Project Analysis and Evaluation Master Class

Solution:

a. Accounting Break-Even:

At the accounting break-even level, net income is zero. The annual operating cash flow (OCF) is equal to the annual depreciation, $D = I/N$.

• IRR: The IRR is the discount rate that makes the NPV equal to zero. In this case, the OCF is $I/N$ for $N$ years, and the initial investment is $I$. The payback period is $N$ years. Since the cash flows exactly recover the initial investment, the IRR is 0%.
• Payback Period: The payback period is the time it takes for cumulative cash flows to equal the initial investment. With OCF = $I/N$ for $N$ years, the total cash inflow is $N \times (I/N) = I$. Thus, the payback period is exactly $N$ years (the project's life).
• NPV: With a required return of $R > 0$ and an IRR of 0%, the NPV must be negative. The present value of the cash flows is $\sum_{t=1}^{N} \frac{I/N}{(1+R)^t} < I$, so $NPV = -I + \text{PV of CF} < 0$.

b. Cash Break-Even:

At the cash break-even level, OCF is zero. There are no cash inflows to the project after the initial investment.

• IRR: The IRR is the rate that makes the present value of the OCFs equal to the initial investment. With zero cash inflows, this can only be achieved at a rate of -100%.
• Payback Period: Since there are no cash inflows after the initial investment, the payback period is never.
• NPV: Since there are no cash inflows, the NPV is simply the negative of the initial investment: $-I$.

c. Financial Break-Even:

At the financial break-even level, the NPV is zero. The project earns exactly its required return, $R$.

• IRR: By definition, the IRR is the discount rate that makes NPV = 0. Therefore, the IRR is equal to the required return, $R$.
• Payback Period: The payback period will be less than $N$ years if $R > 0$. However, the discounted payback period will be exactly $N$ years.
• NPV: By definition, the NPV is exactly zero.

Solution:

The sensitivity of OCF to changes in quantity sold is the change in OCF for a one-unit change in quantity. This is equal to the contribution margin on an after-tax basis. First, find the contribution margin per unit: $P-v = \$28 - \$16 = \$12$.

Now, adjust for the tax rate: $$ \text{OCF Sensitivity} = (P-v) \times (1-T_C) $$

$$ \text{OCF Sensitivity} = (\$12) \times (1 - 0.21) = \$12 \times 0.79 = \textbf{\$9.48 per unit} $$

Solution:

The formula for the degree of operating leverage (DOL) is: $$ DOL = 1 + \frac{FC}{OCF} $$

DOL at the given level of output (61,000 units):

First, calculate the OCF at 61,000 units. Depreciation is $D = \$655,000 / 4 = \$163,750$.

$$ \text{OCF} = [(P-v)Q - FC] \times (1-T_C) + D \times T_C $$

$$ \text{OCF} = [(\$28 - \$16) \times 61,000 - \$245,000] \times (1-0.21) + \$163,750 \times 0.21 $$

$$ \text{OCF} = [\$732,000 - \$245,000] \times 0.79 + \$34,387.50 $$

$$ \text{OCF} = \$487,000 \times 0.79 + \$34,387.50 = \$384,730 + \$34,387.50 = \$419,117.50 $$

Now, calculate DOL: $$ DOL = 1 + \frac{\$245,000}{\$419,117.50} = 1 + 0.585 = \textbf{1.585} $$

DOL at the accounting break-even level:

At the accounting break-even point, OCF = Depreciation ($D = \$163,750$).

$$ DOL = 1 + \frac{FC}{D} = 1 + \frac{\$245,000}{\$163,750} = 1 + 1.496 = \textbf{2.496} $$

Solution:

a. Upper/Lower Bounds, Base Case, and Scenario Analysis

First, calculate annual depreciation: $D = \$1,675,000 / 4 = \$418,750$.

Next, find the upper and lower bounds for the variables:

• Unit Sales: $195 \pm 10\% \implies [175.5, 214.5]$
• Variable Cost: $\$9,400 \pm 10\% \implies [\$8,460, \$10,340]$
• Fixed Costs: $\$550,000 \pm 10\% \implies [\$495,000, \$605,000]$

Base-Case NPV:

$$ \text{OCF} = [(P-v)Q - FC](1-T_C) + D \times T_C $$

$$ \text{OCF} = [(\$16,300-\$9,400) \times 195 - \$550,000](1-0.21) + \$418,750 \times 0.21 $$

$$ \text{OCF} = [(\$6,900) \times 195 - \$550,000] \times 0.79 + \$87,937.50 $$

$$ \text{OCF} = [\$1,345,500 - \$550,000] \times 0.79 + \$87,937.50 = \$795,500 \times 0.79 + \$87,937.50 = \textbf{\$716,332.50} $$

The 4-year annuity factor at 12% is 3.0373. $$ \text{NPV} = -\$1,675,000 + \$716,332.50 \times 3.0373 = \textbf{\$499,353.79} $$

Best-Case NPV: (High sales, low costs)

$$ \text{OCF} = [(\$16,300-\$8,460) \times 214.5 - \$495,000] \times 0.79 + \$87,937.50 $$

$$ \text{OCF} = [\$7,840 \times 214.5 - \$495,000] \times 0.79 + \$87,937.50 = \textbf{\$1,241,863.40} $$

$$ \text{NPV} = -\$1,675,000 + \$1,241,863.40 \times 3.0373 = \textbf{\$2,096,938.90} $$

Worst-Case NPV: (Low sales, high costs)

$$ \text{OCF} = [(\$16,300-\$10,340) \times 175.5 - \$605,000] \times 0.79 + \$87,937.50 $$

$$ \text{OCF} = [\$5,960 \times 175.5 - \$605,000] \times 0.79 + \$87,937.50 = \textbf{-\$21,059.50} $$

$$ \text{NPV} = -\$1,675,000 + (-\$21,059.50) \times 3.0373 = \textbf{-\$1,739,014.28} $$

b. Sensitivity to Fixed Costs:

The impact on OCF from a change in fixed costs is: $$ \text{OCF change} = -\Delta FC \times (1-T_C) = -(-\$1) \times 0.79 = \$0.79 $$

The sensitivity of NPV to fixed costs is: $$ \text{NPV Sensitivity} = \$0.79 \times 3.0373 = \textbf{\$2.40 per dollar of fixed costs} $$

c. Cash Break-Even (ignoring taxes):

$$ Q_{Cash} = \frac{FC}{P-v} = \frac{\$550,000}{\$16,300 - \$9,400} = \frac{\$550,000}{\$6,900} = \textbf{79.71 units} $$

d. Accounting Break-Even & DOL:

$$ Q_{Acc} = \frac{FC + D}{P - v} = \frac{\$550,000 + \$418,750}{\$6,900} = \frac{\$968,750}{\$6,900} = \textbf{140.4 units} $$

At the accounting break-even point, OCF = $D = \$418,750$. $$ DOL = 1 + \frac{FC}{OCF} = 1 + \frac{\$550,000}{\$418,750} = 1 + 1.313 = \textbf{2.313} $$

Solution:

Step 1: Identify Incremental Cash Flows.

The marketing study and R&D costs are sunk costs and are irrelevant. The relevant cash flows are the new project's sales, erosion, synergy, fixed costs, and initial investment.

Incremental Revenue:

• New Clubs: $55,000 \times \$815 = \$44,825,000$
• Erosion (High-priced): $-10,000 \times \$1,345 = -\$13,450,000$
• Synergy (Cheap clubs): $12,000 \times \$445 = \$5,340,000$
• Total Incremental Revenue: $\$44,825,000 - \$13,450,000 + \$5,340,000 = \$36,715,000$

Incremental Variable Costs:

• New Clubs: $55,000 \times \$365 = \$20,075,000$
• Erosion (High-priced): $-10,000 \times \$730 = -\$7,300,000$
• Synergy (Cheap clubs): $12,000 \times \$210 = \$2,520,000$
• Total Incremental Variable Costs: $\$20,075,000 - \$7,300,000 + \$2,520,000 = \$15,295,000$

Step 2: Calculate OCF.

Depreciation: $D = \$39,200,000 / 7 = \$5,600,000$ per year.

$$ \text{OCF} = [(\text{Inc. Rev.} - \text{Inc. VC}) - FC - D] \times (1-T_C) + D $$

$$ \text{OCF} = [(\$36,715,000 - \$15,295,000) - \$9,450,000 - \$5,600,000] \times (1-0.25) + \$5,600,000 $$

$$ \text{OCF} = [\$21,420,000 - \$9,450,000 - \$5,600,000] \times 0.75 + \$5,600,000 $$

$$ \text{OCF} = [\$6,370,000] \times 0.75 + \$5,600,000 = \$4,777,500 + \$5,600,000 = \textbf{\$10,377,500} $$

Step 3: Calculate Initial and Terminal Cash Flows.

Initial Outlay (Year 0): $$ \text{Initial Investment} = -(\text{Plant} + \text{NWC}) = -(\$39,200,000 + \$1,850,000) = \textbf{-\$41,050,000} $$

Terminal Cash Flow (Year 7): The NWC is recovered. There is no salvage value. $$ \text{Terminal CF} = \text{NWC Recovery} = \textbf{\$1,850,000} $$

Step 4: Calculate Payback, NPV, and IRR.

The annual OCF is \$10,377,500. The payback period is the initial investment divided by the annual OCF. $$ \text{Payback} = \frac{\$41,050,000}{\$10,377,500} = \textbf{3.96 \text{ years}} $$

For NPV and IRR, we use the cash flows:

• Year 0: -\$41,050,000
• Years 1-6: \$10,377,500
• Year 7: \$10,377,500 + \$1,850,000 = \$12,227,500

The NPV at a 10% required return is: $$ \text{NPV} = -\$41,050,000 + \sum_{t=1}^{6} \frac{\$10,377,500}{(1.10)^t} + \frac{\$12,227,500}{(1.10)^7} = \textbf{\$4,992,200} $$

The IRR is the rate that makes NPV = 0, which is approximately 12.5%. Since NPV > 0 and IRR > required return, the project is acceptable.

Solution:

The variables with $\pm10\%$ uncertainty are the unit sales figures and the fixed costs.

Best-Case Scenario:

This means higher new club sales, lower erosion, higher synergy, and lower fixed costs.

• New Club Sales: $55,000 \times 1.10 = 60,500$
• Erosion (High-priced): $10,000 \times 0.90 = 9,000$ (loss)
• Synergy (Cheap clubs): $12,000 \times 1.10 = 13,200$ (gain)
• Fixed Costs: $\$9.45 \text{ million} \times 0.90 = \$8.505 \text{ million}$

$$ \text{OCF} = [(\text{Inc. Rev.} - \text{Inc. VC}) - FC - D] \times (1-T_C) + D $$

Incremental Revenue = $(60,500 \times \$815) - (9,000 \times \$1,345) + (13,200 \times \$445) = \$49,307,500 - \$12,105,000 + \$5,874,000 = \$43,076,500$

Incremental Variable Cost = $(60,500 \times \$365) - (9,000 \times \$730) + (13,200 \times \$210) = \$22,082,500 - \$6,570,000 + \$2,772,000 = \$18,284,500$

$$ \text{OCF}_{Best} = [(\$43,076,500 - \$18,284,500) - \$8,505,000 - \$5,600,000] \times 0.75 + \$5,600,000 = \textbf{\$12,231,000} $$

$$ \text{Best-Case NPV} = -\$41,050,000 + \$12,231,000 \times A_{F, 10\%, 6} + \frac{\$12,231,000 + \$1,850,000}{1.10^7} = \textbf{\$15,446,801} $$

Worst-Case Scenario:

This means lower new club sales, higher erosion, lower synergy, and higher fixed costs.

• New Club Sales: $55,000 \times 0.90 = 49,500$
• Erosion (High-priced): $10,000 \times 1.10 = 11,000$ (loss)
• Synergy (Cheap clubs): $12,000 \times 0.90 = 10,800$ (gain)
• Fixed Costs: $\$9.45 \text{ million} \times 1.10 = \$10.395 \text{ million}$

Incremental Revenue = $(49,500 \times \$815) - (11,000 \times \$1,345) + (10,800 \times \$445) = \$40,392,500 - \$14,795,000 + \$4,806,000 = \$30,403,500$

Incremental Variable Cost = $(49,500 \times \$365) - (11,000 \times \$730) + (10,800 \times \$210) = \$18,067,500 - \$8,030,000 + \$2,268,000 = \$12,305,500$

$$ \text{OCF}_{Worst} = [(\$30,403,500 - \$12,305,500) - \$10,395,000 - \$5,600,000] \times 0.75 + \$5,600,000 = \textbf{\$8,527,500} $$

$$ \text{Worst-Case NPV} = -\$41,050,000 + \$8,527,500 \times A_{F, 10\%, 6} + \frac{\$8,527,500 + \$1,850,000}{1.10^7} = \textbf{-\$6,967,314} $$

Solution:

Sensitivity to Price:

The NPV is sensitive to changes in the price of the new clubs through the change in incremental revenue, which then affects OCF. The change in OCF for a \$1 change in price is: $$ \text{OCF Change per \$1} = 1 \times Q \times (1-T_C) = 55,000 \times 0.75 = \textbf{\$41,250} $$

The sensitivity of NPV to price is the change in OCF per dollar multiplied by the present value of an annuity factor (PVAF) for 7 years at 10%. The 7-year PVAF at 10% is 4.8684. $$ \text{NPV Sensitivity to Price} = \$41,250 \times 4.8684 = \textbf{\$200,742} $$

Sensitivity to Quantity Sold:

The change in OCF for a one-unit change in quantity sold of the new clubs is the after-tax contribution margin of the new clubs. $$ \text{OCF Change per unit} = (P_{new}-v_{new}) \times (1-T_C) = (\$815-\$365) \times 0.75 = \$450 \times 0.75 = \textbf{\$337.50} $$

The sensitivity of NPV to quantity is the change in OCF per unit multiplied by the 7-year PVAF at 10%. $$ \text{NPV Sensitivity to Quantity} = \$337.50 \times 4.8684 = \textbf{\$1,643.08} $$

Solution:

a. Break-even miles per year (no time value of money):

Total difference in cost = $(\text{Initial Price Diff}) + (\text{Annual Cost Diff} \times \text{Years})$

Total difference in cost = $\$5,900 + (\$250 \times 6) = \$5,900 + \$1,500 = \$7,400$

Fuel consumption per mile for hybrid: $1/39$ gallons/mile. Fuel consumption for gasoline SUV: $1/30$ gallons/mile.

Cost difference per mile = $\$2.85 \times (1/30 - 1/39) = \$2.85 \times (0.03333 - 0.02564) = \$2.85 \times 0.00769 = \$0.02192$

Break-even total miles = $\frac{\$7,400}{\$0.02192} = 337,500$ miles. Break-even miles per year = $\frac{337,500}{6} = \textbf{56,250 miles per year}$

b. Break-even price per gallon (no time value of money):

Total miles driven = $15,000 \times 6 = 90,000$ miles.

Total gallons saved = $90,000 \times (1/30 - 1/39) = 90,000 \times 0.00769 = 692.31$ gallons.

Break-even price per gallon = $\frac{\text{Total Cost Difference}}{\text{Total Gallons Saved}} = \frac{\$7,400}{692.31} = \textbf{\$10.69 per gallon}$

c. Reworking with time value of money (10%):

First, find the equivalent annual cost (EAC) of the initial investment and fixed costs. Initial investment difference is \$5,900. The 6-year annuity factor at 10% is 4.3553.

Annual savings from hybrid = $(\text{Miles per year} \times \text{Cost diff per mile}) - \text{Annual Cost Diff}$

$$ \text{EAC of Costs} = \frac{\text{Initial Investment Diff}}{\text{Annuity Factor}} + \text{Annual Cost Diff} = \frac{\$5,900}{4.3553} + \$250 = \$1,354.79 + \$250 = \$1,604.79 $$

Break-even annual fuel savings needed = \$1,604.79. Let $M$ be the miles per year. $$ \text{Annual fuel savings} = M \times (\$2.85 \times (1/30 - 1/39)) = M \times \$0.02192 $$

$$ 0.02192 M = \$1,604.79 \Rightarrow M = \textbf{73,211 miles per year} $$

For part (b), with 15,000 miles per year, let $P$ be the price per gallon. $$ \text{Annual fuel savings} = 15,000 \times P \times (1/30-1/39) = 15,000 \times P \times 0.00769 = 115.385 P $$

$$ 115.385 P = \$1,604.79 \Rightarrow P = \textbf{\$13.91 per gallon} $$

d. Assumption about resale value:

The analysis assumes the resale value of each car is equal at the end of the six-year period. If the hybrid had a higher resale value, it would make the hybrid more financially attractive.

Solution:

a. Cash flow per plane (accounting break-even):

At the accounting break-even point, OCF is equal to depreciation. The total initial investment is \$13 billion. The accounting break-even is for the project as a whole, so the total OCF must be equal to the total depreciation. We assume the total number of units sold (249) is for the project's life.

$$ \text{Total Depreciation} = \$13 \text{ billion} $$

$$ \text{Total OCF} = \$13 \text{ billion} $$

$$ \text{Cash flow per plane} = \frac{\$13 \text{ billion}}{249} = \textbf{\$52,208,835} $$

b. Per-year sales for a perpetual stream (financial break-even):

For a perpetual stream of cash flows (a perpetuity), the NPV is: $$ NPV = -I + \frac{OCF}{R} $$

For a zero NPV, the required OCF is: $$ OCF = I \times R = \$13 \text{ billion} \times 0.20 = \$2.6 \text{ billion per year} $$

This is the required annual OCF. Now we need to find the OCF per plane. From part (a), the OCF per plane is \$52,208,835. So the number of planes per year is: $$ \text{Planes per year} = \frac{\$2.6 \text{ billion}}{\$52,208,835} = \textbf{49.8 \text{ or 50 planes per year}} $$

c. Per-year sales for a 10-year project:

We need to find the annual OCF that results in a zero NPV over 10 years at a 20% required return. The 10-year annuity factor at 20% is 4.1925. $$ OCF = \frac{I}{A_F} = \frac{\$13 \text{ billion}}{4.1925} = \$3,099,224,792 \text{ per year} $$

Now, find the number of planes per year: $$ \text{Planes per year} = \frac{\$3,099,224,792}{\$52,208,835} = \textbf{59.36 \text{ or 60 planes per year}} $$

Solution:

a. General relationship with taxes:

We start with the tax shield approach for OCF: $$ OCF = [(P-v)Q - FC] \times (1-T_C) + D \times T_C $$

Rearrange to solve for $(P-v)Q - FC$: $$ OCF - D \times T_C = [(P-v)Q - FC] \times (1-T_C) $$

$$ \frac{OCF - D \times T_C}{1-T_C} = (P-v)Q - FC $$

Now, isolate Q: $$ (P-v)Q = FC + \frac{OCF - D \times T_C}{1-T_C} $$

$$ Q = \frac{FC + \frac{OCF - D \times T_C}{1-T_C}}{P-v} $$

b. Break-even points for Wettway (21% tax rate):

Wettway data: $P = \$40,000, v = \$20,000, FC = \$500,000, D = \$700,000$.

Cash Break-Even ($OCF = 0$):

$$ Q = \frac{\$500,000 + \frac{0 - \$700,000 \times 0.21}{1-0.21}}{\$20,000} = \frac{\$500,000 - \$186,076}{20,000} = \textbf{15.696 units or 16 planes} $$

Accounting Break-Even ($OCF = D$):

$$ Q = \frac{\$500,000 + \frac{\$700,000 - \$700,000 \times 0.21}{1-0.21}}{\$20,000} = \frac{\$500,000 + \$700,000}{20,000} = \textbf{60 units} $$

Financial Break-Even ($NPV=0$):

First, find the required OCF for a zero NPV. $$ OCF = \frac{I}{A_F} = \frac{\$3,500,000}{2.9906} = \$1,170,300.94 $$

$$ Q = \frac{\$500,000 + \frac{\$1,170,300.94 - \$700,000 \times 0.21}{1-0.21}}{\$20,000} = \frac{\$500,000 + \frac{\$1,023,300.94}{0.79}}{20,000} = \frac{\$500,000 + \$1,295,317.65}{20,000} = \textbf{89.77 units or 90 planes} $$

c. Why accounting break-even is the same:

Algebraically, when OCF = D, the numerator of the complex fraction becomes: $$ \frac{D - D \times T_C}{1-T_C} = \frac{D(1-T_C)}{1-T_C} = D $$

The formula simplifies to $Q = (FC+D) / (P-v)$, which is the original formula without taxes. The accounting break-even is unaffected by taxes because, at that point, pretax income is zero, so taxes are also zero.

Solution:

Derivation:

We start with the definition of DOL: $$ DOL = \frac{\% \Delta OCF}{\% \Delta Q} $$

The change in OCF for a one-unit change in Q is: $$ \Delta OCF = (P-v) \times (1-T_C) $$

The change in Q is 1 unit. We also know that OCF is: $$ OCF = [(P-v)Q - FC - D] \times (1-T_C) + D $$

Substitute these into the DOL formula and simplify.

$$ DOL = \frac{\frac{\Delta OCF}{OCF}}{\frac{\Delta Q}{Q}} = \frac{\frac{(P-v)(1-T_C)}{OCF}}{\frac{1}{Q}} = \frac{(P-v)(1-T_C)Q}{OCF} $$

Substitute the after-tax OCF formula: $$ DOL = \frac{[(P-v)Q]}{[(P-v)Q - FC - D](1-T_C)+D} \times (1-T_C) $$

We can rewrite the numerator using the after-tax operating profit as $[(P-v)Q - FC](1-T_C) + FC(1-T_C)$. Then, we can simplify the expression to the given formula. $$ DOL = \frac{[(P-v)Q - FC](1-T_C) + FC(1-T_C)}{[(P-v)Q - FC - D](1-T_C)+D} = 1 + \frac{FC(1-T_C) - DT_C}{OCF} $$

Interpretation:

This formula shows that the degree of operating leverage is a function of the after-tax fixed costs. The term $FC(1-T_C)$ represents the after-tax portion of fixed costs. The term $DT_C$ represents the depreciation tax shield. This means that a project's cash flow is leveraged by the after-tax fixed costs, but this leverage is partially offset by the tax savings from depreciation. When there are no taxes ($T_C = 0$), the formula simplifies to $1 + FC/OCF$, which is the original formula.

Solution:

a. Base Case OCF and NPV:

Depreciation: $D = \$3,100,000 / 5 = \$620,000$. After-tax salvage: The book value is zero, so the entire \$400,000 is a taxable gain. $400,000 \times (1-0.22) = \$312,000$.

$$ \text{OCF} = [(P-v)Q - FC](1-T_C) + D \times T_C $$

$$ \text{OCF} = [(\$295-\$185) \times 20,000 - \$925,000] \times (1-0.22) + \$620,000 \times 0.22 $$

$$ \text{OCF} = [\$110 \times 20,000 - \$925,000] \times 0.78 + \$136,400 $$

$$ \text{OCF} = [\$2,200,000 - \$925,000] \times 0.78 + \$136,400 = \$1,275,000 \times 0.78 + \$136,400 = \textbf{\$1,129,900} $$

The initial investment includes NWC: $$ \text{Initial Outlay} = -(\$3,100,000 + \$380,000) = -\$3,480,000 $$

The 5-year annuity factor at 13% is 3.5172. $$ \text{NPV} = -\$3,480,000 + \$1,129,900 \times 3.5172 + \frac{\$380,000 + (\$400,000(1-0.22))}{1.13^5} $$

$$ \text{NPV} = -\$3,480,000 + \$3,975,417.28 + \frac{\$692,000}{1.8424} = -\$3,480,000 + \$3,975,417.28 + \$375,699.04 = \textbf{\$871,116.32} $$

Yes, the project should be pursued as the NPV is positive.

b. Worst-Case and Best-Case Scenario:

Worst-Case: High costs, low price, low salvage, high initial investment.

Initial cost: $\$3,100,000 \times 1.15 = \$3,565,000$. Salvage: $\$400,000 \times 0.85 = \$340,000$. Price: $\$295 \times 0.90 = \$265.50$. NWC: $\$380,000 \times 1.05 = \$399,000$.

Depreciation: $\$3,565,000 / 5 = \$713,000$. OCF: $$ \text{OCF} = [(\$265.50-\$185) \times 20,000 - \$925,000] \times 0.78 + \$713,000 \times 0.22 = \textbf{\$868,140} $$

Initial Outlay: $-(\$3,565,000 + \$399,000) = -\$3,964,000$. Terminal CF: $\$399,000 + \$340,000(1-0.22) = \$664,200$.

$$ \text{Worst-Case NPV} = -\$3,964,000 + \$868,140 \times 3.5172 + \frac{\$664,200}{1.13^5} = \textbf{\$58,349.52} $$

Best-Case: Low costs, high price, high salvage, low initial investment.

Initial cost: $\$3,100,000 \times 0.85 = \$2,635,000$. Salvage: $\$400,000 \times 1.15 = \$460,000$. Price: $\$295 \times 1.10 = \$324.50$. NWC: $\$380,000 \times 0.95 = \$361,000$.

Depreciation: $\$2,635,000 / 5 = \$527,000$. OCF: $$ \text{OCF} = [(\$324.50-\$185) \times 20,000 - \$925,000] \times 0.78 + \$527,000 \times 0.22 = \textbf{\$1,363,200} $$

Initial Outlay: $-(\$2,635,000 + \$361,000) = -\$2,996,000$. Terminal CF: $\$361,000 + \$460,000(1-0.22) = \$720,800$.

$$ \text{Best-Case NPV} = -\$2,996,000 + \$1,363,200 \times 3.5172 + \frac{\$720,800}{1.13^5} = \textbf{\$2,442,887.89} $$

Even in the worst-case scenario, the NPV is positive. Yes, you should still pursue the project.

Solution:

OCF Sensitivity:

The sensitivity of OCF to a change in quantity is the after-tax contribution margin. $$ \text{OCF Sensitivity} = (P-v) \times (1-T_C) = (\$295-\$185) \times (1-0.22) = \$110 \times 0.78 = \textbf{\$85.80 per ton} $$

NPV Sensitivity:

The sensitivity of NPV to a change in quantity is the OCF sensitivity multiplied by the 5-year annuity factor at 13% (3.5172). $$ \text{NPV Sensitivity} = \$85.80 \times 3.5172 = \textbf{\$301.89 per ton} $$

Minimum level of output:

To find the minimum level of output, we can calculate the financial break-even quantity. This is the point at which the NPV is zero. The required annual OCF for a zero NPV is: $$ OCF^* = \frac{\text{Initial Outlay}}{A_F} = \frac{\$3,480,000 - \frac{\$380,000 + \$312,000}{1.13^5}}{3.5172} = \frac{\$3,480,000 - \$375,699}{3.5172} = \$882,492 $$

Now, we can find the quantity using the OCF formula: $$ \$882,492 = [(\$295-\$185)Q - \$925,000](0.78) + \$620,000(0.22) $$

$$ \$882,492 = [\$110Q - \$925,000](0.78) + \$136,400 $$

$$ \$746,092 = \$85.8Q - \$721,500 $$

$$ \$1,467,592 = \$85.8Q \Rightarrow Q = \textbf{17,105 tons} $$

You wouldn't want to operate below this level because it would result in a negative NPV, meaning the project would not be generating a return equal to its cost of capital.

Solution:

First, we need the formula from Problem 25: $$ Q = \frac{FC + \frac{OCF - T_C \times D}{1-T_C}}{P-v} $$

From Problem 27, we have: $P-v = \$110$, $FC = \$925,000$, $D = \$620,000$, and $T_C = 0.22$.

Cash Break-Even ($OCF = 0$):

$$ Q = \frac{\$925,000 + \frac{0 - 0.22 \times \$620,000}{1-0.22}}{\$110} = \frac{\$925,000 - \$174,051.28}{\$110} = \textbf{6,826.8 units or 6,827 tons} $$

Accounting Break-Even ($OCF = D$):

As verified in Problem 25, the formula simplifies to: $$ Q = \frac{FC + D}{P-v} = \frac{\$925,000 + \$620,000}{\$110} = \frac{\$1,545,000}{\$110} = \textbf{14,045.45 units or 14,046 tons} $$

Financial Break-Even ($NPV = 0$):

From Problem 28, the OCF for a zero NPV is \$882,492. Using the formula from Problem 25: $$ Q = \frac{\$925,000 + \frac{\$882,492 - 0.22 \times \$620,000}{1-0.22}}{\$110} = \frac{\$925,000 + \frac{\$746,092}{0.78}}{\$110} = \frac{\$925,000 + \$956,528.2}{110} = \textbf{17,104.8 units or 17,105 tons} $$

Solution:

Degree of Operating Leverage (DOL) at Base Case:

From Problem 27, the base-case OCF is \$1,129,900. Using the formula from Problem 26: $$ DOL = 1 + \frac{FC \times (1-T_C) - T_C \times D}{OCF} $$

$$ DOL = 1 + \frac{\$925,000(1-0.22) - 0.22(\$620,000)}{\$1,129,900} $$

$$ DOL = 1 + \frac{\$721,500 - \$136,400}{\$1,129,900} = 1 + \frac{\$585,100}{\$1,129,900} = 1 + 0.5178 = \textbf{1.5178} $$

Comparison to Sensitivity:

The DOL of 1.5178 means that a 1% change in quantity will result in a 1.5178% change in OCF. From Problem 28, the OCF sensitivity was \$85.80 per ton, and the base case quantity is 20,000 tons. A 1% change in quantity is 200 tons. The change in OCF would be $200 \times \$85.80 = \$17,160$. The base case OCF is \$1,129,900. The percentage change is $\$17,160 / \$1,129,900 = 1.518\%$. The numbers are very close, with the minor difference due to rounding.

Verification with a New Quantity:

Let's use a new quantity of 21,000 tons (a 5% increase from the base case of 20,000). The DOL method predicts a 7.589% increase in OCF ($1.5178 \times 5\% = 7.589\%$). The new OCF should be $1.07589 \times \$1,129,900 = \$1,215,690$.

The sensitivity method predicts an increase of $1,000 \times \$85.80 = \$85,800$. The new OCF would be $\$1,129,900 + \$85,800 = \$1,215,700$. Both approaches yield essentially the same result, confirming their consistency.