CHAPTER 6

Making Investment Decisions with the Net Present Value Rule

Answers to Practice Questions

1. See the table below. We begin with the cash flows given in the text, Table 6.6, line 8, and utilize the following relationship from Chapter 3:

Real cash flow = nominal cash flow/(1 + inflation rate)^t

Here, the nominal rate is 20 percent, the expected inflation rate is 10 percent, and the real rate is given by the following:

(1 + rnominal)	= (1 + rreal) ´ (1 + inflation rate)
1.20	= (1 + rreal) ´ (1.10)
rreal	= 0.0909 = 9.09%

As can be seen in the table, the NPV is unchanged (to within a rounding error).

		Year 0	Year 1	Year 2	Year 3	Year 4	Year 5	Year 6	Year 7
	Net Cash Flows/Nominal	-12,600	-1,484	2,947	6,323	10,534	9,985	5,757	3,269
	Net Cash Flows/Real	-12,600	-1,349	2,436	4,751	7,195	6,200	3,250	1,678
NPV of Real Cash Flows (at 9.09%) = $3,804

2. No, this is not the correct procedure. The opportunity cost of the land is its value in its best use, so Mr. North should consider the $45,000 value of the land as an outlay in his NPV analysis of the funeral home.

3. Unfortunately, there is no simple adjustment to the discount rate that will resolve the issue of taxes. Mathematically:

and

4. Even when capital budgeting calculations are done in real terms, an inflation forecast is still required because:

a. Some real flows depend on the inflation rate, e.g., real taxes and real proceeds from collection of receivables; and,

b. Real discount rates are often estimated by starting with nominal rates and “taking out” inflation, using the relationship:

(1 + rnominal)

= (1 + rreal) ´ (1 + inflation rate)

5. Investment in working capital arises as a forecasting issue only because accrual accounting recognizes sales when made, not when cash is received (and costs when incurred, not when cash payment is made). If cash flow forecasts recognize the exact timing of the cash flows, then there is no need to also include investment in working capital.

6. If the $50,000 is expensed at the end of year 1, the value of the tax shield is:

If the $50,000 expenditure is capitalized and then depreciated using a five-year MACRS depreciation schedule, the value of the tax shield is:

If the cost can be expensed, then the tax shield is larger, so that the after-tax cost is smaller.

7. a.

NPV_B = -Investment + PV(after-tax cash flow) + PV(depreciation tax shield)

NPV_B = -$4,127

Another, perhaps more intuitive, way to do the Company B analysis is to first calculate the cash flows at each point in time, and then compute the present value of these cash flows:

		t = 0	t = 1	t = 2	t = 3	t = 4	t = 5	t = 6
	Investment	100,000
	Cash In		26,000	26,000	26,000	26,000	26,000
	Depreciation		20,000	32,000	19,200	11,520	11,520	5,760
	Taxable Income		6,000	-6,000	6,800	14,480	14,480	-5,760
	Tax		2,100	-2,100	2,380	5,068	5,068	-2,016
	Cash Flow -100,000		23,900	28,100	23,620	20,932	20,932	2,016
NPV (at 8%) = -$4,127

b. IRR_A = 9.43%

IRR_B = 6.39%

Effective tax rate =

8. Assume the following:

a. The firm will manufacture widgets for at least 10 years.

b. There will be no inflation or technological change.

c. The 15 percent cost of capital is appropriate for all cash flows and is a real, after-tax rate of return.

d. All operating cash flows occur at the end of the year.

Note: Since purchasing the lids can be considered a one-year ‘project,’ the two projects have a common chain life of 10 years.

Compute NPV for each project as follows:

NPV(purchase) =

NPV(make) =

Thus, the widget manufacturer should make the lids.

9. a. Capital Expenditure

1. If the spare warehouse space will be used now or in the future, then the project should be credited with these benefits.

2. Charge opportunity cost of the land and building.

3. The salvage value at the end of the project should be included.

Research and Development

1. Research and development is a sunk cost.

Working Capital

1. Will additional inventories be required as volume increases?

2. Recovery of inventories at the end of the project should be included.

3. Is additional working capital required due to changes in receivables, payables, etc.?

Revenues

1. Revenue forecasts assume prices (and quantities) will be unaffected by competition, a common and critical mistake.

Operating Costs

1. Are percentage labor costs unaffected by increase in volume in the early years?

2. Wages generally increase faster than inflation. Does Reliable expect continuing productivity gains to offset this?

Overhead

1. Is “overhead” truly incremental?

Depreciation

1. Depreciation is not a cash flow, but the ACRS deprecation does affect tax payments.

2. ACRS depreciation is fixed in nominal terms. The real value of the depreciation tax shield is reduced by inflation.

Interest

1. It is bad practice to deduct interest charges (or other payments to security holders). Value the project as if it is all equity-financed.

Taxes

1. See comments on ACRS depreciation and interest.

2. If Reliable has profits on its remaining business, the tax loss should not be carried forward.

Net Cash Flow

1. See comments on ACRS depreciation and interest.

2. Discount rate should reflect project characteristics; in general, it is not equivalent to the company’s borrowing rate.

b. 1. Potential use of warehouse.

2 Opportunity cost of building.

3. Other working capital items.

4. More realistic forecasts of revenues and costs.

5. Company’s ability to use tax shields.

6. Opportunity cost of capital.

c. The table on the next page shows a sample NPV analysis for the project. The analysis is based on the following assumptions:

1. Inflation: 10 percent per year.

2. Capital Expenditure: $8 million for machinery; $5 million for market value of factory; $2.4 million for warehouse extension (we assume that it is eventually needed or that electric motor project and surplus capacity cannot be used in the interim). We assume salvage value of $3 million in real terms less tax at 35 percent.

3. Working Capital: We assume inventory in year t is 9.1 percent of expected revenues in year (t + 1). We also assume that receivables less payables, in year t, is equal to 5 percent of revenues in year t.

4. Depreciation Tax Shield: Based on 35 percent tax rate and 5-year ACRS class. This is a simplifying and probably inaccurate assumption; i.e., not all the investment would fall in the 5-year class. Also, the factory is currently owned by the company and may already be partially depreciated. We assume the company can use tax shields as they arise.

5. Revenues: Sales of 2,000 motors in 2000, 4,000 motors in 2001, and 10,000 motors thereafter. The unit price is assumed to decline from $4,000 (real) to $2,850 when competition enters in 2002. The latter is the figure at which new entrants’ investment in the project would have NPV = 0.

6. Operating Costs: We assume direct labor costs decline progressively from $2,500 per unit in 2000, to $2,250 in 2001 and to $2,000 in real terms in 2002 and after.

7. Other Costs: We assume true incremental costs are 10 percent of revenue.

8. Tax: 35 percent of revenue less costs.

9. Opportunity Cost of Capital: Assumed 20 percent.

Practice Question 9
	1999	2000	2001	2002	2003	2004
Capital Expenditure	(15,400)
Changes in Working Capital
Inventories	(801)	(961)	(1,690)	(345)	(380)	(418)
Receivables – Payables		(440)	(528)	(929)	(190)	(209)
Depreciation Tax Shield		1,078	1,725	1,035	621	621
Revenues		8,800	19,360	37,934	41,727	45,900
Operating Costs		(5,500)	(10,890)	(26,620)	(29,282)	(32,210)
Other costs		(880)	(1,936)	(3,793)	(4,173)	(4,590)
Tax		(847)	(2,287)	(2,632)	(2,895)	(3,185)
Net Cash Flow	(16,201)	1,250	3,754	4,650	5,428	5,909

		2005	2006	2007	2008	2009	2010
Capital Expenditure						5,058
Changes in Working Capital
Inventories		(459)	(505)	(556)	(612)	6,727
Receivables – Payables		(229)	(252)	(278)	(306)	(336)	3,696
Depreciation Tax Shield		310
Revenues		50,489	55,538	61,092	67,202	73,922
Operating Costs		(35,431)	(38,974)	(42,872)	(47,159)	(51,875)
Other costs		(5,049)	(5,554)	(6,109)	(6,720)	(7,392)
Tax		(3,503)	(3,854)	(4,239)	(4,663)	(5,129)
Net Cash Flow		6,128	6,399	7,038	7,742	20,975	3,696
	NPV (at 20%) = $5,991

10. The table below shows the real cash flows. The NPV is computed using the real rate, which is computed as follows:

(1 + rnominal)	= (1 + rreal) ´ (1 + inflation rate)
1.09	= (1 + rreal) ´ (1.03)
rreal	= 0.0583 = 5.83%

	t = 0	t = 1	t = 2	t = 3	t = 4	t = 5	t = 6	t = 7	t = 8
Investment	-35,000.0								15,000.0
Savings		7,410.0	7,410.0	7,410.0	7,410.0	7,410.0	7,410.0	7,410.0	7,410.0
Insurance		-1,200.0	-1,200.0	-1,200.0	-1,200.0	-1,200.0	-1,200.0	-1,200.0	-1,200.0
Fuel		-526.5	-526.5	-526.5	-526.5	-526.5	-526.5	-526.5	-526.5
Net Cash Flow	-35,000.0	5,683.5	5,683.5	5,683.5	5,683.5	5,683.5	5,683.5	5,683.5	20,683.5
NPV (at 5.83%) =	$10,064.9

11.

	t = 0	t = 1	t = 2	t = 3	t = 4	t = 5	t = 6	t = 7	t = 8
Sales		4,200.0	4,410.0	4,630.5	4,862.0	5,105.1	5,360.4	5,628.4	5,909.8
Manufacturing Costs		3,780.0	3,969.0	4,167.5	4,375.8	4,594.6	4,824.3	5,065.6	5,318.8
Depreciation		120.0	120.0	120.0	120.0	120.0	120.0	120.0	120.0
Rent		100.0	104.0	108.2	112.5	117.0	121.7	126.5	131.6
Earnings Before Taxes		200.0	217.0	234.9	253.7	273.5	294.4	316.3	339.4
Taxes		70.0	76.0	82.2	88.8	95.7	103.0	110.7	118.8
Cash Flow
Operations		180.0	240.1	250.6	261.8	273.5	285.84	298.8	1,247.4
Working Capital	350.0	420.0	441.0	463.1	486.2	510.5	536.0	562.8	0.0
Increase in W.C.	350.0	70.0	21.0	22.1	23.2	24.3	25.5	26.8	-562.8
Rent (after tax)		65.0	67.6	70.3	73.1	76.0	79.1	82.2	85.5
Initial Investment	1,200.0
Sale of Plant									400.0
Tax on Sale									56.0
Net Cash Flow -1,550.0		180.0	240.1	250.6	261.8	273.5	285.8	298.8	1,247.4	1247.4
NPV(at 12%) =	$85.8

12. Note: There are several different calculations of pre-tax profit and taxes given in Section 6.2, based on different assumptions; the solution below is based on Table 6.6 in the text.

See the table on the next page. With full usage of the tax losses, the NPV of the tax payments is $4,779. With tax losses carried forward, the NPV of the tax payments is $5,741. Thus, with tax losses carried forward, the project’s NPV decreases by $962, so that the value to the company of using the deductions immediately is $962.

Tax Cash Flows
	t = 0	t = 1	t = 2	t = 3	t = 4	t = 5	t = 6	t = 7
Pretax Profit	-4,000	-4,514	748	9,807	16,940	11,579	5,539	1,949
Full usage of tax losses
Immediately (Table 6.6)	-1,400	-1,580	262	3,432	5,929	4,053	1,939	682
NPV at 20%	$4,779
Tax loss carry-forward	0	0	0	714	5,929	4,053	1,939	682
NPV (at 20%) =	$5,741

13. (Note: Row numbers in the table below refer to the rows in Table 6.8.)

		t = 0	t = 1	t = 2	t = 3	t = 4	t = 5	t = 6	t = 7	t = 8
1. Capital investment		83.5								-12.0
4. Working capital		2.3	4.4	7.6	6.9	5.3	3.2	2.5	0.0	0.0
Change in W.C.			2.1	3.2	-0.7	-1.6	-2.1	-0.7	-2.5	0.0
9. Depreciation			11.9	11.9	11.9	11.9	11.9	11.9	11.9	11.9
12. Profit after tax			-5.8	3.9	25.0	21.8	14.3	4.7	1.5	7.2
Cash Flow		-85.8	4.0	12.6	37.6	35.3	28.3	17.3	15.9	7.2
NPV (at 11.0%) =	$15.60

14. In order to solve this problem, we calculate the equivalent annual cost for each of the two alternatives. (All cash flows are in thousands.)

Alternative 1 – Sell the new machine: If we sell the new machine, we receive the cash flow from the sale, pay taxes on the gain, and pay the costs associated with keeping the old machine. The present value of this alternative is:

The equivalent annual cost for the five-year period is computed as follows:

PV₁ = EAC₁ ´ [annuity factor, 5 time periods, 12%]

-93.80 = EAC₁ ´ [3.605]

EAC₁ = -26.02, or an equivalent annual cost of $26,020

Alternative 2 – Sell the old machine: If we sell the old machine, we receive the cash flow from the sale, pay taxes on the gain, and pay the costs associated with keeping the new machine. The present value of this alternative is:

The equivalent annual cost for the ten-year period is computed as follows:

PV₂ = EAC₂ ´ [annuity factor, 10 time periods, 12%]

-127.51 = EAC₂ ´ [5.650]

EAC₂ = -22.57, or an equivalent annual cost of $22,570

Thus, the least expensive alternative is to sell the old machine because this alternative has the lowest equivalent annual cost.

One key assumption underlying this result is that, whenever the machines have to be replaced, the replacement will be a machine that is as efficient to operate as the new machine being replaced.

15. The current copiers have net cost cash flows as follows:

Year	Before-Tax Cash Flow	After-Tax Cash Flow	Net Cash Flow
1	-2,000	(-2,000 ´ .65) + (.35 ´ .0893 ´ 20,000)	-674.9
2	-2,000	(-2,000 ´ .65) + (.35 ´ .0893 ´ 20,000)	-674.9
3	-8,000	(-8,000 ´ .65) + (.35 ´ .0893 ´ 20,000)	-4,574.9
4	-8,000	(-8,000 ´ .65) + (.35 ´ .0445 ´ 20,000)	-4,888.5
5	-8,000	(-8,000 ´ .65)	-5,200.0
6	-8,000	(-8,000 ´ .65)	-5,200.0

These cash flows have a present value, discounted at 7 percent, of -$15,857. Using the annuity factor for 6 time periods at 7 percent (4.767), we find an equivalent annual cost of $3,326. Therefore, the copiers should be replaced only when the equivalent annual cost of the replacements is less than $3,326.

When purchased, the new copiers will have net cost cash flows as follows:

Year	Before-Tax Cash Flow	After-Tax Cash Flow	Net Cash Flow
0	-25,000	-25,000	-25,000.0
1	-1,000	(-1,000 ´ .65) + (.35 ´ .1429 ´ 25,000)	600.0
2	-1,000	(-1,000 ´ .65) + (.35 ´ .2449 ´ 25,000)	1,493.0
3	-1,000	(-1,000 ´ .65) + (.35 ´ .1749 ´ 25,000)	880.0
4	-1,000	(-1,000 ´ .65) + (.35 ´ .1249 ´ 25,000)	443.0
5	-1,000	(-1,000 ´ .65) + (.35 ´ .0893 ´ 25,000)	131.0
6	-1,000	(-1,000 ´ .65) + (.35 ´ .0893 ´ 25,000)	131.0
7	-1,000	(-1,000 ´ .65) + (.35 ´ .0893 ´ 25,000)	131.0
8	-1,000	(-1,000 ´ .65) + (.35 ´ .0445 ´ 25,000)	-261.0

These cash flows have a present value, discounted at 7 percent, of -$21,969. The decision to replace must also take into account the resale value of the machine, as well as the associated tax on the resulting gain (or loss). Consider three cases:

a. The book (depreciated) value of the existing copiers is now $6,248. If the existing copiers are replaced now, then the present value of the cash flows is:

b.

-21,969 + 8,000 – [0.35 ´ (8,000 – 6,248)] = -$14,582

Using the annuity factor for 8 time periods at 7 percent (5.971), we find that the equivalent annual cost is $2,442.

c. Two years from now, the book (depreciated) value of the existing copiers will be $2,676. If the existing copiers are replaced two years from now, then the present value of the cash flows is:

(-674.9/1.07¹) + (-674.9/1.07²) + (-21,969/1.07²) +

{3,500 – [0.35 ´ (3,500 – 2,676)]}/1.07² = -$17,604

Using the annuity factor for 10 time periods at 7 percent (7.024), we find that the equivalent annual cost is $2,506.

d. Six years from now, both the book value and the resale value of the existing copiers will be zero. If the existing copiers are replaced six years from now, then the present value of the cash flows is:

-15,857+ (-21,969/1.07⁶) = -$30,496

Using the annuity factor for 14 time periods at 7 percent (8.745), we find that the equivalent annual cost is $3,487.

The copiers should be replaced immediately.

16. a.

	Year 1	Year 2	Year 3	Year 4	Year 5	Year 6	Year 7	Year 8	Year 9	Year 10	Year 11
MACRS Percent	10.00%	18.00%	14.40%	11.52%	9.22%	7.37%	6.55%	6.55%	6.55%	6.55%	3.29%
MACRS Depr.	40.00	72.00	57.60	46.08	36.88	29.48	26.20	26.20	26.20	26.20	13.16
Tax Shield	15.60	28.08	22.46	17.97	14.38	11.50	10.22	10.22	10.22	10.22	5.13
Present Value (at 7%) = $114.57 million

The equivalent annual cost of the depreciation tax shield is computed by dividing the present value of the tax shield by the annuity factor for 25 years at 7%:

Equivalent annual cost = $114.57 milliion/11.654 = $9.83 million

The equivalent annual cost of the capital investment is:

$34.3 million – $9.83 million = $24.47 million

b. The extra cost per gallon (after tax) is:

$24.47 million/900 million gallons = $0.0272 per gallon

The pre-tax charge = $0.0272/0.65 = $0.0418 per gallon

17. Since the growth in value of both timber and land is less than the cost of capital after year 8, it must pay to cut by that time. The table below shows that PV is maximized if you cut in year 8. Therefore, if we cut in year 8, the NPV of the offer is: $140,000 – 109,900 = $30,100

		Year 1	Year 2	Year 3	Year 4	Year 5
Future Value:	Timber	48.3	58.2	70.2	84.7	97.8
	Land	52.0	54.1	56.2	58.5	60.8
	Total	100.3	112.3	126.4	143.2	158.6
Present Value:		92.0	94.5	97.6	101.4	103.1
		Year 6	Year 7	Year 8	Year 9
Future Value:	Timber	112.9	130.3	150.5	162.7
	Land	63.3	65.8	68.4	71.2
	Total	176.2	196.1	218.9	233.9
Present Value:		105.1	107.3	109.9	107.7

18. a.

PV_A = $66,730 (Note that this is a cost.)

PV_B = $77,721 (Note that this is a cost.)

Equivalent annual cost (EAC) is found by:

PVA =	EACA ´ [annuity factor, 6%, 3 time periods]
66,730 =	EACA ´ 2.673
EACA =	$24,964 per year rental
PVB =	EACB ´ [annuity factor, 6%, 4 time periods]
77,721 =	EACB ´ 3.465
EACB =	$22,430 per year rental

b. Annual rental is $24,964 for Machine A and $22,430 for Machine B. Borstal should buy Machine B.

c. The payments would increase by 8 percent per year. For example, for Machine A, rent for the first year would be $24,964; rent for the second year would be ($24,964 ´ 1.08) = $26,961; etc.

19. NOTE: The following answer is in error. Can you spot the error and correct it?

Because the cost of a new machine now decreases by 10 percent per year, the rent on such a machine also decreases by 10 percent per year. Therefore:

PV_A = $61,820 (Note that this is a cost.)

PV_B = $71,613 (Note that this is a cost.)

Equivalent annual cost (EAC) is found as follows:

PVA =	EACA ´ [annuity factor, 6%, 3 time periods]
61,820 =	EACA ´ 2.673
EACA =	$23,128, a reduction of 7.35%
PVB =	EACB ´ [annuity factor, 6%, 4 time periods]
71,613 =	EACB ´ 3.465
EACB =	$20,668, a reduction of 7.86%

20. With a 6-year life, the equivalent annual cost (at 8 percent) of a new jet is: ($1,100,000/4.623) = $237,941. If the jet is replaced at the end of year 3 rather than year 4, the company will incur an incremental cost of $237,941 in year 4. The present value of this cost is:

$237,941/1.08⁴ = $174,894

The present value of the savings is:

The president should allow wider use of the present jet because the present value of the savings is greater than the present value of the cost.

Challenge Questions

1. a.

		Year 0	Year 1	Year 2	Year 3	Year 4	Year 5	Year 6	Year 7
	Pre-Tax Flows	-14,000.0	-3,064.0	3,209.0	9,755.0	16,463.0	14,038.0	7,696.0	3,444.0
	IRR = 33.3%
	Post-Tax Flows	-12,600.0	-1,630.0	2,381.0	6,205.0	10,685.0	10,136.0	6,110.0	3,444.0
	IRR = 26.8%
Effective Tax Rate = 19.5%

b. If the depreciation rate is accelerated, this has no effect on the pretax IRR, but it increases the after-tax IRR. Therefore, the numerator decreases and the effective tax rate decreases.

If the inflation rate increases, we would expect pretax cash flows to increase at the inflation rate, while after tax cash flows increase at a slower rate. After tax cash flows increase at a slower rate than the inflation rate because depreciation expense does not increase with inflation. Therefore, the numerator of T_E becomes proportionately larger than the denominator and the effective tax rate increases.

c.

Hence, if the up-front investment is deductible for tax purposes, then the effective tax rate is equal to the statutory tax rate.

2. a. With a real rate of 6 percent and an inflation rate of 5 percent, the nominal rate, r, is determined as follows:

(1 + r) =	(1 + 0.06) ´ (1 + 0.05)
r =	0.113 = 11.3%

For a three-year annuity at 11.3 percent, the annuity factor (using the annuity formula from Chapter 3) is 2.4310; for a two-year annuity, the annuity factor is 1.7057.

For a three-year annuity with a present value of $28.37, the nominal annuity is: ($28.37/2.4310) = $11.67

For a two-year annuity with a present value of $21.00, the nominal annuity is: ($21.00/1.7057) = $12.31

These nominal annuities are not realistic estimates of equivalent annual costs because the appropriate rental cost (i.e., the equivalent annual cost) must take into account the effects of inflation.

b. With a real rate of 6 percent and an inflation rate of 25 percent, the nominal rate, r, is determined as follows:

(1 + r) =	(1 + 0.06) ´ (1 + 0.25)
r =	0.325 = 32.5%

For a three-year annuity at 32.5 percent, the annuity factor (using the annuity formula from Chapter 3) is 1.7542; for a two-year annuity, the annuity factor is 1.3243.

For a three-year annuity with a present value of $28.37, the nominal annuity is: ($28.37/1.7542) = $16.17

For a two-year annuity with a present value of $21.00, the nominal annuity is: ($21.00/1.3243) = $15.86

With an inflation rate of 5 percent, Machine A has the lower nominal annual cost ($11.67 compared to $12.31). With inflation at 25 percent, Machine B has the lower nominal annual cost ($15.86 compared to $16.17). Thus it is clear that inflation has a significant impact on the calculation of equivalent annual cost, and hence, the warning in the text to do these calculations in real terms. The rankings change because, at the higher inflation rate, the machine with the longer life (here, Machine A) is affected more.

3. a. The cash outflow in Period 0 becomes -$10,426,000 and NPV = $5,693,684. The format is advantageous since it recognizes additional cash flows created by the tax-deductibility of depreciation. However, it may also be disadvantageous because several assumptions are made here. We are assuming:

1. The tax rate remains constant.

2. The depreciation method remains constant.

3. The company’s ability to generate taxable income continues so the tax shield can be used.

b. Since the cash flows are relatively safe, they should probably be discounted at an after-tax borrowing or lending rate.

c. The discount rate for the other cash flows should not change since it must represent the opportunity cost of funds in a project of similar risk.