (28 in a non leap year; 29 in a leap year) and
D1.M1.Y1 is the last day of February, Set D2 = 30
2. If D1 is the last day of February, Set D1 = 30
3. If D2 = 31 and D1 = 30 or 31, Set D2 = 30
4. If D1 = 31, Set D1 = 30
Den = 360

Below are few of the examples chosen to highlight the differences between the stated conventions

##### Example 1

Let us assume D1.M1.Y1 = 28/12/2007 and D2.M2.Y2 = 28/2/2008 (Remember Y2 is Leap).

Table 3: DCF calculations (1/4)

Convention Calculation DCF
Act/Act 4/365+58/366 0.16942884946478
Act/365F 62/365 0.16986301369863
Act/360 62/360 0.172222222222222
Act/365A 62/365 0.16986301369863
Act/365L 62/366 0.169398907103825
NL/365 62/365 0.16986301369863
30/360 ISDA 60/360 0.166666666666667
30E/360 60/360 0.166666666666667
30E+/360 60/360 0.166666666666667
30/360 German 60/360 0.166666666666667
30/360 US 60/360 0.166666666666667
##### Example 2

Now let us suppose D1.M1.Y1 = 28/12/2007 and D2.M2.Y2 = 29/2/2008 (Remember Y2 is Leap).

Table 4: DCF calculations (2/4)

Convention Calculation DCF
Act/Act 4/365+59/366 0.172161089901939
Act/365F 63/365 0.172602739726027
Act/360 63/360 0.175
Act/365A 63/366 0.172131147540984
Act/365L 63/366 0.172131147540984
NL/365 62/365 0.16986301369863
30/360 ISDA 61/360 0.169444444444444
30E/360 61/360 0.169444444444444
30E+/360 61/360 0.169444444444444
30/360 German 62/360 0.172222222222222
30/360 US 61/360 0.169444444444444
##### Example 3

Now let us suppose D1.M1.Y1 = 31/10/2007 and D2.M2.Y2 = 30/11/2008 (Remember Y2 is Leap).

Table 5: DCF calculations (3/4)

Convention Calculation DCF
Act/Act 62/365+334/366 1.08243131970956
Act/365F 396/365 1.08493150684932
Act/360 396/360 1.1000000000000
Act/365A 396/366 1.08196721311475
Act/365L 396/366 1.08196721311475
NL/365 395/365 1.08219178082192
30/360 ISDA 390/360 1.08333333333333
30E/360 390/360 1.08333333333333
30E+/360 390/360 1.08333333333333
30/360 German 390/360 1.08333333333333
30/360 US 390/360 1.08333333333333
##### Example 4

Let's take one last example. D1.M1.Y1 = 2/1/2008 and D2.M2.Y2 = 5/31/2009

Table 6: DCF calculations (4/4)

Convention Calculation DCF
Act/Act 335/366+150/365 1.32625945055768
Act/365F 485/365 1.32876712328767
Act/360 485/360 1.34722222222222
Act/365A 485/366 1.32513661202186
Act/365L 485/365 1.32876712328767
NL/365 484/365 1.32602739726027
30/360 ISDA 480/360 1.33333333333333
30E/360 479/360 1.33055555555556
30E+/360 480/360 1.33333333333333
30/360 German 479/360 1.33055555555556
30/360 US 480/360 1.33333333333333