We describe the relationship between an initial investment or present value (PV), which earns a rate of return (the interest rate per period) denoted as r, and its future value (FV), which will be received N years or periods from today.

The following example illustrates this concept. Suppose you invest \$ 100 (PV=\$ 100) in an interest-bearing bank account paying 5 percent annually. At the end of the first year, you will have the $\$ 100$ plus the interest earned, \(0.05 \times \$ 100=\$ 5\), for a total of \$ 105. To formalize this one-period example, we define the following terms:

$\mathrm{PV}=$ present value of the investment

$\mathrm{FV}_{N}=$ future value of the investment $N$ periods from today

$r=$ rate of interest per period

For $N=1$, the expression for the future value of amount PV is

Equation (1)

$$

\mathrm{FV}_{1}=\mathrm{PV}(1+r)

$$

For this example, we calculate the future value one year from today as \(\mathrm{FV}_{1} =\$ 100(1.05)=\$ 105\)

Now suppose you decide to invest the initial \$ 100 for two years with interest earned and credited to your account annually (annual compounding).

At the end of the first year (the beginning of the second year), your account will have \$ 105, which you will leave in the bank for another year. Thus, with a beginning amount of \(\$ 105(PV=\$ 105)\), the amount at the end of the second year will be \(\$ 105(1.05)=\$ 110.25\).

Note that the \$ 5.25 interest earned during the second year is 5 percent of the amount invested at the beginning of Year 2 .