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 Beforetax income 
950,500 

Aftertax income 
570,300 

Depreciation 
1,000,000 

Aftertax 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)+(.6x1000000(.6x1000000 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 
Aftertax income 
Depr 
Initial Inv and Salvage 
Aftertax 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) 
Year 
Sales 
Pretax Operating margin 
Aftertax 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 .
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.
.
The initial costs equal 150,000, for a NPV of $211,614. Hence, it is worthwhile to offer the service. (We use the aftertax expense of $36,000 instead of the pretax expense of $ 60,000)
6.
7. a. The PV of the aftertax cash inflows = . The initial investment is $50(100,000) = $5m. The PV of the $5m. sales price in 10 years = 5/1.15^{10} = 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 = .25^{2} = 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(d_{1})  3,000,000e^{(0.07)(25)}N(d_{2} ), where
d_{1} =[ ln(3,309,755.06/3,000,000) + (0.070.04 + .0625/2)25]/(0.0625x25)^{0.5} = 1.30
d_{2} = 0.053
N(d_{1}) =0.9038; N(d_{2}) = 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(d_{1})  200e^{(0.08)(5)}N(d_{2} ), where
d_{1} =[ ln(250/200) + (0.080.05 + .04/2)5]/(0.04x5)^{0.5} = 1.06
d_{2} = 0.61
N(d_{1}) = 0.85; N(d_{2}) = 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.
The option value equals 250e^{0.1(10)} N(d_{1})  500e^{(0.06)(10)}N(d_{2} ), where
d_{1} =[ ln(250/500) + (0.06 0.10 + .36/2)5]/(0.36x10)^{0.5} = 0.3725
d_{2} = 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.
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.