Day count conventions
The day count convention determines how interest accrues over time in a variety of transactions, including bonds, swaps, bills and loans. Interest is usually expressed to accrue at a rate per annum (the reference period). It is often due and payable at shorter intervals, usually a number of months (the interest period).
The day count convention regulates how the parties are to calculate the amount of interest payable at the end of each interest or other period. It is commonly expressed as a fraction. The numerator will be the convention for the number of days in the period - usually actual or a notional 30. The denominator is the convention for the number of days in the reference period - often 360 or 365.
Conventions vary depending on the market type, location and the currency in question. For example, euro-denominated bonds are usually calculated on an actual/actual basis, while fixed rate non-euro denominated bonds are often calculated on a 30/360 basis. The London interbank market, on the other hand, operates on the basis of actual/360, except where the currency is sterling, for which the London interbank convention is actual/365.
Commonly used day count conventions are considered below, but first it is important to bear in mind the various business day conventions that may apply. These regulate the start and end date of each period.
Business day conventions
No date adjustment
Cycle dates are not adjusted for weekends or holidays and are forced to land within a cycle month.
Following business day/next good business day
Dates are adjusted for weekends and holidays to the next good business day.
Preceding/previous good business day
Dates are adjusted for weekends and holidays to the previous good business day.
Modified following/modified business day
Dates are adjusted to the next good business day unless that day falls in the next calendar month in which case the date is adjusted to the previous good business day.
End of month – no adjustment
Dates are adjusted to land on last day of the month.
End of month – previous good business day
Dates are adjusted to the last day of the month but if that day is a weekend or holiday, then it is adjusted backward to the previous good business day.
Two business days prior to third Wednesday of month
Dates generated are two business days prior to the third Wednesday of the month (used in conjunction with Eurodollar futures).
Deposit rollover method
Each date is set so it occurs on the same day of the month as the previous date. Each date is set to the next good business day but no dates may be adjusted past the last good business day of the month.
Day count conventions and conversions
Interest is calculated as the principal times the interest rate times the day-count fraction, where the day-count fraction is defined by the day-count convention associated with the interest rate.
Money market basis (actual/360)
This basis is commonly used for all Eurocurrency Libor rates, except sterling. The day count fraction is defined as the actual number of days in the period over 360.
Actual/365 fixed
This basis is commonly used for all sterling interest rates, including Libor. The day count fraction is defined as the actual number of days in the period over 365. It is also used for money markets in Australia, Canada and New Zealand. This basis is sometimes confused with actual/365, which is defined next.
Actual/365 or actual/actual
This basis is commonly used for all sterling bonds, Euro denominated bonds, US Treasury bonds and for some USD interest rate swaps. In this case, the day-count fraction is the number of days in the period in a normal year over 365 or the number of days in the period in a leap year over 366.
Eurobond basis (30E/360)
30E/360 is used for calculating accrued interest on some legacy currency pre Euro Eurobonds and on bonds in Sweden and Switzerland. This method assumes that all months have 30 days, even February, and that a year is 360 days. Effectively if the start date d1 is 31 then it changes to 30, and if the second date d2 is 31 it too changes to 30. The day count fraction is defined as the number of days in the period (Δ360) over where (Δ360) is calculated as if every month had 30 days, as described in Figure 1.
Bond basis
This basis is used for calculating accrued interest on domestic US bonds (e.g. Yankee bonds, federal agencies, corporate and municipal bonds). Each month is assumed to have 30 days, with an exception that if the last day is the 31st and the first day is not 30th or 31st then that month has 31 days. So the rule is if d1 is 31 it changes to 30, and if d2 is 31 change it to 30 but only if d1 is either 30 or 31.
Floating rate notes
FRNs always use actual/360 or 365 in the case of sterling.
Fixed coupon
This basis is commonly used for Eurobonds, and the daycount fraction is just one divided by the number of interest payments per year. Thus the coupon payments are always the same and any small difference in the number of days between successive coupon payments is ignored.
Conversion between different day count conventions
Compounding conventions
Simple or periodic interest
Simple or periodic interest does not offer the opportunity to earn interest on interest, ie, there is no compounding of interest. Simple interest is typically used for instruments with a maturity of less than one year.
<math>Discount factor (DF) = \frac{1}{(1+ R\frac{n}{365})}</math>
where:
- R = simple interest rate on an actual/365 fixed basis; and
- n = number of days in period.
(Note: If R is on an actual/360 basis, replace 365 with 360 in the above formula.)
Compound interest
Offers the opportunity for interest payments to be reinvested in order to earn interest on interest. Compound interest is typically used for instruments greater than one year.
<math>Discount factor (DF) = (1+\frac{R}{f})^{-ft}</math>
where:
- R = compound interest rate;
- f = compounding frequency in number of periods per year; and
- t = number of years to maturity.
Continuous compounding
Compounding can be taken to an extreme in which the interest is continuously compounded, i.e. where f in the previous formula tends towards infinity. Continuously compounded rates are rarely quoted in practice, although they are used extensively in derivatives pricing systems.
<math>Discount factor (DF) = e^{-Rt}</math>
where:
- R = continuously compounded interest rate;
- t = number of years to maturity; and
- e = 2.71828 (to five decimal places).
Converting between different compounding frequencies
Settlement day conventions
The settlement day is the day on which traded bonds or securities actually change ownership and are paid for. This is often a few business days after the transaction date, T. Interest rate calculations normally start on the settlement day. Settlement in Euro denominated bonds and in most markets is on T+3, although for us and UK government bonds and bills it is T+1, and for sterling bonds it is T+5.
