TIME LINES

The first step in time value analysis is to set up a time line, which will help you visualize what’s happening in a particular problem. To illustrate, consider the following diagram, where PV represents $100 that is on hand today and FV is the value that will be in the account on a future date:

0 21 35%
Periods
Cash PV = $100 FV = ?

The intervals from 0 to 1, 1 to 2, and 2 to 3 are time periods such as years or months. Time 0 is today, and it is the beginning of Period 1; Time 1 is one period from today, and it is both the end of Period 1 and the beginning of Period 2; and so on. Although the periods are often years, periods can also be quarters or months or even days. Note that each tick mark corresponds to both the end of one period and the beginning of the next one. Thus, if the periods are years, the tick mark at Time 2 represents both the end of Year 2 and the beginning of Year 3.

Cash flows are shown directly below the tick marks, and the relevant inter- est rate is shown just above the time line. Unknown cash flows, which you are trying to find, are indicated by question marks. Here the interest rate is 5 per- cent; a single cash outflow, $100, is invested at Time 0; and the Time 3 value is an unknown inflow. In this example, cash flows occur only at Times 0 and 3, with no flows at Times 1 or 2. Note that in our example the interest rate is constant for all three years. That condition is generally true, but if it were not then we would show different interest rates for the different periods.

Time lines are essential when you are first learning time value concepts, but even experts use them to analyze complex finance problems, and we use them throughout the book. We begin each problem by setting up a time line to show what’s happening, after which we provide an equation that must be solved to find the answer, and then we explain how to use a regular calculator, a financial calculator, and a spreadsheet to find the answer.

Indeed, time value analysis is used throughout the book, so it is vital that you understand this chapter before continuing.
You need to understand basic time value concepts, but conceptual knowledge will do you little good if you can’t do the required calculations.
Therefore, this chapter is heavy on calculations. Also, most students studying finance have a financial or scientific calculator, and some also own or have access to a computer. Moreover, one of these tools is necessary to work many finance problems in a “reasonable” length of time. However, when they start on this chapter, many students don’t know how to use the time value functions in their calculator or computer. If you are in that situation, you will find yourself simultaneously studying concepts and trying to learn to use your calculator, and you will need more time to cover this chapter than you might expect. 1

1 Calculator manuals tend to be long and complicated, partly because they cover a number of topics that aren’t required in the basic finance course. Therefore, we provide, on the Thomson NOW Web site, tutorials for the most commonly used calculators. The tutorials are keyed to this chapter, and they show exactly how to do the required calculations. If you don’t know how to use your calculator, go to the ThomsonNOW Web site, get the relevant tutorial, and go through it as you study the chapter.

2 A fifth procedure, using tables that show “interest factors,” was used before financial calculators and computers became available. Now, though, calculators and spreadsheets such as Excel are programmed to calculate the specific factor needed for a given problem and then to use it to find the FV. This is much more efficient than using the tables. Moreover, calculators and spreadsheets can handle fractional periods and fractional interest rates, like the FV of $100 after 3.75 years when the interest rate is 5.375 percent, whereas tables provide numbers only for specific periods and rates. For these reasons, tables are not used in business today; hence we do not discuss them in the text.
  • Do time lines deal only with years or could other periods be used?
  • Set up a time line to illustrate the following situation: You currently have $2,000 in a three-year certificate of deposit (CD) that pays a guaranteed 4 percent annually.