Compound Interest
A special case of the exponential growth formula is the compound interest formula. We use this formula for investments when interest is added more than one time per year.
A special case of the exponential growth formula is the compound interest formula. We use this formula for investments when interest is added more than one time per year.
Compound interest formula : \(y=P\left(1+\frac{r}{n}\right)^{nt}\)
In the formula, \(P\) represents the initial investment (value at time = \(0\)), \(r\) represents the annual rate of interest (this must be converted to a decimal by moving the decimal two spots to the left), \(n\) represents the number of times per year interest is added to the account, \(t\) represents a period of time (years), and \(y\) represents the ending value of the investment after a certain number of growth periods (which is represented by \(nt\))
In the formula, \(P\) represents the initial investment (value at time = \(0\)), \(r\) represents the annual rate of interest (this must be converted to a decimal by moving the decimal two spots to the left), \(n\) represents the number of times per year interest is added to the account, \(t\) represents a period of time (years), and \(y\) represents the ending value of the investment after a certain number of growth periods (which is represented by \(nt\))
Values for \(n\) if interest is compounded:
- Yearly: \(n=1\) Semi-Annually: \(n=2\) Quarterly: \(n=4\)
- Monthly: \(n=12\) Weekly: \(n=52\) Daily: \(n=365\)
Let’s think about this formula before we begin to apply it. If an account earns \(8\%\) annual interest, compounded quarterly, this does NOT mean that you will receive \(8\%\) interest \(4\) times per year. It means you will receive \(2\%\) interest each compounding period, for an effective annual rate of \(8\%\) per year. In addition, we multiply the length of time of the investment by the compounding period in order to determine the exponent, which represents the number of growth periods over the course of the investment.
Example 1: Find the value of a \(\$1500\) investment after \(8\) years if the annual interest rate is \(2.5\%\) compounded monthly.
\(\begin{align}&y=P\left(1+\frac{r}{n}\right)^{nt}\ \ & \ \ \ \ \ \ &\text{1) Write the general formula.}\\
&P=1500\ \ \ r=2.5\%=.025\ \ \ n=12\ \ \ t=8\ \ & \ \ \ \ \ \ &\text{2) Identify the values of the variables.}\\
&y=1500\left(1+\frac{.025}{12}\right)^{12\cdot8}\ \ & \ \ \ \ \ \ &\text{3) Substitute the values into the formula.}\\
&y=1831.723018\ \ & \ \ \ \ \ \ &\text{4) Enter the values into the calculator.}\end{align}\)
The investment has a value of \(\$1831.72\) after \(8\) years.
Example 2: How long will it take an investment earning an annual interest rate of \(4.25\%\), compounded quarterly, to double? Assume there are no additional deposits or withdrawals.