**1.** Forecasting risk is the risk that a poor decision is made because of errors in projected cash flows. The danger is greatest with a new product because the cash flows are probably harder to predict.

**2.** With a sensitivity analysis, one variable is examined over a broad range of values. With a scenario analysis, all variables are examined for a limited range of values.

**3.** It is true that if average revenue is less than average cost, the firm is losing money. This much of the statement is therefore correct. At the margin, however, accepting a project with marginal revenue in excess of its marginal cost clearly acts to increase operating cash flow.

**4.** It makes wages and salaries a fixed cost, driving up operating leverage.

**5.** Fixed costs are relatively high because airlines are relatively capital intensive (and airplanes are expensive). Skilled employees such as pilots and mechanics mean relatively high wages which, because of union agreements, are relatively fixed. Maintenance expenses are significant and relatively fixed as well.

**6.** From the shareholder perspective, the financial break-even point is the most important. A project can exceed the accounting and cash break-even points but still be below the financial break-even point. This causes a reduction in shareholder (your) wealth.

**7.** The project will reach the cash break-even first, the accounting break-even next and finally the financial break-even. For a project with an initial investment and sales after, this ordering will always apply. The cash break-even is achieved first since it excludes depreciation. The accounting break-even is next since it includes depreciation. Finally, the financial break-even, which includes the time value of money, is achieved.

**8.** Soft capital rationing implies that the firm as a whole isn’t short of capital, but the division or project does not have the necessary capital. The implication is that the firm is passing up positive NPV projects. With hard capital rationing the firm is unable to raise capital for a project under any circumstances. Probably the most common reason for hard capital rationing is financial distress, meaning bankruptcy is a possibility.

**9.** The implication is that they will face hard capital rationing.

**Solutions to Questions and Problems**

*NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.*

* Basic*

**1.** *a.* The total variable cost per unit is the sum of the two variable costs, so:

Total variable costs per unit = $1.43 + 2.44

Total variable costs per unit = $3.87

** ***b.* The total costs include all variable costs and fixed costs. We need to make sure we are including all variable costs for the number of units produced, so:

Total costs = Variable costs + Fixed costs

Total costs = $3.87(320,000) + $650,000

Total costs = $1,888,400

*c.* The cash breakeven, that is the point where cash flow is zero, is:

Q_{C} = $650,000 / ($10.00 – 3.87)

Q_{C} = 106,036 units

And the accounting breakeven is:

Q_{A} = ($650,000 + 190,000) / ($10.00 – 3.87)

Q_{A} = 137,031 units

**2.** The total costs include all variable costs and fixed costs. We need to make sure we are including all variable costs for the number of units produced, so:

Total costs = ($16.15 + 18.50)(150,000) + $800,000

Total costs = $5,997,500

The marginal cost, or cost of producing one more unit, is the total variable cost per unit, so:

Marginal cost = $16.15 + 18.50

Marginal cost = $34.65

The average cost per unit is the total cost of production, divided by the quantity produced, so:

Average cost = Total cost / Total quantity

Average cost = $5,997,500/150,000

Average cost = $39.98

Minimum acceptable total revenue = 10,000($34.65)

Minimum acceptable total revenue = $346,500

Additional units should be produced only if the cost of producing those units can be recovered.

**3.** The base-case, best-case, and worst-case values are shown below. Remember that in the best-case, sales and price increase, while costs decrease. In the worst-case, sales and price decrease, and costs increase.

Unit

Scenario Unit Sales Unit Price Variable Cost Fixed Costs

Base 105,000 $1,900.00 $170.00 $6,000,000

Best 120,750 $2,185.00 $144.50 $5,100,000

Worst 89,250 $1,615.00 $195.50 $6,900,000

**4.** An estimate for the impact of changes in price on the profitability of the project can be found from the sensitivity of NPV with respect to price: DNPV/DP. This measure can be calculated by finding the NPV at any two different price levels and forming the ratio of the changes in these parameters. Whenever a sensitivity analysis is performed, all other variables are held constant at their base-case values.

**5.** *a*. To calculate the accounting breakeven, we first need to find the depreciation for each year. The depreciation is:

Depreciation = $896,000/8

Depreciation = $112,000 per year

And the accounting breakeven is:

Q_{A} = ($900,000 + 112,000)/($38 – 25)

Q_{A} = 77,846 units

To calculate the accounting breakeven, we must realize at this point (and only this point), the OCF is equal to depreciation. So, the DOL at the accounting breakeven is:

DOL = 1 + FC/OCF = 1 + FC/D

DOL = 1 + [$900,000/$112,000]

DOL = 9.036

*b.* We will use the tax shield approach to calculate the OCF. The OCF is:

OCF_{base} = [(P – v)Q – FC](1 – t_{c}) + t_{c}D

OCF_{base} = [($38 – 25)(100,000) – $900,000](0.65) + 0.35($112,000)

OCF_{base} = $299,200

Now we can calculate the NPV using our base-case projections. There is no salvage value or NWC, so the NPV is:

NPV_{base} = –$896,000 + $299,200(PVIFA_{15%,8})

NPV_{base} = $446,606.60

To calculate the sensitivity of the NPV to changes in the quantity sold, we will calculate the NPV at a different quantity. We will use sales of 105,000 units. The NPV at this sales level is:

OCF_{new} = [($38 – 25)(105,000) – $900,000](0.65) + 0.35($112,000)

OCF_{new} = $341,450

And the NPV is:

NPV_{new} = –$896,000 + $341,450(PVIFA_{15%,8})

NPV_{new} = $636,195.93

So, the change in NPV for every unit change in sales is:

DNPV/DS = ($636,195.93 – 446,606.60)/(105,000 – 100,000)

DNPV/DS = +$37.918

If sales were to drop by 500 units, then NPV would drop by:

NPV drop = $37.918(500) = $18,958.93

You may wonder why we chose 105,000 units. Because it doesn’t matter! Whatever sales number we use, when we calculate the change in NPV per unit sold, the ratio will be the same.

*c.* To find out how sensitive OCF is to a change in variable costs, we will compute the OCF at a variable cost of $24. Again, the number we choose to use here is irrelevant: We will get the same ratio of OCF to a one dollar change in variable cost no matter what variable cost we use. So, using the tax shield approach, the OCF at a variable cost of $24 is:

OCF_{new} = [($38 – 24)(100,000) – 900,000](0.65) + 0.35($112,000)

OCF_{new} = $364,200

So, the change in OCF for a $1 change in variable costs is:

DOCF/Dv = ($299,200 – 364,200)/($25 – 24)

DOCF/Dv = –$65,000

If variable costs decrease by $1 then, OCF would increase by $65,000

**6.** We will use the tax shield approach to calculate the OCF for the best- and worst-case scenarios. For the best-case scenario, the price and quantity increase by 10 percent, so we will multiply the base case numbers by 1.1, a 10 percent increase. The variable and fixed costs both decrease by 10 percent, so we will multiply the base case numbers by .9, a 10 percent decrease. Doing so, we get:

OCF_{best} = {[($38)(1.1) – ($25)(0.9)](100K)(1.1) – $900K(0.9)}(0.65) + 0.35($112K)

OCF_{best} = $892,650

The best-case NPV is:

NPV_{best} = –$896,000 + $892,650(PVIFA_{15%,8})

NPV_{best} = $3,109,607.54

For the worst-case scenario, the price and quantity decrease by 10 percent, so we will multiply the base case numbers by .9, a 10 percent decrease. The variable and fixed costs both increase by 10 percent, so we will multiply the base case numbers by 1.1, a 10 percent increase. Doing so, we get:

OCF_{worst} = {[($38)(0.9) – ($25)(1.1)](100K)(0.9) – $900K(1.1)}(0.65) + 0.35($112K)

OCF_{worst} = –212,350

The worst-case NPV is:

NPV_{worst} = –$896,000 – $212,350(PVIFA_{15%,8})

NPV_{worst} = –$1,848,882.72

**7.** The cash breakeven equation is:

Q_{C} = FC/(P – v)

And the accounting breakeven equation is:

Q_{A} = (FC + D)/(P – v)

Using these equations, we find the following cash and accounting breakeven points:

(1): Q_{C} = $15M/($3,000 – 2,275) Q_{A} = ($15M + 6.5M)/($3,000 – 2,275)

Q_{C} = 20,690 Q_{A} = 29,655

(2): Q_{C} = $73,000/($39 – 27) Q_{A} = ($73,000 + 140,000)/($39 – 27)

Q_{C} = 6,083 Q_{A} = 17,750

(3): Q_{C} = $1,200/($8 – 3) Q_{A} = ($1,200 + 840)/($8 – 3)

Q_{C} = 240 Q_{A} = 408

**8.** We can use the accounting breakeven equation:

Q_{A} = (FC + D)/(P – v)

to solve for the unknown variable in each case. Doing so, we find:

(1): Q_{A} = 130,200 = ($820,000 + D)/($41 – 30)

D = $612,200

(2): Q_{A} = 135,000 = ($3.2M + 1.15M)/(P – $56)

P = $88.22

(3): Q_{A} = 5,478 = ($160,000 + 105,000)/($105 – v)

v = $56.62

**9.** The accounting breakeven for the project is:

Q_{A} = [$6,000 + ($12,000/4)]/($70 – 37)

Q_{A} = 273

And the cash breakeven is:

Q_{C} = $6,000/($70 – 37)

Q_{C} = 182

At the financial breakeven, the project will have a zero NPV. Since this is true, the initial cost of the project must be equal to the PV of the cash flows of the project. Using this relationship, we can find the OCF of the project must be:

NPV = 0 implies $12,000 = OCF(PVIFA_{15%,4})

OCF = $4,203.18

Using this OCF, we can find the financial breakeven is:

Q_{F} = ($6,000 + $4,203.18)/($70 – 37) = 309

And the DOL of the project is:

DOL = 1 + ($6,000/$4,203.18) = 2.427

**10.** In order to calculate the financial breakeven, we need the OCF of the project. We can use the cash and accounting breakeven points to find this. First, we will use the cash breakeven to find the price of the product as follows:

Q_{C} = FC/(P – v)

13,000 = $120,000/(P – $23)

P = $32.23

Now that we know the product price, we can use the accounting breakeven equation to find the depreciation. Doing so, we find the annual depreciation must be:

Q_{A} = (FC + D)/(P – v)

19,000 = ($120,000 + D)/($32.23 – 23)

Depreciation = $55,385

We now know the annual depreciation amount. Assuming straight-line depreciation is used, the initial investment in equipment must be five times the annual depreciation, or:

Initial investment = 5($55,385) = $276,923

The PV of the OCF must be equal to this value at the financial breakeven since the NPV is zero, so:

$276,923 = OCF(PVIFA_{16%,5})

OCF = $84,574.91

We can now use this OCF in the financial breakeven equation to find the financial breakeven sales figure is:

Q_{F} = ($120,000 + 84,574.91)/($32.23 – 23)

Q_{F} = 22,162

**11.** We know that the DOL is the percentage change in OCF divided by the percentage change in quantity sold. Since we have the original and new quantity sold, we can use the DOL equation to find the percentage change in OCF. Doing so, we find:

DOL = %DOCF / %DQ

Solving for the percentage change in OCF, we get:

%DOCF = (DOL)(%DQ)

%DOCF = 2.5[(47,000 – 40,000)/40,000]

%DOCF = 43.75%

The new level of operating leverage is lower since FC/OCF is smaller.

**12.** Using the DOL equation, we find:

DOL = 1 + FC / OCF

2.5 = 1 + $150,000/OCF

OCF = $100,000

The percentage change in quantity sold at 35,000 units is:

%ΔQ = (35,000 – 40,000) / 40,000

%ΔQ = –.1250 or –12.50%

So, using the same equation as in the previous problem, we find:

%ΔOCF = 2.5(–12.5%)

%ΔQ = –.3125 or –31.25%

So, the new OCF level will be:

New OCF = (1 – .3125)($100,000)

New OCF = $68,750

And the new DOL will be:

New DOL = 1 + ($150,000/$68,750)

New DOL = 3.182

**13.** The DOL of the project is:

DOL = 1 + ($45,000/$71,000)

DOL = 1.6338

If the quantity sold changes to 8,500 units, the percentage change in quantity sold is:

%DQ = (8,500 – 8,000)/8,000

%ΔQ = .0625 or 6.25%

So, the OCF at 8,500 units sold is:

%DOCF = DOL(%DQ)

%ΔOCF = 1.6338(.0625)

%ΔOCF = .1021 or 10.21%

This makes the new OCF:

New OCF = $71,000(1.1021)

New OCF = $78,250.00

And the DOL at 8,500 units is:

DOL = 1 + ($45,000/$78,250.00)

DOL = 1.5751

**14.** We can use the equation for DOL to calculate fixed costs. The fixed cost must be:

DOL = 2.75 = 1 + FC/OCF

FC = (2.75 – 1)$16,000

FC = $28,000

If the output rises to 11,000 units, the percentage change in quantity sold is:

%DQ = (11,000 – 10,000)/10,000

%ΔQ = .10 or 10.00%

The percentage change in OCF is:

%DOCF = 2.75(.10)

%ΔOCF = .2750 or 27.50%

So, the operating cash flow at this level of sales will be:

OCF = $16,000(1.275)

OCF = $20,400

If the output falls to 9,000 units, the percentage change in quantity sold is:

%DQ = (9,000 – 10,000)/10,000

%ΔQ = –.10 or –10.00%

The percentage change in OCF is:

%DOCF = 2.75(–.10)

%ΔOCF = –.2750 or –27.50%

So, the operating cash flow at this level of sales will be:

OCF = $16,000(1 – .275)

OCF = $11,600

**15.** Using the equation for DOL, we get:

DOL = 1 + FC/OCF

At 11,000 units

DOL = 1 + $28,000/$20,400

DOL = 2.3725

At 9,000 units

DOL = 1 + $28,000/$11,600

DOL = 3.4138

* Intermediate*

**16.** *a*. At the accounting breakeven, the IRR is zero percent since the project recovers the initial investment. The payback period is N years, the length of the project since the initial investment is exactly recovered over the project life. The NPV at the accounting breakeven is:

NPV = I [(1/N)(PVIFA_{R%,N}) – 1]

*b*. At the cash breakeven level, the IRR is –100 percent, the payback period is negative, and the NPV is negative and equal to the initial cash outlay.

*c*. The definition of the financial breakeven is where the NPV of the project is zero. If this is true, then the IRR of the project is equal to the required return. It is impossible to state the payback period, except to say that the payback period must be less than the length of the project. Since the discounted cash flows are equal to the initial investment, the undiscounted cash flows are greater than the initial investment, so the payback must be less than the project life.

**17.** Using the tax shield approach, the OCF at 110,000 units will be:

OCF = [(P – v)Q – FC](1 – t_{C}) + t_{C}(D)

OCF = [($28 – 19)(110,000) – 190,000](0.66) + 0.34($420,000/4)

OCF = $563,700

We will calculate the OCF at 111,000 units. The choice of the second level of quantity sold is arbitrary and irrelevant. No matter what level of units sold we choose, we will still get the same sensitivity. So, the OCF at this level of sales is:

OCF = [($28 – 19)(111,000) – 190,000](0.66) + 0.34($420,000/4)

OCF = $569,640

The sensitivity of the OCF to changes in the quantity sold is:

Sensitivity = DOCF/DQ = ($569,640 – 563,700)/(111,000 – 110,000)

DOCF/DQ = +$5.94

OCF will increase by $5.94 for every additional unit sold.

**18.** At 110,000 units, the DOL is:

DOL = 1 + FC/OCF

DOL = 1 + ($190,000/$563,700)

DOL = 1.3371

The accounting breakeven is:

Q_{A} = (FC + D)/(P – v)

QA = [$190,000 + ($420,000/4)]/($28 – 19)

Q_{A} = 32,777

And, at the accounting breakeven level, the DOL is:

DOL = 1 + ($190,000/$105,000)

DOL = 2.8095

**19.** *a*. The base-case, best-case, and worst-case values are shown below. Remember that in the best-case, sales and price increase, while costs decrease. In the worst-case, sales and price decrease, and costs increase.

Scenario Unit sales Variable cost Fixed costs

Base 190 $15,000 $225,000

Best 209 $13,500 $202,500

Worst 171 $16,500 $247,500

Using the tax shield approach, the OCF and NPV for the base case estimate is:

OCF_{base} = [($21,000 – 15,000)(190) – $225,000](0.65) + 0.35($720,000/4)

OCF_{base} = $657,750

NPV_{base} = –$720,000 + $657,750(PVIFA_{15%,4})

NPV_{base} = $1,157,862.02

The OCF and NPV for the worst case estimate are:

OCF_{worst} = [($21,000 – 16,500)(171) – $247,500](0.65) + 0.35($720,000/4)

OCF_{worst} = $402,300

NPV_{worst} = –$720,000 + $402,300(PVIFA_{15%,4})

NPV_{worst} = +$428,557.80

And the OCF and NPV for the best case estimate are:

OCF_{best} = [($21,000 – 13,500)(209) – $202,500](0.65) + 0.35($720,000/4)

OCF_{best} = $950,250

NPV_{best} = –$720,000 + $950,250(PVIFA_{15%,4})

NPV_{best} = $1,992,943.19

*b*. To calculate the sensitivity of the NPV to changes in fixed costs we choose another level of fixed costs. We will use fixed costs of $230,000. The OCF using this level of fixed costs and the other base case values with the tax shield approach, we get:

OCF = [($21,000 – 15,000)(190) – $230,000](0.65) + 0.35($720,000/4)

OCF = $654,500

And the NPV is:

NPV = –$720,000 + $654,500(PVIFA_{15%,4})

NPV = $1,148,583.34

The sensitivity of NPV to changes in fixed costs is:

DNPV/DFC = ($1,157,862.02 – 1,148,583.34)/($225,000 – 230,000)

DNPV/DFC = –$1.856

For every dollar FC increase, NPV falls by $1.86.

*c*. The cash breakeven is:

Q_{C} = FC/(P – v)

QC = $225,000/($21,000 – 15,000)

Q_{C} = 38

*d*. The accounting breakeven is:

Q_{A} = (FC + D)/(P – v)

QA = [$225,000 + ($720,000/4)]/($21,000 – 15,000)

Q_{A} = 68

At the accounting breakeven, the DOL is:

DOL = 1 + FC/OCF

DOL = 1 + ($225,000/$180,000) = 2.2500

For each 1% increase in unit sales, OCF will increase by 2.2500%.

**20.** The marketing study and the research and development are both sunk costs and should be ignored. We will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs and gain sales of the cheap clubs, these must be accounted for as erosion. The total sales for the new project will be:

| Sales | |

New clubs | $700 ´ 55,000 = $38,500,000 | |

Exp. clubs | $1,100 ´ (–13,000) = –14,300,000 | |

Cheap clubs | $400 ´ 10,000 = 4,000,000 | |

$28,200,000 |

For the variable costs, we must include the units gained or lost from the existing clubs. Note that the variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will save these variable costs, which is an inflow. So:

Var. costs | ||

New clubs | –$320 ´ 55,000 = –$17,600,000 | |

Exp. clubs | –$600 ´ (–13,000) = 7,800,000 | |

Cheap clubs | –$180 ´ 10,000 = –1,800,000 | |

–$11,600,000 |

The pro forma income statement will be:

| Sales | $28,200,000 |

Variable costs | 11,600,000 | |

Costs | 7,500,000 | |

Depreciation | 2,600,000 | |

EBT | 6,500,000 | |

Taxes | 2,600,000 | |

Net income | $ 3,900,000 |

Using the bottom up OCF calculation, we get:

OCF = NI + Depreciation = $3,900,000 + 2,600,000

OCF = $6,500,000

So, the payback period is:

Payback period = 2 + $6.15M/$6.5M

Payback period = 2.946 years

The NPV is:

NPV = –$18.2M – .95M + $6.5M(PVIFA_{14%,7}) + $0.95M/1.14^{7}

NPV = $9,103,636.91

And the IRR is:

IRR = –$18.2M – .95M + $6.5M(PVIFA_{IRR%,7}) + $0.95M/IRR^{7}

IRR = 28.24%

**21.** The upper and lower bounds for the variables are:

Base Case Lower Bound Upper Bound

Unit sales (new) 55,000 49,500 60,500

Price (new) $700 $630 $770

VC (new) $320 $288 $352

Fixed costs $7,500,000 $6,750,000 $8,250,000

Sales lost (expensive) 13,000 11,700 14,300

Sales gained (cheap) 10,000 9,000 11,000

Best-case

We will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs and gain sales of the cheap clubs, these must be accounted for as erosion. The total sales for the new project will be:

| Sales | |

New clubs | $770 ´ 60,500 = $46,585,000 | |

Exp. clubs | $1,100 ´ (–11,700) = – 12,870,000 | |

Cheap clubs | $400 ´ 11,000 = 4,400,000 | |

$38,115,000 |

For the variable costs, we must include the units gained or lost from the existing clubs. Note that the variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will save these variable costs, which is an inflow. So:

Var. costs | ||

New clubs | $288 ´ 60,500 = $17,424,000 | |

Exp. clubs | $600 ´ (–11,700) = – 7,020,000 | |

Cheap clubs | $180 ´ 11,000 = 1,980,000 | |

$12,384,000 |

The pro forma income statement will be:

| Sales | $38,115,000 |

Variable costs | 12,384,000 | |

Costs | 6,750,000 | |

Depreciation | 2,600,000 | |

EBT | 16,381,000 | |

Taxes | 6,552,400 | |

Net income | $9,828,600 |

Using the bottom up OCF calculation, we get:

OCF = Net income + Depreciation = $9,828,600 + 2,600,000

OCF = $12,428,600

And the best-case NPV is:

NPV = –$18.2M – .95M + $12,428,600(PVIFA_{14%,7}) + .95M/1.14^{7}

NPV = $34,527,280.98

Worst-case

We will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs and gain sales of the cheap clubs, these must be accounted for as erosion. The total sales for the new project will be:

| Sales | |

New clubs | $630 ´ 49,500 = $31,185,000 | |

Exp. clubs | $1,100 ´ (– 14,300) = – 15,730,000 | |

Cheap clubs | $400 ´ 9,000 = 3,600,000 | |

$19,055,000 |

For the variable costs, we must include the units gained or lost from the existing clubs. Note that the variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will save these variable costs, which is an inflow. So:

Var. costs | ||

New clubs | $352 ´ 49,500 = $17,424,000 | |

Exp. clubs | $600 ´ (– 14,300) = – 8,580,000 | |

Cheap clubs | $180 ´ 9,000 = 1,620,000 | |

$10,464,000 |

The pro forma income statement will be:

| Sales | $19,055,000 | |

Variable costs | 10,464,000 | ||

Costs | 8,250,000 | ||

Depreciation | 2,600,000 | ||

EBT | – 2,259,000 | ||

Taxes | 903,600 | *assumes a tax credit | |

Net income | –$1,355,400 |

Using the bottom up OCF calculation, we get:

OCF = NI + Depreciation = –$1,355,400 + 2,600,000

OCF = $1,244,600

And the worst-case NPV is:

NPV = –$18.2M – .95M + $1,244,600(PVIFA_{14%,7}) + .95M/1.14^{7}

NPV = –$13,433,120.34

**22.** To calculate the sensitivity of the NPV to changes in the price of the new club, we simply need to change the price of the new club. We will choose $750, but the choice is irrelevant as the sensitivity will be the same no matter what price we choose.

We will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs and gain sales of the cheap clubs, these must be accounted for as erosion. The total sales for the new project will be:

| Sales | |

New clubs | $750 ´ 55,000 = $41,250,000 | |

Exp. clubs | $1,100 ´ (– 13,000) = –14,300,000 | |

Cheap clubs | $400 ´ 10,000 = 4,000,000 | |

$30,950,000 |

Var. costs | ||

New clubs | $320 ´ 55,000 = $17,600,000 | |

Exp. clubs | $600 ´ (–13,000) = –7,800,000 | |

Cheap clubs | $180 ´ 10,000 = 1,800,000 | |

$11,600,000 |

The pro forma income statement will be:

| Sales | $30,950,000 |

Variable costs | 11,600,000 | |

Costs | 7,500,000 | |

Depreciation | 2,600,000 | |

EBT | 9,250,000 | |

Taxes | 3,700,000 | |

Net income | $ 5,550,000 |

Using the bottom up OCF calculation, we get:

OCF = NI + Depreciation = $5,550,000 + 2,600,000

OCF = $8,150,000

And the NPV is:

NPV = –$18.2M – 0.95M + $8.15M(PVIFA_{14%,7}) + .95M/1.14^{7}

NPV = $16,179,339.89

So, the sensitivity of the NPV to changes in the price of the new club is:

DNPV/DP = ($16,179,339.89 – 9,103,636.91)/($750 – 700)

DNPV/DP = $141,514.06

For every dollar increase (decrease) in the price of the clubs, the NPV increases (decreases) by $141,514.06.

To calculate the sensitivity of the NPV to changes in the quantity sold of the new club, we simply need to change the quantity sold. We will choose 60,000 units, but the choice is irrelevant as the sensitivity will be the same no matter what quantity we choose.

| Sales | |

New clubs | $700 ´ 60,000 = $42,000,000 | |

Exp. clubs | $1,100 ´ (– 13,000) = –14,300,000 | |

Cheap clubs | $400 ´ 10,000 = 4,000,000 | |

$31,700,000 |

Var. costs | ||

New clubs | $320 ´ 60,000 = $19,200,000 | |

Exp. clubs | $600 ´ (–13,000) = –7,800,000 | |

Cheap clubs | $180 ´ 10,000 = 1,800,000 | |

$13,200,000 |

The pro forma income statement will be:

| Sales | $31,700,000 |

Variable costs | 13,200,000 | |

Costs | 7,500,000 | |

Depreciation | 2,600,000 | |

EBT | 8,400,000 | |

Taxes | 3,360,000 | |

Net income | $ 5,040,000 |

Using the bottom up OCF calculation, we get:

OCF = NI + Depreciation = $5,040,000 + 2,600,000

OCF = $7,640,000

The NPV at this quantity is:

NPV = –$18.2M – $0.95M + $7.64(PVIFA_{14%,7}) + $0.95M/1.14^{7}

NPV = $13,992,304.43

So, the sensitivity of the NPV to changes in the quantity sold is:

DNPV/DQ = ($13,992,304.43 – 9,103,636.91)/(60,000 – 55,000)

DNPV/DQ = $977.73

For an increase (decrease) of one set of clubs sold per year, the NPV increases (decreases) by $977.73.

* Challenge*

**23.*** a.* The tax shield definition of OCF is:

OCF = [(P – v)Q – FC ](1 – t_{C}) + t_{C}D

Rearranging and solving for Q, we find:

(OCF – t_{C}D)/(1 – t_{C}) = (P – v)Q – FC

Q = {FC + [(OCF – t_{C}D)/(1 – t_{C})]}/(P – v)

*b.* The cash breakeven is:

Q_{C} = $500,000/($40,000 – 20,000)

Q_{C} = 25

And the accounting breakeven is:

Q_{A} = {$500,000 + [($700,000 – $700,000(0.38))/0.62]}/($40,000 – 20,000)

Q_{A} = 60

The financial breakeven is the point at which the NPV is zero, so:

OCF_{F} = $3,500,000/PVIFA_{20%,5}

OCF_{F} = $1,170,328.96

So:

Q_{F} = [FC + (OCF – t_{C} × D)]/(P – v)

Q_{F} = {$500,000 + [$1,170,328.96 – .35($700,000)]}/($40,000 – 20,000)

Q_{F} = 97.93 » 98

*c.* At the accounting break-even point, the net income is zero. This using the bottom up definition of OCF:

OCF = NI + D

We can see that OCF must be equal to depreciation. So, the accounting breakeven is:

Q_{A} = {FC + [(D – t_{C}D)/(1 – t)]}/(P – v)

Q_{A} = (FC + D)/(P – v)

Q_{A} = (FC + OCF)/(P – v)

The tax rate has cancelled out in this case.

**24.** The DOL is expressed as:

DOL = %DOCF / %DQ

DOL = {[(OCF_{1 }– OCF_{0})/OCF_{0}] / [(Q_{1} – Q_{0})/Q_{0}]}

The OCF for the initial period and the first period is:

OCF_{1} = [(P – v)Q_{1} – FC](1 – t_{C}) + t_{C}D

OCF_{0} = [(P – v)Q_{0} – FC](1 – t_{C}) + t_{C}D

The difference between these two cash flows is:

OCF_{1 }– OCF_{0} = (P – v)(1 – t_{C})(Q_{1 }– Q_{0})

Dividing both sides by the initial OCF we get:

(OCF_{1 }– OCF_{0})/OCF_{0} = (P – v)( 1– t_{C})(Q_{1 }– Q_{0}) / OCF_{0}

Rearranging we get:

[(OCF_{1 }– OCF_{0})/OCF0][(Q_{1 }– Q_{0})/Q_{0}] = [(P – v)(1 – t_{C})Q_{0}]/OCF_{0} = [OCF_{0 }– t_{C}D + FC(1 – t)]/OCF_{0}

DOL = 1 + [FC(1 – t) – t_{C}D]/OCF_{0}

**25.** *a*. Using the tax shield approach, the OCF is:

OCF = [($230 – 210)(40,000) – $450,000](0.62) + 0.38($1,700,000/5)

OCF = $346,200

And the NPV is:

NPV = –$1.7M – 450K + $346,200(PVIFA_{13%,5}) + [$450K + $500K(1 – .38)]/1.13^{5}

NPV = –$519,836.99

*b*. In the worst-case, the OCF is:

OCF_{worst} = {[($230)(0.9) – 210](40,000) – $450,000}(0.62) + 0.38($1,955,000/5)

OCF_{worst} = –$204,820

And the worst-case NPV is:

NPV_{worst} = –$1,955,000 – $450,000(1.05) + –$204,820(PVIFA_{13%,5}) +

[$450,000(1.05) + $500,000(0.85)(1 – .38)]/1.13^{5}

NPV_{worst} = –$2,748,427.99

The best-case OCF is:

OCF_{best} = {[$230(1.1) – 210](40,000) – $450,000}(0.62) + 0.38($1,445,000/5)

OCF_{best} = $897,220

And the best-case NPV is:

NPV_{best} = – $1,445,000 – $450,000(0.95) + $897,220(PVIFA_{13%,5}) +

[$450,000(0.95) + $500,000(1.15)(1 – .38)]/1.13^{5}

NPV_{best} = $1,708,754.02

**26.** To calculate the sensitivity to changes in quantity sold, we will choose a quantity of 41,000. The OCF at this level of sale is:

OCF = [($230 – 210)(41,000) – $450,000](0.62) + 0.38($1,700,000M/5)

OCF = $358,600

The sensitivity of changes in the OCF to quantity sold is:

DOCF/DQ = ($358,600 – 346,200)/(41,000 – 40,000)

DOCF/DQ = +$12.40

The NPV at this level of sales is:

NPV = –$1.7M – $450,000 + $358,600(PVIFA_{13%,5}) + [$450K + $500K(1 – .38)]/1.13^{5}

NPV = –$476,223.32

And the sensitivity of NPV to changes in the quantity sold is:

DNPV/DQ = (–$476,223.32 – (–519,836.99))/(41,000 – 40,000)

DNPV/DQ = +$43.61

You wouldn’t want the quantity to fall below the point where the NPV is zero. We know the NPV changes $43.61 for every unit sale, so we can divide the NPV for 40,000 units by the sensitivity to get a change in quantity. Doing so, we get:

–$519,836.99 = $43.61(DQ)

DQ = –11,919

For a zero NPV, we need to increase sales by 11,919 units, so the minimum quantity is:

Q_{Min} = 40,000 + 11,919

Q_{Min} = 51,919

**27.** At the cash breakeven, the OCF is zero. Setting the tax shield equation equal to zero and solving for the quantity, we get:

OCF = 0 = [($230 – 210)Q_{C} – $450,000](0.62) + 0.38($1,700,000/5)

Q_{C} = 12,081

The accounting breakeven is:

Q_{A} = [$450,000 + ($1,700,000/5)]/($230 – 210)

Q_{A} = 39,500

From Problem 26, we know the financial breakeven is 51,919 units.

**28.** Using the tax shield approach to calculate the OCF, the DOL is:

DOL = 1 + [$450,000(1 – 0.38) – 0.38($1,700,000/5)]/ $346,200

DOL = 1.43270

Thus a 1% rise leads to a 1.43270% rise in OCF. If Q rises to 41,000, then

The percentage change in quantity is:

DQ = (41,000 – 40,000)/40,000 = .0250 or 2.50%

So, the percentage change in OCF is:

%DOCF = 2.50%(1.43270)

%DOCF = 3.5817%

From Problem 26:

DOCF/OCF = ($358,600 – 346,200)/$346,200

DOCF/OCF = 0.035817

In general, if Q rises by 1 unit, OCF rises by 3.5817%.