Calculating the Future Value of Money: Compounding
The simplest example of money growing over time (compounding) is a savings account at a bank. Individuals choose to deposit money in the bank for a variety of reasons, including the knowledge that money invested in the bank grows because interest is earned on the money. For example, if an initial deposit of $100 is made in a bank that promises to pay 2% interest annually, at the end of 12 months a total of $102 is available. The original deposit has grown or compounded from $100 to $102. The additional $2 is the interest earned on the deposit.
If the money is left in the bank for another year, assuming no change in the interest rate, at the end of year 2 a total of $104.04 is available in the account. The original $100 has grown or compounded by $4.04 as a result of interest earnings on the account. Thus, using conventional terminology, the future value of $100 invested for 2 years at an interest rate of 2% compounded annually is $104.04. The increase in the account in the second year is $2.04, not $2.00, as was earned in year 1. This is because the amount in the account at the end of year 1 (i.e., the amount that is compounded for an additional [second] year) was $102 (the original $100 deposit + $2 of interest earned to that point in time). Thus, during year 2, 2% was earned on $102, not $100.
In situations involving compounding or discounting, it is helpful to create a “picture” or timeline of the investment scenario. A timeline is used to indicate the present and future value of money, the applicable interest rate, and the length of time involved. In fact, always beginning a time value of money analysis with a timeline may be a prerequisite for accurate calculations.
For example, Figure 4-1 depicts the situation just described. Note that time period “0” refers to the present time (i.e., now), and that the future value refers to the size of account at the end of the time period indicated. The compounding (interest) rate is shown on the timeline for the appropriate time periods. Creating a timeline is a simple, yet helpful tool to organize the “facts” of the investment opportunity and to help ensure that mangers have all the information required for decision making.
In general terms, compounding is represented by the following equation:
Future Value = Present Value + Interest Earned (I)
where
Interest Earned = Present Value * Interest Rate (i)
Figure 4-1 Timeline Showing Present Value (PV) and Future Value (FV) of $100 Invested for 2 Years at a 2% Annual Rate
Stated another way:
FV = PV + I
which is the same as:
FV = PV + (PV*i)
Simplifying:
FV = PV(1 + i) Equation 4-1
Using Equation 4-1 for the preceding example described, where the present value is the amount of the original deposit (i.e., $100) and the interest rate is 2%, the future value at the end of year 1 is calculated as:
FVyear 1 = $100(1 + 0.02)
= $100(1.02)
= $102
In the second year, another year’s interest is earned. To reflect this second year of interest, using Equation 4-1:
FVyear 2 = PV(1 + i)(1 + i) Equation 4-1
= PV(1 + i)2
This is the same as:
FVyear 2 = FVyear 1(1 + i)
Each term (1 + i), known as the compounding factor, indicates an additional period during which interest is being earned. In this example, it is said that interest is compounded for two periods. Substituting numbers in the equation, the future value at the end of year 2 is calculated to be:
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