Financial Exponential Functions - Compound Interest

Compound Interest and Several Examples of Exponential Functions

Here, we will go over some examples of Compound Interest. It is important to know that Compound Interest is just a exponential function that can be solved using simple algebra. 

Labeled Formula of Compound Interest

P = the principal amount, otherwise known as the original/ initial amount.

r = rate at which the money grows per year; interest rate (written in decimals)

m = number of times compounded in a year (ex. daily, biweekly, weekly, monthly, semimonthly...)

t = after how many years (amount of years)

A = the total amount after 't' years

If 'm' equals the following numbers, that is the way we are compounding in a year.

m = 1 only counted once per year, every year

m = 2 semiannually, because it is only counted 2 times a year; meaning after every 6 months (12/2=6)

m = 4 quarterly, because it is only counted every 4 times a year; meaning after every 3 months (12/4=3)

m = 12 monthly, because there are 12 months in a year, so after every month for 12 total counts in a year.

m = 52 weekly, because there are 52 weeks in a year, so after every week for 52 total counts in a year.

m = 365 daily, because there are 365 days in a year, so after every day for 365 total counts in a year.

Great, let's try an example.

So, we have to solve for x in terms of 'm'.

Let's rephrase the question into a formula.

Original formula: A = P(1 + r/m)^mt

Our equation: A = 200(x)^mt 

where x = (1 + r/m)

We know that 'r' or rate has to be written as a decimal. So the rate HAS to be a decimal.

The answer is A.)

See? No math required on that one!

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Here's another example:

Peter starts with $5,000. That is his principal value. His rate is written as a percentage; 9.2%. Since we are given percentage, we would need to divide by 100 = (9.2/100). Rate is 0.092. Now we just plug the numbers we have to find total amount (A)

Original formula: A = P(1+r/m)^mt

A = 5000(1+0.092/1)^1*2

P= 5000

r= 0.092

m= 1, it's calculated per year

t= 2, total amount after 2 years

A= 5000(1.092)^2

A = 5000(1.192464)

A = $5962.32