### Compounding Intervals and Interest Rate

Theoretically there are two types of interest rates, simple and compounding. However, in finance the word interest usually refers to compound interest. Simple interest almost never factors in financial calculations. In all calculations related to present values and future values, compound interest is used. However, as a student of corporate finance, it is essential to know the difference that compounding intervals have on the effective interest rate that is paid on the investment. This article explains the same:
##### Simple Interest vs. Compound Interest
We are all aware of the difference between simple and compound interest. However, just to reiterate, the principal amount never changes in a simple interest calculation. So if \$100 are lent for 3 years at 10% simple interest, the interest paid in each of the 3 years would be \$10.
But if \$100 were lent at 10% for 3 years and compounding happens annually, the interest payments would be \$10, \$11 and \$13.1 for years 1,2 and 3 respectively. This is because at the end of each period the accrued interest gets added to the principal and therefore the interest in the next period is a little bit more.
##### Annual vs. Semi-Annual Compounding
In case of compound interest 10% compounded annually and 10% compounded semi-annually i.e. twice a year do not means the same thing. Let’s understand this with the help of an example:
Annual Compounding: \$100 @10%, Interest = \$10
Semi-Annual Compounding: \$100 @10%, Interest \$5 after 6 months and %5.25 after another 6 months. Hence the total interest would be \$10.25 as opposed to \$10 on an annual basis.
Rates Increase As Compounding Intervals Grow Smaller:
As we can see from the above example that semi-annual rates give more interest than the annual rates. We can extend this logic further and say that monthly rates will provide more interest as compared to semi-annual rates and weekly rates will provide more interest than monthly rates.
As a thumb rule, we can say that the smaller the compounding intervals, the higher the interest rates will be. As far as investments are concerned, most rates are compounded annually or semi-annually. Smaller compounding frequencies are not used. In common usage, only in the case of credit cards are the rates expressed as monthly compounding interest rates.
##### Continuous Compounding
Until now, we have considered discrete intervals at which interest was being paid. We could bring the intervals down to hours, minutes or even seconds and yet they will be discrete. Theoretically it is possible that interest be paid continuously over a given period of time. This is not possible in reality. However, continuously compounded interest rates provide some ease in mathematical calculations. It is for this reason that they are often used in finance. Compounded interest rates can be converted into continuously compounded interest rates by multiplying them with — ert
Where:
e = 2.718
r = annually compounded rate of interest
t = number of time periods