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The Basics of Capital Budgeting

Evaluating Cash Flows

ANSWERS TO END-OF-CHAPTER QUESTIONS

13-1 a. The capital budget outlines the planned expenditures on fixed assets. Capital budgeting is the whole process of analyzing projects and deciding whether they should be included in the capital budget. This process is of fundamental importance to the success or failure of the firm as the fixed asset investment decisions chart the course of a company for many years into the future. Strategic business plan is a long-run plan which outlines in broad terms the firm’s basic strategy for the next 5 to 10 years.

b. The payback, or payback period, is the number of years it takes a firm to recover its project investment. Payback may be calculated with either raw cash flows (regular payback) or discounted cash flows (discounted payback). In either case, payback does not capture a project's entire cash flow stream and is thus not the preferred evaluation method. Note, however, that the payback does measure a project's liquidity, and hence many firms use it as a risk measure.

c. Mutually exclusive projects cannot be performed at the same time. We can choose either Project 1 or Project 2, or we can reject both, but we cannot accept both projects. Independent projects can be accepted or rejected individually.

d. The net present value (NPV) and internal rate of return (IRR) techniques are discounted cash flow (DCF) evaluation techniques. These are called DCF methods because they explicitly recognize the time value of money. NPV is the present value of the project's expected future cash flows (both inflows and outflows), discounted at the appropriate cost of capital. NPV is a direct measure of the value of the project to shareholders.

e. The internal rate of return (IRR) is the discount rate that equates the present value of the expected future cash inflows and outflows. IRR measures the rate of return on a project, but it assumes that all cash flows can be reinvested at the IRR rate.

f. The modified internal rate of return (MIRR) assumes that cash flows from all projects are reinvested at the cost of capital as opposed to the project's own IRR. This makes the modified internal rate of return a better indicator of a project's true profitability. The profitability index is found by dividing the project’s PV of future cash flows by its initial cost. A profitability index greater than 1 is equivalent to a positive NPV project.

g. An NPV profile is the plot of a project's NPV versus its cost of capital. The crossover rate is the cost of capital at which the NPV profiles for two projects intersect.

h. Capital projects with nonnormal cash flows have a large cash outflow either sometime during or at the end of their lives. A common problem encountered when evaluating projects with nonnormal cash flows is multiple IRRs. A project has normal cash flows if one or more cash outflows (costs) are followed by a series of cash inflows.

i. The hurdle rate is the project cost of capital, or discount rate. It is the rate used in discounting future cash flows in the NPV method, and it is the rate that is compared to the IRR.

j. The mathematics of the NPV method imply that project cash flows are reinvested at the cost of capital while the IRR method assumes reinvestment at the IRR. Since project cash flows can be replaced by new external capital which costs k, the proper reinvestment rate assumption is the cost of capital, and thus the best capital budget decision rule is NPV.

k. The post-audit is the final aspect of the capital budgeting process. The post-audit is a feedback process in which the actual results are compared with those predicted in the original capital budgeting analysis. The post-audit has several purposes, the most important being to improve forecasts and improve operations.

13-2 Project classification schemes can be used to indicate how much analysis is required to evaluate a given project, the level of the executive who must approve the project, and the cost of capital that should be used to calculate the project's NPV. Thus, classification schemes can increase the efficiency of the capital budgeting process.

13-3 The NPV is obtained by discounting future cash flows, and the discounting process actually compounds the interest rate over time. Thus, an increase in the discount rate has a much greater impact on a cash flow in Year 5 than on a cash flow in Year 1.

13-4 This question is related to Question 13-3 and the same rationale applies. With regard to the second part of the question, the answer is no; the IRR rankings are constant and independent of the firm's cost of capital.

13-5 The NPV and IRR methods both involve compound interest, and the mathematics of discounting requires an assumption about reinvestment rates. The NPV method assumes reinvestment at the cost of capital, while the IRR method assumes reinvestment at the IRR. MIRR is a modified version of IRR which assumes reinvestment at the cost of capital.

13-6 The statement is true. The NPV and IRR methods result in conflicts only if mutually exclusive projects are being considered since the NPV is positive if and only if the IRR is greater than the cost of capital. If the assumptions were changed so that the firm had mutually exclusive projects, then the IRR and NPV methods could lead to different conclusions. A change in the cost of capital or in the cash flow streams would not lead to conflicts if the projects were independent. Therefore, the IRR method can be used in lieu of the NPV if the projects being considered are independent.

13-7 Yes, if the cash position of the firm is poor and if it has limited access to additional outside financing. But even here, the relationship between present value and cost would be a better decision tool.

13-8 a. In general, the answer is no. The objective of management should be to maximize value, and as we point out in subsequent chapters, stock values are determined by both earnings and growth. The NPV calculation automatically takes this into account, and if the NPV of a long-term project exceeds that of a short-term project, the higher future growth from the long-term project must be more than enough to compensate for the lower earnings in early years.

b. If the same $100 million had been spent on a short-term project--one with a faster payback--reported profits would have been higher for a period of years. This is, of course, another reason why firms sometimes use the payback method.

13-8 Generally, the failure to employ common life analysis in such situations will bias the NPV against the shorter project because it "gets no credit" for profits beyond its initial life, even though it could possibly be "renewed" and thus provide additional NPV.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS

13-1 $52,125/$12,000 = 4.3438, so the payback is about 4 years.

13-2 NPV = -$52,125 + $12,000[(1/i)-(1/(i*(1+i)n)]

= -$52,125 + $12,000[(1/0.12)-(1/(0.12*(1+0.12)8)]

= -$52,125 + $12,000(4.9676) = $7,486.20.

Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 12, and then solve for NPV = $7,486.68.

13-3 Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for IRR = 16%.

13-4 Project K's discounted payback period is calculated as follows:

Annual Discounted @12%

Period Cash Flows Cash Flows Cumulative

0 ($52,125) ($52,125.00) ($52,125.00)

1 12,000 10,714.80 (41,410.20)

2 12,000 9,566.40 (31,843.80)

3 12,000 8,541.60 (23,302.20)

4 12,000 7,626.00 (15,676.20)

5 12,000 6,808.80 (8,867.40)

6 12,000 6,079.20 (2,788.20)

7 12,000 5,427.60 2,639.40

8 12,000 4,846.80 7,486.20

The discounted payback period is 6 + years, or 6.51 years.

Alternatively, since the annual cash flows are the same, one can divide $12,000 by 1.12 (the discount rate = 12%) to arrive at CF1 and then continue to divide by 1.12 seven more times to obtain the discounted cash flows (Column 3 values). The remainder of the analysis would be the same.

13-5 MIRR: PV Costs = $52,125.

FV Inflows:

PV FV

0 1 2 3 4 5 6 7 8

| | | | | | | | |

12,000 12,000 12,000 12,000 12,000 12,000 12,000 12,000

13,440

15,053

16,859

18,882

21,148

23,686

26,528

52,125 MIRR = 13.89% 147,596

Financial calculator: Obtain the FVA by inputting N = 8, I = 12, PV = 0, PMT = 12000, and then solve for FV = $147,596. The MIRR can be obtained by inputting N = 8, PV = -52125, PMT = 0, FV = 147596, and then solving for I = 13.89%.

13-6 Project A:

Using a financial calculator, enter the following:

CF0 = -15000000

CF1 = 5000000

CF2 = 10000000

CF3 = 20000000

I = 10; NPV = $12,836,213.

Change I = 10 to I = 5; NPV = $16,108,952.

Change I = 5 to I = 15; NPV = $10,059,587.

Project B:

Using a financial calculator, enter the following:

CF0 = -15000000

CF1 = 20000000

CF2 = 10000000

CF3 = 6000000

I = 10; NPV = $15,954,170.

Change I = 10 to I = 5; NPV = $18,300,939.

Change I = 5 to I = 15; NPV = $13,897,838.

13-7 Using a financial calculator, enter the following:

CF0 = -200

CF1 = 235

CF2 = -65

CF3 = 300

I = 11.5; NPV = $174.90.

13-8 Truck:

NPV = -$17,100 + $5,100(PVIFA14%,5)

= -$17,100 + $5,100(3.4331) = -$17,100 + $17,509

= $409. (Accept)

Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 14, and then solve for NPV = $409.

Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for IRR = 14.99% ? 15%.

MIRR: PV Costs = $17,100.

FV Inflows:

PV FV

0 1 2 3 4 5

| | | | | |

5,100 5,100 5,100 5,100 5,100

5,814

6,628

7,556

8,614

17,100 MIRR = 14.54% (Accept) 33,712

Financial calculator: Obtain the FVA by inputting N = 5, I = 14, PV = 0, PMT = 5100, and then solve for FV = $33,712. The MIRR can be obtained by inputting N = 5, PV = -17100, PMT = 0, FV = 33712, and then solving for I = 14.54%.

Pulley:

NPV = -$22,430 + $7,500(3.4331) = -$22,430 + $25,748

= $3,318. (Accept)

Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 14, and then solve for NPV = $3,318.

Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for IRR = 20%.

MIRR: PV Costs = $22,430.

FV Inflows:

PV FV

0 1 2 3 4 5

| | | | | |

7,500 7,500 7,500 7,500 7,500

8,550

9,747

11,112

12,667

22,430 MIRR = 17.19% (Accept) 49,576

Financial calculator: Obtain the FVA by inputting N = 5, I = 14, PV = 0, PMT = 7500, and then solve for FV = $49,576. The MIRR can be obtained by inputting N = 5, PV = -22430, PMT = 0, FV = 49576, and then solving for I = 17.19%.

13-9 Electric-powered:

NPVE = -$22,000 + $6,290 [(1/i)-(1/(i*(1+i)n)]

= -$22,000 + $6,290 [(1/0.12)-(1/(0.12*(1+0.12)6)]

= -$22,000 + $6,290(4.1114) = -$22,000 + $25,861 = $3,861.

Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 12, and then solve for NPV = $3,861.

Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for IRR = 18%.

Gas-powered:

NPVG = -$17,500 + $5,000 [(1/i)-(1/(i*(1+i)n)]

= -$17,500 + $5,000 [(1/0.12)-(1/(0.12*(1+0.12)6)]

= -$17,500 + $5,000(4.1114) = -$17,500 + $20,557 = $3,057.

Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 12, and then solve for NPV = $3,057.

Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for IRR = 17.97% ? 18%.

The firm should purchase the electric-powered forklift because it has a higher NPV than the gas-powered forklift. The company gets a high rate of return (18% > k = 12%) on a larger investment.

13-10 Financial calculator solution, NPV:

Project S

Inputs 5 12 3000 0

Output = -10,814.33

NPVS = $10,814.33 - $10,000 = $814.33.

Project L

Inputs 5 12 7400 0

Output = -26,675.34

NPVL = $26,675.34 - $25,000 = $1,675.34.

Financial calculator solution, IRR:

Input CF0 = -10000, CF1 = 3000, Nj = 5, IRRS = ? IRRS = 15.24%.

Input CF0 = -25000, CF1 = 7400, Nj = 5, IRRL = ? IRRL = 14.67%.

Financial calculator solution, MIRR:

Project S

Inputs 5 12 0 3000

Output = -19,058.54

PV costsS = $10,000.

FV inflowsS = $19,058.54.

Inputs 5 -10000 0 19058.54

Output = 13.77

MIRRS = 13.77%.

Project L

Inputs 5 12 0 7400

Output = -47,011.07

PV costsL = $25,000.

FV inflowsL = $47,011.07.

Inputs 5 -25000 0 47011.07

Output = 13.46

MIRRL = 13.46%.

PIS = = 1.081. PIL = = 4.067.

Thus, NPVL > NPVS, IRRS > IRRL, MIRRS > MIRRL, and PIS > PIL. The scale difference between Projects S and L result in the IRR, MIRR, and PI favoring S over L. However, NPV favors Project L, and hence L should be chosen.

13-11 a. The IRRs of the two alternatives are undefined. To calculate an IRR, the cash flow stream must include both cash inflows and outflows.

b. The PV of costs for the conveyor system is -$556,717, while the PV of costs for the forklift system is -$493,407. Thus, the forklift system is expected to be -$493,407 - (-$556,717) = $63,310 less costly than the conveyor system, and hence the forklifts should be used.

Note: If the PVIFA interest factors are used, then PVC = -$556,720 and PVF = -$493,411.

13-12 Project X: 0 1 2 3 4

| | | | |

-1,000 100 300 400 700.00

448.00

376.32

140.49

1,664.81

1,000 13.59% = MIRRX`

$1,000 = $1,664.81/(1 + MIRRX)4.

Project Y: 0 1 2 3 4

| | | | |

-1,000 1,000 100 50 50.00

56.00

125.44

1,404.93

1,636.37

1,000 13.10% = MIRRY

$1,000 = $1,636.37/(1 + MIRRY)4.

Thus, since MIRRX > MIRRY, Project X should be chosen.

Alternative step: You could calculate NPVs, see that Project X has the higher NPV, and just calculate MIRRX.

NPVX = $58.02 and NPVY = $39.94.

13-13 Input the appropriate cash flows into the cash flow register, and then calculate NPV at 10% and the IRR of each of the projects:

Project S: NPVS = $39.14; IRRS = 13.49%.

Project L: NPVL = $53.55; IRRL = 11.74%.

Since Project L has the higher NPV, it is the better project

13-14 Step 1: Determine the PMT:

1 10

| | · · · |

1,000 PMT PMT

With a financial calculator, input N = 10, I = 12, PV = -1000, and FV = 0 to obtain PMT = $176.98.

Step 2: Calculate the project's MIRR:

PMT TV

176.98 2,820.61

MIRR = 10.93%

FV of inflows: With a financial calculator, input N = 10, I = 10, PV = 0, and PMT = 176.98 to obtain FV = $2,820.61. Then input

N = 10, PV = -1000, PMT = 0, and FV = 2820.61 to obtain

I = MIRR = 10.93%.

13-15 a. Purchase price $ 900,000

Installation 165,000

Initial outlay $1,065,000

CF0 = -1065000; CF1-5 = 350000; I = 14; NPV = ?

NPV = $136,578; IRR = 19.22%.

b. Ignoring environmental concerns, the project should be undertaken because its NPV is positive and its IRR is greater than the firm's cost of capital.

c. Environmental effects could be added by estimating penalties or any other cash outflows that might be imposed on the firm to help return the land to its previous state (if possible). These outflows could be so large as to cause the project to have a negative NPV--in which case the project should not be undertaken.

13-16 a. Year Sales Royalties Marketing Net

0 (20,000) (20,000)

1 75,000 (5,000) (10,000) 60,000

2 52,500 (3,500) (10,000) 39,000

3 22,500 (1,500) 21,000

Payback period = $20,000/$60,000 = 0.33 years.

NPV = $60,000/(1.11)1 + $39,000/(1.11)2 + $21,000/(1.11)3 - $20,000

= $81,062.35.

IRR = 261.90%.

b. Finance theory dictates that this investment should be accepted. However, ask your students "Does this service encourage cheating?" If yes, does a businessperson have a social responsibility not to make this service available?

13-17 Facts: 5 years remaining on lease; rent = $2,000/month; 60 payments left, payment at end of month.

New lease terms: $0/month for 9 months; $2,600/month for 51 months.

Cost of capital = 12% annual (1% per month).

a. 0 1 2 59 60

| | | · · · | |

2,000 2,000 2,000 2,000

PV cost of old lease: N = 60; I = 1; PMT = 2000; FV = 0; PV = ?

PV = -$89,910.08.

0 1 9 10 59 60

| | · · · | | · · · | |

0 0 2,600 2,600 2,600

PV cost of new lease: CF0 = 0, CF1-9 = 0; CF10-60 = 2,600; I = 1.

NPV = -$94,611.45.

Sharon should not accept the new lease because the present value of its cost is $94,611.45 - $89,910.08 = $4,701.37 greater than the old lease.

b. 0 1 2 9 10 59 60

| | | · · · | | · · · | |

2,000 2,000 2,000 PMT PMT PMT

FV = ?

FV of first 9 months' rent under old lease:

N = 9; I = 1; PV = 0; PMT = 2000; FV = ? FV = $18,737.05.

The FV of the first 9 months' rent is equivalent to the PV of the 51-period annuity whose payments represent the incremental rent during months 10-60. To find this value:

N = 51; I = 1; PV = -18737.05; FV = 0; PMT = ? PMT = $470.80.

Thus, the new lease payment that will make her indifferent is $2,000 + $470.80 = $2,470.80.

Check:

0 1 9 10 59 60

| | · · · | | · · · | |

0 0 2,470.80 2,470.80 2,470.80

PV cost of new lease: CF0 = 0; CF1-9 = 0; CF10-60 = 2470.80; I = 1.

NPV = -$89,909.99.

Except for rounding; the PV cost of this lease equals the PV cost of the old lease.

c. Period Old Lease New Lease D Lease

0 0 0 0

1-9 2,000 0 2,000

10-60 2,000 2,600 -600

CF0 = 0; CF1-9 = 2000; CF10-60 = -600; IRR = ? IRR = 1.9113%. This is the periodic rate. To obtain the nominal cost of capital, multiply by 12: 12(0.019113) = 22.94%.

Check: Old lease terms:

N = 60; I = 1.9113; PMT = 2000; FV = 0; PV = ? PV = -$71,039.17.

New lease terms:

CF0 = 0; CF1-9 = 0; CF10-60 = 2600; I = 1.9113; NPV = ? NPV = -$71,038.98.

Except for rounding differences; the costs are the same.

13-18 a.

k

NPVA

NPVB

0.0%

$890

$399

10.0

283

179

12.0

200

146

18.1

0

62

20.0

(49)

41

24.0

(138)

0

30.0

(238)

(51)

b. IRRA = 18.1%; IRRB = 24.0%.

c. At k = 10%, Project A has the greater NPV, specifically $200.41 as compared to Project B's NPV of $145.93. Thus, Project A would be selected. At k = 17%, Project B has an NPV of $63.68 which is higher than Project A's NPV of $2.66. Thus, choose Project B if k = 17%.

d. Here is the MIRR for Project A when k = 10%:

PV costs = $300 + $387/(1.10)1 + $193/(1.10)2

+ $100/(1.10)3 + $180/(1.10)7 = $978.82.

TV inflows = $600(1.10)3 + $600(1.10)2 + $850(1.10)1 = $2,459.60.

Now, MIRR is that discount rate which forces the TV of $2,459.60 in

7 years to equal $978.82:

$952.00 = $2,547.60(1+MIRR)7.

MIRRA = 14.07%.

Similarly, MIRRB = 15.89%.

At k = 17%,

MIRRA = 17.57%.

MIRRB = 19.91%.

e. To find the crossover rate, construct a Project D which is the difference in the two projects' cash flows:

Project D =

Year CFA - CFB

0 $105

1 (521)

2 (327)

3 (234)

4 466

5 466

6 716

7 (180)

IRRD = Crossover rate = 14.53%.

Projects A and B are mutually exclusive, thus, only one of the projects can be chosen. As long as the cost of capital is greater than the crossover rate, both the NPV and IRR methods will lead to the same project selection. However, if the cost of capital is less than the crossover rate the two methods lead to different project selections--a conflict exists. When a conflict exists the NPV method must be used.

Because of the sign changes and the size of the cash flows, Project D has multiple IRRs. Thus, a calculator's IRR function will not work. One could use the trial and error method of entering different discount rates until NPV = $0. However, an HP can be "tricked" into giving the roots. After you have keyed Project Delta's cash flows into the g register of an HP-10B, you will see an "Error-Soln" message. Now enter 10 ? STO ? IRR/YR and the 14.53% IRR is found. Then enter 100 ? STO ? IRR/YR to obtain IRR = 456.22%. Similarly, Excel or Lotus 1-2-3 can also be used.

13-19 a. Incremental Cash

Year Plan B Plan A Flow (B - A)

0 ($10,000,000) ($10,000,000) $ 0

1 1,750,000 12,000,000 (10,250,000)

2-20 1,750,000 0 1,750,000

If the firm goes with Plan B, it will forgo $10,250,000 in Year 1, but will receive $1,750,000 per year in Years 2-20.

b. If the firm could invest the incremental $10,250,000 at a return of 16.07%, it would receive cash flows of $1,750,000. If we set up an amortization schedule, we would find that payments of $1,750,000 per year for 19 years would amortize a loan of $10,250,000 at 16.0665%.

Financial calculator solution:

Inputs 19 -10250000 1750000 0

Output = 16.0665

c. Yes, assuming (1) equal risk among projects, and (2) that the cost of capital is a constant and does not vary with the amount of capital raised.

d. See graph. If the cost of capital is less than 16.07%, then Plan B should be accepted; if k > 16.07%, then Plan A is preferred.

13-20 a. Financial calculator solution:

Plan A

Inputs 20 10 8000000 0

Output = -68,108,510

NPVA = $68,108,510 - $50,000,000 = $18,108,510.

Plan B

Inputs 20 10 3400000 0

Output = -28,946,117

NPVB = $28,946,117 - $15,000,000 = $13,946,117.

Plan A

Inputs 20 -50000000 8000000 0

Output = 15.03

IRRA = 15.03%.

Plan B

Inputs 20 -15000000 3400000 0

Output = 22.26

IRRB = 22.26%.

b. If the company takes Plan A rather than B, its cash flows will be (in millions of dollars):

Cash Flows Cash Flows Project D

Year from A from B Cash Flows

0 ($50) ($15.0) ($35.0)

1 8 3.4 4.6

2 8 3.4 4.6

. . . .

. . . .

. . . .

20 8 3.4 4.6

So, Project D has a "cost" of $35,000,000 and "inflows" of $4,600,000 per year for 20 years.

Inputs 20 10 4600000 0

Output = -39,162,393

NPVD = $39,162,393 - $35,000,000 = $4,162,393.

Inputs 2 -35000000 4600000 0

Output = 11.71

IRRD = 11.71%.

Since IRRD > k, and since we should accept D. This means accept the larger project (Project A). In addition, when dealing with mutually exclusive projects, we use the NPV method for choosing the best project.

c.

d. The NPV method implicitly assumes that the opportunity exists to reinvest the cash flows generated by a project at the cost of capital, while use of the IRR method implies the opportunity to reinvest at the IRR. If the firm's cost of capital is constant at 10 percent, all projects with an NPV > 0 will be accepted by the firm. As cash flows come in from these projects, the firm will either pay them out to investors, or use them as a substitute for outside capital which costs 10 percent. Thus, since these cash flows are expected to save the firm 10 percent, this is their opportunity cost reinvestment rate.

The IRR method assumes reinvestment at the internal rate of return itself, which is an incorrect assumption, given a constant expected future cost of capital, and ready access to capital markets.

13-21 a. The project's expected cash flows are as follows (in millions of dollars):

Time Net Cash Flow

0 ($ 4.4)

1 27.7

2 (25.0)

We can construct the following NPV profile:

Discount Rate NPV

0% ($1,700,000)

9 (29,156)

10 120,661

50 2,955,556

100 3,200,000

200 2,055,556

300 962,500

400 140,000

410 70,204

420 2,367

430 (63,581)

The table above was constructed using a financial calculator with the following inputs: CF0 = -4400000, CF1 = 27700000, CF2 = -25000000, and I = discount rate to solve for the NPV.

b. If k = 8%, reject the project since NPV < 0. But if k = 14%, accept the project because NPV > 0.

c. Other possible projects with multiple rates of return could be nuclear power plants where disposal of radioactive wastes is required at the end of the project's life, or leveraged leases where the borrowed funds are repaid at the end of the lease life. (See Chapter 19 for more information on leases.)

d. Here is the MIRR for the project when k = 8%:

PV costs = $4,400,000 + $25,000,000/(1.08)2 = $25,833,470.51.

TV inflows = $27,700,000(1.08)1 = $29,916,000.00.

Now, MIRR is that discount rate which forces the PV of the TV of $29,916,000 over 2 years to equal $25,833,470.51:

$25,833,470.51 = $29,916,000(PVIFk,2).

Inputs 2 -25833470.51 0 19916000

Output = 7.61

MIRR = 7.61%.

At k = 14%,

Inputs 2 -23636688.21 0 31578000

Output = 15.58

MIRR = 15.58%.

PV costs = $4,400,000 + $25,000,000/(1.14)2 = $23,636,688.21.

TV inflows = $27,700,000(1.14)1 = $31,578,000.

Now, MIRR is that discount rate which forces the PV of the TV of $31,578,000 over 2 years to equal $23,636,688.21:

$23,636,688.21 = $31,578,000(PVIFk,2).

Yes. The MIRR method leads to the same conclusion as the NPV method. Reject the project if k = 8%, which is greater than the corresponding MIRR of 7.61%, and accept the project if k = 14%, which is less than the corresponding MIRR of 15.58%.

13-22 Determine cash flows:

t = 0: The firm must borrow the entire $2,000,000 in order to invest in the casino project, since the casino will not generate any funds until the end of the second year. However, the loan must be repaid at the end of the first year, therefore, the firm must use the extra $1 million to provide the funds needed to repay the loan.

t = 1: Repay $2 million loan, plus 10 percent interest ($200,000), plus $700,000 fee: Net cash out flow = $2.9 million.

t = 2: Receive $2 million from sale of casino.

Work out NPV profile:

NPV = +$1,000,000 - .

Solve at different values of k:

k NPV

0% $100,000

10 16,529

13 (78) ? 0

15 (9,452)

25 (40,000)

35 (50,754)

50 (44,444)

77 (32) ? 0

100 50,000

150 160,000

The table above was constructed using a financial calculator with the following inputs: CF0 = 1000000, CF1 = -2900000, CF2 = 2000000, and

I = discount rate to solve for the NPV.

As the graph indicates, the NPV is positive at any k less than 13 percent or greater than 77 percent; within that range, the NPV is negative.

The deal really amounts to a loan plus a construction project. If the firm could borrow at low rates (less than 13 percent), then the project would be profitable because the profit on the sale of the casino ($1 million) would more than cover the interest and fee on the loan. Or, if the firm had such good investment opportunities that the firm could make over 76 percent on the $1 million made available by the deal, it would be profitable. In between, it is not a good project.

13-23 a. The IRRs of the two alternatives are undefined. To calculate an IRR, the cash flow stream must include both cash inflows and outflows.

b. The PV of costs for the conveyor system is ($911,067), while the PV of costs for the forklift system is ($838,834). Thus, the forklift system is expected to be ($838,834) - ($911,067) = $72,233 less costly than the conveyor system, and hence the forklift trucks should be used.

Financial calculator solution:

Input: CF0 = -500000, = -120000, Nj = 4, CF2 = -20000, I = 8,

NPVC = ? NPVC = -911,067.

Input: CF0 = -200000, = -160000, = 5, I = 8, = ? = -838,834.

13-24 a. Payback A (cash flows in thousands):

Annual

Period Cash Flows Cumulative

0 ($25,000) ($25,000)

1 5,000 (20,000)

2 10,000 (10,000)

3 15,000 5,000

4 20,000 25,000

PaybackA = 2 + $10,000/$15,000 = 2.67 years.

Payback B (cash flows in thousands):

Annual

Period Cash Flows Cumulative

0 ($25,000) $25,000)

1 20,000 (5,000)

2 10,000 5,000

3 8,000 13,000

4 6,000 19,000

PaybackB = 1 + $5,000/$10,000 = 1.50 years.

b. Discounted payback A (cash flows in thousands):

Annual Discounted @10%

Period Cash Flows Cash Flows Cumulative

0 ($25,000) ($25,000.00) ($25,000.00)

1 5,000 4,545.45 ( 20,454.55)

2 10,000 8,264.46 ( 12,190.09)

3 15,000 11,269.72 ( 920.37)

4 20,000 13,660.27 12,739.90

Discounted PaybackA = 3 + $920.37/$13,660.27 = 3.07 years.

Discounted payback B (cash flows in thousands):

Annual Discounted @10%

Period Cash Flows Cash Flows Cumulative

0 ($25,000) ($25,000.00) ($25,000.00)

1 20,000 18,181.82 ( 6,818.18)

2 10,000 8,264.46 1,446.28

3 8,000 6,010.52 7,456.80

4 6,000 4,098.08 11,554.88

Discounted PaybackB = 1 + $6,818.18/$8,264.46 = 1.825 years.

c. NPVA = $12,739,908; IRRA = 27.27%.

NPVB = $11,554,880; IRRB = 36.15%.

Both projects have positive NPVs, so both projects should be undertaken.

d. At a discount rate of 5%, NPVA = $18,243,813.

At a discount rate of 5%, NPVB = $14,964,829.

At a discount rate of 5%, Project A has the higher NPV; consequently, it should be accepted.

e. At a discount rate of 15%, NPVA = $8,207,071.

At a discount rate of 15%, NPVB = $8,643,390.

At a discount rate of 15%, Project B has the higher NPV; consequently, it should be accepted.

f. Project D =

Year CFA - CFB

0 $ 0

1 (15)

2 0

3 7

4 14

IRRD = Crossover rate = 13.5254% ? 13.53%.

g. Use 3 steps to calculate MIRRA @ k = 10%:

Step 1: Calculate the NPV of the uneven cash flow stream, so its FV can then be calculated. With a financial calculator, enter the cash flow stream into the cash flow registers, then enter I = 10, and solve for NPV = $37,739,908.

Step 2: Calculate the FV of the cash flow stream as follows:

Enter N = 4, I = 10, PV = -37739908, and PMT = 0 to solve for FV = $55,255,000.

Step 3: Calculate MIRRA as follows:

Enter N = 4, PV = -25000000, PMT = 0, and FV = 55255000 to solve for I = 21.93%.

Use 3 steps to calculate MIRRB @ k = 10%:

Step 1: Calculate the NPV of the uneven cash flow stream, so its FV can then be calculated. With a financial calculator, enter the cash flow stream into the cash flow registers, then enter I = 10, and solve for NPV = $36,554,880.

Step 2: Calculate the FV of the cash flow stream as follows:

Enter N = 4, I = 10, PV = -36554880, and PMT = 0 to solve for FV = $53,520,000.

Step 3: Calculate MIRRB as follows:

Enter N = 4, PV = -25000000, PMT = 0, and FV = 53520000 to solve for I = 20.96%.

According to the MIRR approach, if the 2 projects were mutually exclusive, Project A would be chosen because it has the higher MIRR. This is consistent with the NPV approach.

13-25 Plane A: Expected life = 5 years; Cost = $100 million; NCF = $30 million; COC = 12%.

Plane B: Expected life = 10 years; Cost = $132 million; NCF = $25 million; COC = 12%.

0 1 2 3 4 5 6 7 8 9 10

A: | | | | | | | | | | |

-100 30 30 30 30 30 30 30 30 30 30

-100

-70

Enter these values into the cash flow register: CF0 = -100; CF1-4 = 30;

CF5 = -70; CF6-10 = 30. Then enter I = 12, and press the NPV key to get NPVA = 12.764 » $12.76 million.

0 1 2 3 4 5 6 7 8 9 10

B: | | | | | | | | | | |

-132 25 25 25 25 25 25 25 25 25 25

Enter these cash flows into the cash flow register, along with the interest rate, and press the NPV key to get NPVB = 9.256 ? $9.26 million.

Project A is the better project and will increase the company's value by $12.76 million.

13-26 0 1 2 3 4 5 6 7 8

A: | | | | | | | | |

-10 4 4 4 4 4 4 4 4

-10

-6

Machine A's simple NPV is calculated as follows: Enter CF0 = -10 and CF1-4 = 4. Then enter I = 10, and press the NPV key to get NPVA = $2.679 million. However, this does not consider the fact that the project can be repeated again. Enter these values into the cash flow register: CF0 = -10; CF1-3 = 4; CF4 = -6; CF5-8 = 4. Then enter I = 10, and press the NPV key to get Extended NPVA = $4.5096 ? $4.51 million.

0 1 2 3 4 5 6 7 8

B: | | | | | | | | |

-15 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5

Enter these cash flows into the cash flow register, along with the interest rate, and press the NPV key to get NPVB = $3.672 ? $3.67 million.

Machine A is the better project and will increase the company's value by $4.51 million.

3-27 a. Using a financial calculator, input the following: CF0 = -190000,

CF1 = 87000, Nj = 3, and I = 14 to solve for NPV190-3 = $11,981.99 ? $11,982 (for 3 years).

Adjusted NPV190-3 = $11,982 + $11,982/(1.14)3 = $20,070.

Using a financial calculator, input the following: CF0 = -360000,

CF1 = 98300, Nj = 6, and I = 14 to solve for NPV360-6 = $22,256.02 ? $22,256 (for 6 years).

Both new machines have positive NPVs, hence the old machine should be replaced. Further, since its adjusted NPV is greater, choose Model 360-6.

13-28 a. NPV of termination after Year t:

NPV0 = -$22,500 + $22,500 = 0.

Using a financial calculator, input the following: CF0 = -22500,

CF1 = 23750, and I = 10 to solve for NPV1 = -$909.09 ? -$909.

Using a financial calculator, input the following: CF0 = -22500,

CF1 = 6250, CF2 = 20250, and I = 10 to solve for NPV2 = -$82.64 ? -$83.

Using a financial calculator, input the following: CF0 = -22500,

CF1 = 6250, Nj = 2, CF3 = 17250, and I = 10 to solve for NPV3 = $1,307.29 ? $1,307.

Using a financial calculator, input the following: CF0 = -22500,

CF1 = 6250, Nj = 3, CF4 = 11250, and I = 10 to solve for NPV4 = $726.73 ? $727.

Using a financial calculator, input the following: CF0 = -22500,

CF1 = 6250, Nj = 5, and I = 10 to solve for NPV5 = $1,192.42 ? $1,192.

The firm should operate the truck for 3 years, NPV3 = $1,307.

b. No. Salvage possibilities could only raise NPV and IRR. The value of the firm is maximized by terminating the project after Year 3.

SOLUTION TO SPREADSHEET PROBLEMS

13-29 The detailed solution for the problem is available both on the instructor’s resource CD-ROM (in the file Solution for Ch 13-29 Build a Model.xls) and on the instructor’s side of the Harcourt College Publishers’ web site, http://www.harcourtcollege.com/finance/theory10e.

13-30 a.

k NPVA NPVB

0.0% $890 $399

10.0 283 179

18.1 0 62

20.0 (49) 41

24.0 (138) 0

30.0 (238) (51)

b. Input: CF0 = -300, CF1 = -387, CF2 = -193, CF3 = -100, CF4 = 600, Nj = 2, CF6 = 850, CF7 = -180, IRRA = ? IRRA = 18.1%.

Input: CF0 = -405, CF1 = 134, Nj = 6, CF7 = 0, IRRB = ? IRRB = 24.0%.

c. At k = 10%, Project A has the greater NPV, specifically $283.34 as compared to Project B's NPV of $178.60. Thus, Project A would be selected. At k = 17%, Project B has an NPV of $75.95 which is higher than Project A's NPV of $31.05. Thus, choose Project B if k = 17%.

d. Here is the MIRR for Project A when k = 10%:

PV costs = $300 + $387/(1.10)1 + $193/(1.10)2 + $100/(1.10)3

+ $180/(1.10)7 = $978.82.

TV inflows = $600(1.10)3 + $600(1.10)2 + $850(1.10)1 = $2,459.60.

Now, MIRR is that discount rate which forces the PV of the TV of $2,459.60 over 7 years to equal $978.82:

$978.82 = $2,459.60(PVIFk,7).

Financial calculator solution:

Inputs 7 -978.82 0 2459.60

Output = 14.07

MIRRA = 14.07%.

Here is the MIRR for Project B when k = 10%:

BEGIN MODE

Inputs 6 10 0 134

Output = -1,137.28

Because the $134 payments occur in Years 1 through 6, but not Year 7, it can be thought of as a 6-year annuity due.

FV inflowsL = $1,137.28.

PV costsB = $405.

END MODE

Inputs 7 -405 0 1137.28

Output = 15.89

MIRRB = 15.89%.

MIRR for Project A at k = 17%:

PV costs = $300 + $387/(1.17)1 + $193/(1.17)2 + $100/(1.17)3

+ $180/(1.17)7 = $894.17.

TV inflows = $600(1.17)3 + $600(1.17)2+ $850(1.17)1 = $2,776.81.

Now, MIRR is that discount rate which forces the PV of the TV of $2,776.81 over 7 years to equal $894.17:

$894.17 = $2,776.81(PVIFk,7).

Financial calculator solution:

Inputs 7 -894.17 0 2776.81

Output = 17.57

MIRRA = 17.57%.

MIRR for Project B at k = 17%:

BEGIN MODE

Inputs 6 17 0 134

Output = -1,443.45

Because the $134 payments occur in Years 1 through 6, but not Year 7, it can be thought of as a 6-year annuity due.

FV inflowsL = $1,443.45.

PV costsB = $405.

END MODE

Inputs 7 -405 0 1443.45

Output = 19.91

MIRRB = 19.91%.

MIRR for Project B at k = 17%:

e. To find the crossover rate, construct a Project D which is the difference in the two projects' cash flows:

Project D

Year CFA - CFB

0 $ 105

1 (521)

2 (327)

3 (234)

4 466

5 466

6 716

7 (180)

Input: CF0 = 105, CF1 = -521, CF2 = -327, CF3 = -234, CF4 = 466, Nj = 2, CF6 = 716, CF7 = -180, IRRD = ? IRRD = 14.53% = Crossover rate.

This is the discount rate at which the NPV profiles of the two projects cross and thus, at which the projects’ NPVs are equal.

Because of the sign changes and the size of the cash flows, Project D has multiple IRRs. Thus, a calculator's IRR function will not work. One could use the trial and error method of entering different discount rates until NPV = $0. However, an HP 10B can be "tricked" into giving the roots. You must enter a guess for the IRR as follows: after keying in Project Delta's cash flows into the CFj register, estimate an IRR (for example, 10), press the gold key, the STO key, the gold key again, and then IRR. The IRR of 14.53% is found. Then guess 100, using the same key strokes as was done for the guess of 10, to obtain IRR = 456.22%. Similarly, a spreadsheet program can also be used.

f. Worst case scenario:

(k = 10%) *Project A: NPV = $221.76 IRR = 16.67% MIRR = 13.08%

Project B: NPV = $101.63 IRR = 19.86% MIRR = 13.05%

(k = 17%) Project A: NPV = ($8.93) IRR = 16.67% MIRR = 16.84%

*Project B: NPV = $25.97 IRR = 19.86% MIRR = 17.93%

Best case scenario:

(k = 10%) *Project A: NPV = $339.79 IRR = 19.30% MIRR = 15.04%

Project B: NPV = $240.18 IRR = 26.54% MIRR = 17.57%

(k = 17%) Project A: NPV = $67.70 IRR = 19.30% MIRR = 18.28%

*Project B: NPV = $115.93 IRR = 26.54% MIRR = 21.28%

*Project choice

g. k = 0%: NPVA = $890 NPVB = $399

k = 5%: NPVA = $540 NPVB = $275

k = 20%: NPVA = ($49) NPVB = $41

k = 400%: NPVA = ($385) NPVB = ($372)

CYBERPROBLEM

13-31 The detailed solution for the cyberproblem is available on the instructor’s side of the Harcourt College Publishers’ web site: http://www.harcourtcollege.com/finance/theory10e.

MINI CASE

ASSUME THAT YOU RECENTLY WENT TO WORK FOR AXIS COMPONENTS COMPANY, A SUPPLIER OF AUTO REPAIR PARTS USED IN THE AFTER-MARKET WITH PRODUCTS FROM CHRYSLER, FORD, AND OTHER AUTO MAKERS. YOUR BOSS, THE CHIEF FINANCIAL OFFICER (CFO), HAS JUST HANDED YOU THE ESTIMATED CASH FLOWS FOR TWO PROPOSED PROJECTS. PROJECT L INVOLVES ADDING A NEW ITEM TO THE FIRM'S IGNITION SYSTEM LINE; IT WOULD TAKE SOME TIME TO BUILD UP THE MARKET FOR THIS PRODUCT, SO THE CASH INFLOWS WOULD INCREASE OVER TIME. PROJECT S INVOLVES AN ADD-ON TO AN EXISTING LINE, AND ITS CASH FLOWS WOULD DECREASE OVER TIME. BOTH PROJECTS HAVE 3-YEAR LIVES, BECAUSE AXIS IS PLANNING TO INTRODUCE ENTIRELY NEW MODELS AFTER 3 YEARS.

HERE ARE THE PROJECTS' NET CASH FLOWS (IN THOUSANDS OF DOLLARS):

EXPECTED NET CASH FLOW

YEAR PROJECT L PROJECT S

0 ($100) ($100)

1 10 70

2 60 50

3 80 20

DEPRECIATION, SALVAGE VALUES, NET WORKING CAPITAL REQUIREMENTS, AND TAX EFFECTS ARE ALL INCLUDED IN THESE CASH FLOWS.

THE CFO ALSO MADE SUBJECTIVE RISK ASSESSMENTS OF EACH PROJECT, AND HE CONCLUDED THAT BOTH PROJECTS HAVE RISK CHARACTERISTICS WHICH ARE SIMILAR TO THE FIRM'S AVERAGE PROJECT. AXIS’S WEIGHTED AVERAGE COST OF CAPITAL IS 10 PERCENT. YOU MUST NOW DETERMINE WHETHER ONE OR BOTH OF THE PROJECTS SHOULD BE ACCEPTED.

A. WHAT IS CAPITAL BUDGETING? ARE THERE ANY SIMILARITIES BETWEEN A FIRM'S CAPITAL BUDGETING DECISIONS AND AN INDIVIDUAL'S INVESTMENT DECISIONS?

ANSWER: CAPITAL BUDGETING IS THE PROCESS OF ANALYZING ADDITIONS TO FIXED ASSETS. CAPITAL BUDGETING IS IMPORTANT BECAUSE, MORE THAN ANYTHING ELSE, FIXED ASSET INVESTMENT DECISIONS CHART A COMPANY'S COURSE FOR THE FUTURE. CONCEPTUALLY, THE CAPITAL BUDGETING PROCESS IS IDENTICAL TO THE DECISION PROCESS USED BY INDIVIDUALS MAKING INVESTMENT DECISIONS.

THESE STEPS ARE INVOLVED:

1. ESTIMATE THE CASH FLOWS--INTEREST AND MATURITY VALUE OR DIVIDENDS IN THE CASE OF BONDS AND STOCKS, OPERATING CASH FLOWS IN THE CASE OF CAPITAL PROJECTS.

2. ASSESS THE RISKINESS OF THE CASH FLOWS.

3. DETERMINE THE APPROPRIATE DISCOUNT RATE, BASED ON THE RISKINESS OF THE CASH FLOWS AND THE GENERAL LEVEL OF INTEREST RATES. THIS IS CALLED THE PROJECT COST OF CAPITAL IN CAPITAL BUDGETING.

4. FIND (A) THE PV OF THE EXPECTED CASH FLOWS AND/OR (B) THE ASSET'S RATE OF RETURN.

5. IF THE PV OF THE INFLOWS IS GREATER THAN THE PV OF THE OUTFLOWS (THE NPV IS POSITIVE), OR IF THE CALCULATED RATE OF RETURN (THE IRR) IS HIGHER THAN THE PROJECT COST OF CAPITAL, ACCEPT THE PROJECT.

B. WHAT IS THE DIFFERENCE BETWEEN INDEPENDENT AND MUTUALLY EXCLUSIVE PROJECTS? BETWEEN PROJECTS WITH NORMAL AND NONNORMAL CASH FLOWS?

ANSWER: PROJECTS ARE INDEPENDENT IF THE CASH FLOWS OF ONE ARE NOT AFFECTED BY THE ACCEPTANCE OF THE OTHER. CONVERSELY, TWO PROJECTS ARE MUTUALLY EXCLUSIVE IF ACCEPTANCE OF ONE IMPACTS ADVERSELY THE CASH FLOWS OF THE OTHER; THAT IS, AT MOST ONE OF TWO OR MORE SUCH PROJECTS MAY BE ACCEPTED. PUT ANOTHER WAY, WHEN PROJECTS ARE MUTUALLY EXCLUSIVE IT MEANS THAT THEY DO THE SAME JOB. FOR EXAMPLE, A FORKLIFT TRUCK VERSUS A CONVEYOR SYSTEM TO MOVE MATERIALS, OR A BRIDGE VERSUS A FERRY BOAT.

PROJECTS WITH NORMAL CASH FLOWS HAVE OUTFLOWS, OR COSTS, IN THE FIRST YEAR (OR YEARS) FOLLOWED BY A SERIES OF INFLOWS. PROJECTS WITH NONNORMAL CASH FLOWS HAVE ONE OR MORE OUTFLOWS AFTER THE INFLOW STREAM HAS BEGUN. HERE ARE SOME EXAMPLES:

INFLOW (+) OR OUTFLOW (-) IN YEAR

0 1 2 3 4 5

NORMAL - + + + + +

- - + + + +

- - - + + +

NONNORMAL - + + + + -

- + + - + -

+ + + - - -

C. 1. WHAT IS THE PAYBACK PERIOD? FIND THE PAYBACKS FOR PROJECTS L AND S.

ANSWER: THE PAYBACK PERIOD IS THE EXPECTED NUMBER OF YEARS REQUIRED TO RECOVER A PROJECT'S COST. WE CALCULATE THE PAYBACK BY DEVELOPING THE CUMULATIVE CASH FLOWS AS SHOWN BELOW FOR PROJECT L (IN THOUSANDS OF DOLLARS):

EXPECTED NCF

YEAR ANNUAL CUMULATIVE

0 ($100) ($100)

1 10 (90)

2 60 (30)

3 80 50

0 1 2 3

| | | |

-100 10 60 80

-90 -30 +50

PROJECT L'S $100 INVESTMENT HAS NOT BEEN RECOVERED AT THE END OF YEAR 2, BUT IT HAS BEEN MORE THAN RECOVERED BY THE END OF YEAR 3. THUS, THE RECOVERY PERIOD IS BETWEEN 2 AND 3 YEARS. IF WE ASSUME THAT THE CASH FLOWS OCCUR EVENLY OVER THE YEAR, THEN THE INVESTMENT IS RECOVERED $30/$80 = 0.375 ? 0.4 INTO YEAR 3. THEREFORE, PAYBACKL = 2.4 YEARS. SIMILARLY, PAYBACKS = 1.6 YEARS.

C. 2. WHAT IS THE RATIONALE FOR THE PAYBACK METHOD? ACCORDING TO THE PAYBACK CRITERION, WHICH PROJECT OR PROJECTS SHOULD BE ACCEPTED IF THE FIRM'S MAXIMUM ACCEPTABLE PAYBACK IS 2 YEARS, AND IF PROJECTS L AND S ARE INDEPENDENT? IF THEY ARE MUTUALLY EXCLUSIVE?

ANSWER: PAYBACK REPRESENTS A TYPE OF "BREAKEVEN" ANALYSIS: THE PAYBACK PERIOD TELLS US WHEN THE PROJECT WILL BREAK EVEN IN A CASH FLOW SENSE. WITH A REQUIRED PAYBACK OF 2 YEARS, PROJECT S IS ACCEPTABLE, BUT PROJECT L IS NOT. WHETHER THE TWO PROJECTS ARE INDEPENDENT OR MUTUALLY EXCLUSIVE MAKES NO DIFFERENCE IN THIS CASE.

C. 3. WHAT IS THE DIFFERENCE BETWEEN THE REGULAR AND DISCOUNTED PAYBACK PERIODS?

ANSWER: DISCOUNTED PAYBACK IS SIMILAR TO PAYBACK EXCEPT THAT DISCOUNTED RATHER THAN RAW CASH FLOWS ARE USED.

SETUP FOR PROJECT L'S DISCOUNTED PAYBACK, ASSUMING A 10% COST OF CAPITAL:

EXPECTED NET CASH FLOWS

YEAR RAW DISCOUNTED CUMULATIVE

0 ($100) ($100.00) ($100.00)

1 10 9.09 (90.91)

2 60 49.59 (41.32)

3 80 60.11 18.79

DISCOUNTED PAYBACKL = 2 + ($41.32/$60.11) = 2.69 = 2.7 YEARS.

VERSUS 2.4 YEARS FOR THE REGULAR PAYBACK.

C. 4. WHAT IS THE MAIN DISADVANTAGE OF DISCOUNTED PAYBACK? IS THE PAYBACK METHOD OF ANY REAL USEFULNESS IN CAPITAL BUDGETING DECISIONS?

ANSWER: REGULAR PAYBACK HAS TWO CRITICAL DEFICIENCIES: (1) IT IGNORES THE TIME VALUE OF MONEY, AND (2) IT IGNORES THE CASH FLOWS THAT OCCUR AFTER THE PAYBACK PERIOD. DISCOUNTED PAYBACK DOES CONSIDER THE TIME VALUE OF MONEY, BUT IT STILL FAILS TO CONSIDER CASH FLOWS AFTER THE PAYBACK PERIOD; HENCE IT HAS A BASIC FLAW. IN SPITE OF ITS DEFICIENCY, MANY FIRMS TODAY STILL CALCULATE THE DISCOUNTED PAYBACK AND GIVE SOME WEIGHT TO IT WHEN MAKING CAPITAL BUDGETING DECISIONS. HOWEVER, PAYBACK IS NOT GENERALLY USED AS THE PRIMARY DECISION TOOL. RATHER, IT IS USED AS A ROUGH MEASURE OF A PROJECT'S LIQUIDITY AND RISKINESS.

D. 1. DEFINE THE TERM NET PRESENT VALUE (NPV). WHAT IS EACH PROJECT'S NPV?

ANSWER: THE NET PRESENT VALUE (NPV) IS SIMPLY THE SUM OF THE PRESENT VALUES OF A PROJECT'S CASH FLOWS:

NPV = .

PROJECT L'S NPV IS $18.79:

0 1 2 3

| | | |

(100.00) 10 60 80

9.09

49.59

60.11

18.79 = NPVL

NPVs ARE EASY TO DETERMINE USING A CALCULATOR WITH AN NPV FUNCTION. ENTER THE CASH FLOWS SEQUENTIALLY, WITH OUTFLOWS ENTERED AS NEGATIVES; ENTER THE COST OF CAPITAL; AND THEN PRESS THE NPV BUTTON TO OBTAIN THE PROJECT'S NPV, $18.78 (NOTE THE PENNY ROUNDING DIFFERENCE). THE NPV OF PROJECT S IS NPVS = $19.98.

D. 2. WHAT IS THE RATIONALE BEHIND THE NPV METHOD? ACCORDING TO NPV, WHICH PROJECT OR PROJECTS SHOULD BE ACCEPTED IF THEY ARE INDEPENDENT? MUTUALLY EXCLUSIVE?

ANSWER: THE RATIONALE BEHIND THE NPV METHOD IS STRAIGHTFORWARD: IF A PROJECT HAS NPV = $0, THEN THE PROJECT GENERATES EXACTLY ENOUGH CASH FLOWS (1) TO RECOVER THE COST OF THE INVESTMENT AND (2) TO ENABLE INVESTORS TO EARN THEIR REQUIRED RATES OF RETURN (THE OPPORTUNITY COST OF CAPITAL). IF NPV = $0, THEN IN A FINANCIAL (BUT NOT AN ACCOUNTING) SENSE, THE PROJECT BREAKS EVEN. IF THE NPV IS POSITIVE, THEN MORE THAN ENOUGH CASH FLOW IS GENERATED, AND CONVERSELY IF NPV IS NEGATIVE.

CONSIDER PROJECT L'S CASH INFLOWS, WHICH TOTAL $150. THEY ARE SUFFICIENT (1) TO RETURN THE $100 INITIAL INVESTMENT, (2) TO PROVIDE INVESTORS WITH THEIR 10 PERCENT AGGREGATE OPPORTUNITY COST OF CAPITAL, AND (3) TO STILL HAVE $18.79 LEFT OVER ON A PRESENT VALUE BASIS. THIS $18.79 EXCESS PV BELONGS TO THE SHAREHOLDERS--THE DEBTHOLDERS' CLAIMS ARE FIXED, SO THE SHAREHOLDERS' WEALTH WILL BE INCREASED BY $18.79 IF PROJECT L IS ACCEPTED. SIMILARLY, AXIS'S SHAREHOLDERS GAIN $19.98 IN VALUE IF PROJECT S IS ACCEPTED.

IF PROJECTS L AND S ARE INDEPENDENT, THEN BOTH SHOULD BE ACCEPTED, BECAUSE THEY BOTH ADD TO SHAREHOLDERS' WEALTH, HENCE TO THE STOCK PRICE. IF THE PROJECTS ARE MUTUALLY EXCLUSIVE, THEN PROJECT S SHOULD BE CHOSEN OVER L, BECAUSE S ADDS MORE TO THE VALUE OF THE FIRM.

D. 3. WOULD THE NPVs CHANGE IF THE COST OF CAPITAL CHANGED?

ANSWER: THE NPV OF A PROJECT IS DEPENDENT ON THE COST OF CAPITAL USED. THUS, IF THE COST OF CAPITAL CHANGED, THE NPV OF EACH PROJECT WOULD CHANGE. NPV DECLINES AS k INCREASES, AND NPV RISES AS k FALLS.

E. 1. DEFINE THE TERM INTERNAL RATE OF RETURN (IRR). WHAT IS EACH PROJECT'S IRR?

ANSWER: THE INTERNAL RATE OF RETURN (IRR) IS THAT DISCOUNT RATE WHICH FORCES THE NPV OF A PROJECT TO EQUAL ZERO:

0 1 2 3

| | | |

CF0 CF1 CF2 CF3

PVCF1

PVCF2

PVCF3

0 = SUM OF PVs = NPV.

EXPRESSED AS AN EQUATION, WE HAVE:

IRR: = $0 = NPV.

NOTE THAT THE IRR EQUATION IS THE SAME AS THE NPV EQUATION, EXCEPT THAT TO FIND THE IRR THE EQUATION IS SOLVED FOR THE PARTICULAR DISCOUNT RATE, IRR, WHICH FORCES THE PROJECT'S NPV TO EQUAL ZERO (THE IRR) RATHER THAN USING THE COST OF CAPITAL (k) IN THE DENOMINATOR AND FINDING NPV. THUS, THE TWO APPROACHES DIFFER IN ONLY ONE RESPECT: IN THE NPV METHOD, A DISCOUNT RATE IS SPECIFIED (THE PROJECT'S COST OF CAPITAL) AND THE EQUATION IS SOLVED FOR NPV, WHILE IN THE IRR METHOD, THE NPV IS SPECIFIED TO EQUAL ZERO AND THE DISCOUNT RATE (IRR) WHICH FORCES THIS EQUALITY IS FOUND.

PROJECT L'S IRR IS 18.1 PERCENT:

0 1 2 3

| | | |

-100.00 10 60 80

8.47

43.02

48.57

$ 0.06 ? $0 IF IRRL = 18.1% IS USED AS THE DISCOUNT RATE.

THEREFORE, IRRL ? 18.1%.

A FINANCIAL CALCULATOR IS EXTREMELY HELPFUL WHEN CALCULATING IRRs. THE CASH FLOWS ARE ENTERED SEQUENTIALLY, AND THEN THE IRR BUTTON IS PRESSED. FOR PROJECT S, IRRS ? 23.6%. NOTE THAT WITH MANY CALCULATORS, YOU CAN ENTER THE CASH FLOWS INTO THE CASH FLOW REGISTER, ALSO ENTER

k = I, AND THEN CALCULATE BOTH NPV AND IRR BY PRESSING THE APPROPRIATE BUTTONS.

E. 2. HOW IS THE IRR ON A PROJECT RELATED TO THE YTM ON A BOND?

ANSWER: THE IRR IS TO A CAPITAL PROJECT WHAT THE YTM IS TO A BONDCIT IS THE EXPECTED RATE OF RETURN ON THE PROJECT, JUST AS THE YTM IS THE PROMISED RATE OF RETURN ON A BOND.

E. 3. WHAT IS THE LOGIC BEHIND THE IRR METHOD? ACCORDING TO IRR, WHICH PROJECTS SHOULD BE ACCEPTED IF THEY ARE INDEPENDENT? MUTUALLY EXCLUSIVE?

ANSWER: IRR MEASURES A PROJECT'S PROFITABILITY IN THE RATE OF RETURN SENSE: IF A PROJECT'S IRR EQUALS ITS COST OF CAPITAL, THEN ITS CASH FLOWS ARE JUST SUFFICIENT TO PROVIDE INVESTORS WITH THEIR REQUIRED RATES OF RETURN. AN IRR GREATER THAN k IMPLIES AN ECONOMIC PROFIT, WHICH ACCRUES TO THE FIRM'S SHAREHOLDERS, WHILE AN IRR LESS THAN k INDICATES AN ECONOMIC LOSS, OR A PROJECT THAT WILL NOT EARN ENOUGH TO COVER ITS COST OF CAPITAL.

PROJECTS' IRRs ARE COMPARED TO THEIR COSTS OF CAPITAL, OR HURDLE RATES. SINCE PROJECTS L AND S BOTH HAVE A HURDLE RATE OF 10 PERCENT, AND SINCE BOTH HAVE IRRs GREATER THAN THAT HURDLE RATE, BOTH SHOULD BE ACCEPTED IF THEY ARE INDEPENDENT. HOWEVER, IF THEY ARE MUTUALLY EXCLUSIVE, PROJECT S WOULD BE SELECTED, BECAUSE IT HAS THE HIGHER IRR.

E. 4. WOULD THE PROJECTS' IRRs CHANGE IF THE COST OF CAPITAL CHANGED?

ANSWER: IRRs ARE INDEPENDENT OF THE COST OF CAPITAL. THEREFORE, NEITHER IRRS NOR IRRL WOULD CHANGE IF k CHANGED. HOWEVER, THE ACCEPTABILITY OF THE PROJECTS COULD CHANGE--L WOULD BE REJECTED IF k WERE ABOVE 18.1%, AND S WOULD ALSO BE REJECTED IF k WERE ABOVE 23.6%.

F. 1. DRAW NPV PROFILES FOR PROJECTS L AND S. AT WHAT DISCOUNT RATE DO THE PROFILES CROSS?

ANSWER: THE NPV PROFILES ARE PLOTTED IN THE FIGURE BELOW.

NOTE THE FOLLOWING POINTS:

1. THE Y-INTERCEPT IS THE PROJECT'S NPV WHEN k = 0%. THIS IS $50 FOR L AND $40 FOR S.

2. THE X-INTERCEPT IS THE PROJECT'S IRR. THIS IS 18.1 PERCENT FOR L AND 23.6 PERCENT FOR S.

3. NPV PROFILES ARE CURVES RATHER THAN STRAIGHT LINES. TO SEE THIS, NOTE THAT THESE PROFILES APPROACH COST = -$100 AS k APPROACHES INFINITY.

4. FROM THE FIGURE BELOW, IT APPEARS THAT THE CROSSOVER POINT IS BETWEEN 8 AND 9 PERCENT. THE PRECISE VALUE IS APPROXIMATELY 8.7 PERCENT. ONE CAN CALCULATE THE CROSSOVER RATE BY (1) GOING BACK TO THE DATA ON THE PROBLEM, FINDING THE CASH FLOW DIFFERENCES FOR EACH YEAR, (2) ENTERING THOSE DIFFERENCES INTO THE CASH FLOW REGISTER, AND (3) PRESSING THE IRR BUTTON TO GET THE CROSSOVER RATE, 8.68% ? 8.7%.

k NPVL NPVS

0% $50 $40

5 33 29

10 19 20

15 7 12

20 (4) 5

F. 2. LOOK AT YOUR NPV PROFILE GRAPH WITHOUT REFERRING TO THE ACTUAL NPVs AND IRRs. WHICH PROJECT OR PROJECTS SHOULD BE ACCEPTED IF THEY ARE INDEPENDENT? MUTUALLY EXCLUSIVE? EXPLAIN. ARE YOUR ANSWERS CORRECT AT ANY COST OF CAPITAL LESS THAN 23.6 PERCENT?

ANSWER: THE NPV PROFILES SHOW THAT THE IRR AND NPV CRITERIA LEAD TO THE SAME ACCEPT/REJECT DECISION FOR ANY INDEPENDENT PROJECT. CONSIDER PROJECT L. IT INTERSECTS THE X-AXIS AT ITS IRR, 18.1 PERCENT. ACCORDING TO THE IRR RULE, L IS ACCEPTABLE IF k IS LESS THAN 18.1 PERCENT. ALSO, AT ANY k LESS THAN 18.1 PERCENT, L'S NPV PROFILE WILL BE ABOVE THE X AXIS, SO ITS NPV WILL BE GREATER THAN $0. THUS, FOR ANY INDEPENDENT PROJECT, NPV AND IRR LEAD TO THE SAME ACCEPT/REJECT DECISION.

NOW ASSUME THAT L AND S ARE MUTUALLY EXCLUSIVE. IN THIS CASE, A CONFLICT MIGHT ARISE. FIRST, NOTE THAT IRRS = 23.6% > 18.1% = THEREFORE, REGARDLESS OF THE SIZE OF k, PROJECT S WOULD BE RANKED HIGHER BY THE IRR CRITERION. HOWEVER, THE NPV PROFILES SHOW THAT NPVL > NPVS IF k IS LESS THAN 8.7 PERCENT. THEREFORE, FOR ANY k BELOW THE 8.7% CROSSOVER RATE, SAY k = 7 PERCENT, THE NPV RULE SAYS CHOOSE L, BUT THE IRR RULE SAYS CHOOSE S. THUS, IF k IS LESS THAN THE CROSSOVER RATE, A RANKING CONFLICT OCCURS.

G. 1. WHAT IS THE UNDERLYING CAUSE OF RANKING CONFLICTS BETWEEN NPV AND IRR?

ANSWER: FOR NORMAL PROJECTS' NPV PROFILES TO CROSS, ONE PROJECT MUST HAVE BOTH A HIGHER VERTICAL AXIS INTERCEPT AND A STEEPER SLOPE THAN THE OTHER. A PROJECT'S VERTICAL AXIS INTERCEPT TYPICALLY DEPENDS ON (1) THE SIZE OF THE PROJECT AND (2) THE SIZE AND TIMING PATTERN OF THE CASH FLOWS--LARGE PROJECTS, AND ONES WITH LARGE DISTANT CASH FLOWS, WOULD GENERALLY BE EXPECTED TO HAVE RELATIVELY HIGH VERTICAL AXIS INTERCEPTS. THE SLOPE OF THE NPV PROFILE DEPENDS ENTIRELY ON THE TIMING PATTERN OF THE CASH FLOWS--LONG-TERM PROJECTS HAVE STEEPER NPV PROFILES THAN SHORT-TERM ONES. THUS, WE CONCLUDE THAT NPV PROFILES CAN CROSS IN TWO SITUATIONS: (1) WHEN MUTUALLY EXCLUSIVE PROJECTS DIFFER IN SCALE (OR SIZE) AND (2) WHEN THE PROJECTS' CASH FLOWS DIFFER IN TERMS OF THE TIMING PATTERN OF THEIR CASH FLOWS (AS FOR PROJECTS L AND S).

G. 2. WHAT IS THE "REINVESTMENT RATE ASSUMPTION”, AND HOW DOES IT AFFECT THE NPV VERSUS IRR CONFLICT?

ANSWER: THE UNDERLYING CAUSE OF RANKING CONFLICTS IS THE REINVESTMENT RATE ASSUMPTION. ALL DCF METHODS IMPLICITLY ASSUME THAT CASH FLOWS CAN BE REINVESTED AT SOME RATE, REGARDLESS OF WHAT IS ACTUALLY DONE WITH THE CASH FLOWS. DISCOUNTING IS THE REVERSE OF COMPOUNDING. SINCE COMPOUNDING ASSUMES REINVESTMENT, SO DOES DISCOUNTING. NPV AND IRR ARE BOTH FOUND BY DISCOUNTING, SO THEY BOTH IMPLICITLY ASSUME SOME DISCOUNT RATE. INHERENT IN THE NPV CALCULATION IS THE ASSUMPTION THAT CASH FLOWS CAN BE REINVESTED AT THE PROJECT'S COST OF CAPITAL, WHILE THE IRR CALCULATION ASSUMES REINVESTMENT AT THE IRR RATE.

G. 3. WHICH METHOD IS THE BEST? WHY?

ANSWER: WHETHER NPV OR IRR GIVES BETTER RANKINGS DEPENDS ON WHICH HAS THE BETTER REINVESTMENT RATE ASSUMPTION. NORMALLY, THE NPV'S ASSUMPTION IS BETTER. THE REASON IS AS FOLLOWS: A PROJECT'S CASH INFLOWS ARE GENERALLY USED AS SUBSTITUTES FOR OUTSIDE CAPITAL, THAT IS, PROJECTS' CASH FLOWS REPLACE OUTSIDE CAPITAL AND, HENCE, SAVE THE FIRM THE COST OF OUTSIDE CAPITAL. THEREFORE, IN AN OPPORTUNITY COST SENSE, A PROJECT'S CASH FLOWS ARE REINVESTED AT THE COST OF CAPITAL. TO SEE THIS GRAPHICALLY, THINK OF THE FOLLOWING SITUATION: ASSUME THE FIRM'S COST OF CAPITAL IS A CONSTANT 10% WITHIN THE RELEVANT RANGE OF FINANCING CONSIDERED, AND IT HAS PROJECTS AVAILABLE AS SHOWN IN THE GRAPH BELOW:

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