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## 7.1 Multiple Cash Flows

PLEASE NOTE: This book is currently in draft form; material is not final.

### Learning Objectives

1. Calculate the NPV of an investment.
2. Evaluate investment choices using NPV.

Jamie was up late watching infomercials, and saw that she could purchase the latest wonder product for either \$43, or three easy monthly installments (the first payment is immediate) of \$15. If she earns 4% APR compounded monthly in her savings account, which option should she pick?

The trick to comparing cash flows usually is to PV them all to the same point in time. When they have been discounted to the same time, then they can be added, subtracted, or compared without worry. Discounting all of the cash flows for an investment to the present, adding inflows and subtracting outflows, is called finding the net present value (NPV)Discounting all of the cash flows for an investment to the present, adding inflows and subtracting outflows.. When deciding among investments, we typically wish to choose the one with the highest (or, in some cases, least negative) NPV.

The NPV of paying \$43 immediately is obviously−\$43. To find the NPV of the installment plan, we should look at the timeline.

Figure 7.1 NPV Example Timeline

4% APR compounded monthly gives a monthly interest rate of 0.3333%. (More discussion of example)

Many financial calculators have an NPV solving functionality to handle multiple cash flows. Please check the appendix for differences between the models, but the standard process is:

• Step 1: Enter Cash Flow worksheet.
• Step 1a: If entering new data (as opposed to editing what is already entered), clear the worksheet.
• Step 2: Enter CF0 (Net cash flows at time 0).
• Step 3a: Advance worksheet to enter C01 (Cash flow at time 1).
• Step 3b: Advance worksheet to enter F01 (The number of times C01 repeats).
• Repeat steps 3a and 3b until all are entered.
• Step 4: Press NPV and enter periodic interest rate.
• Step 5: Compute NPV.

In our example problem, the initial cash flow (CF0) is −\$15. Both of the next two cash flows are −\$15, so we can either enter them separately (C01: −15; F01: 1; C02: −15; F02: 1) or tell the calculator to repeat the entry (C01: −15; F01: 2). Both methods will give the proper answer. Here are the precise keystrokes for the latter method:

<CF> <2ND> <CLR WORK> −15 <ENTER> <DOWN ARROW> −15 <ENTER> <DOWN ARROW> 2 <ENTER> <NPV> 0.3333 <ENTER> <DOWN ARROW> <CPT>

Remember that interest rate should be entered as a percentage (0.3333 in this case, not 0.003333).

=NPV(periodic rate, cash flows from period 1 to n) + net of initial cash flows

Note that the spreadsheet function expects cash flows to start in the first period: any initial cash flows need to be netted and added to the result. Like the financial calculator, positive values should be used for cash inflows, and negative values for outflows.

Figure 7.2 NPV in a Spreadsheet

### Key Takeaways

• The net present value (NPV) of an investment can be found by summing the PVs of all of its cash flows.
• Comparing investments is most easily done by comparing their NPVs: the higher the NPV, the better the investment.

### Exercises

1. An investment costs \$1,000 today, but will produce cash flows of \$400 in year 1, \$500 in year 2, and \$600 in year 3. If interest rates are 10%, what is the NPV of this investment?
2. Investment A costs \$2,000 today, with cash inflows of \$400, \$400, \$400, and \$1,200 in years 1–4 respectively. Investment B costs \$1,000 today and will have one cash inflow of \$1,100 in 1 year. If interest rates are 8%, what are the NPVs of the investments? At this interest rate, which is the better investment? If interest rates are 5%, what are the NPVs? At this rate, which is the better investment?