Chapter 5: Measuring Return on Investments
1. The aftertax earnings is 120000(10.34) = 79200. The average book value of capital invested is $250,000, since the book value is depreciated from 500,000 to zero in 10 years. Hence, the aftertax return on capital equals 79200/250000 = 19.80%
2. a.
Year 
Beginning Value 
Ending value 
Average Book Value 
Aftertax earnings 
Aftertax ROC 
1 
1200 
800 
1000 
132 
0.132 
2 
800 
400 
600 
132 
0.22 
3 
400 
0 
200 
132 
0.66 
4 
0 
0 
0 
132 
n/a 
5 
0 
0 
0 
132 
n/a 
Average 
360 
132 
The market value is not used, since it is irrelevant for the purpose of defining the bookvalue of the investment. For the last two years, the denominator is zero, and hence the ROC is undefined. To get around this problem, we use the average book value and after tax earnings over the 5 years.
Return on Capital = 132/360 = 36.67%
b. The geometric average cannot be defined, since the aftertax ROC for the last two years is undefined: the book value for the denominator being zero.
c. Using the return on capital of 36.67% estimated from using the averages, we would accept the project since it is high enough to exceed a cost of capital of 25%.
3. If we compute the average return on equity over the entire period, we have an average equity investment of $80,000 [$1m.x(10.40) = $400,000 going down to zero in 5 years]. The yearly net income equals 50,000. Hence the beforetax return on equity = 50000/80000 = 66.67%.
4. If the debtequity ratio is 100%, the debttocapital ratio is 50%. Hence, we need
Minimum return on capital = (0.5)(aftertax interest rate) + (0.5)(minimum return on equity). Solving, the implied minimum return on equity = 19%.
5. a.
Year 
Cash Flow 
Cumulated cash flow 
1 
250000 
250000 
2 
500000 
750000 
3 
750000 
1500000 
4 
750000 
2250000 
5 
750000 
3000000 
6 
750000 
3750000 
7 
750000 
4500000 
In year 5, the cumulated cash flow equals the initial investment of $3m. Hence, the payback period = 5 years.
b. The net present value = present value of the inflows  $3b. = 5,724,015.7  3 = $2.72b.
6.
Year 
FCFF 
PV @ 10% 
PV @ 15% 
0 
(2,000,000.00) 
2,000,000.00) 
(2,000,000.00) 
1 
100,000.00 
90,909.09 
86,956.52 
2 
300,000.00 
247,933.88 
226,843.10 
3 
300,000.00 
225,394.44 
197,254.87 
4 
300,000.00 
204,904.04 
171,525.97 
5 
300,000.00 
186,276.40 
149,153.02 
6 
300,000.00 
169,342.18 
129,698.28 
7 
300,000.00 
153,947.44 
112,781.11 
8 
300,000.00 
139,952.21 
98,070.53 
9 
300,000.00 
127,229.29 
85,278.72 
10 
300,000.00 
115,662.99 
74,155.41 
NPV 
(338,448.05) 
(668,282.46) 
The project should not be accepted at either discount rate.
7. The present value of the annual free cash flow to equity can be computed using the annuity formula: . This would be the maximum initial investment.
8. The entire benefit of the NPV should accrue to the shareholders. Hence the share price should rise by $2 m./1 m. = $2 per share. However, to the extent that such projects have already been foreseen by the market and incorporated into the stock price, there will be no current impact.
9., 10. Assuming that the discount rates given only apply to the corresponding year, the present values of the flows would be 300,000/(1.1) = $272,727.27and 350,000/(1.1)(1.12) = $284,090.91. The NPV = $56818.18
11.
Year 
Project A Cash flows 
Project B Cash flows 
NPV(A) @ 5% 
NPV(B) @ 5% 
NPV(A) @ 7.5% 
NPV(B) @ 7.5% 
0 
500 
2000 
500 
2000 
500 
2000 
1 
50 
190 
47.61905 
180.9524 
44.29679 
168.3278 
2 
50 
190 
45.35147 
172.3356 
39.24411 
149.1276 
3 
50 
190 
43.19188 
164.1291 
34.76776 
132.1175 
4 
50 
190 
41.13512 
156.3135 
30.802 
117.0476 
5 
50 
190 
39.17631 
148.87 
27.2886 
103.6967 
6 
50 
190 
37.31077 
141.7809 
24.17594 
91.86858 
7 
50 
190 
35.53407 
135.0295 
21.41833 
81.38966 
8 
50 
190 
33.84197 
128.5995 
18.97527 
72.10601 
9 
50 
190 
32.23045 
122.4757 
16.81087 
63.8813 
10 
50 
190 
30.69566 
116.6435 
14.89335 
56.59473 
11 
50 
190 
29.23396 
111.0891 
13.19455 
50.13929 
12 
50 
190 
27.84187 
105.7991 
11.68952 
44.42019 
13 
50 
190 
26.51607 
100.7611 
10.35617 
39.35344 
14 
50 
190 
25.2534 
95.96291 
9.1749 
34.86462 
15 
50 
190 
24.05085 
91.39325 
8.128372 
30.88781 
16 
50 
190 
22.90558 
87.04119 
7.201215 
27.36462 
17 
50 
190 
21.81483 
82.89637 
6.379814 
24.24329 
18 
50 
190 
20.77603 
78.94892 
5.652106 
21.478 
19 
50 
190 
19.7867 
75.18945 
5.007402 
19.02813 
20 
100 
340 
37.68895 
128.1424 
8.872474 
30.16641 
IRR 
8% 
7% 

NPV 
141.955 
424.3534 
141.67 
641.897 
The IRR for project A is 8%
The IRR for project B is 7%.
According to the IRR rule, project A should be accepted.
The NPVs for the two projects at a cost of capital of 5% are 141.96 and 424.35 respectively. Hence, project B should be accepted.
The NPVs for the two projects at a cost of capital of 7.5% are —141.67 and —641.90 respectively. Hence, project A should be accepted.
Clearly, the IRR and the NPV rules don’t always reach the same conclusions. However, the NPV rule is more consistent with the objective of maximizing shareholder wealth.
12. a. Using straight line depreciation, the depreciation each year = (153)/10 = $1.2 m. At a tax rate of 40%, this results in a tax saving of $0.48m. a year, for a total nominal value of $4.8 m. The present value can be computed using the annuity formula:
b., c. Using doubledeclining balance depreciation, the nominal value does not change. However, the depreciation is higher in earlier years, and the present value increases.
Year 
Depr 
Nominal Tax savings 
PV 
Doubledeclining Depreciation 
Yearend book value 
Nominal Tax saving 
PV 
0 
15.000 

1 
1.200 
0.480 
0.429 
3.000 
12.000 
1.200 
1.071 
2 
1.200 
0.480 
0.383 
2.400 
9.600 
0.960 
0.765 
3 
1.200 
0.480 
0.342 
1.920 
7.680 
0.768 
0.547 
4 
1.200 
0.480 
0.305 
1.536 
6.144 
0.614 
0.390 
5 
1.200 
0.480 
0.272 
1.229 
4.915 
0.492 
0.279 
6 
1.200 
0.480 
0.243 
0.983 
3.932 
0.393 
0.199 
7 
1.200 
0.480 
0.217 
0.786 
3.146 
0.315 
0.142 
8 
1.200 
0.480 
0.194 
0.146 
3.000 
0.058 
0.024 
9 
1.200 
0.480 
0.173 
0.000 
3.000 
0.000 
0.000 
10 
1.200 
0.480 
0.155 
0.000 
3.000 
0.000 
0.000 
4.800 
2.712 
4.800 
3.418 
The present value is $3.418 m.
13. a., b.
Year 
ACRS Rate 
Depreciation 
Tax Benefit 
PV of Tax Benefit 
1 
20% 
0.40 
0.16 
0.15 
2 
32% 
0.64 
0.26 
0.21 
3 
19.20% 
0.38 
0.15 
0.12 
4 
11.50% 
0.23 
0.09 
0.06 
5 
11.50% 
0.23 
0.09 
0.06 
6 
5.80% 
0.12 
0.05 
0.03 
Present Value of Tax Benefits from Deprecn = 
$0.62m. 
c. Tax Benefits from Expensing Asset Immediately = $2.5 (0.4) = $1 million; hence the additional saving = 10.62 = $0.38m.
14. In problem 12, if salvage value is ignored, the PV of Tax Savings from Straight line Depreciation = $ 1.5 (PVA,12%,10 years) = $3.39.
The PV of the Capital Gains Taxes on Salvage = 3 (0.2)/1.12^{10 }= 0.19.
Hence the PV of the tax savings from ignoring salvage = 3.39  0.19 = $3.20. This is 0.488m. higher than the PV with salvage considered (3.2  2.712)
In problem 13, if salvage value is ignored, the PV of the tax benefit is:
Year 
ACRS Rate 
Depreciation 
Tax Benefit 
PV of Tax Benefit 
1 
20% 
0.50 
0.20 
0.18 
2 
32% 
0.80 
0.32 
0.26 
3 
19.20% 
0.48 
0.19 
0.14 
4 
11.50% 
0.29 
0.12 
0.08 
5 
11.50% 
0.29 
0.12 
0.07 
6 
5.80% 
0.15 
0.06 
0.03 
The present value of tax benefits from depreciation less capital gains taxes from salvage = 0.77  0.5*0.2/1.1^{5} = 0.77 0.06 = 0.71.
15.a. The Straightline method provides the higher nominal tax savings.
b. The Doubledeclining method provides a higher present value of tax benefits.
Year 
Depr. 
Tax rate 
Nominal Tax savings 
PV 
Doubledeclining Depreciation 
Nominal Tax saving 
PV 
0.000 

1.000 
2.000 
0.200 
0.400 
0.357 
4.000 
0.800 
0.71 
2.000 
2.000 
0.250 
0.500 
0.399 
2.400 
0.600 
0.48 
3.000 
2.000 
0.300 
0.600 
0.427 
1.440 
0.432 
0.31 
4.000 
2.000 
0.350 
0.700 
0.445 
1.08 
0.302 
0.24 
5.000 
2.000 
0.400 
0.800 
0.454 
1.08 
0.518 
0.25 
3.000 
2.082 
2.653 
1.99 
I switched to straight line depreciation in the last two years.
Problem 16
a. The aftertax operating cash flow is computed as
Revenues 
$ 5.00 
COGS (w/o depr.) 
$ 1.50 
Depreciation 
$ 2.00 
EBIT 
$ 1.50 
EBIT (1t) 
$ 0.90 
+ Depreciation 
$ 2.00 
ATCF 
$ 2.90 
b. Using the annuity formula, we have = 10.72 as the present value of the operating cashflows. Deducting the initial investment of $10m., we get an NPV of $0.72m.
c. The yearly increment to cashflow due to depreciation is the savings in taxes, which is 2(0.4) = 0.8m. The PV of this flow = $2.96m.
d.
1 
2 
3 
4 
5 

Revenues 
5.00 
5.00 
5.00 
5.00 
5.00 
COGS 
1.50 
1.50 
1.50 
1.50 
1.50 
Depreciation 
2.00 
2.00 
2.00 
2.00 
2.00 
EBIT 
1.50 
1.50 
1.50 
1.50 
1.50 
 Taxes 
 
 
 
2.40 
0.60 
EBIT ( 1t) 
1.50 
1.50 
1.50 
(0.90) 
0.90 
+ Deprec'n 
2.00 
2.00 
2.00 
2.00 
2.00 
ATCF 
3.50 
3.50 
3.50 
1.10 
2.90 
PV of ATCF 
3.15 
2.84 
2.56 
0.72 
1.72 
The sum of the PVs = $11.00. The NPV of the project = 11  10 = $1m.
17. a. To compute the appropriate discount rate, we need to figure out the beta. The unlevered beta for NukNuk and Gerber are computed as 1.3/(1+(10.4(0.5)) and 1.5/(1+(10.5(1.0)) respectively or 1.0 and 1.0 respectively. The discount rate therefore is .115 + 1.0(.055) = 17%.
b. The yearly aftertax operating cash flow equals:
(Revenues — Manufacturing Costs — Depreciation — Opportunity Cost of Garage)(1tax rate) + Depreciation = 11,600.
c. To compute the NPV, we also need to factor in the outflow of $7500 in inventory setup at time zero and the inflow of $6000 in year 10. The present value of this working capital cost = 7500  6000/(1.17)^{10} = 6251.78.
The present value of the aftertax operating cash flow equals
Hence the NPV = 54039.80  50000  6251.78 = $2211.98 < 0.
18. If the facility were sold, capital gains tax would have to be paid on the gain of 100,000  60,000 = $40,000 at 25%, i.e. a tax of $10,000. The cost of the smaller facility is $40,000. However, it would be possible to obtain a tax gain from the depreciation. This would amount to (40%)(40,000/10) = 1600 per year for 10 years. At 10%, the PV of this is $9831.30. On the other hand, depreciation from the old facility would be lost. This would amount to (40%)($60,000/10) = 2400 per year for 10 years. At 10%, this works out to $14,746.96. The net cashflow = 10,000 + 9831.30  14,746.96 + 100,000  40,000 = $45084.35
Assuming nothing else would be done with the facility if it were kept, so that there are no other hidden opportunity costs, the next opportunity cost of using the existing facility instead of selling it and buying a new facility is 45084.35.
However, in the absence of other information, the optimal course would seem to be to actually sell the facility and buy a smaller facility. On the other hand, the existing facility would allow for greater flexibility, thus arguing for keeping it. Then, we may consider the $45084.35 the cost of that additional flexibility, since by using the existing facility, we are forgoing an additional cashflow of $45084.35.
19. a., b. The annual aftertax cashflows from the project are:
Revenues 
500 x 500 = 
250000 
Cost of instructors 
24000 x 5 
120000 
Rent 
48000 

Depreciation 
50000/10 
5000 
Net Income 
77,000 

Aftertax Income 
77000(10.4) 
46200 
Depreciation 
+5000 

Aftertax cashflow 
51,200 
The present value of an annuity of 51,200 for 10 years at 15% is 256960.95. The NPV = $206,960.95.
The IRR is 102%; hence the investment is worthwhile using either decision rule.
20. If the warehouse is rented out, it would bring in $100,000 per year; aftertax, this works out to (10.4)100000 = $60,000.
The tax advantage from depreciation would be 0.4(500000/10) = $20,000 a year. The PV of this at 15% is $100,375.37. However, this would be available irrespective of what the premises would be used for. Hence, this is not relevant for the decision. Consequently, the opportunity cost would simply be the PV of an annuity of $60,000 for 10 years = $301,126.12.
Problem 21
The annual cashflows are
Revenues 
1m. bottles at $1 each 
$1,000,000 
Variable costs 
1m. bottles at 50 cents each 
$500,000 
Fixed costs 
$200,000 

Depreciation 
550,000/5 
$110,000 
Net Income 
$190,000 

After tax income 
190000(10.50) 
$95,000 
Depreciation 
$110,000 

Total aftertax cashflow 
$205,000 
Assumes licensing costs can be capitalized and depreciated.
The PV of this cashflow at 10% for 5 years is $777,111. The investment tax credit adds 500,000(0.10) = $50,000 to the current cashflow. Hence the NPV = $777,111 + 50,000  $550,000 = $277,111. This has to be compared to the present value of the salary foregone. If the $ 75,000 is pretax, and the tax rate on this income is also 50% (It might be lower).
Present Value of Salary foregone = $ 75,000 (1.5) (PV of Annuity, 10%, 5 years) = $142,154
Take the project. It has a net present value greater than $ 142,154.
Problem 22
The annual cashflows are
1 
2 
3 
4 
5 

Revenues 
600000.00 
679800.00 
770213.40 
872651.78 
988714.47 
Software specialists 
250000.00 
257500.00 
265225.00 
273181.75 
281377.20 
Rent 
50000.00 
51500.00 
53045.00 
54636.35 
56275.44 
Depreciation 
20000.00 
20000.00 
20000.00 
20000.00 
20000.00 
Marketing and selling costs 
100000.00 
103000.00 
106090.00 
109272.70 
112550.88 
Cost of materials 
120000.00 
135960.00 
154042.68 
174530.36 
197742.89 
Net Income 
60000.00 
111840.00 
171810.72 
241030.63 
320768.05 
After tax income 
36000.00 
67104.00 
103086.43 
144618.38 
192460.83 
+ Depreciation 
20000.00 
20000.00 
20000.00 
20000.00 
20000.00 
Change in WC 
7980.00 
9041.34 
10243.84 
11606.27 
98,871.45 
ATCF 
48020.00 
78062.66 
112842.59 
153012.11 
311,332.28 
Working Capital 
60000.00 
67980.00 
77021.34 
87265.18 
98871.45 
Working capital is fully salvaged in the last year.
There is an initial investment of 100,000 plus an initial outlay of $60,000 for working capital. Taking these into account, the NPV = $ 299,325
The project has a positive NPV and should be accepted.
Problem 23
Year 
Excess Capacity 
Encroachment 
Cash flow from racquets 
1 
22500.00 
0.00 
0 
2 
19750.00 
250.00 
9000 
3 
16725.00 
3275.00 
117900 
4 
13397.50 
6602.50 
237690 
5 
9737.25 
10262.75 
369459 
6 
5710.97 
14289.03 
514404.9 
7 
1282.07 
18717.93 
673845.39 
8 
0 
20000.00 
720000 
9 
0 
20000.00 
720000 
10 
0 
20000.00 
720000 
NPV 
$2,042,752.63 
The opportunity cost is $2,042,752.63.
Problem 24.
a. There is no opportunity cost to using the employees for the first three years, since they must be paid their salaries whether or not they are used on the project. However, their salaries for the last years is an opportunity cost. This equals 80,000(10.4) /1.1^{4} + 80000(10.4)/1.1^{5} = $62,589.
b. The opportunity cost of the packaging plant = (250,000)/1.1^{4}  (250,000)/1.1^{8} = $54,126.52.
c. The depreciation tax advantage can be reaped whether or not the van is used for the current project. The opportunity cost is simply the present value of the aftertax rental income, which is equal to (3000/.1)(1(1.1)^{5})(10.4) = $6,823.42.
d. The annual cash flows equal (Revenues — Cost of Goods Sold — Depreciation)(10.4) + Depreciation = (400000 — 160000 — 100000)(10.4) + 100000 = 184000.
The PV of the aftertax operating cash flow =
The NPV of the project = 697505  500000  54127  6823  62,589 = $73966.
Problem 25.
a. The initial investment is $10 m. + additional working capital at the beginning of 0.10(10,000,000) = $1m; hence total initial investment = $11m.
b.
Current level 
New level 
Increment 

Revenue 
$100m.(.10) 
10,000,000 
20,000,000 
$10,000,000 
Fixed Costs 
2,000,000 
2,000,000 
0 

Variable Costs 
4,000,000 
8,000,000 
4,000,000 

Advertising 
1,000,000 
1,000,000 

Depreciation 
1,000,000 
1,000,000 

Beforetax income 
4,000,000 

Aftertax income 
2,400,000 

Depreciation 
1,000,000 

Aftertax Operating Cashflow 
3,400,000 
The present value of the aftertax operating cash flow =
Additional working capital at the beginning equals 0.10(10,000,000) = $1m, which will be recouped at the end. The present value consequence of this is $1m(11.08^{10}) = $536,806.50. The NPV of the project = 22,814,277  10,000,000  536,806.50 = $12,277,470.
Problem 26.
Year 
Current use; old prod 
current use; new prod 
total need 
restriction of old product 
cost of restricting old product 
1 
50.00% 
30.00% 
80.00% 
0.00% 
0.00 
2 
52.50% 
33.00% 
85.50% 
0.00% 
0.00 
3 
55.13% 
36.30% 
91.43% 
0.00% 
0.00 
4 
57.88% 
39.93% 
97.81% 
0.00% 
0.00 
5 
60.78% 
43.92% 
104.70% 
4.70% 
2818987.50 
6 
63.81% 
48.32% 
112.13% 
12.13% 
7277626.88 
7 
67.00% 
53.15% 
120.15% 
20.15% 
12090967.22 
8 
70.36% 
58.46% 
128.82% 
28.82% 
17289920.48 
9 
73.87% 
64.31% 
138.18% 
38.18% 
22908261.89 
10 
77.57% 
70.74% 
148.30% 
48.30% 
28982904.92 
NPV 
$41,018,357.39 
a. The projects will run out of capacity in year 5.
b. Assuming that the old product can be continued to be produced after the end of the new product’s life, we simply compare the relative flows of the old product versus the new product at the margin. The marginal aftertax operating profit for the old product currently is $50(10.4)/50m = $0.6m, while it is $36(10.4)/30m. = $21.6/30 = 0.72m. for the new product per % unit of capacity. This does not change over time. Hence, it would be appropriate to go with the new product in year 5 to the extent of 44% of capacity, and restrict the old product to 56% of capacity. Similarly, in future years, we would restrict the old product further to allow the new product to expand. The extent of the restriction is shown in column 5 in the table above. The NPV of restricting the capacity usage of the old product in years 5 through 10 is equal to $41.018m.
c. The old product itself is growing at the rate of 5% per year. Hence, the existing facility would be insufficient even for the old product in n years, where n is the highest integer that satisfies 50(1.05)^{n} £ 100, or n = 14. Hence a new facility would have to be built even without the new product line, in 14 years. If we assume that the cost of a new facility is still $50m. (as given), then the opportunity cost assigned to the new facility if we decided to build it in year 5 is simply the difference in present values of building it in year 5 instead of year 14, which can be computed as 50/1.1^{5}  50/1.1^{14} = $17.88m. However, in this case, we also postpone the depreciation tax advantages from year 5 to year 14. The difference in present values of this advantage are $2m.(0.40)[1(1.1)^{25}]/0.1 x (1/1.1^{5}  1/1.1^{14}) = $2.60m. The net opportunity cost, therefore, is 17.88  2.60 = 15.28m.
Problem 27.
a. Cash flow at time zero is the sum of the installation cost of $10m. and the change in working capital. Existing working capital = $5m. (0.50) = $2.5m. New working capital requirements are $8m. (0.25) = $2m. Hence there will be a reduction of $0.5m., and the net cash flow at time zero = $9.5m.
b.
Annual flow 
Existing system 
New system 
Operating cost aftertax 
0.9 
0.3 
Reduction in taxes due to Depreciation (Annual Depr. 0f $1m. x Tax rate) 
0.4 

Profits after tax [Profit margin x (1Tax rate)] 
1.5 
2.4 
0.6 
2.5 
c. The NPV of this project = 9.5 = 3.249m.
{Since this project requires an investment in working capital at the beginning, a reasonable argument can be made that that cash inflow should be reversed in year 5 — working capital increased by $ 0.5 million . If this is done, the net present value of this project will be only $ 3.017 million.]
Problem 28.
Country 
Cash flow before taxes 
Marginal tax rate 
After tax flow 
A 
20 
0.6 
8 
B 
15 
0.5 
7.5 
C 
10 
0.4 
6 
D 
5 
0.4 
3 
E 
3 
0.35 
1.95 
Total 
53 
26.45 
The marginal tax rate is the weighted average of the tax rates in column 3 weighted by the relative weights of the cash flows in column 2, which works out to 0.500943. Alternatively, solve for t in 26.45 = 53(1t).
Problem 29.
Year 
Cash flow before taxes 
Marginal tax rate 
After tax flow 
1 
10 
0.25 
7.5 
2 
20 
0.3 
14 
3 
50 
0.3 
35 
4 
50 
0.3 
35 
5 
100 
0.4 
60 
The present value of these flows at 12% = $99.06m. The NPV = 20.94m.