### 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 — e

^{rt}
Where:

- e = 2.718

r = annually compounded rate of interest

t = number of time periods