### 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. |

Act/365F | DCF = Num/Den Num = Actual number of days within the accrual period |

Act/360 | DCF = Num/Den Num = Actual number of days within the accrual period |

Act/365A | DCF = Num/Den Num = Actual number of days within the accrual period |

Act/365L | DCF = Num/Den Num = Actual number of days within the accrual period |

NL/365 | DCF = Num/Den Num: If the Leap day (29th Feb) does not fall within the accrual period Den=365 |

30/360 ISDA | DCF = Num/Den Num: |

30E/360 | DCF = Num/Den Num: |

30E+/360 | DCF = Num/Den Num: |

30/360 German | DCF = Num/Den |

30/360 US | DCF = Num/Den |

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 |