Equivalent Interest Rate Calculator

How the Equivalent Interest Rate Calculator Works:

The Equivalent Interest Rate Calculator allows you to convert an interest rate with one compounding frequency into an equivalent interest rate with a different compounding frequency. This helps when interest is compounded at different frequencies and you want to compare or convert rates accurately.

Inputs:

1. Interest Rate (R): The nominal interest rate per period, expressed as a percentage. In this example, we use 3%.
2. Compounding (m): The number of times the interest is compounded per period in the original scenario. For this example, 11 times per period.
3. New Compounding (q): The new number of times the interest is compounded per period. For this example, 2 times per period.

Output:

• New Nominal Interest Rate (Iₙ): The equivalent nominal interest rate for the new compounding frequency.
• Original and New Effective Interest Rates (Rₑ, Iₑ): The actual interest rate considering the compounding frequency.

Formulas:

The calculator uses the following formulas to convert the interest rates between different compounding frequencies:

1. Effective Interest Rate (Rₑ):

   $$Rₑ = \left(1 + \frac{R}{m}\right)^m – 1$$

   where:

   • $R$ is the nominal interest rate (as a decimal),
   • $m$ is the original compounding frequency.

2. Equivalent Nominal Interest Rate for New Compounding (Iₙ):

   $$Iₙ = q \times \left(\left(1 + Rₑ\right)^{\frac{1}{q}} – 1\right)$$

   where:

   • $q$ is the new compounding frequency,
   • $Rₑ$ is the effective interest rate.

3. New Effective Interest Rate (Iₑ):

   $$Iₑ = \left(1 + \frac{Iₙ}{q}\right)^q – 1$$

   where $Iₙ$ is the new nominal interest rate.

Example Calculation:

Input:

• Interest Rate (R): 3% per period
• Compounding Frequency (m): 11 times per period
• New Compounding Frequency (q): 2 times per period

Step-by-Step Calculation:

1. Original Effective Interest Rate (Rₑ):

   $$Rₑ = \left(1 + \frac{0.03}{11}\right)^{11} – 1 = 0.0304105$$

   So, the original effective interest rate is 3.04105%.

2. New Nominal Interest Rate (Iₙ):

   $$Iₙ = 2 \times \left(\left(1 + 0.0304105\right)^{\frac{1}{2}} – 1\right) = 0.0301926$$

   So, the equivalent nominal interest rate for compounding twice per period is 3.01926%.

3. New Effective Interest Rate (Iₑ):

   $$Iₑ = \left(1 + \frac{0.0301926}{2}\right)^2 – 1 = 0.0304105$$

   So, the new effective interest rate is 3.04105%, which matches the original effective rate.