Visualizing Compound Interest

To illustrate the compounding of interest in the calculation of a future value, we will assume that a single amount of $10,000 will be deposited into an account on January 1, 2012 and will remain on deposit for one year. The depositor may select one of three accounts and each of the accounts pays interest of 8% per year. However, the three accounts will differ in the compounding of interest as noted here:

• Account #1. The interest will be added to the account at the end of the year.
  (Interest is compounded annually on December 31.)
• Account #2. The interest will be added to the account at the end of each six-month period.
  (Interest is compounded semiannually on June 30 and December 31.)
• Account #3. The interest will be added to the account at the end of each calendar quarter.
  (Interest is compounded quarterly on March 31, June 30, September 30, and December 31.)

The following timelines will allow us to visualize the compounding of interest and its effect on each account’s ending balance. 



Account #1. A single amount of $10,000 is deposited on January 1, 2012 and will remain in the account until December 31, 2012. The account will earn interest of 8% per yearcompounded annually. The timeline showing this information appears here:

	PV =			FV = $10,800
Principal added:       $10,000
Interest added:					$800
			← 1 year →
	01/01/12				12/31/12
Period No.	0				1
	n = 1 annual period; i = 8% per year

The timeline shows the single deposit of $10,000 as the present value and occurring at time period 0. (Time period 0 is the present time and it is also the beginning of the first time period.) The timeline also shows that the interest earned during the year 2012 is $800 ($10,000 x 8%) and it is added on December 31, 2012. The result is a future value at December 31, 2012 of $10,800.



Account #2. A single amount of $10,000 is deposited on January 1, 2012 and will remain in the account until December 31, 2012. The account will earn interest at 8% per year but the interest is compounded semiannually. Because interest will be compounded semiannually, the variables n and i must be stated in six-month or semiannual terms as shown in the following timeline:

	PV =			FV = $10,816
Principal added:       $10,000
Interest added:			$400		$416
		←6 months→		←6 months→
	01/01/12		06/30/12		12/31/12
Period No.	0		1		2
	n = 2 semiannual periods; i = 4% per semiannual period

Again, the $10,000 is the present value shown at time period 0. The timeline also shows that $400 ($10,000 x 4%; or $10,000 x 8% x 0.5 year) of interest is added to the account on June 30, 2012. After the interest is added to the account, the new balance of $10,400 will earn interest during the second half of the year–resulting in interest of $416 ($10,400 x 4% = $416) added on December 31, 2012. The result is a future value at December 31, 2012 of $10,816. 



Account #3. A single amount of $10,000 is deposited on January 1, 2012 and will remain in the account until December 31, 2012. The account will earn interest at 8% per year but the interest is compounded quarterly. As a result the variables n and i must be thought of in terms of quarters or three-month periods as shown in the following timeline:

	PV =			FV = $10,824
Principal added:      $10,000
Interest added:		$200	$204	$208	$212
		←3 months→	←3 months→	←3 months→	←3 months→
	01/01/12	03/31/12	06/30/12	09/30/12	12/31/12
Period No.	0	1	2	3	4
	n = 4 quarterly periods; i = 2% per quarter

The present value of $10,000 will be earning compounded interest every three months. During the first quarter, the account will earn $200 ($10,000 x 2%; or $10,000 x 8% x 3/12 of a year.) and will result in a balance of $10,200 on March 31. During the second quarter of 2012 the account will earn interest of $204 based on the account balance as of March 31, 2012 ($10,200 x 2% per quarter). The interest for the third quarter is $208 ($10,404 x 2%) and the interest for the fourth quarter is $212 ($10,612 x 2%). The result is a future value of $10,824 at December 31, 2012.