Recognise the most widely used day count conventions, the markets in which each normally applies, and how to use each in an interest calculation.

FRM® Part I · Financial Markets and Products · Interest Rate Futures · FRM1.FM.19.a

The idea

A day count convention fixes how many days of interest accrue between two dates and how long a year is. In US practice, Treasury bonds use Actual/Actual: actual days elapsed over actual days in the coupon period. Corporate and municipal bonds use 30/360, where each month counts as 30 days and the year as 360. Days = 360 × (Y2 − Y1) + 30 × (M2 − M1) + (D2 − D1). Before using it, set a start day of 31 to 30, and set an end day of 31 to 30 only if the start day is 30 or 31. Money market instruments such as bills and commercial paper use Actual/360. Interest = principal × rate × day count fraction. At the same quoted rate, Actual/360 pays slightly more than Actual/365. The usual error is applying one convention to every instrument, so match the convention to the market first.

Formulas

Day count under 30/360

days = 360 × (Y₂ − Y₁) + 30 × (M₂ − M₁) + (D₂ − D₁)

Y₁, Y₂: the start and end years
M₁, M₂: the start and end months
D₁, D₂: the start and end days, after adjusting 31s

Set a start day of 31 to 30; set an end day of 31 to 30 only if the start day is 30 or 31. Used for US corporate and municipal bonds.

Interest accrued under a day count convention

interest = principal × rate × day count fraction

principal: the amount on which interest accrues
rate: the quoted annual rate
day count fraction: days accrued ÷ year length, both as the convention defines them

Match the convention to the market first: Actual/Actual for Treasuries, 30/360 for corporates, Actual/360 for money market instruments.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which of the following markets most commonly uses the 30/360 day count convention?

US Treasury bondsWhy not: Treasury bonds accrue on Actual/Actual, not 30/360.
US Treasury billsWhy not: Bills are money market instruments quoted on an Actual/360 basis.
Overnight interbank depositsWhy not: Short-term deposits accrue on actual days (typically Actual/360), not on 30-day months.
US corporate bonds - correct

Corporate and municipal bonds in the US accrue interest on 30/360, treating every month as 30 days. Treasury bonds use Actual/Actual and money market instruments use Actual/360.

A harder one

A corporate bond with a 7% annual coupon and face value USD 1,000,000 last paid a coupon on 15 March. Settlement is 31 May. Use 30/360 with the rule that a day 31 at the end becomes 30 only if the start day is 30 or 31. Accrued interest is closest to:

USD 14,777.78 - correct
USD 14,972.22Why not: Counts actual days (77) over 360, mixing Actual with a 360-day year.
USD 14,767.12Why not: Counts actual days over 365 (Actual/365) instead of the 30/360 convention.
USD 14,583.33Why not: Changes the end day 31 to 30 even though the start day (15) is not 30 or 31, giving 75 days instead of 76.

30/360 days = 30 x (5 - 3) + (31 - 15) = 76, because the end day stays 31 when the start day is 15. Accrued = 70,000 x 76/360 = 14,777.78.

The hardest tier

A US Treasury bond with a 3% annual coupon, face value USD 10,000,000, pays coupons on 15 March and 15 September. Settlement is 24 June (a non-leap year). Accrued interest is closest to:

USD 82,500.00Why not: Uses 30/360 (99 days of 180); Treasury bonds accrue on Actual/Actual.
USD 82,336.96 - correct
USD 83,013.70Why not: Uses Actual/365 on the annual coupon instead of actual days over actual days in the coupon period.
USD 84,166.67Why not: Uses Actual/360 on the annual coupon, a money market convention that does not apply to Treasury bonds.

Actual/Actual: semiannual coupon 150,000 x 101 days elapsed / 184 days in the period = 82,336.96.

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