Day count convention


In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, mortgages, medium-term notes, swaps, and forward rate agreements. This determines the number of days between two coupon payments, thus calculating the amount transferred on payment dates and also the accrued interest for dates between payments. The day count is also used to quantify periods of time when discounting a cash-flow to its present value. When a security such as a bond is sold between interest payment dates, the seller is eligible to some fraction of the coupon amount.
The day count convention is used in many other formulas in financial mathematics as well.

Development

The need for day count conventions is a direct consequence of interest-earning investments. Different conventions were developed to address often conflicting requirements, including ease of calculation, constancy of time period and the needs of the accounting department. This development occurred long before the advent of computers.
There is no central authority defining day count conventions, so there is no standard terminology, however the International Swaps and Derivatives Association and the International Capital Market Association have done work gathering and documenting conventions. Certain terms, such as "30/360", "Actual/Actual", and "money market basis" must be understood in the context of the particular market.
The conventions have evolved, and this is particularly true since the mid-1990s. Part of it has simply been providing for additional cases or clarification.
There has also been a move towards convergence in the marketplace, which has resulted in the number of conventions in use being reduced. Much of this has been driven by the introduction of the euro.

Definitions

;Interest: Amount of interest accrued on an investment.
;CouponFactor: The Factor to be used when determining the amount of interest paid by the issuer on coupon payment dates. The periods may be regular or irregular.
;CouponRate: The interest rate on the security or loan-type agreement, e.g., 5.25%. In the formulas this would be expressed as 0.0525.
;Date1 : Starting date for the accrual. It is usually the coupon payment date preceding Date2.
;Date2 : Date through which interest is being accrued. You could word this as the "to" date, with Date1 as the "from" date. For a bond trade, it is the settlement date of the trade.
;Date3 : Is the next coupon payment date, usually it is close to Date2. This would be the maturity date if there are no more interim payments to be made.
;Days: Function returning the number of days between StartDate and EndDate on a Julian basis. For instance, Days returns 31.
;EOM: Indicates that the investment always pays interest on the last day of the month. If the investment is not EOM, it will always pay on the same day of the month.
;DayCountFactor: Figure representing the amount of the CouponRate to apply in calculating Interest. It is often expressed as "days in the accrual period / days in the year". If Date2 is a coupon payment date, DayCountFactor is zero. DayCountFactor is also known as year fraction, abbreviated YearFrac.
;Freq: The coupon payment frequency. 1 = annual, 2 = semi-annual, 4 = quarterly, 12 = monthly, etc.
;Principal: Par value of the investment.. In the case of an amortizing bond, it is the unpaid principal = outstanding principal amount = principal balance. In the case of an accreting bond, where the principal increases with the accumulation of notional coupons that are not paid, Principal means principal balance. The latter is the most general denomination because it accommodates, for example, a bond that accumulates interest to the principal at the beginning of its life and, after that, amortizes the principal in installments.
For all conventions, the Interest is calculated as:

30/360 methods

All conventions of this class calculate the DayCountFactor as:
They calculate the CouponFactor as:
This is the same as the DayCountFactor calculation, with Date2 replaced by Date3. In the case that it is a regular coupon period, this is equivalent to:
The conventions are distinguished by the manner in which they adjust Date1 and/or Date2 for the end of the month. Each convention has a set of rules directing the adjustments.
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.

30/360 Bond Basis

This convention is exactly as 30U/360 below, except for the first two rules. Note that the order of calculations is important:
  • .
  • If then
Other names:
  • 30A/360.
Sources:
  • ISDA 2006 Section 4.16.

    30/360 US

Date adjustment rules :
  • If the investment is EOM and and, then change D2 to 30.
  • If the investment is EOM and, then change D1 to 30.
  • If D2 is 31 and D1 is 30 or 31, then change D2 to 30.
  • If D1 is 31, then change D1 to 30.
This convention is used for US corporate bonds and many US agency issues. It is most commonly referred to as "30/360", but the term "30/360" may also refer to any of the other conventions of this class, depending on the context.
Other names:
  • 30U/360 - 30U/360 is not strictly the same as 30/360, it is used for the Euribor curve and Euro denominated swaps, with the distinction that under 30/360, each day in a 31-day month accrues 30/31 of interest, whereas in 30U/360 payment occurs on the 30th and the 31st is considered to be part of the next month. - Bloomberg
  • 30/360
Sources:
  • ISDA 2006 Section 4.16, though the first two rules are not included.

  • 30E/360

Date adjustment rules:
  • If D1 is 31, then change D1 to 30.
  • If D2 is 31, then change D2 to 30.
Other names:
  • 30/360 ICMA
  • 30/360 ISMA
  • 30S/360
  • Eurobond basis
  • Special German
Sources:
  • ICMA Rule 251.1, 251.2.
  • ISDA 2006 Section 4.16.

    30E/360 ISDA

Date adjustment rules:
  • If D1 is the last day of the month, then change D1 to 30.
  • If D2 is the last day of the month, then change D2 to 30.
Other names:
  • 30E/360 ISDA
  • Eurobond basis
  • German
Sources:
  • ISDA 2006 Section 4.16.

    Actual methods

The conventions of this class calculate the number of days between two dates as the Julian day difference. This is the function Days.
The conventions are distinguished primarily by the amount of the CouponRate they assign to each day of the accrual period.

Actual/Actual ICMA

Formulas:
For regular coupon periods where Date2 and Date3 are equal:
For irregular coupon periods, the period has to be divided into one or more quasi-coupon periods that match the normal frequency of payment dates. The interest in each such period is then computed, and then the amounts are summed over the number of quasi-coupon periods. For details, see or the ISDA paper.
This method ensures that all coupon payments are always for the same amount.
It also ensures that all days in a coupon period are valued equally. However, the coupon periods themselves may be of different lengths; in the case of semi-annual payment on a 365-day year, one period can be 182 days and the other 183 days. In that case, 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.
This is the convention used for US Treasury bonds and notes, among other securities.
Other names:
  • Actual/Actual
  • Act/Act ICMA
  • ISMA-99
  • Act/Act ISMA
Sources:
  • ICMA Rule 251.1.
  • ISDA 2006 Section 4.16.
  • Actual/Actual comparison, EMU and Market Conventions: Recent Developments.

    Actual/Actual ISDA

Formulas:
This convention accounts for days in the period based on the portion in a leap year and the portion in a non-leap year.
The days in the numerators are calculated on a Julian day difference basis. In this convention the first day of the period is included and the last day is excluded.
The CouponFactor uses the same formula, replacing Date2 by Date3. In general, coupon payments will vary from period to period, due to the differing number of days in the periods. The formula applies to both regular and irregular coupon periods.
Other names are:
  • Actual/Actual
  • Act/Act
  • Actual/365
  • Act/365
Sources:
  • ISDA 2006 Section 4.16.

    Actual/365 Fixed

Formulas:
Each month is treated normally and the year is assumed to be 365 days. For example, in a period from February 1, 2005, to April 1, 2005, the Factor is considered to be 59 days divided by 365.
The CouponFactor uses the same formula, replacing Date2 by Date3. In general, coupon payments will vary from period to period, due to the differing number of days in the periods. The formula applies to both regular and irregular coupon periods.
Other names:
  • Act/365 Fixed
  • A/365 Fixed
  • A/365F
  • English
Sources:
  • ISDA 2006 Section 4.16.

  • Actual/360

Formulas:
This convention is used in money markets for short-term lending of currencies, including the US dollar and Euro, and is applied in ESCB monetary policy operations. It is the convention used with Repurchase agreements. For example, in a period from February 1, 2005, to April 1, 2005, the Factor is considered to be 59 days divided by 360.
The CouponFactor uses the same formula, replacing Date2 by Date3. In general, coupon payments will vary from period to period, due to the differing number of days in the periods. The formula applies to both regular and irregular coupon periods.
Other names:
  • Act/360
  • A/360
  • French
Sources:
  • ICMA Rule 251.1.
  • ISDA 2006 Section 4.16.
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