Chapter 6 Solutions

Problem 4

The issue here is what outflow per year for the aluminum siding makes the decision moot.

Approach one would be to use the present value of the costs of the wood siding and solve the present value of the aluminum siding that has the same present value of costs. This does not set them on equal lives as the aluminum siding has a “forever” horizon and thus you can’t match lives with a replicating costs for the two choices until you get to a year where both choices are starting a new cycle. So you will need to use the equivalent annuity approach. Step one is to determine the present value of the wood siding.

Present value of wood siding: $5.000 + $1,000 x (1 – (1/(1+0.10)10)) / (0.10)

$5,000 + $1,000 x (6.14457) = $11,144.57

Step two is to find the equivalent annuity of the wood siding.

Equivalent annuity of $11,144.57 is, $11,144.57 x (0.10) / (1 – (1.10)-10)

= $11,144.57 x 0.162745 = $1,813.73

Step three is to use this equivalent annuity for the aluminum siding.

Equivalent annuity of aluminum siding:

Present Value of Costs x ((0.10) / (1 – (1.10)-∞) = $1,813.72

Present Value of Costs = $1,813.73 / ((0.10) / (1 – (1.10)-∞)

Present Value of Costs = $1,813.73 / (0.10) = $18,137.27

Present Value of Costs = $15,000 + Maintenance / 0.10 = $18,137.27

Maintenance Costs = ($18,137.27 - $15,000) x 0.10 = $313.73

Problem 5

Magazine subscription assumes that you will renew each year for the next three years…at the beginning of the period (annuity due type of charge).

Equal Lives and replicating approach…means you need to go out to year six so all the choices have equal lives.

On a time line we have…

Annual renewal = $20 x ((1 – (1/(1+0.20)6)) / (0.20) x (1.20) = $79.81

Renew every other year = $36 + $36 /(1.20)2 + $36 /(1.20)4 = $78.36

Renewal every three years = $45 + $45 /(1.20)3 = $71.04

Renew every three years….but you lose the option to cancel at the end of the other periods of time and the value of this option has not been included in the incremental cash flow.

Problem 7

The cost of capital is missing from this problem (so use 10%). This is a replacement now or later question. We do not know the value of the option to replace later so will answer the question, keep existing machine or replace with new $2,000.000 machine.

Step one: Find the salvage value if we replace. The old machine has a current book value of $500,000 but will be sold off for only $300,000. We have a loss on disposal of $200,000. With a tax rate of 40% we have a tax credit of:

Tax credit at salvage of old machine = $200,000 x 0.40 = $80,000

After tax cash flow for salvage = $300,000 + $80,000 = $380,000

Step two: Determine the present value of the depreciation tax shields for the two machines. For old machine it is a lost tax shield (cost) and for the new machine it is a tax shield (reduction in cost or revenue).

Tax Shield of Old Machine = $100,000 x 0.40 x (1 – (1/(1+0.10)5)) / (0.10)

Tax Shield of Old Machine = $40,000 x 3.7908 = $151,631

Tax Shield of New Machine = $200,000 x (0.40) x (1 – (1/(1+0.10)10)) / (0.10)

Tax Shield of New Machine = $80,000 x 6.1446 = $491,565

Step Three: Determine the PV of all but the maintenance savings costs.

PV = -$2,000,000 + $380,000 - $151,631 + $491,565 = -$1,280,066

Step Four: Find the after-tax annuity (maintenance savings) that will make the NPV positive for his replacement.

NPV = -$1,280,066 + after-tax annuity x (1 – (1/(1+0.10)10)) / (0.10)

After Tax Annutiy (at NPV = 0) = $1,280.066 / 6.1446 = $208,325

Step Five: Find the Before Tax Annuity

$208,325 / (1 – 0.40) = $347,208

Problem 15

This is a capital rationing problem where the company will only spend $20,000,000 on capital projects this year. Given the following projects, which set (portfolio) is the best choice?

Choices:

ProjectInitial InvestmentNPVIRRPI

I $10$ 3.021%0.30

II $ 5$ 2.528%0.50

III $15$ 4.019%0.27

IV $10$ 4.024%0.40

V $ 5$ 2.020%0.40

Pure Ranking without regard to Capital Constraint

ProjectInitial InvestmentNPVIRRPI

I $10 3 3 4

II $ 5 4 1 1

III $15 2 5 5

IV $10 1 2 3

V $ 5 5 4 2

With Capital Constraint of $20,000,000

using PI you pick – II, V, and IV, and you have a NPV total of $8.5 million
using IRR you pick, II, IV, and V, and you have a NPV total of $8.5 million (note you pick V over I because I would have put you over the $20 million and with V at only $5 million investment you can substitute
In this case they don’t but…is there a better combination…check out all possible combinations that stay within the constraint of $20,000,000

Project PortfoliosTotal Initial InvestmentTotal NPV

I, II, and V$20,000,000$7,500,000

I and IV$20,000,000$7,000,000

II and III$20,000,000$6,500,000

II, IV, and V$20,000,000$8,500,000

III and V$20,000,000$6,000,000

From the table you can see that the best portfolio is II, IV, and V…the same answer as above. But the portfolio approach will always give you the highest NPV within the capital constraint.

Problem 16

Should you open the gardening store knowing that there will be erosion from the hardware store? This is a simple NPV problem where the lost cash flow from the hardware store is part of the incremental cash flow of the gardening center.

First find the after-tax cash flow of the lost sales in the hardware store (because you have the after-tax cash flow for the gardening store).

(Sales – Costs) x (1 – 0.40) → 40% after-tax margin → [$3,000 (0.60)] = $1800

Now find the NPV of the gardening store:

NPV = -$50,000 + ($10,000 - $1800) x (1 – (1/(1+0.14)10)) / (0.14)

NPV = -$50,000 + ($8,200) x (5.21612) = -$7,227.82

With the erosion to the hardware store sales the answer is do not open the gardening center.

Failing to account for the erosion…

NPV = -$50,000 + ($10,000) x (1 – (1/(1+0.14)10)) / (0.14) = $2,161

Problem 18 (Key, must be in like state at the end)

Lease versus buy decision on replacing computers. First find the cash flow associated with the buy decision. Again, we have no cost of capital so we will use 10%.

Purchase of computers = 500 x $2,500 = $1,250,000

Depreciation (three year life, with salvage of $500), PV of Annual savings from depreciation,

$2,500/3 x 0.40 x (1 – (1/(1+0.10)3)) / (0.10) = $829 per computer

500 x $829 = $414,475

Cash flow from Salvage:

Book value at three years is zero. Gain from sale is $500 and the after tax cash flow from the gain is:

$500 x (1 -0.40) = $300 per computer

500 x $300 = $150,000

Present Value of Salvage is $150,000 / (1.10)3 = $112,697

The PV of the buy is:

$-1,250,000 + $414.475 + $112,697 = -$722,828

Lease choice is $5,000,000 x (1 – 0.04) = $3,000,000 per year…

Answer seems obvious…buy your own computers. There was an error on this from the author…there were suppose to be 5,000 employees. Redo this on your own but use 5,000 employees instead of the 500 in the problem.

If we use 5,000

Lease versus buy decision on replacing computers. First find the cash flow associated with the buy decision. Again, we have no cost of capital so we will use 10%.

Purchase of computers = 5,000 x $2,500 = $12,500,000

Depreciation (three year life, with salvage of $500), PV of Annual savings from depreciation,

$2,500/3 x 0.40 x (1 – (1/(1+0.10)3)) / (0.10) = $829 per computer

5,000 x $829 = $4,144,753

Cash flow from Salvage:

Book value at three years is zero. Gain from sale is $500 and the after tax cash flow from the gain is:

$500 x (1 -0.40) = $300 per computer

5000 x $300 = $1,500,000

Present Value of Salvage is $1,500,000 / (1.10)3 = $1,126,972

The PV of the buy is:

$-12,500,000 + $4,144,753 + $1,126,972 = -$7,228,275

Lease choice is $5,000,000 x (1 – 0.04) = $3,000,000 per year…

Take the equivalent annuity for the three years…

EA = $3,000,000 x (1 – (1/(1+0.10)3)) / (0.10) = $7,460,556

Do it yourself…the lease is more expensive by $232,281

Problem 21

How much of the excess capacity do you take up each year such that tennis rackets can not be made? First look at the unabated growth of the tennis rackets and their contribution margin (after-tax).

YearExcess capacity Additional Tennis Production Δ Gross Margin

125,0002,500(27,500)$150,000

222,5002,750(30,250)$165,000

319,7503,025(33,275)$181,500

416,7253,327(36,602)$199,620

513,3983,660(40,262)$219,600

6 9,7384,026 (44,288)$241,560

7 5,7124,428 (48,716)$265,680

8 1,2841,284 (50,000)$ 77,040

9 0 0 (50,000)$ 0

10 0 0 (50,000)$ 0

Note the Gross Margins are the addition each year to the gross margins from the prior year.

If you add the squash rackets you will tie up 40% of the total capacity or 20,000. The additional tennis rackets are now:

YearExcess capacity Additional Tennis ProductionGross Margin

1 5,0002,500(27,500)$150,000

2 2,5005,000(30,000)$150,000

3 0 0(30,000)$ 0

The opportunity cost is now the additional gross margin from the tennis sales that can not take place due to production ceiling of 30,000 rackets.

YearΔ Gross MarginW/O SquashWith Squash

1$ 0$ 150,000$150,000

2$ 15,000$ 315,000$300,000

3$ 196,500$ 496,500$300,000

4$ 396,120$ 696,120$300,000

5$ 615,720$ 915,720$300,000

6$ 857,280$1,157,280$300,000

7$1,122,960$1,422,960$300,000

8$1,200,000$1,500,000$300,000

9$1,200,000$1,500,000$300,000

10$1,200,000$1,500,000$300,000

The opportunity cost is now the Δ Gross Margin x 0.40 / ( 1.10)year

YearΔ Gross Margin1/ (1.1)yearPV Δ Gross Margin

1$ 00.9090$ 0

2$ 15,0000.8264$ 12,397

3$ 196,5000.7513$ 147,633

4$ 396,1200.6830$ 270,555

5$ 615,7200.6209$ 382,313

6$ 857,2800.5645$ 483,912

7$1,122,9600.5132$ 576,256

8$1,200,0000.4665$ 559,809

9$1,200,0000.4241$ 508,917

10$1,200,0000.3855$ 462,652

$3,403,891

This is the before tax gross margin…

After-tax gross margin is the opportunity cost…

Opportunity Cost = $3,403,891 x (1 – 0.40) = $2,042,335