What are Accrual Range, Accrual Period and Accrual Year?

Accrual Range - Accrual Range is a given date range for which the Accrued Interest is being calculated.

Accrual Period - Accrual Period refers to the Interest Period (or the the Compound Period) within which an Accrual Range is being given to calculate the Accrued Interest.

Accrual Year - Accrual Year refers to the year within which an Accrual Range is being given to calculate the Accrued Interest.

Let's use a 1-year term deposit with a simple interest rate and monthly interest payments as an example to demonstrate how Accrual Range, Accrual Period and Accrual Year are related to each other.

Interest Method:    Simple Interest 
Interest Frequency: 12 (Monthly) 
Interest Rate:      5.00%
Term:               1 Year 
  Start Date:       2026-01-01 
  Maturity Date:    2027-01-01

Now if you invest a Principal of $1,000.00 to this term deposit, and want to calculate the Accrued Interest for the first 15 days of your investment, then the following diagram shows you how Accrual Range, Accrual Period and Accrual Year are related to each other:

   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

where: 
  T1 = (Y1,M1,D1) = (2026,01,01)
     = Starting date (inclusive) of an Accrual Range 
     = Starting date (inclusive) of an Accrual Period 
     = Starting date (inclusive) of an Accrual Year 

  T2 = (Y2,M2,D2) = (2026,01,16)
     = Ending date (exclusive) of the Accrual Range 

  T3 = (Y3,M3,D3) = (2026,02,01)
     = Ending date (exclusive) of the Accrual Period 

  T4 = (Y4,M4,D4) = (2027,01,01)
     = Ending date (exclusive) of the Accrual Year 

In this example, the starting dates (inclusive) of Accrual Range, Accrual Period and Accrual Year are aligned to the same date, represented by the T1 = (Y1, M1,D1) = (2026,01,16) notation. Their ending dates (exclusive) are represented by T2 = (Y2,M2,D2) = (2026,01,16), T3 = (Y3,M3,D3) = (2026,02,01), and T4 = (Y4,M4,D4) = (2027,01,01) notations respectively.

If T1 and T2 moves to a different Accrual Period, T3 and T4 will move accordingly. In other words, only T1 and T2 are independent parameters. So now we can define the Accrued Interest and other related calculations as functions of (T1,T2) or (Y1,M1,D1,Y2,M2,D2):

Days in [T1,T2): 
  Days_in_Range(T1,T2) = Days_in_Range(Y1,M1,D1,Y2,M2,D2)

Days in [T1,T3):
  Days_in_Period(T1,T2) = Days_in_Period(Y1,M1,D1,Y2,M2,D2)
  
Days in [T1,T4):
  Days_in_Year(T1,T2) = Days_in_Year(Y1,M1,D1,Y2,M2,D2)

Day Count Factor for [T1,T2):
  Day_Count_Factor(T1,T2) = Days_in_Range(T1,T2) / Days_in_Year(T1,T2)

Accrued Interest for [T1,T2):  
  Accrued_Interest(T1,T2) 
    = Principal × Interest_rate × Day_Count_Factor(T1,T2)

With these notations, we can redefine Day Count Conventions as rules to construct the Day_Count_Factor(T1,T2) function, which can be expressed in 2 formats:

Day_Count_Factor(T1,T2) 
  = Days_in_Range(T1,T2) / Days_in_Year(T1,T2)

Day_Count_Factor(Y1,M1,D1,Y2,M2,D2)
   = Days_in_Range(Y1,M1,D1,Y2,M2,D2) / Days_in_Year(Y1,M1,D1,Y2,M2,D2)

where: 
  (Y1,M1,D1) are Year, Month and Day components of T1 
  (Y2,M2,D2) are Year, Month and Day components of T2

We can also short the function names as:

DiR() = Days_in_Range()
DiP() = Days_in_Period() 
DiY() = Days_in_Year() 

Day_Count_Factor(T1,T2) 
  = DiR(T1,T2) / DiR(T1,T2)

Day_Count_Factor(Y1,M1,D1,Y2,M2,D2)
   = DiR(Y1,M1,D1,Y2,M2,D2) / DiY(Y1,M1,D1,Y2,M2,D2)

⇒ List of Day Count Conventions

⇐ Interest Period and Interest Frequency

⇑ Introduction to Day Count Convention

⇑⇑ Day Count Conventions