# Net present value profile a graph showing the

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Net Present Value Profile A graph showing the relationship between a project's NPV and the firm's cost of capital.
384 Part 4 Investing in Long-Term Assets: Capital Budgeting Crossover Rate The cost of capital at which the NPV profiles of two projects cross and, thus, at which the proj- ects ' NPVs are equal. Project 8 NPV: -\$1,000 + \$1 00 /( 1.1 0) 1 + \$1 ,300 /( 1.1 0) 2 = \$165 . 29 . Alternatively, enter the cash flows into the financial calculator as follows: CF 0 = -1000; CF 1 = 1 00; CF 2 = 1300; 1/YR = 1 0; NPV = \$165.29 . IRR : Enter the cash flows into the financial calculator as follows: CF 0 = -1000; CF 1 = 100; CF 2 = 1300; .I RR = 19.13% . MIRR: 0 2 -\$ 1,000 \$100 \$1,300 I X 1.10 \$ 110 - \$1,000 TV= \$1,410 Using a financial calculator, enter the following data: N = 2; PV = -1000; PMT = 0; FV = 141 0; and solve for 1/YR = MIRR = 18.74% . b. Here's a summary of the results. The project chosen under each method is highlighted. Project A Project B NPV \$128.10 1 s 165 .2 91 IRR 123.12% 1 19.13% MIRR 16.83% 1 18.74 %1 Using the NPV and MIRR criteria, you would select Project B; however, if you use the IRR criteria you would select Project A. Because Project B adds the most value to the firm, B should be chosen. The IRRs are fixed, and S has the higher IRR regardless of the cost of capital. However, the NPVs vary depending on the actual cost of capital. The two NPV profile lines cross at a cost of capital of 11.975%, which is called the crossover rate. The crossover rate can be found by calculating the IRR of the differences in the projects' cash flows, as demonstrated below: 0 2 3 4 ProjectS -\$1,000 \$500 \$400 \$300 \$1 00 -Project L -\$1,000 \$100 \$300 \$400 \$675 6 = CFs- CFL \$ 0 \$400 \$100 -\$100 -\$575 IRR 6 = 11.975% = Crossover rate Project L has the higher NPV if the cost of capital is less than the crossover rate, butS has the higher NPV if the cost of capital is greater than that rate. Notice that Project L has the steeper slope, indicating that a given increase in the cost of capital causes a larger decline in NPVL than in NPV s. To see why this is so, recall that L's cash flows come in later than those of S. Therefore, L is a long- term project and S is a short-term project. Next, recall the equation for the NPV:
Chapter 1 1 The Basics of Capital Budgeting 385 FIGURE 11.5 NPV Profile for Project S NPV (\$) 5 At r= 10 %, NP V > 0, NPV= 0, so IRR ,= 14.489% IRR >r= 10 %, so accept I I I I 20 Cost of Capital (%) Cost of Capital NPV 5 0% \$300.00 5 180.42 10 78.82 IRR s = 14.489 0.00 15 -8.33 20 - 83.72 CF 1 CF 2 CFN NPV = CF 0 +---+---+ ··· +--- ( 1 + r) 1 ( 1 + r) 2 ( 1 + r)N Now recognize that the impact of an increase in the cost of capital is much greater on distant than near-term cash flows, as we demonstrate here: Effect of doubling ron a Year 1 cash flow: . \$100 PV of \$100 due m 1 year @ r = 5% :--- 1 = \$95.24 (1.05) . \$100 PV of \$100 due m 1 year @ r = 10% : --- 1 = \$90. 91 (1.10) d I . d h' h \$95.24 - \$90. 91 Percentage ec me ue to 19 er r = \$ = 4.5% 95.24 Effect of doubling ron a Year 20 cash flow: . \$100 PV of \$100 due m 20 years @ r = 5% : 20 = \$37.69 (1.05) . \$100 PV of \$100 due m 20 years @ r = 10 %: 20 = \$14.86 (1.10) P d r d h' h \$37.69 - \$14.86 ercentage ec me ue to 19 err = \$ 37 . 69 = 60.6% Thus, a doubling of the discount rate results in only a 4.5% decline in the PV of a Year 1 cash flow, but the same discount rate increase causes the PV of a Year 20 cash flow to fall by more than 60%.