Reading: Lesson 5 - Uneven or Irregular Cash Flows

4.5.A - Uneven or Irregular Cash Flows

1. Uneven, or Irregular, Cash Flows

1. The definition of an annuity includes the term constant payment, which suggests that annuities involve a set of identical payments over a given number of periods. Although many financial decisions do involve constant payments, many others involve cash flows that are uneven or irregular. For example, the dividends on common stocks are typically expected to increase over time, and the investments that companies make in new products, expanded production capacity, and replacement machinery almost always generate cash flows that vary from year to year. Throughout the book, we use the term payment (PMT) in situations where the cash flows are constant and thus an annuity is involved; if different cash flows occur in different time periods, t, then we use the term CFt to designate the cash flow in period t. There are two important classes of uneven cash flows: (1) those in which the cash flow stream consists of a series of annuity payments plus an additional final lump sum in Year N, and (2) all other uneven streams. Bonds are an instance of the first type, while stocks and capital investments illustrate the second type. Here’s an example of each type.
   

Following the step-by-step procedure, to find the PV of either stream. However, as we shall see, the solution process differs significantly for the two types.

2. Annuity Plus Additional Final Payment

1. First, consider Stream 1 and notice that it is a 5-year, 12%, ordinary annuity plus a final payment of $1,000. We can find the PV of the annuity, find the PV of the final payment, and then sum them to get the PV of the stream. Financial calculators are programmed to do this for us—we use all five time value of money (TVM) keys, entering the data for the four known values as shown below, and then pressing the PV key to get the answer, $927.90:
   

Similarly, we could use Excel’s PV function: =PV(I,N,PMT,FV) = PV(0.12,5,100,1000) = −$927.90. Note that the process is similar to that for annuities, except we now have a nonzero value for FV.

3. Irregular Cash Flow Stream

1. Now consider the irregular stream, which is analyzed in the figure below. The top section shows the basic time line, which contains the inputs, and we first use the step-by-step approach to find PV = $1,016.35. Note that we show the PV of each cash flow directly below the cash flow, and then we sum these PVs to find the PV of the stream. This setup saves space as compared with showing the individual PVs in a column, and it is also transparent and thus easy to understand.
   
2. Now consider the financial calculator approach. The cash flows don’t form an annuity, so you can’t use the annuity feature on the calculator. You could, of course, use the calculator in the step-by-step procedure, but financial calculators have a feature—the cash flow register— that allows you to find the present value more efficiently. First, you input the individual cash flows, in chronological order, into the cash flow register. Cash flows are designated CF0, CF1, CF2, CF3, and so on, up to the last cash flow, CFN. Next, you enter the interest rate, I/YR. At this point, you have substituted in all the known values of Equation 4-10, so when you press the NPV key you get the PV of the stream. The calculator finds the PV of each cash flow and sums them to find the PV of the entire stream. To input the cash flows for this problem, enter 0 (because CF0 = 0), 100, 300, 300, 300, and 500 in that order into the cash flow register, enter I/YR = 12, and then press NPV to obtain the answer, $1,016.35.
   
3. Two points should be noted. First, when dealing with the cash flow register, the calculator uses the term “NPV” rather than “PV.” The N stands for “net,” so NPV is the abbreviation for “net present value,” which is simply the net present value of a series of positive and negative cash flows, including any cash flow at time zero. The NPV function will be used extensively when we get to capital budgeting, where CF0 is generally the cost of the project.
4. The second point to note is that repeated cash flows with identical values can be entered into the cash flow register more efficiently on some calculators by using the Nj key. In this illustration, you would enter CF0 = 0, CF1 = 100, CF2 = 300, Nj = 3 (which tells the calculator that the 300 occurs 3 times), and CF5 = 500.11 Then enter I = 12, press the NPV key, and 1,016.35 will appear in the display. Also, note that numbers entered into the cash flow register remain in the register until they are cleared. Thus, if you previously worked a problem with eight cash flows and then moved to one with only four cash flows, the calculator would simply add the cash flows from the second problem to those of the first problem, and you would get an incorrect answer. Therefore, you must be sure to clear the cash flow register before starting a new problem.
   
5. Spreadsheets are especially useful for solving problems with uneven cash flows. You enter the cash flows in the spreadsheet as shown in the figure below on Row 471. To find the PV of these cash flows without going through the step-by-step process, you would use the NPV function. First put the cursor on the cell where you want the answer to appear, Cell G481, click the function wizard, choose Financial, scroll down to NPV, and click OK to get the dialog box. Then enter C467 (or 0.12) for Rate and enter either the individual cash flows or the range of cells containing the cash flows, C471:G471, for Value 1. Be very careful when entering the range of cash flows. With a financial calculator, you begin by entering the Time-0 cash flow. With Excel, you do not include the Time-0 cash flow; instead, you begin with the Time-1 cash flow. Now, when you click OK, you get the PV of the stream, $1,016.35. Note that you can use the PV function if the payments are constant, but you must use the NPV function if the cash flows are not constant. Finally, note that Excel has a major advantage over financial calculators in that you can see the cash flows, which makes it easy to spot data-entry errors. With a calculator, the numbers are buried in the machine, making it harder to check your work.
   

4. Future Value of an Uneven Cash Flow Stream

1. The future value of an uneven cash flow stream (sometimes called the terminal, or horizon, value) is found by compounding each payment to the end of the stream and then summing the future values:
   

The future value of our illustrative uneven cash flow stream is $1,791.15.

Most financial calculators have a net future value (NFV) key, which, after the cash flows and interest rate have been entered, can be used to obtain the future value of an uneven cash flow stream. If your calculator doesn’t have the NFV feature, you can first find the net present value of the stream, and then find its net future value as NFV = NPV(1 + I)N. In the illustrative problem, we find PV = 1,016.35 using the cash flow register and I = 12. Then we use the TVM register, entering N = 5, I = 12, PV = −1016.35, and PMT = 0. When we press FV, we find FV = 1,791.15, which is the same as the value shown on the time line in the figure above. As the figure below also shows, the same procedure can be used with Excel.