How to apply the "Act/365", or "Actual/365" Day Count Convention to Daily Compound Interest?

Most banks in US offer term deposits (or CD - Certificate of Deposit) with daily compound interest using the Act/365 Day Count Convention.

In this case, even the Daily Interest Rate is a fixed value, we can not use the following formula to calculate Accrued Interest for any Accrual Range longer than 1 day. This is because the Principal is changing every day.

Accrued_Interest(T1,T2)
  = Principal × Interest_Rate × Day_Count_Factor(T1,T2)
  = Principal × Interest_Rate × Calendar_Days(T1,T2) / 365 
  = Principal × Daily_Interest_Rate × Calendar_Days(T1,T2)

# Not valid, if [T1,T2) is longer than 1 day

To calculate the Accrued Interest in an Accrual Range longer than 1 day, we need to split the Accrual Range into multiple 1-day parts.

T1 = (Y1,M1,D1): Starting date (inclusive)
T2 = (Y2,M2,D2): Ending date (exclusive) of the Accrual Range 

R1 = [T1, T1 + 1 day)
R2 = [T1 + 1 day, T1 + 2 day)
R3 = [T1 + 2 day, T1 + 3 day)
... 

Calculating the Accrued Interest of the first day is easy:

Accrued_Interest(R1)
  = Principal × Daily_Interest_Rate × Calendar_Days(R1)
  = Principal × Daily_Interest_Rate × 1
  = Principal × Daily_Interest_Rate

For the Accrued Interest of the second day, and additional days, we need to compound the interest of the previous day into the principal:

Accrued_Interest(R2)
  = (Principal + Accrued_Interest(R1)) × Daily_Interest_Rate

Accrued_Interest(R3)
  = (Principal + Accrued_Interest(R1) + Accrued_Interest(R2)) 
    × Daily_Interest_Rate

...

Obviously, we can re-write these formulas into a recursive form:

Accrued_Interest(R1) = Principal × Daily_Interest_Rate
Principal(R1) = Principal + Accrued_Interest(R1)

Accrued_Interest(R2) = Principal(R1) × Daily_Interest_Rate
Principal(R2) = Principal(R1) + Accrued_Interest(R2)

Accrued_Interest(R3) = Principal(R2) × Daily_Interest_Rate
Principal(R3) = Principal(R2) + Accrued_Interest(R3)

...

We can also re-write these compounded Principals in an exponential form:

Principal(R1) = Principal + Accrued_Interest(R1)
  = Principal + Principal × Daily_Interest_Rate
  = Principal × (1 + Daily_Interest_Rate) ** 1

Principal(R2) = Principal(R1) + Accrued_Interest(R2)
  = Principal(R1) + Principal(R1) × Daily_Interest_Rate
  = Principal(R1) × (1 + Daily_Interest_Rate)
  = Principal × (1 + Daily_Interest_Rate) ** 2

Principal(R3) = Principal(R2) + Accrued_Interest(R3)
  = Principal(R2) + Principal(R2) × Daily_Interest_Rate
  = Principal(R2) × (1 + Daily_Interest_Rate)
  = Principal × (1 + Daily_Interest_Rate) ** 3

...

Now we have derived the generic formula to calculate the compounded Principal at the end of an Accrual Range of [T1, T2):

Principal(T2) 
  = Principal(T1) × (1 + Daily_Interest_Rate) ** Calendar_Days(T1,T2)
  = Principal(T1) × (1 + Interest_Rate/365) ** Calendar_Days(T1,T2)

where: 
  Principal(T1) is the Principal on date T1. 

From the compounded Principal, we can derive the generic formula to calculate the Accrued Interest of an Accrual Range of [T1, T2):

Accrued_Interest(T1,T2) = Principal(T2) - Principal(T1) 
  = Principal(T1) × (1 + Rate/365) ** Days - Principal(T1)
  = Principal(T1) × ((1 + Rate/365) ** Days - 1)

where: 
  Rate = Interest_Rate 
  Days = Calendar_Days(T1,T2)

⇒ "Act/365" - Monthly Compound Interest

⇐ "Act/365" - Leap Years

⇑ Day Count Convention - "Act/365"

⇑⇑ Day Count Conventions