Chapter 6: Estimating Side Costs and Benefits

1. At a market share of 14.9175%, we find that we are indifferent between investing in the new distribution system or not.

Current level

New level

Increment

Revenue

10,000,000

14,917,500

$4,917,500

Fixed Costs

2,000,000

2,000,000

$0

Variable Costs

4,000,000

5,967,000

$1,967,000

Advertising

1,000,000

$1,000,000

Depreciation

1,000,000

$1,000,000

Incremental Before-tax income

950,500

After-tax income

570,300

Depreciation

1,000,000

After-tax Operating Cashflow

1,570,300

Present value of working capital flows (increase of $1m. today and decrease of $1m. in 10 years)

536806.51

Initial Distribution system cost

10,000,000

NPV

34

This can be solved algebraically through the following equation:

(-10000000 &emdash;1,000,000)+(.6x-1000000-(.6x-1000000 1000000)*.4)(PVA,10yrs,8%)+1,000,000/1.08^10=0

Solving for X, we get X = 4,917,000

If we make the initial working capital investment a function of the revenues, we get a lower breakeven point of 4,802,025

2. The existing machine has an annual depreciation tax advantage = 500000(0.40)/5 = 40,000. The present value of this annuity equals

The new machine has an annual depreciation tax advantage = 2000000(0.40)/10 = 80,000. The present value of this annuity equals . However, it will be necessary to spend an additional 1.7m. to acquire the new machine.

Net Cost of the New Machine = -1,700,000 + 491,565 &emdash; 151,531 = $1,360,066

. Solving, for the annual savings that we would need each year for the next 10 years,

Annual Savings = $ 1,360,066 (Annuity given PV, 10 years, 10%) = $221,344

3.

Year

Revenues

Operating Expenses

Depr

Taxable Income

After-tax income

Depr

Initial Inv and Salvage

After-tax cashflow

0

-50000

-50000

1

15000

7500

8000

-500

-300

8000

7700

2

15750

7875

8000

-125

-75

8000

7925

3

16537.5

8268.75

8000

268.75

161.25

8000

8161.25

4

17364.4

8682.19

8000

682.188

409.313

8000

8409.3125

5

18232.6

9116.3

8000

1116.3

669.778

8000

10000

18669.7781

NPV

($15,060.22)

  1. The net present value without the additional sales is negative.

Year

Sales

Pre-tax Operating margin

After-tax operating margin

0

1

20000

8000

4800

2

22000

8800

5280

3

24200

9680

5808

4

26620

10648

6388.8

5

29282

11712.8

7027.68

NPV (@12%)

$20,677

The present value of the cashflows accruing from the additional book sales equals $20,677 .

  1. The net effect is equal to $20,677 - $15,060 = $ 5,617. Hence, the coffee shop should be opened.

4. The present value of the cashflows from the gardening shop is -50000 + PV(annuity of 10000 for 10 yrs at 14%) = -50000 + - 50000 = 2,161.16. However the present value of the lost sales due to the parking conflict equals . Since this outweighs the present value of the flows from the gardening shop, it would not be optimal to open the gardening shop.

  1. The annual after-tax operating flows from the service is 5,000,000(.20)(0.1)-36,000 = 64,000. The present value of these flows is

.

The initial costs equal 150,000, for a NPV of $211,614. Hence, it is worthwhile to offer the service. (We use the after-tax expense of $36,000 instead of the pre-tax expense of $ 60,000)

6.

7. a. The PV of the after-tax cash inflows = . The initial investment is $50(100,000) = $5m. The PV of the $5m. sales price in 10 years = 5/1.1510 = 1,235,923.50. The NPV = -5,000,000 + 1,235,923.50 + 2,509,384.30 = - $1,254,692.20 < 0. Hence from a standard capital budgeting perspective, the project would not be accepted.

b. The standard deviation of prices per square foot can be estimated, using the provided data as:

Year

Price

% change

Squared Deviation

-6

20

-5

30

0.5

0.08019603

-4

55

0.83333

0.38009983

-3

70

0.27273

0.00312663

-2

55

-0.21429

0.18584435

-1

50

-0.09091

0.09469163

0

50

0

0.047007

Variance

0.1582

The option is to buy at today’s price which is $ 50 per square foot. Thus,

S = $ 50

K = $ 50

Riskless rate = 6%

T = 5

Variance = 0.1582

Value of the call option per square foot = $ 22.20

Total Value of Call option =100,000 * $22.20 = $2,220,000

Problem 8

In the absence of better information, we use the 25 year bond rate of 7% as the discount rate. This would be acceptable to the extent that the risk in the mine is diversifiable; and in fact, there is some evidence that commodity futures betas are close to zero. We also assume that the tax rate is zero.

The traditional method of computing the value of the mine using a discount rate of 7% yields a Net Present Value of $309,755.06, as shown below.

Year

Revenue

Cost

Net profit

PV

0

-3000000

-3000000

1

340000

160000

180000

168224.2991

2

353600

164800

188800

164905.2319

3

367744

169744

198000

161626.9796

4

382454

174836.32

207617.44

158390.3509

5

397752

180081.41

217670.501

155196.0588

6

413662

185483.852

228178.135

152044.7259

7

430208

191048.367

239160.099

148936.8898

8

447417

196779.818

250636.986

145873.0081

9

465313

202683.213

262630.264

142853.4625

10

483926

208763.709

275162.307

139878.5639

11

503283

215026.621

288256.436

136948.5563

12

523414

221477.419

301936.96

134063.6211

13

544351

228121.742

316229.212

131223.8805

14

566125

234965.394

331159.598

128429.4018

15

588770

242014.356

346755.636

125680.2001

16

612321

249274.787

363046.005

122976.2426

17

636814

256753.03

380060.593

120317.4507

18

662286

264455.621

397830.547

117703.7038

19

688778

272389.29

416388.325

115134.8417

20

716329

280560.968

435767.751

112610.6678

21

744982

288977.798

456004.071

110130.9508

22

774781

297647.131

477134.012

107695.428

23

805772

306576.545

499195.844

105303.8074

24

838003

315773.842

522229.443

102955.7695

25

871523

325247.057

546276.359

100650.9699

The NPV = $309,755.06

b. The option value of the mine can be computed using the following inputs:

interest rate = 7%, variance = .252 = 0.0625, the dividend rate = 1/25 = 4%, exercise price = $3m., the value of the underlying asset = 3,309,755.06, option maturity = 25 years.

Using these inputs, the option value can be computed as:

3,309,755.06e-0.04(25) N(d1) - 3,000,000e(-0.07)(25)N(d2 ), where

d1 =[ ln(3,309,755.06/3,000,000) + (0.07-0.04 + .0625/2)25]/(0.0625x25)0.5 = 1.30

d2 = 0.053

N(d1) =0.9038; N(d2) = 0.5214

The option value equals $828,674.

c. The two values are different because in the traditional method, we have not taken into account the ability to delay the project.

9. a. The value of the project based on traditional NPV = $250m. - $200m. = $50m.

b. There is an additional value based on the option to delay the project for up to 5 years. The inputs to this option valuation are: the value of the underlying asset = $250m.; the exercise price = $200m.; the maturity of the option = 5 years; the variance = .04; the yearly payment can be modeled as a dividend payment, which is equal to 12.5/250 = 5%; the riskfree rate = 8%.

Using these inputs, the option value can be computed as:

250e-0.05(5) N(d1) - 200e(-0.08)(5)N(d2 ), where

d1 =[ ln(250/200) + (0.08-0.05 + .04/2)5]/(0.04x5)0.5 = 1.06

d2 = 0.61

N(d1) = 0.85; N(d2) = 0.73

The option value equals 68.68.

c. The two values are different because in the traditional method, we have not taken into account the ability to delay the project. The value of this depends mainly on the variance of the cashflows.

  1. The present value of the asset = 250m; the exercise price = 500m.; the life of the option = 10 years; the dividend rate = 1/10 = 10%; the variance = 0.36; the riskfree rate = 6%

The option value equals 250e-0.1(10) N(d1) - 500e(-0.06)(10)N(d2 ), where

d1 =[ ln(250/500) + (0.06 -0.10 + .36/2)5]/(0.36x10)0.5 = 0.3725

d2 = -1.53

The option value equals $39.35 million

This value will be reduced by the present value of $10 million in research that the firm has to invest each year to keep its patent alive.

If the variance increases, the value of this option will increase. Consequently, it can be argued that patents in technologically volatile areas will have much more value than patents in stable businesses.

  1. a. False.

b. Generally true; there must be some comparative advantage, such that the project will not be taken up by competitors if the company fails to act on it immediately.

c. Not necessarily true. The expected growth rate may be set high enough to allow for the effect of these options on future earnings.

  1. False

  2. True.