## Net Present Value and Net Future Value

You may recognize that discounting is the inverse of the familiar idea of compounding. Compoui and forecasts a future value based on an interest rate or capital factor rate. Discounting begins w which a discount factor is applied to obtain a present value of the amount. In the same project, t discount rate are the same rate. Therefore, it is relatively easy to work from the present to the fu present.

Most of us who have had a savings or investment account are familiar with the compounding for compounding as a lead-in to discounting. Let's use "k" as the compounding (or discounting facto end of the compounding period:

Future value (FV) = Present value (PV) * (1 + k)N where N is the number of periods to be compounded. N takes on values from N = 0 to some val To take a simple numerical example, if "k" = 8% and "N" = 3, then we can calculate the future v

Thus, if we had an opportunity to invest \$10,000 for three years, we have an expectation that \$1. end of three years, given that the compounding factor is 8%. The return on our investment is the return: \$12,597 - \$10,000 = \$2,597.

If we had other opportunities to evaluate, the opportunity costs would be the difference in return! factor 1.2597 is called the future value factor. Future value factors are distinguished by always I than 1. Future value factors are easily calculated in Excel® using the "power" function or they ca finance books. Table 5-6 is an abridged collection of future value factors.

Table 5-6: Future Value Factors

 Discount Rate Year 0 Year 1 Year 2 Year 3 5% 1 1.05 1.1025 1.157625 7% 1 1.07 1.1449 1.225043 9% 1 1.09 1.1881 1.295029

These factors are easily calculated using a spreadsheet. The formula in any particular cell is: F where N is the number of the year, 0 to 4._

In Excel, the "power" function can be used to calculate the Factor equation. There are two argi is (1 + discount rate), and the second is N._

Now, about discounting: discounting begins in the future with forecasted flows; the DCF factor "k We solve for present value using the future value formula given above:

where N is the number of periods to be compounded.

The present value factors are the inverse of the future value factors. For instance, for m = 3, the 1/1.2597, or 0.7938. All present value factors will be numbers less than or equal to 1. Table 5-7 values.

 Discount Rate Year 0 Year 1 Year 2 Year 3 5% 1 0.952381 0.907029 0.863838 7% 1 0.934579 0.873439 0.816298 9% 1 0.917431 0.84168 0.772183 These factors are easily calculated using a spreadsheet. The formula in any particular cell is: F rate)N where N is the number of the year, 0 to 4. In Excel, the "power" function can be used to calculate the denominator in the Factor equation in "power": the first is (1 + discount rate), and the second is N.

Using the 8% factors and m = 3, suppose we had a forecast for future flows due to a present in\

N = 0, \$10,000 present-time investment outflow and future forecasted inflows:

The present value of these flows given a discount of 8% is given by:

PV(inflows) = \$3,000 * 0.9259 + \$6,000 * 0.8573 + \$7,000 * 0.7938 = \$13,478

NPV of return = -\$10,000 + \$13,478 = \$3,478 at the end of three periods.

Figure 5-3 provides a graphical presentation of the cash flows. It is usually simpler to calculate t "outflows" as negative numbers and all "inflows" as positive numbers. In the figure, outflows are inflows are arrows pointing up. With this convention, the above would be written as:

NPV = PV (all inflows) - PV (all outflows) NPV = PV (all inflows - all outflows)

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Figure 5-3: NPV Example. At 8% discount rate:

Figure 5-3: NPV Example. At 8% discount rate:

NPV = -\$10,000 + \$3,000 * 0.9259 + \$6,000 * 0.8573 + \$7,000 * 0.7938 = \$3,478

NPV is our first risk-adjusted measure of project financial performance. If the outflows are the c; the inflows are the cash received because of the project, then in the case illustrated above it ma because the NPV = \$0. If the NPV is negative, then more money is going out of the business th. risk-adjusted basis; it makes no financial sense to do projects of negative NPV:

Decision policy: Only projects with NPV = \$0 will be selected for ex 