Calculating Compound Interest Rates
Revised March, 2014

nominal rate - Basically, this is the rate before it is compounded.
compounded rate - Rate after it has been compounded.



8 per cent interest compounded semi-annually equals what nominal rate?
We know the nominal rate is 8 per cent so:
Compounded Interest Rate = (1 + [.08 ÷ 2])2 -1

Compounded Interest Rate = (1 + .04)2 -1

Compounded Interest Rate = 1.0816 -1

Compounded Interest Rate = .0816
which equals 8.16 per cent.

You probably have concluded that:
n = 4 for quarterly compounded interest
n = 12 for monthly compounded interest and
n = 365 for daily compounded interest.




If we have a monthly compounded interest rate of .072290080856235 (or 7.2290080856235%), what was the rate before compounding?
(Or what is the nominal rate?)

Since we are dealing with monthly compounding, n=12.
Putting the numbers into the formula, we see that the nominal rate equals:

12 * [(1 + .072290080856235)(1 ÷ 12)-1)]
= 12 * [(1.072290080856235)(.08333333...)-1)]
= 12 * (1.00583333333333... -1)
= 12 * (.00583333333333)
nominal rate = .07 or 7 per cent




Continuously Compounded Interest
For the first time, we have a formula that uses the mathematical constant e which equals 2.71828182845904523536....
So, let's comptue the continuously compounded rate of a nominal rate of 9 per cent.
continuously compounded rate = er -1
continuously compounded rate = (2.71828...).09-1
continuously compounded rate = 1.09417428370521 -1
continuously compounded rate = .09417428370521
continuously compounded rate = 9.417428370521%



A bank account yields 6% interest when compounded continuously.
What is the nominal interest rate?

nominal rate = natural log [1 + .06]

nominal rate = 0.0582689081239758

rate = 5.82689081239758% before compounding



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