Day count convention

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In finance, a day count convention is a method to calculate the fraction of a year between two dates. It is used in the calculation of accrued interest, and in many other formulae in financial mathematics. Often, the particular day count convention is agreed upon for a particular security which has discrete payout dates of fixed amounts. When this security is sold before the payout date, the seller is then eligible to some fraction of the fixed amount.

For ease of explanation:

• datediff(units, Date_x, Date_y) returns the number of integer increments of the given unit that can be made to Date_x without exceeding Date_y
• dd(Date_x) returns Date_x's day of month, 1..31
• mm(Date_x) returns Date_x's month of year, 1..12
• yy(Date_x) returns Date_x's Gregorian year 1753..inf

This is the most common convention and is used by most currencies including the US dollar and Euro. Each month is treated normally and the year is assumed to be 360 days e.g. in a period from February 1, 2005 to April 1, 2005, the fraction is considered to be 59 days divided by 360. The days between Date₁ and Date₂ are counted as follows:

Contents 1 30/360 2 ACT/ACT (actual/actual) 3 ACT/365F (actual/365 fixed) 4 Others 5 See also

[edit] 30/360

Each month is treated as having 30 days, so a period from February 1, 2005 to April 1, 2005 is considered to be 60 days. The year is considered to have 360 days. This convention is frequently chosen for ease of calcuation: the payments tend to be regular and at predictable amounts. The days between Date₁ and Date₂ are counted as follows:

With these definitions:

Variants:

• 30E/360 (30/360 European) — Date_b = Date₂ - 1day if dd(Date₂) = 31
• 30+/360 — Date_b = Date₂ + 1day if dd(Date₂) = 31, that is, Date_b becomes the 1st of the following month in this case.
• 30/360 ISDA — (needs description here)
• End-of-Month rule (PSA) — this rule can be added to either 30/360 or any of its variants. Replace dd(Date_a) with 30 if Date₁ is the last day of February, 28th usually or 29th on a leap year. Also, replace dd(Date_b) with 30 if both Date₁ and Date₂ are at the ends of February.

Rationale:

• Treating a month as 30 days and a year as 360 days was devised for its ease of calculation by hand compared with manually calculating the actual days between two dates. Also, because 360 is highly factorable, payment frequencies of semi-annual and quarterly and monthly will be 180, 90, and 30 days of a 360 day year, meaning the payment amount will not change between payment periods.
• The 30/360 clauses that ensure dd returns in the inclusive range 1..30 modify the dates such that the period is never altered more than by 1 day. That is, it is impossible for Date_a = Date₁ - 1day and Date_b = Date₂ + 1day at the same time. Most of the time, the period is lengthened by 1 day because the condition of Date₁'s reduction by 1day (elongating the period by 1day) is statistically easier to satisfy than Date₂'s reduction by 1day (shrinking the period by 1day). The only time the period is reduced is the 30th to the 31st which is treated as the 30th to the 30th.
• The 30E/360 variant makes the condition when Date₂ is reduced by 1day statistically as easily satisfiable as when Date₁ is reduced by 1day. This means that the 1st to the 31st is treated as the 1st to the 30th, which is 29 days, helping the payer pay less than with 30/360.
• The 30+/360 variant means that period is never shortened, only elongated. This means that the 30th to the 31st is treated as 1 day, helping the payee receive more than with 30/360.

[edit] ACT/ACT (actual/actual)

• (1) Each month is treated normally, and the year has the usual number of days. For example, a period from February 1, 2005 to April 1, 2005 is considered to be 59 days. In this convention leap years do affect the final result.
• (2) As used by UK gilts. Each month is treated normally, and the year is the number of days in the current coupon period multiplied by the number of coupons in a year e.g. if the coupon is payable 1st February and August then on April 1, 2005 the days in the year is 362 i.e. 181 (the number of days between 1 February and 1 August 2005) x 2 (semi-annual). The formula is as follows:

With these definitions:

Variants:

• ACT/ACT ISDA - (needs description here)

Rationale:

• When determining what fraction of the next coupon's payment the holder is entitled to when selling an instrument before its next coupon payment date, the ACT/ACT value is multiplied by the CouponFrequency and then multiplied by the amount of the coupon's payment. Thus, the holder is entitled to CouponAmount * datediff(days, Date₁, Date₂)/CouponPeriod. With annual coupons, this expression simplifies to either datediff(days, Date₁, Date₂}/365 or datediff(days, Date₁, Date₂)/366, depending on whether a leap day occurs between Date_LastPayment and Date_NextPayment.
• This method prevents any day in a coupon period from being more or less valuable than another day in the same period. However, the coupon periods themselves may be uneven; in the case of semi-annual payment on a 365 day year, one period will be 182 days and the other 183 days. Thus, all the days in one period will be valued 1/182nd of the payment amount and all the days in the other period will be valued 1/183rd of the payment amount.

[edit] ACT/365F (actual/365 fixed)

Several major currencies currently adopt this convention including the UK pound, Canadian dollar, Australian dollar, as well as the New Zealand dollar. Each month is treated normally, and the year is assumed to have 365 days, regardless of leap year status. For example, a period from February 1, 2005 to April 1, 2005 is considered to be 59 days. This convention results in periods having slightly different lengths.

Variants:

• ACT/365 — widely used synonym for Actual/365 Fixed

[edit] Others

• NL/365 (NLY/365) — "No Leap Year" logic extension to ACT/365 where leap days are subtracted, ensuring the quotient never exceeds 1.
• 30/365 — Rare extension to 30/360 where the denominator is set to 365.
• ACT/366 (actual/366) — Very rare logic extension to ACT/365 where the denominator is set to 366, ensuring the quotient never exceeds 1.
• ACT/252 (bus/252, business days/252) — common in South American instruments - calculated as the number of business days in a nominal year of 252 business days (in Brazil). Weekends and holidays are excluded; thus, Friday to Monday would be considered 1 day.

[edit] See also

• ISDA - standards body governing day count convention alongside ISMA.
• ISDA - ISDA PDF discussion of ISDA/ISMA/AFB Actual/Actual day count conventions.
• Pricing of Game Options (in a market with stochastic interest rates) - Section Chapter II: A Little Bit of Finance, Section 1: Brief introduction to Financial Securities, from pages 26 to 33, formally mention day count conventions.
• Practical Issues Arising from the Introduction of the Euro - Issue 7 (March 1998) - Chapter 4: Financial Markets and Exchanges: discusses European nations' day-count conventions and changes required to unify the day-count conventions for the EU member states.
• jFin pure java open source implementation of financial date arithmetic