4.3 Compound Interest Formula

The procedure for adding interest each period can always be used to find the future value of a loan or deposit, but the following general formula gives the future value more directly.

[latex]FV = PV(1+i)^n = PV \left(1+\frac{j_m}{m}\right)^n[/latex]

where:

• FV = future value of the loan
• PV = present value of the loan (principal)
• i = periodic interest rate
• n = number of compounding periods

Example 4.3.1

To see how the formula is developed, consider the $5,000 loan at 8% compounded semi-annually for two years.

First, [latex]i = \frac{0.08}{2}=0.04[/latex]

The balances would be:

At 6 months:

[latex]$5,000(1+0.04) =$5,000(1.04)=$5,200[/latex]

At 1 year:

[latex]$5,200 (1.04)=$5,000(1.04)(1.04) =$5,000 (1.04)^2 = $5,408[/latex]

At 18 months:

[latex]$5,408(1.04)=$5,000(1.04)^2 (1.04) =$5,000(1.04)^3 = $5,624.32[/latex]

At 2 years:

[latex]$5,624.32(1.04)=$5,000(1.04)^3 (1.04) =$5,000(1.04)^4 = $5,849.29[/latex]

This last calculation for the two-year balance is the general formula for FV with:

• PV = $5,000
• i = 0.04
• n = 4 = 2× 2

In general, the values for i and n are found by:

[latex]i = \frac{j_m}{m}[/latex]

and

[latex]n = (\text{number of years})\times (\text{number of periods per year})=t\times m[/latex]

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