How does "Act/Act Bond" convention works on an Accrual Range that crosses one or more Accrual Period boundaries?

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.

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 [2024-11-30, 2024-05-31) using the Act/Act Bond convention:

1. Split the Accrual Range into multiple parts

R1 = [2024-11-30, 2025-05-15) within 
  P1 =[2024-11-15, 2025-05-15) Accrual Period 
R2 = [2025-05-15, 2025-05-31) within 
  P2 =[2025-05-15, 2025-11-15) Accrual Period 

2. Calculate Accrued Interest for each part

Accrued_Interest(R1) 
  = $1,000.00 × 4.5% × Day_Count_Factor(R1)
  = $1,000.00 × 4.5% × 0.5 × Calendar_Days(R1)/Calendar_Days(P1)
  = $1,000.00 × 4.5% × 0.5 × 166/181 
  = $20.635359116022

Accrued_Interest(R2) 
  = $1,000.00 × 4.5% × Day_Count_Factor(R2)
  = $1,000.00 × 4.5% × 0.5 × Calendar_Days(R2)/Calendar_Days(P2)
  = $1,000.00 × 4.5% × 0.5 × 16/184 
  = $1.9565217391304 

3. Sum up Accrued Interests from every part

Accrued_Interest 
  = Accrued_Interest(R1) + Accrued_Interest(R2)  
  = $20.635359116022 + $1.9565217391304 
  = $22.591880855152
  = $22.59

⇒ "Act/Act Bond" - Daily Compound Interest

⇐ "Act/Act Bond" - Whole Accrual Period

⇑ Day Count Convention - "Act/Act Bond"

⇑⇑ Day Count Conventions