Calculating Compound Interest Rates
Revised March, 2014
annual (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 annual (nominal) rate?
We know the annual (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 annual (nominal) rate?)
Since we are dealing with monthly compounding, n=12.
Putting the numbers into the formula, we see that the annual (nominal) rate equals:
12 * [(1 + .072290080856235)^{(1 ÷ 12)}-1)]
= 12 * [(1.072290080856235)^{(.08333333...)}-1)]
= 12 * (1.00583333333333... -1)
= 12 * (.00583333333333)
annual (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 an annual (nominal) rate of 9 per cent.
continuously compounded rate = e^{r} -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 annual (nominal) interest rate?
annual (nominal) rate = natural log [1 + .06]
annual (nominal) rate = 0.0582689081239758
rate = 5.82689081239758% before compounding