Compound Interest with Daily Rest

In the UK banking environment, compound interest with daily rest is a prevalent method for calculating interest on various financial products, particularly mortgages and loans. This method ensures that interest is precisely accounted for based on the daily fluctuations of the outstanding balance.

Component definitions

• Compound interest: This refers to the calculation of interest not only on the initial principal amount but also on the accumulated interest from previous periods. Unlike simple interest, which is calculated only on the original principal, compound interest allows for interest to compound, or grow on itself.
• Daily rest (or daily compounding): This is the frequency at which accrued interest is added to your principal balance. With daily rest, interest is calculated each day based on the closing balance of the previous day. This balance already includes any interest that accrued and was added from that prior day's calculation. This continuous, daily addition means that the balance, which forms the basis for the current day's interest calculation, is constantly changing, leading to an exponential growth effect on your overall interest.

Exponential calculation of interest

The exponential nature of compound interest with daily rest arises because the balance on which interest is charged is updated every day.

The following formulas are used to calculate interest:

• Daily interest rate: The annual nominal interest rate is converted into a daily rate based on the actual number of days in the year . Daily IR = Annual Interest Rate​/Number of Days in Year
• **Daily interest accrual: Each day, interest is calculated on the current outstanding balance.
  • Daily accrued interest = Current Opening Balance * Daily IR
  • Closing balance for today = Opening balance + Interest accrued
  • Closing balance for today = Opening balance used in interest computation from tomorrow
• Compounding effect over a period: To determine the total amount after a specific number of days, the formula demonstrates its exponential growth:
  • *Opening balance ((1+Daily IR)^(Number of days in interval)-1)
  • This formula effectively shows how interest is earned on previously accrued interest over the specified period.

Implications for Payment (PMT) calculation

While interest is compounded daily, loan payments (PMT) in UK banking are typically scheduled monthly. The PMT calculation for a loan with daily rest compounding accurately factors in this daily accrual.

• PMT recalculation: For loans with a fixed repayment schedule (e.g., Declining Balance Equal Instalment loans), the monthly PMT is calculated to ensure the loan is repaid over its term, accounting for the daily compounding. The interest component of each PMT will specifically reflect the actual interest accrued over the exact number of days in that month's payment period.
• PMT=PMT(Daily IR ,No of installments/12Days in year ,-Loan amount)Days in year/12

The following document shows some of the calculations for compound interest: