4.1 Introduction to Compound Interest

Compound interest is the interest paid on previously earned interest as well as on the original principal.

Example 4.1.1

Consider a loan of $1,000 for two years at 10% per year interest. If simple interest is used, the lender would receive at the end of the loan.

[latex]$1,000\times (1 + 0.1\times 2) = $1,200[/latex]

But at the end of the first year, the lender could reasonably consider the value of the loan to be

[latex]$1,000\times (1 + 0.1\times 1) = $1,100[/latex]

and feel that the interest for the second year should be based on this amount as principal. The lender would then receive

[latex]$1,100\times (1 + 0.1\times 1) = $1,210[/latex]

at the end of the second year. This is described as interest compounded annually, or converted (changed to principal) annually.

Compounding takes effect in accounts where interest is regularly added to the balance.

While it is possible to consider the principal of a loan as continuously increasing because of interest, the usual way to state and calculate compound interest is to follow the idea above and increase the principal by the interest at regular intervals. The interval to be used is stated in the rate. Commonly used intervals are given by the terms:

• Annually
• semi-annually (every half year = every six months)
• quarterly (every three months)
• monthly
• biweekly (every two weeks or fortnight)

Occasionally loans are compounded weekly or daily.

The interest rate in the example on the previous page would be described as 10% compounded annually. Compounding takes effect in accounts in which the interest is added to the balance at regular intervals.

Example 4.1.2

Imagine a trust account set up with an initial balance of $5,000 and an earning of 8% compounded semi-annually. After two years the account would look like this:

The interest calculations for the account would be as follows: At 6 months:

[latex]Interest =$5,000\times 0.08\times 0.5 =$5,000\times 0.04 = $200[/latex]

Note: 8% compounded semi-annually means 4% every half year. At 1 year:

[latex]Interest=$5,200\times 0.04 = $208[/latex]

At 18 months:

[latex]Interest=$5,408 \times 0.04 = $216.32[/latex]

At 2 years:

[latex]Interest=$5,624.32 \times .04 = $224.97[/latex]

The total interest is $849.29 vs $800 at 8% simple interest.

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