Chapter 5: Measuring Return on Investments

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.

 Year Beginning Value Ending value Average Book Value After-tax earnings After-tax 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 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.

 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 = (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.

 Year Depr Nominal Tax savings PV Double-declining Depreciation Year-end 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 = 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:

 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*0.2/1.15 = 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.

 Year Depr. Tax rate Nominal Tax savings PV Double-declining 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 after-tax operating cash flow is computed as

 Revenues \$ 5.00 COGS (w/o depr.) \$ 1.50 Depreciation \$ 2.00 EBIT \$ 1.50 EBIT (1-t) \$ 0.90 + Depreciation \$ 2.00 ATCF \$ 2.90

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.

 1 2 3 4 5 Revenues 5.00 5.00 5.00 5.00 5 COGS 1.50 1.50 1.50 1.50 1.5 Depreciation 2.00 2.00 2.00 2.00 2 EBIT 1.50 1.50 1.50 1.50 1.5 - Taxes - - - 2.40 0.6 EBIT ( 1-t) 1.50 1.50 1.50 (0.90) 0.9 + Deprec'n 2.00 2.00 2.00 2.00 2 ATCF 3.50 3.50 3.50 1.10 2.9 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 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:

 Revenues 500 x 500 = 250000 Cost of instructors 24000 x 5 -120000 Rent -48000 Depreciation 50000/10 -5000 Net Income 77,000 After-tax Income 77000(1-0.4) 46200 Depreciation +5000 After-tax 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; 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

 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(1-0.50) \$95,000 Depreciation \$110,000 Total after-tax 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 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

 1 2 3 4 5 Revenues 600000 679800 770213 872652 988714.47 Software specialists 250000 257500 265225 273182 281377.20 Rent 50000 51500 53045 54636.3 56275.44 Depreciation 20000 20000 20000 20000 20000.00 Marketing and selling costs 100000 103000 106090 109273 112550.88 Cost of materials 120000 135960 154043 174530 197742.89 Net Income 60000 111840 171811 241031 320768.05 After tax income 36000 67104 103086 144618 192460.83 + Depreciation 20000 20000 20000 20000 20000.00 Change in WC -7980 -9041.34 -10243.8 -11606.3 98,871.45 ATCF 48020 78062.7 112843 153012 311,332.28 Working Capital 60000 67980 77021.3 87265.2 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(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.

 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 Before-tax income 4,000,000 After-tax income 2,400,000 Depreciation 1,000,000 After-tax Operating Cashflow 3,400,000

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.

 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 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.

 Annual flow Existing system New system Operating cost after-tax -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 (1-Tax 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(1-t).

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.