What Is "Act/Act Bond"

Q

What is the "Act/Act Bond", or "Actual/Actual Bond" Day Count Convention?

✍: FYIcenter.com

A

Act/Act Bond (Actual/Actual Bond) - Act/Act Bond is a Day Count Convention used by the US Treasury Department for their bonds and other securities. This convention ensures that every Accrual Period generates the same amount of interest. And every day generates the same amount of interest in the same accrual period.

Act/Act Bond is also referred as Act/Act ICMA, Act/Act ISMA.

Act/Act Bond can be formally defined as:

Day_Count_Factor(T1,T2) = DiR(T1,T2) / DiY(T1,T2)
DiR(T1,T2) = Calendar_Days(T1,T2)
DiY(T1,T2) = Accrual_Frequency × Calendar_Days(T1,T3)

where: 
  T1 = (Y1,M1,D1): Starting date (inclusive)  
  T2 = (Y2,M2,D2): Ending date (exclusive) of the Accrual Range  
  T3 = (Y3,M3,D3): Ending date (exclusive) of the Accrual Period 
  T4 = (Y4,M4,D4): Ending date (exclusive) of the Accrual Year 

Example:
   2026-01-01   2025-01-16   2026-02-01            2027-01-01
---T1-----------T2-----------T3-----------...------T4--- 

  [T1-----------T2): Accrual Range
  [T1------------------------T3): Accrual Period
  [T1-------------------------------------...------T4): Accrual Year

Here is an example of US bond security:

CUSIP:                912810UE6
Issuer:               US Treasury
Interest rate:        4.5%
Interest frequency:   Semiannually (2 times in a year)
Day Count Convention: Act/Act Bond
Term:                 30 Years 
  Start Date:         2024-11-15
  Maturity Date:      2054-11-15

If you bought $1,000.00 of this US bond, here is how you can calculate Accrued Interest in the date range of [T1,T2) = [2024-11-15, 2024-11-30):

T1 = 2024-11-15: Starting date (inclusive)
T2 = 2024-11-30: Ending date (exclusive) of the Accrual Range 
T3 = 2025-05-15: Ending date (exclusive) of the Accrual Period  
T4 = 2025-11-15: Ending date (exclusive) of the Accrual Year

DiR(T1,T2) = DiR(T1,T2) 
  = Calendar_Days(T1,T2) 
  = 15

DiY(T1,T2) = DiY(T1,T2) 
  = Accrual_Frequency × Calendar_Days(T1,T3)
  = 2 × 181 
  = 362 

Day_Count_Factor(T1,T2) = DiR(T1,T2) / DiY(T1,T2)
  = 15 / 362 
  = 0.041436464088398

Accrued_Interest = Principal × Interest_Rate × Day_Count_Factor
  = $1,000.00 × 4.5% × 0.041436464088398
  = $1.8646408839779
  = $1.86

Act/Act Bond convention has several main properties. See next tutorials for more details.

The Day Count Factor for a whole Accrual Period is a constant.
A fixed amount of Accrued Interest is paid out for each Accrual Period:
Daily Interest Rate varies from one Accrual Period to another Accrual Period.
Leap Day reduces the Daily Interest Rate.
When an Accrual Range crosses one or more Accrual Period boundaries, it needs to be split into multiple smaller parts, so that each part can fit into a single period.
Not applicable for daily Compound Interest.
Day Count Factor is not additive for consecutive and non-overlapping Accrual Ranges.

References: