MANAGEMENT ACCOUNTING CONCEPTS AND TECHNIQUES

By Dennis Caplan, University at Albany (State University of New York)

CHAPTER 19: Capital Budgeting

Solutions

19-1:

A) A project requires an initial cash outlay of $800, and returns $1,000 at the end of year 3 (nothing at the end of years 1 or 2). What is the approximately Net Present Value of this project, using a cost of capital of 10%?

($1,000 x 0.7513) - $800 = negative $49

B) A project requires an initial cash outlay of $5,000, and returns $1,000 at the end of each year from Year 1 through Year 10. What is the project’s approximate Internal Rate of Return.

$5,000 ÷ $1,000 = 5

From Table 2, row 10, the factor 5.0188 is found in the column for 15%.

Hence, the IRR is approximately 15%.

C) Refer to the project in part (B). What is the Payback Period of the Project? Also, what is the Accounting Rate of Return on the average net investment, assuming that the $5,000 purchase price is for a machine that is depreciated using straight-line depreciation over 10 years, with zero salvage value?

Payback period = $5,000 ÷ $1,000 = 5 years

Accounting Rate of Return

= ($1,000 - $500 depreciation) ÷ $2,500 average book investment = 20%

19-2: A machine costs $4,000 (paid out at the beginning of year 1), and generates year-end net cash inflows of $2,000 per year for five years (this is the useful life of the machine). The machine has zero salvage value. The company uses straight-line depreciation and a 10% cost of capital.

Required:

A) What is the payback period for this project?

$4,000 ÷ $2,000 = two years

B) What is the net present value of this project?

Present value of future cash inflows: ($2,000 x 3.7908) = $7,582

Net present value: $7,582 – $4,000 = $3,582

C) What is the accounting rate of return for this project?

Depreciation expense = $4,000 ÷5 years = $800 per year

Average book investment = $4,000 ÷ 2 = $2,000

Accounting rate of return = ($2,000 – $800) ÷ $2,000 = 60%

19-3: A company purchases an asset for $40,000. The asset has a useful life of seven years. The salvage value is expected to be $10,000. For purposes of computing the accounting rate of return (i.e., the book rate of return), what is the average net investment in the asset, if the salvage value is used to reduce the depreciable basis of the asset?

($40,000 + $10,000) ÷ 2 = $25,000

19-4: What is the net present value of a project that requires an initial cash outlay of $1,000, and returns $7,000 at the end of seven years, using a discount rate of 12%?

($7,000 x 0.4523) - $1,000 = $2,166

19-5: An investment of $400 will yield a single, lump-sum payoff of $1,000 after 10 years. At a discount rate of 7%, what is the Net Present Value of this project?

($1,000 x 0.5083) - $400 = $108

19-6: A machine that costs $1,000 will save $200 in operating costs every year for the next seven years. What is the approximate Internal Rate of Return of this capital project?

$1,000 ¸ $200 = 5 Þ approximately 9% from table 2

19-7: A project requires an initial investment of $500, and will return $100 per year for 11 years. Using a discount rate of 9%, what is this project’s Net Present Value?

($100 x 6.8052) - $500 = $181

19-8: A machine costs $50,000 (paid out at the beginning of year 1), and generates year-end net cash inflows of $8,834 per year for 12 years (this is the useful life of the machine). The machine has zero salvage value. The company uses straight-line depreciation.

Required:

A) What is the internal rate of return?

$50,000 ÷ $8,834 = 5.66 → 14% from Table 2

B) What is the net present value, using a discount rate of 10%?

($8,834 x 6.8137) – $50,000 = $10,192

C) What is the accounting rate of return?

Depreciation expense = $50,000 ÷ 12 years = $4,167 per year

Income = $8,834 – $4,167 = $4,667

Average book investment = $50,000 ÷ 2 = $25,000

Accounting rate of return = $4,667 ÷ $25,000 = 18.7%

19-9: Using an 11% discount rate, what is the net present value of a project that requires cash outlays of $10,000 at the beginning of years one, three and five, and provides cash inflows of $20,000 at the end of years one, three and five? You may assume that the present value of a cashflow at the end of year X is equivalent to the cashflow of an equivalent amount at the beginning of year X + 1 (e.g., December 31, 2005 is the same as January 1, 2006).

Event Date

Cash Flow

Discount Factor

Present Value

Beginning of year 1

End of year 1

End of year 2

End of year 3

End of year 4

End of year 5

Total

-$10,000

$20,000

-$10,000

$20,000

-$10,000

$20,000

1.0000

0.9009

0.8116

0.7312

0.6587

0.5935

-$10,000

18,018

-8,116

14,624

-6,587

11,870

$19,809

19-10: The Seven Flags over the Land of Enchantment amusement park plans to build a new roller coaster that will be faster, higher, scarier and more thrilling than its existing roller coasters. Seven Flags uses a 12% discount rate to evaluate capital expenditures. The cost to prepare the site and construct the new coaster is $550,000, and this expenditure will be incurred evenly throughout 2007 and 2008 (which is equivalent to a single cash outlay of $550,000 incurred on December 31, 2007). The new coaster will be finished at the beginning of 2009. The operating costs for the new coaster will be $30,000 per year beginning January 1, 2009 (assume that the annual operating cost is incurred at the beginning of each year). The additional revenue due to additional ticket sales are projected to be $200,000 per year for eight years (the projected life of the coaster) from the time the coaster opens at the beginning of 2009 through 2016 (assume, for simplicity, that the annual revenue is received at the end of each year). At the end of 2016, the amusement park will pay $150,000 to remove the coaster.

Required: Calculate the net present value of the new roller coaster, as of January 1, 2007.

Construction cost: $550,000 x 0.8929 = ($491,095)

Operating costs:

NPV as of 1/1/08: $30,000 x 4.9676 = $149,028

NPV as of 1/1/07: $149,028 x 0.8929 = ($133,067)

Revenue:

NPV as of 1/1/09: $200,000 x 4.9676 = $993,520

NPV as of 1/1/07: $993,520 x 0.7972 = $ 792,034

Removal costs: $150,000 x 0.322 = ($ 48,300)

Total $ 119,572

19-11: Consider a capital project with a one-year life. The cash outlay for the equipment occurs at the beginning of year one, and a single cash in-flow occurs at the end of year one. The equipment has zero salvage value. Straight-line depreciation is used. Indicate which of the following statements are true in this specific setting.

Required:

A) If the payback period is less than one, the net present value will be greater than zero.

False

B) The internal rate of return is half of the accounting (book) rate of return.

True

C) The inverse of the payback period equals the internal rate of return plus one.

True

D) The payback period is the inverse of the internal rate of return.

False

E) If the internal rate of return is greater than the discount rate, the net present value will be greater than zero.

True

19-12: Consider the following two possible capital projects:

Project	Initial Cost (incurred at beginning of year 1)	Project Life	Salvage Value	Positive annual cash flow (received at the end of each year)
A	$200,000	16 years	$0	$28,000
B	$56,000	14 years	$0	$9,783

Using the above information, it can be shown that the Internal Rate of Return of Project B is higher than the Internal Rate of Return of project A.

Project A: $200,000 ÷ $28,000 = 7.14 → 11.5%

Project B: $ 56,000 ÷ $ 9,783 = 5.72 → 15.0% ¬

Required:

A) Is the NPV for Project A higher than, equal to, or lower than, the NPV for Project B, assuming a 10% discount rate?

Project A: ($28,000 x 7.8237) – $200,000 = $19,064 ¬

Project B: ($ 9,783 x 7.3667) – $ 56,000 = $16,068

B) Is the Payback Period for Project A better than, equal to, or worse than, the Payback Period for Project B?

Project A: $200,000 ÷ $28,000 = 7.14 years

Project B: $ 56,000 ÷ $ 9,783 = 5.72 years ¬

C) Is the Accounting Rate of Return for Project A higher than, equal to, or lower than, the Accounting Rate of Return for Project B, assuming straight-line depreciation?

Project A: ($28,000 - $12,500) ÷ $100,000 = 15.5%

Project B: ($ 9,783 - $ 4,000) ÷ $ 28,000 = 20.7% ¬

D) If the discount rate is 11% instead of 10%, which project has the higher NPV?

Project A: ($28,000 x 7.3792) – $200,000 = $ 6,618

Project B: ($ 9,783 x 6.9819) – $ 56,000 = $12,304 ¬

E) If the discount rate is 10%, and both projects have a salvage value of $66,000 (i.e., the equipment for Project B actually appreciates), which project would have the higher NPV?

Project A: ($28,000 x 7.8237) – $200,000 = $19,064 + ($66,000 x 0.2176) = $33,425

Project B: ($ 9,783 x 7.3667) – $ 56,000 = $16,068 + ($66,000 x 0.2633) = $33,446 ¬

19-13: A machine with a useful life of five years and a salvage value of $4,000 is purchased for $20,000. The benefit of the machine is that it reduces normal cash operating expenses by $5,000 per year during the first two years of the machine’s life, and by $4,000 for each of the following three years.

Required:

A) Calculate the accounting rate of return for the project, assuming that the full $20,000 purchase price is depreciated using the straight-line method, so that at the end of year five, the machine has a book value of zero, and the salvage value is treated as income in year five.

Average book investment = ($20,000 + $0) ÷ 2 = $10,000

Depreciation expense = $20,000 ÷ 5 = $4,000

Average annual increase in income

= ($5,000 + $5,000 + $4,000 + $4,000 + $4,000 + $4,000 salvage value) ÷ 5 = $5,200

Accounting rate of return = ($5,200 – $4,000) ÷ $10,000 = 12%

B) Calculate the accounting rate of return for the project, assuming that the net cost of the machine (purchase price less salvage value) is depreciated using the straight-line method.

Average book investment = ($20,000 + $4,000) ÷ 2 = $12,000

Depreciation expense = $16,000 ÷ 5 = $3,200

Average annual increase in income

= ($5,000 + $5,000 + $4,000 + $4,000 + $4,000) ÷ 5 = $4,400

Accounting rate of return = ($4,400 – $3,200) ÷ $12,000 = 10%

19-14: Plain Vanilla Industries purchases a machine for $120,000. The machine has a six year life and a salvage value of zero. The company depreciates the machine using the sum-of-the-years-digits method, which results in depreciation expense in each year as follows:

Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

$34,286

28,571

22,857

17,143

11,429

5,714

$120,000

The machine increases cash inflows to the firm by $30,000 in year one, $35,000 in year two, $40,000 in year three, $35,000 in year four, and $25,000 in year five.

Required: Calculate the accounting rate of return from the investment in the machine.

Cumulative increase in net income over all six years:

$165,000 cash inflows – $120,000 in depreciation expense = $45,000

Average annual increase in net income: $45,000 ÷ 6 years = $7,500 per year

Average book investment:

[$120,000 + $85,714 + $57,143 + $34,286 + $17,143 + $5,714 + $0] ÷ 7

= $320,000 ÷ 7

= $45,714

Accounting rate of return = $7,500 ÷ $45,714 = 16.4%

19-15: A machine can be purchased for $120,000 that increases cash flows by $20,000 each year for the next six years. In addition, the machine has a salvage value of $20,000, which is used to reduce the depreciable basis of the asset. Assume the purchase price is paid at the beginning of the first year, and that all cash inflows are received at the end of each year. The company uses sum-of-the-years-digits depreciation, which results in the following depreciation schedule:

Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

$28,571

23,810

19,048

14,286

9,524

4,761

Required: Compute the Accounting Rate of Return.

The net PP&E (historical cost less accumulated depreciation expense) is calculated as follows:

Year 0

Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

$120,000

$120,000 – $28,571 = $ 91,429

$91,429 – $23,810 = $ 67,619

$67,619 – $19,048 = $ 48,571

$48,571 – $14,286 = $ 34,285

$34,285 – $9,524 = $ 24,761

$24,761 – $4,761 = $ 20,000

$406,665

Average book investment: $406,665 ÷ 7 = $58,095

Cumulative income = ($20,000 x 6) – 100,000 = $20,000

Average annual income = $20,000 ÷ 6 = $3,333.33

Accounting Rate of Return = $3,333.33 ÷ $58,095 = 5.74%

19-16: A company is considering purchasing a machine that will reduce its operating costs. The purchase requires a down payment of $5,000, and three equal annual payments of $3,000, due at the beginning of the second, third and fourth years of the life of the machine. The machine will reduce operating expenses in an amount equivalent to $2,000 received at the end of each year for eight years. The machine has a useful life of eight years, at the end of which it has a salvage value of $1,500. The company records the purchase of the machine at $14,000 (i.e., interest expense is not imputed), and treats the salvage value as a reduction in the depreciable basis of the asset. The asset is depreciated using straight-line depreciation. The discount rate is 8%.

Required: Calculate the payback period, the net present value, and the average accounting rate of return.

Cash flows can be summarized as follows:

Year	Cash flow
0	$ –5,000
1	–1,000
2	–1,000
3	–1,000
4	2,000
5	2,000
6	2,000
7	2,000
8	3,500

Payback period:

The payback period is 7 years.

Net present value:

($1,500 x 0.5403) + ($2,000 x 5.7466) – ($3,000 x 2.5771) – $5,000

= $810.45 + $11,493.20 – $7,731.30 – $5,000

= – $427.65

Accounting rate of return:

Depreciation expense = ($14,000 – $1,500) ÷ 8 years = $1,562.50 per year

Average annual income = $2,000 – $1,562.50 = $437.50

Average book investment = ($14,000 + $1,500) ÷ 2 = $7,750

ARR = $437.50 ÷ $7,750 = 5.645%

PRESENT VALUE TABLES:

Table 1: Present value of $1 received (or paid) n years from now

10%

11%

12%

13%

14%

15%

20%

0.9434

0.8900

0.8396

0.7921

0.7473

0.7050

0.6651

0.6274

0.5919

0.5584

0.5268

0.4970

0.4688

0.4423

0.4173

0.3936

0.3714

0.3503

0.3305

0.3118

0.9346

0.8734

0.8163

0.7629

0.7130

0.6663

0.6227

0.5820

0.5439

0.5083

0.4751

0.4440

0.4150

0.3878

0.3624

0.3387

0.3166

0.2959

0.2765

0.2584

0.9259

0.8573

0.7938

0.7350

0.6806

0.6302

0.5835

0.5403

0.5002

0.4632

0.4289

0.3971

0.3677

0.3405

0.3152

0.2919

0.2703

0.2502

0.2317

0.2145

0.9174

0.8417

0.7722

0.7084

0.6499

0.5963

0.5470

0.5019

0.4604

0.4224

0.3875

0.3555

0.3262

0.2992

0.2745

0.2519

0.2311

0.2120

0.1945

0.1784

0.9091

0.8264

0.7513

0.6830

0.6209

0.5645

0.5132

0.4665

0.4241

0.3855

0.3505

0.3186

0.2897

0.2633

0.2394

0.2176

0.1978

0.1799

0.1635

0.1486

0.9009

0.8116

0.7312

0.6587

0.5935

0.5346

0.4817

0.4339

0.3909

0.3522

0.3173

0.2858

0.2575

0.2320

0.2090

0.1883

0.1696

0.1528

0.1377

0.1240

0.8929

0.7972

0.7118

0.6355

0.5674

0.5066

0.4523

0.4039

0.3606

0.3220

0.2875

0.2567

0.2292

0.2046

0.1827

0.1631

0.1456

0.1300

0.1161

0.1037

0.8850

0.7831

0.6931

0.6133

0.5428

0.4803

0.4251

0.3762

0.3329

0.2946

0.2607

0.2307

0.2042

0.1807

0.1599

0.1415

0.1252

0.1108

0.0981

0.0868

0.8772

0.7695

0.6750

0.5921

0.5194

0.4556

0.3996

0.3506

0.3075

0.2697

0.2366

0.2076

0.1821

0.1597

0.1401

0.1229

0.1078

0.0946

0.0829

0.0728

0.8696

0.7561

0.6575

0.5718

0.4972

0.4323

0.3759

0.3269

0.2843

0.2472

0.2149

0.1869

0.1625

0.1413

0.1229

0.1069

0.0929

0.0808

0.0703

0.0611

0.8333

0.6944

0.5787

0.4823

0.4019

0.3349

0.2791

0.2326

0.1938

0.1615

0.1346

0.1122

0.0935

0.0779

0.0649

0.0541

0.0451

0.0376

0.0313

0.0261

Table 2: Present value of an annuity of $1 received (or paid) each year for the next n years

10%

11%

12%

13%

14%

15%

20%

0.9434

1.8334

2.6730

3.4651

4.2124

4.9173

5.5824

6.2098

6.8017

7.3601

7.8869

8.3838

8.8527

9.2950

9.7122

10.1059

10.4773

10.8276

11.1581

11.4699

0.9346

1.8080

2.6243

3.3872

4.1002

4.7665

5.3893

5.9713

6.5152

7.0236

7.4987

7.9427

8.3577

8.7455

9.1079

9.4466

9.7632

10.0591

10.3356

10.5940

0.9259

1.7833

2.5771

3.3121

3.9927

4.6229

5.2064

5.7466

6.2469

6.7101

7.1390

7.5361

7.9038

8.2442

8.5595

8.8514

9.1216

9.3719

9.6036

9.8181

0.9174

1.7591

2.5313

3.2397

3.8897

4.4859

5.0330

5.5348

5.9952

6.4177

6.8052

7.1607

7.4869

7.7862

8.0607

8.3126

8.5436

8.7556

8.9501

9.1285

0.9091

1.7355

2.4869

3.1699

3.7908

4.3553

4.8684

5.3349

5.7590

6.1446

6.4951

6.8137

7.1034

7.3667

7.6061

7.8237

8.0216

8.2014

8.3649

8.5136

0.9009

1.7125

2.4437

3.1024

3.6959

4.2305

4.7122

5.1461

5.5370

5.8892

6.2065

6.4924

6.7499

6.9819

7.1909

7.3792

7.5488

7.7016

7.8393

7.9633

0.8929

1.6901

2.4018

3.0373

3.6048

4.1114

4.5638

4.9676

5.3282

5.6502

5.9377

6.1944

6.4235

6.6282

6.8109

6.9740

7.1196

7.2497

7.3658

7.4694

0.8850

1.6681

2.3612

2.9745

3.5172

3.9975

4.4226

4.7988

5.1317

5.4262

5.6869

5.9176

6.1218

6.3025

6.4624

6.6039

6.7291

6.8399

6.9380

7.0248

0.8772

1.6467

2.3216

2.9137

3.4331

3.8887

4.2883

4.6389

4.9464

5.2161

5.4527

5.6603

5.8424

6.0021

6.1422

6.2651

6.3729

6.4674

6.5504

6.6231

0.8696

1.6257

2.2832

2.8550

3.3522

3.7845

4.1604

4.4873

4.7716

5.0188

5.2337

5.4206

5.5831

5.7245

5.8474

5.9542

6.0472

6.1280

6.1982

6.2593

0.8333

1.5278

2.1065

2.5887

2.9906

3.3255

3.6046

3.8372

4.0310

4.1925

4.3271

4.4392

4.5327

4.6106

4.6755

4.7296

4.7746

4.8122

4.8435

4.8696

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July 2019

For a printer-friendly version, contact Dennis Caplan at dcaplan@albany.edu