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Summarise how interest on a fixed-income security is computed under the common day-count conventions.

FRM® Part I · Valuation and Risk Models · Pricing Conventions, Discounting, and Arbitrage · FRM1.VR.9.g

The idea

Day-count conventions fix how interest accrues. Actual/Actual is used for US Treasury notes and bonds. It counts the actual days elapsed over the actual days in the coupon period. Accrued interest = coupon per period × elapsed days ÷ days in period. 30/360 is used for US corporate, agency and municipal bonds. It treats every month as 30 days and the year as 360, so a semiannual period always has 180 days. Days = 360 × (year difference) + 30 × (month difference) + (day difference). A day 31 is treated as 30. Actual/360 is standard in money markets and for many floating-rate loans. Interest = principal × rate × actual days ÷ 360. That is a little more than Actual/365 would pay. Use the coupon per period, not the annual coupon, in an accrual formula. Always match the convention to the instrument.

Formulas

Accrued interest between coupon dates

AI = Cp × (t ÷ T)

Cp
the coupon per period, not the annual coupon
t
days since the last coupon date, counted under the bond's convention
T
days in the coupon period under the same convention

Actual/Actual for Treasuries uses calendar days; 30/360 for corporates, agencies and municipals makes a semiannual period 180 days. Using the annual coupon is the usual error.

Day count under 30/360

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

Y1
the year of the start date
Y2
the year of the end date
M1
the month of the start date
M2
the month of the end date
D1
the day of the start date
D2
the day of the end date

Every month counts as 30 days and a day 31 is treated as 30. Do not use it for Treasuries, which accrue on Actual/Actual.

Interest under Actual/360

Interest = N × r × t ÷ 360

N
the principal
r
the annual interest rate, as a decimal
t
the actual number of days in the period

Standard in money markets and many floating-rate loans. Dividing by 360 rather than 365 pays slightly more for the same quoted rate.

A practice question

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

Which day-count convention is used to compute accrued interest on US Treasury notes and bonds?

  1. 30/360Why not: Describes the convention for US corporate, agency and municipal bonds.
  2. Actual/360Why not: Describes the money-market convention used for bills and many floating-rate loans.
  3. 30E/360Why not: Is a European bond-market convention, not the one used for Treasuries.
  4. Actual/Actual - correct

Treasury accrued interest uses actual days elapsed over actual days in the coupon period. 30/360 is for corporates and Actual/360 for money markets.

A harder one

A US Treasury note pays a 4% coupon semiannually on 15 May and 15 November. A buyer settles on 9 August of a non-leap year. The coupon period contains 184 days and 86 days have elapsed since 15 May. Accrued interest per 100 face value is closest to:

  1. 0.93 - correct
  2. 1.87Why not: Uses the annual coupon of 4.00 with the semiannual day fraction.
  3. 0.47Why not: Uses the semiannual coupon of 2.00 over a 365-day year.
  4. 1.07Why not: Counts the 98 days remaining to the next coupon instead of the days elapsed.

Actual/Actual accrual is the semiannual coupon x elapsed days / days in the period = 2.00 x 86 / 184 = 0.93.

The hardest tier

A Treasury note pays a 3% coupon semiannually on 15 February and 15 August; the coupon period from 15 February to 15 August contains 181 days. It is quoted at 99-08 (99 and 8/32) per 100 face value for USD 1 million face value settling on 30 April of that non-leap year. The invoice amount is closest to:

  1. USD 998,633 - correct
  2. USD 996,933Why not: Reads 99-08 as 99.08 rather than 99 and 8/32.
  3. USD 995,583Why not: Accrues on Actual/360 over 74 days, 15,000 x 74 / 360, instead of Actual/Actual.
  4. USD 1,004,765Why not: Uses the annual coupon of 30,000 in the accrued interest.

Clean = 99.25% x 1,000,000 = 992,500. Accrued = 15,000 x 74 / 181 = 6,133. Invoice = clean + accrued = 998,633.

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