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.

• The exercise price is the initial investment, which is \$300 m.
• The Value of the Underlying Asset, (S in the model) is the present value of the cashflows, which is
• The time to expiration is 10 years.
• The standard deviation is 20%.
• If we assume that the value of the project decreases over the 10 years, the equivalent dividend yield is 1/20 = 5%.
• The riskfree rate is 12%, since that corresponds to the life of the option.

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.