## Discounted Cash Flow Analysis: NPV and IRR

The HP-12C provides functions for the two most widely-used methods of discounted cash flow analysis: NPV (net present value) and IRR (internal rate of return). These functions enable you to analyze financial problems involving cash flows (money paid out or recieved) occurring at regular intervals. As in compound interest calculations, the interval between cash flows can be any time period; however, the amounts of these cash flows need not be equal.

To understand how to use NPV and IRR, let’s consider the cash flow for an investment that requires an initial cash outlay (CF0) and generates a cash flow (CF1) at the end of the first year, and so on up the final cash flow (CFj).

NPV is calculated by adding the initial investment (represented as a negative cash flow) to the present value of the anticipated future cash flows. The interest rate, i, will be referred to in this discussion of NPV and IRR as the rate of return. The value of NPV indicates the result of the investment: If NPV is positive, the financial value of the investor’s assets would be increased--the investment is financially attractive. If NPV is zero, the financial value of the investor’s assets would not change--the investor is indifferent toward the investment. If NPV is negative, the financial value of the investor’s assets would be decreased--the investment is not financially attractive.

A comparison of the NPV’s of alternative investment possibilities indicates which of them is most desirable: the greater the NPV, the greater the increase in the financial value of the investor’s assets.

IRR is the rate of return at which the discounted future cash flows equal the initial cash outlay: IRR is the discount rate at which NPV is zero. The value of IRR relative to the present value discount rate also indicates the result of the investment: If IRR is greater than the desired rate of return, the investment is financially attractive. If IRR is equal to the desired rate of return, the investor is indifferent toward the investment. If IRR is less than the desired rate of return, the investment is not financially attractive.

Calculating Net Present Value (NPV)

Calculating NPV for Ungrouped Cash Flows

Calculates the net present value of up to 20 uneven cash flows (in addition to the initial investment CFo). The amount of the the initial investment (CFo) is entered into the calculator using the CFo key. Pressing g CFo stores CFo in storage register R0 and also stores the number 0 in the n register.

The amounts of the subsequent cash flows are stored--in the order they occur--in the remaining storage registers: CF1 thru CF9 in R1 thru R9, and CF10 thru CF19 in R.0 thru R.9, respectively. If there is a CF20, that amount is stored in the FV register. Each of these cash flows (CF1, CF2, etc.) is designated CFj, where j takes on values from 1 up to the number of the final cash flow. The amounts of these cash flows are all entered using the CFj key. Each time g CFj is pressed, the amount in the display is stored in the next available storage register, and the number in the n register is increased by 1. This register therefore counts how many cash flow amounts (in addition to the initial investment CFo) have been entered.

When entering cash flow amounts--including the initial investment CFo--remember to observe the cash flow sign convention by pressing CHS after keying in a negative cash flow.

In summary, to enter the cash flow amounts: Press f CLEAR REG to clear the financial and storage registers. Key in the amount of the initial investment, press CHS if that cash flow is negative, then press g CFo. If there is no initial investment, press 0 g CFo. Key in the amount of the next cash flow, press CHS if the cash flow is negative, then press g CFj. If the cash flow amount is zero in the next period, press 0 g CFj. Repeat last step for each cash flow until all have been entered. Enter the interest rate, using i or 12/. Press f NPV.

The calculated value of NPV appears in the display and also is automatically stored in the PV register.

Calculating NPV for Grouped Cash Flows

A maximum of 20 cash flow amounts (in addition to initial investment CFo) can be stored in the HP-12C. However, problems involving more than 20 cash flows can be handled if among the cash flows there are equal consecutive cash flows. For such problems, you merely enter along with the amounts of the cash flows the number of times--up to 99--each amount occurrs consecutively. This number is designated Nj, corresponding to cash flow amount CFj, and is entered using the Nj key. Each Nj is stored in a special register inside the calculator.

This method can, of course, be used for problems involving fewer than 20 cash flows.

In summary, to enter the cash flow amounts: Press f CLEAR REG to clear the financial and storage registers. Key in the amount of the initial investment, press CHS if that cash flow is negative, then press g CFo. If there is no initial investment, press 0 g CFo. If the initial investment consists of more than one cash flow of the amount entered, key in the number of those cash flows, then press g Nj. If g Nj is not pressed, the calculator assumes that N0 is 1. Key in the amount of the next cash flow, press CHS if the cash flow is negative, then press g CFj. If the cash flow amount is zero in the next period, press 0 g CFj. If the amount entered occurs more than once consecutively, key in the number of times that cash flow amount occurs consecutively, then press g Nj. If g Nj is not pressed, the calculator assumes that Nj is 1 for the CFj just entered. Repeat last step for each cash flow until all have been entered. Enter the interest rate, using i or 12/. Press f NPV.

The calculated value of NPV appears in the display and also is automatically stored in the PV register.

Calculating Internal Rate of Return (IRR)

Enter the cash flows using the method described for NPV. Press f IRR. The calculated value of IRR appears in the display and also is automatically stored in the i register.

If you get an Error 3 message in the display, you can try the procedure outlined in Error Conditions regarding an Error 3 message to resolve this condition by entering a new guess for IRR.

Reviewing Cash Flow Entries

To display a single cash flow amount, press RCL, then key in the number of the register containing the cash flow amount to be displayed. Alternatively, store the number of that cash flow amount (that is, the value of j for the CFj desired) in the n register, then press RCL g CFj.

To review all the cash flow amounts, press RCL g CFj repeatedly. This displays the cash flow amounts in reverse order--that is, beginning with the final cash flow and proceeding to CF0.

To display the number of times a cash flow amount occurs consecutively--that is, to display the Nj for a CFj--store the number of that cash flow amount (that is, the value of j) in the n register, the press RCL g Nj.

To review all the cash flow amounts together with the number of times each cash flow amount occurs consecutively (that is, to review each CFj and Nj pair), press RCL g Nj RCL g CFj repeatedly. This displays Nj followed by CFj beginning with the final cash flow amount and proceeding to N0 and CF0.