# Nominal interest rate and effective interest rate converter

This tool converts nominal interest rates to effective interest rates and vice versa.
The compounding period and payment period can be set at different frequencies.
Besides nominal interest rates and effective interest rates this tool also calculates the periodic interest rate.
To convert the interest rates the following equations are used:

Compound period equals payment period:
Re = (1 + (R / N))N - 1
R = N * ((1 + Re)(1 / N) - 1)
r = R / N

Compound period is not equal to payment period:
Re = (1 + (R / N))N - 1
r = ((1 + Re) ^ (1 / M) - 1)
R = r * M

where:

 R Nominal annual interest rate (as a decimal, not in percentage) Is the annual rate of interest without taking into account the compounding of interest within that year. It does not truly represents the amount of interest earned in a year. The nominal annual interest rate is also known as: nominal rate stated rate quoted rate annual percentage rate (APR) Re Effective annual interest rate (as a decimal, not in percentage) Is the annual rate of interest while taking into account the compounding of interest within that year. It truly represents the amount of interest earned in a year. The effective annual interest rate is also known as: annual percentage yield (APY) equivalent annual rate (EAR) annual equivalent rate (AER) N Compounding period The compounding period is the number of times per year, where the amount of interest is determined and then added to the capital. Compounding period = daily (N=360,364,365,366), weekly (N=52), bi-weekly (N=26), semi-monthly (N=24), monthly (N=12), bi-monthly (N=6), quarterly (N=4), semi-annually (N=2) or annually (N=1). M Payment period The payment period is the number of times per year, where payments are made. Payment period = daily (N=360,364,365,366), weekly (N=52), bi-weekly (N=26), semi-monthly (N=24), monthly (N=12), bi-monthly (N=6), quarterly (N=4), semi-annually (N=2) or annually (N=1). Depending on the financial product the payment period and compounding period may or may not coincide. For example, payments can occur quartely while compounding occurs semi-annually. r Periodic interest rate Compound period equals payment period: The periodic interest rate is computed by dividing the nominal rate by the number of compounding periods per year. Thus a 6% nominal rate compounded monthly is equivalent to a periodic rate of 0.5% per month. Compound period is not equal to payment period: The effective interest rate per payment period is calculated.

Example 1:
The nominal annual interest rate is 4.67% compounded quarterly.

Question:
What is the effective annual interest rate?

Solution:
Re = (1 + (R / N))N - 1 = (1 + (0.0467 / 4))4 - 1 = 0.047524

Example 2:
The effective annual interest rate is 3.5% compounded weekly.

Question:
What is the nominal annual interest rate?

Solution:
R = N * ((1 + Re)(1 / N) - 1) = 52 * ((1 + 0.035)(1 / 52) - 1) = 0.034413

Example 3:
Saving bank A pays a nominal annual interest rate (APR) of 10% compounded semi-annually.
Savings bank B pays a nominal annual interest rate (APR) of 9.5% compounded quarterly.

Question:
Which bank offers the best effective annual interest rate (EAR)?

Solution:
Bank A:
Re = (1 + (R / N))N - 1 = (1 + (0.10 / 2))2 - 1 = 0.1025

Bank B:
Re = (1 + (R / N))N - 1 = (1 + (0.095 / 4))4 - 1 = 0.1027

Bank B offers the best savings return or Equivalent Annual Rate (EAR).

Example 4:
A savings bank pays 2.5% interest every 3 months.

Question:
What are the nominal and effective interest rates per year?

Solution:
R = r * N = 0.025 * 4 = 0.1
Re = (1 + (R / N))N - 1 = (1 + (0.1 / 4))4 - 1 = 0.1038

Example 5:
Bank A offers an effective annual interest rate (Re) of 6%.
Bank B offers a nominal interest rate of 1.5% per quarter.

Question:
Which of these two banks offers the best return?

Solution:
Bank B:
R = r * N = 0.015 * 4 = 0.06
Re = (1 + (R / N))N - 1 = (1 + (0.06 / 4))4 - 1 = 0.061364
Bank B offers a better return.

Example 6:
The nominal interest rate of 10% compounded monthly.

Question:
Find the effective interest rate per payment period if the payment period is:
• quarterly
• semi-annually
• annually
Solution:
Re = (1 + (R / N))N - 1 = (1 + (0.10 / 12))12 - 1 = 0.104713
• quarterly
r = ((1 + Re)(1 / M) - 1) = ((1 + 0.104713)(1 / 4) - 1) = 0.025209 per quarter
• semi-annually
r = ((1 + Re)(1 / M) - 1) = ((1 + 0.104713)(1 / 2) - 1) = 0.051053 per half year
• annually
r = ((1 + Re)(1 / M) - 1) = ((1 + 0.104713)(1 / 1) - 1) = 0.104713 per year

## Input nominal interest rate and effective interest rate converter:

 Calculate: Effective annual interest rate (Re)Nominal annual interest rate (R) Nominal annual interest rate (R) *: % Effective annual interest rate (Re) *: % Compounding period (N): dailyweeklybi-weekly (every two weeks)semi-monthly (twice a month)monthlybi-monthly (every two months)quarterlysemi-annually (twice a year)annually Payment period (M): dailyweeklybi-weekly (every two weeks)semi-monthly (twice a month)monthlybi-monthly (every two months)quarterlysemi-annually (twice a year)annually Days in year: 360364365366 * = required