1. The after-tax earnings is 120000(1-0.34) = 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 after-tax return on capital equals 79200/250000 = 19.80%

2. a.

The market value is not used, since it is irrelevant for the purpose of defining the book-value 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 after-tax 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(1-0.40) = $400,000 going down to zero in 5 years]. The yearly net income equals 50,000. Hence the before-tax return on equity = 50000/80000 = 66.67%.

4. If the debt-equity ratio is 100%, the debt-to-capital ratio is 50%. Hence, we need

Minimum return on capital = (0.5)(after-tax interest rate) + (0.5)(minimum return on equity). Solving, the implied minimum return on equity = 19%.

5. a.

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.

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.

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 = (15-3)/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 double-declining balance depreciation, the nominal value does not change. However, the depreciation is higher in earlier years, and the present value increases.

The present value is $3.418 m.

13. a., b.

c. Tax Benefits from Expensing Asset Immediately = $2.5 (0.4) = $1 million; hence the additional saving = 1-0.62 = $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)/1.12¹⁰ = 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:

The present value of tax benefits from depreciation less capital gains taxes from salvage = 0.77 - 0.5*0.2/1.1⁵ = 0.77 -0.06 = 0.71.

15.a. The Straight-line method provides the higher nominal tax savings.

b. The Double-declining method provides a higher present value of tax benefits.

I switched to straight line depreciation in the last two years.

Problem 16

a. The after-tax operating cash flow is computed as

b. Using the annuity formula, we have = 10.72 as the present value of the operating cash-flows. 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.

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 Nuk-Nuk and Gerber are computed as 1.3/(1+(1-0.4(0.5)) and 1.5/(1+(1-0.5(1.0)) respectively or 1.0 and 1.0 respectively. The discount rate therefore is .115 + 1.0(.055) = 17%.

b. The yearly after-tax operating cash flow equals:

(Revenues — Manufacturing Costs — Depreciation — Opportunity Cost of Garage)(1-tax 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 after-tax 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 after-tax cashflows from the project are:

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; after-tax, this works out to (1-0.4)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

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 pre-tax, 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

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

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(1-0.4) /1.1⁴ + 80000(1-0.4)/1.1⁵ = $62,589.

b. The opportunity cost of the packaging plant = (250,000)/1.1⁴ - (250,000)/1.1⁸ = $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 after-tax rental income, which is equal to (3000/.1)(1-(1.1)^-5)(1-0.4) = $6,823.42.

d. The annual cash flows equal (Revenues — Cost of Goods Sold — Depreciation)(1-0.4) + Depreciation = (400000 — 160000 — 100000)(1-0.4) + 100000 = 184000.

The PV of the after-tax 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.

The present value of the after-tax 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(1-1.08¹⁰) = $536,806.50. The NPV of the project = 22,814,277 - 10,000,000 - 536,806.50 = $12,277,470.

Problem 26.

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 after-tax operating profit for the old product currently is $50(1-0.4)/50m = $0.6m, while it is $36(1-0.4)/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 50/1.1⁵ - 50/1.1¹⁴ = $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 (1/1.1⁵ - 1/1.1¹⁴) = $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.

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.

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(1-t).

Problem 29.

The present value of these flows at 12% = $99.06m. The NPV = -20.94m.