Compound Interest

Recall the formula for simple interest $I=Prt$, where $P$ is principal (amount deposited),$r$ is annual rate of interest expressed as a decimal, and $t$ is time in years that the principal earns interest. Suppose $t = 1$ yr.  Then at the end of the year the amount has grown to $P+Pr=P(1+r)$.

If this balance earns interest at the same interest rate for another year, the balance at the end of that year will increase as follows:

$\left[P\left(1+r\right)\right]+\left[P\left(1+r\right)\right]r=P\left(1+r\right)^2$

Continuing in this way produces a formula for interest compounded annually.

Formula for Interest Compounded Annually

$A=P\left(1+r\right)^t$

If P dollars are deposited in an account paying an annual rate of interest r compounded (paid) n times per year, then after t years the account will contain A dollars, according to the following formula.

General Compound Interest Formula

$A=P\left(1+\frac{r}{n}\right)^{tn}$

Example

Suppose $1000 is deposited in an account paying 4% interest per year compounded quarterly (four times per year).

1. Find the amount in the account after 10 years with no withdrawals.
2. How much interest was earned over the 10 year period.

Solution

1. Find the amount in the account after 10 years with no withdrawals.
   $A=P\left(1+\frac{r}{n}\right)^{tn}=1000\left(1+\frac{0.04}{4}\right)^{10\left(4\right)}\:\:because\:P=1000,\:r=0.04,\:\&\:t=10$
   $A=1488.86$
   Thus, $1488.86 is in the account after 10 years.
2. How much interest was earned over the 10 year period.
   The interest earned for that period is:
   $1488.86-$1000=$488.86

Finding the Present Value

Here is an example.

Example

Becky must pay a lump sum of $6000 in 5 years.

1. What amount deposited today (present value) at 3.1% compounded annually will grow to $6000 in 5 years?
2. If only $5000 is available to deposit now, what annual interest rate is necessary for the money to increase to $6000 in 5 years?

Solution

1. What amount deposited today (present value) at 3.1% compounded annually will grow to $6000 in 5 years?
   If Becky leaves $5150.60 for 5 years in an account paying 3.1% compounded annually, she will have $6000 in 5 years.
2. If only $5000 is available to deposit now, what annual interest rate is necessary for the money to increase to $6000 in 5 years?
   $A=P\left(1+\frac{r}{n}\right)^{tn}$
   $6000=5000\left(1+r\right)^5\:\:\:because\:A=6000,\:P=5000,\:n=1,\:\&\:t=5$
   $\frac{6}{5}=\left(1+r\right)^5$
   $\left(\frac{6}{5}\right)^{\frac{1}{5}}=1+r$
   $\left(\frac{6}{5}\right)^{\frac{1}{5}}-1=r$
   Using a calculator.
   $r\approx0.0371$
   An interest rate of 3.71% will produce enough interest to increase the $5000 to $6000 by the end of 5 years.