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.

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.

The balances would be:

At 6 months:

$\$5,000(1+0.04)=\$5,000\left(1.04\right)=\$5,200$

At 1 year:

$\$5,200\left(1.04\right)=\$5,000\left(1.04\right)\left(1.04\right)=\$5,000(1.04{)}^2=\$5,408$

At 18 months:

$\$5,408\left(1.04\right)=\$5,000\left(1.04{)2}^{}\right(1.04)=\$5,000(1.04{)3}^{}=\$5,624.32$

At 2 years:

$\$5,624.32\left(1.04\right)=\$5,000\left(1.04{)3}^{}\right(1.04)=\$5,000(1.04{)4}^{}=\$5,849.29$

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:

 

and

$n=\left(\text{number of years}\right)\times \left(\text{number of periods per year}\right)=t\times m$

 

 

 

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