Understanding Effective Rate of Interest

The Effective Rate of Interest (ERI) is the actual interest earned or paid on an investment or loan after accounting for compounding over a given period. It differs from the nominal interest rate because it considers the effects of compounding.

Formula for Effective Rate of Interest

If the nominal annual interest rate is r and the number of compounding periods per year is n, the Effective Rate of Interest (ERI) is calculated as:

ERI = (1 + r/n)ⁿ - 1

where:

• r = nominal annual interest rate (as a decimal, e.g., 5% = 0.05)
• n = number of compounding periods per year
  • Annual compounding: n = 1
  • Semi-annual compounding: n = 2
  • Quarterly compounding: n = 4
  • Monthly compounding: n = 12
  • Daily compounding: n = 365

Example Calculation

Suppose a bank offers a nominal annual interest rate of 10% (0.10), compounded quarterly (n = 4).

ERI = (1 + 0.10/4)⁴ - 1
ERI = (1 + 0.025)⁴ - 1
ERI = (1.025)⁴ - 1
ERI = 1.10381 - 1
ERI = 0.10381 or 10.38%

So, the effective rate of interest is 10.38%, meaning the actual return on the investment is higher than the nominal rate due to compounding.

Special Case: Continuous Compounding

If interest is compounded continuously, the effective interest rate is calculated using:

ERI = e^r - 1

where e is Euler’s number (approximately 2.718).

For example, if r = 10% = 0.10:

ERI = e^0.10 - 1
ERI = 1.10517 - 1 = 0.10517 or 10.52%

This means the effective interest rate for continuous compounding is 10.52%.