Project Analysis and Evaluation Master Class

Solution:

a. Variable cost per unit

$$ v = \text{Variable Materials Cost} + \text{Variable Labor Cost} $$

$$ v = \$12.14 + \$6.89 = \textbf{\$19.03 per unit} $$

b. Total costs for the year

$$ \text{Total Costs} = (\text{Variable Cost per Unit} \times \text{Quantity}) + \text{Fixed Costs} $$

$$ \text{Total Costs} = (\$19.03 \times 210,000) + \$845,000 $$

$$ \text{Total Costs} = \$3,996,300 + \$845,000 = \textbf{\$4,841,300} $$

c. Cash break-even and accounting break-even

Cash Break-Even:

The cash break-even point occurs when OCF = 0. Since taxes are not mentioned, we can use the formula $Q = FC / (P-v)$.

$$ Q = \frac{\$845,000}{\$49.99 - \$19.03} = \frac{\$845,000}{\$30.96} = \textbf{27,294 units} $$

The company needs to sell 27,294 units to break even on a cash basis. Since total production is 210,000 units, the company is well above its cash break-even point.

Accounting Break-Even:

The accounting break-even point occurs when Net Income = 0. The formula is $Q = (FC + D) / (P-v)$.

$$ Q = \frac{\$845,000 + \$450,000}{\$49.99 - \$19.03} = \frac{\$1,295,000}{\$30.96} = \textbf{41,827 units} $$

Solution:

Total production costs:

First, find the variable cost per pair:

$$ v = \$45.17 + \$29.73 = \$74.90 $$

Now, calculate total costs:

$$ \text{Total Costs} = (v \times Q) + FC $$

$$ \text{Total Costs} = (\$74.90 \times 155,000) + \$2,150,000 = \$11,600,000 + \$2,150,000 = \textbf{\$13,750,000} $$

Marginal cost per pair:

The marginal cost is the cost to produce one additional unit. This is equal to the variable cost per pair.

$$ \text{Marginal Cost} = \textbf{\$74.90} $$

Average cost per pair:

$$ \text{Average Cost} = \frac{\text{Total Costs}}{\text{Quantity}} = \frac{\$13,750,000}{155,000} = \textbf{\$88.71} $$

Minimum acceptable total revenue for a one-time order:

The minimum acceptable revenue is the total variable cost of the additional units. The fixed costs are already covered by the existing production and are considered sunk for this decision.

$$ \text{Minimum Revenue} = \text{Marginal Cost} \times \text{Extra Quantity} $$

$$ \text{Minimum Revenue} = \$74.90 \times 5,000 = \textbf{\$374,500} $$

Solution:

Best-Case Scenario:

For the best-case, we use the most optimistic values. This means a higher price and quantity, and lower costs.

• Price: $\$1,220 \times (1 + 0.15) = \textbf{\$1,403}$
• Variable Costs: $\$380 \times (1 - 0.15) = \textbf{\$323}$
• Fixed Costs: $\$3.75 \text{ million} \times (1 - 0.15) = \textbf{\$3.1875 million}$
• Quantity: $90,000 \times (1 + 0.15) = \textbf{103,500 units}$

Worst-Case Scenario:

For the worst-case, we use the most pessimistic values. This means a lower price and quantity, and higher costs.

• Price: $\$1,220 \times (1 - 0.15) = \textbf{\$1,037}$
• Variable Costs: $\$380 \times (1 + 0.15) = \textbf{\$437}$
• Fixed Costs: $\$3.75 \text{ million} \times (1 + 0.15) = \textbf{\$4.3125 million}$
• Quantity: $90,000 \times (1 - 0.15) = \textbf{76,500 units}$

Solution:

You would address this concern by performing a **sensitivity analysis** on the price. This involves holding all other variables constant at their **base-case** values while varying the price within its pessimistic and optimistic range.

To calculate the impact, you would first determine the project's NPV at the base-case price of \$1,220. Then, you would calculate the NPV at the worst-case price of \$1,037 and the best-case price of \$1,403. You could also plot these values on a chart to visualize the sensitivity of the NPV to price changes. The steeper the line, the more sensitive the project's profitability is to price.

Solution:

a. Accounting Break-Even and DOL

First, calculate straight-line depreciation: $$ D = \frac{\$845,000}{8} = \$105,625 $$

Next, calculate the accounting break-even point ($Q_A$):

$$ Q_A = \frac{FC + D}{P - v} = \frac{\$950,000 + \$105,625}{\$53 - \$27} = \frac{\$1,055,625}{\$26} = \textbf{40,601 units} $$

At the accounting break-even point, OCF equals depreciation, so $OCF_A = D = \$105,625$. Now, calculate DOL at this point:

$$ DOL = 1 + \frac{FC}{OCF} = 1 + \frac{\$950,000}{\$105,625} = 1 + 8.99 = \textbf{9.99} $$

b. Base-Case Cash Flow, NPV, and Sensitivity to Quantity

First, calculate base-case OCF:

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

$$ \text{OCF} = [(\$53-\$27) \times 51,000 - \$950,000] \times (1-0.22) + \$105,625 \times 0.22 $$

$$ \text{OCF} = [\$1,326,000 - \$950,000] \times 0.78 + \$23,237.50 $$

$$ \text{OCF} = \$376,000 \times 0.78 + \$23,237.50 = \$293,280 + \$23,237.50 = \textbf{\$316,517.50} $$

Next, calculate NPV. The 8-year annuity factor at 10% is 5.3349.

$$ \text{NPV} = -I + OCF \times A_F = -\$845,000 + \$316,517.50 \times 5.3349 = \textbf{\$841,850} $$

To find the sensitivity of NPV to quantity sold, we calculate the change in OCF for a one-unit change in quantity. This is the contribution margin after-tax.

$$ \text{Change in OCF} = (P - v) \times (1 - T_C) = (\$53 - \$27) \times (1 - 0.22) = \$26 \times 0.78 = \$20.28 $$

The sensitivity of NPV to quantity is the change in OCF multiplied by the annuity factor.

$$ \text{NPV Sensitivity} = \$20.28 \times 5.3349 = \textbf{\$108.20} $$

A 500-unit decrease in quantity sold would result in an NPV decrease of $500 \times \$108.20 = \textbf{\$54,100}$.

c. Sensitivity of OCF to changes in variable cost

A \$1 decrease in variable costs is a positive change for the project. The impact on OCF is the tax-adjusted dollar change.

$$ \text{OCF Sensitivity} = -\text{Change in } v \times (1 - T_C) = -(-\$1) \times (1 - 0.22) = \textbf{\$0.78} $$

This means for every \$1 decrease in variable costs, the annual OCF increases by \$0.78. The NPV would increase by $\$0.78 \times 5.3349 = \$4.16$.

Solution:

Best-Case Scenario:

• Quantity: $51,000 \times 1.10 = 56,100$ units
• Price: $\$53 \times 1.10 = \$58.30$
• Variable Cost: $\$27 \times 0.90 = \$24.30$
• Fixed Costs: $\$950,000 \times 0.90 = \$855,000$

Now calculate OCF for the best case:

$$ \text{OCF} = [(\$58.30-\$24.30) \times 56,100 - \$855,000] \times (1-0.22) + \$105,625 \times 0.22 $$

$$ \text{OCF} = [\$34 \times 56,100 - \$855,000] \times 0.78 + \$23,237.50 $$

$$ \text{OCF} = [\$1,907,400 - \$855,000] \times 0.78 + \$23,237.50 $$

$$ \text{OCF} = \$1,052,400 \times 0.78 + \$23,237.50 = \$820,872 + \$23,237.50 = \$844,109.50 $$

$$ \text{Best-Case NPV} = -\$845,000 + \$844,109.50 \times 5.3349 = \textbf{\$3,656,530} $$

Worst-Case Scenario:

• Quantity: $51,000 \times 0.90 = 45,900$ units
• Price: $\$53 \times 0.90 = \$47.70$
• Variable Cost: $\$27 \times 1.10 = \$29.70$
• Fixed Costs: $\$950,000 \times 1.10 = \$1,045,000$

Now calculate OCF for the worst case:

$$ \text{OCF} = [(\$47.70-\$29.70) \times 45,900 - \$1,045,000] \times (1-0.22) + \$105,625 \times 0.22 $$

$$ \text{OCF} = [(\$18 \times 45,900) - \$1,045,000] \times 0.78 + \$23,237.50 $$

$$ \text{OCF} = [\$826,200 - \$1,045,000] \times 0.78 + \$23,237.50 $$

$$ \text{OCF} = (-\$218,800) \times 0.78 + \$23,237.50 = -\$170,664 + \$23,237.50 = -\$147,426.50 $$

$$ \text{Worst-Case NPV} = -\$845,000 - \$147,426.50 \times 5.3349 = \textbf{-\$1,630,227} $$

Solution:

Formulas:

$$ \text{Cash Break-Even } (Q_{Cash}) = \frac{FC}{P - v} $$

$$ \text{Accounting Break-Even } (Q_{Acc}) = \frac{FC + D}{P - v} $$

Case A:

$$ Q_{Cash} = \frac{\$7,200,000}{\$2,980 - \$1,640} = \frac{\$7,200,000}{\$1,340} = \textbf{5,373 units} $$

$$ Q_{Acc} = \frac{\$7,200,000 + \$2,900,000}{\$1,340} = \frac{\$10,100,000}{\$1,340} = \textbf{7,537 units} $$

Case B:

$$ Q_{Cash} = \frac{\$275,000}{\$44 - \$39} = \frac{\$275,000}{\$5} = \textbf{55,000 units} $$

$$ Q_{Acc} = \frac{\$275,000 + \$183,000}{\$5} = \frac{\$458,000}{\$5} = \textbf{91,600 units} $$

Case C:

$$ Q_{Cash} = \frac{\$5,800}{\$8 - \$3} = \frac{\$5,800}{\$5} = \textbf{1,160 units} $$

$$ Q_{Acc} = \frac{\$5,800 + \$1,100}{\$5} = \frac{\$6,900}{\$5} = \textbf{1,380 units} $$

Solution:

We use the accounting break-even formula: $$ Q_{Acc} = \frac{FC + D}{P - v} $$

Case 1: Find Depreciation (D)

$$ 143,286 = \frac{\$820,000 + D}{\$39 - \$30} $$

$$ 143,286 \times 9 = \$820,000 + D $$

$$ \$1,289,574 = \$820,000 + D $$

$$ D = \$1,289,574 - \$820,000 = \textbf{\$469,574} $$

Case 2: Find Unit Price (P)

$$ 104,300 = \frac{\$975,000 + \$2,320,000}{P - \$27} $$

$$ 104,300(P - \$27) = \$3,295,000 $$

$$ P - \$27 = \frac{\$3,295,000}{104,300} \approx \$31.59 $$

$$ P = \$31.59 + \$27 = \textbf{\$58.59} $$

Case 3: Find Unit Variable Cost (v)

$$ 24,640 = \frac{\$237,000 + \$128,700}{\$92 - v} $$

$$ 24,640(\$92 - v) = \$365,700 $$

$$ \$92 - v = \frac{\$365,700}{24,640} \approx \$14.84 $$

$$ v = \$92 - \$14.84 = \textbf{\$77.16} $$

Solution:

First, determine depreciation (D):

$$ D = \frac{\text{Initial Investment}}{\text{Life}} = \frac{\$46,200}{4} = \$11,550 $$

Accounting Break-Even ($Q_{Acc}$):

$$ Q_{Acc} = \frac{FC + D}{P - v} = \frac{\$31,460 + \$11,550}{\$53 - \$22} = \frac{\$43,010}{\$31} = \textbf{1,387 units} $$

Cash Break-Even ($Q_{Cash}$):

$$ Q_{Cash} = \frac{FC}{P - v} = \frac{\$31,460}{\$31} = \textbf{1,015 units} $$

Financial Break-Even ($Q_{Fin}$):

First, find the required OCF for a zero NPV. The 4-year annuity factor at 12% is 3.0373.

$$ NPV = 0 = -I + OCF \times A_F $$

$$ 0 = -\$46,200 + OCF \times 3.0373 $$

$$ OCF = \frac{\$46,200}{3.0373} = \$15,211.57 $$

Now, solve for the quantity:

$$ Q_{Fin} = \frac{FC + OCF}{P - v} = \frac{\$31,460 + \$15,211.57}{\$31} = \frac{\$46,671.57}{\$31} = \textbf{1,506 units} $$

Degree of Operating Leverage (DOL) at financial break-even:

At the financial break-even point, OCF is \$15,211.57. The formula for DOL is $DOL = 1 + FC/OCF$.

$$ DOL = 1 + \frac{\$31,460}{\$15,211.57} = 1 + 2.068 = \textbf{3.068} $$

Solution:

Step 1: Find the contribution margin ($P-v$) using cash break-even.

$$ Q_{Cash} = \frac{FC}{P - v} $$

$$ 9,700 = \frac{\$205,000}{P-v} \Rightarrow P-v = \frac{\$205,000}{9,700} = \$21.134 $$

Step 2: Find depreciation (D) using accounting break-even.

$$ Q_{Acc} = \frac{FC + D}{P-v} $$

$$ 14,300 = \frac{\$205,000 + D}{\$21.134} $$

$$ 14,300 \times \$21.134 = \$205,000 + D $$

$$ \$302,108.20 = \$205,000 + D \Rightarrow D = \$97,108.20 $$

Step 3: Find the initial investment ($I$).

$$ I = D \times \text{Life} = \$97,108.20 \times 5 = \$485,541 $$

Step 4: Find the required OCF for a zero NPV.

The 5-year annuity factor at 12% is 3.6048.

$$ OCF = \frac{I}{A_F} = \frac{\$485,541}{3.6048} = \$134,693.30 $$

Step 5: Find the financial break-even quantity.

$$ Q_{Fin} = \frac{FC + OCF}{P-v} = \frac{\$205,000 + \$134,693.30}{\$21.134} = \frac{\$339,693.30}{\$21.134} = \textbf{16,074 units} $$

Solution:

Step 1: Find the percentage change in output.

$$ \text{Change in } Q = \frac{44,000 - 40,000}{40,000} = \frac{4,000}{40,000} = \textbf{0.10 \text{ or } 10\%} $$

Step 2: Find the percentage change in OCF using DOL.

$$ \% \Delta \text{OCF} = DOL \times \% \Delta Q = 3.19 \times 0.10 = 0.319 \text{ or } \textbf{31.9\%} $$

New level of operating leverage:

The new level of operating leverage will be **lower**. As output (and OCF) increases, fixed costs become a smaller percentage of the total operating cash flow. The formula for DOL is $DOL = 1 + FC/OCF$. Since OCF increases, the ratio $FC/OCF$ decreases, and therefore, the DOL decreases.

Solution:

Step 1: Find the OCF at 40,000 units.

We use the DOL formula at 40,000 units to find the OCF at that level: $$ DOL = 1 + \frac{FC}{OCF} $$

$$ 3.19 = 1 + \frac{\$183,000}{OCF_{40,000}} \Rightarrow 2.19 = \frac{\$183,000}{OCF_{40,000}} \Rightarrow OCF_{40,000} = \$83,561.64 $$

Step 2: Find the contribution margin ($P-v$).

The change in OCF for a one-unit change in quantity is the contribution margin. We can use the information from the previous problem to find the total change in OCF from a 10% increase in quantity (40,000 to 44,000 units), which was 31.9% or $0.319 \times \$83,561.64 = \$26,666.67$. This is for a change of 4,000 units.

$$ P-v = \frac{\$26,666.67}{4,000} = \$6.67 $$

Step 3: Find the OCF at 38,000 units.

The change from 40,000 units to 38,000 units is -2,000 units. The change in OCF is:

$$ \Delta OCF = (P-v) \times \Delta Q = \$6.67 \times (-2,000) = -\$13,340 $$

$$ OCF_{38,000} = OCF_{40,000} + \Delta OCF = \$83,561.64 - \$13,340 = \textbf{\$70,221.64} $$

Step 4: Find the DOL at 38,000 units.

$$ DOL = 1 + \frac{FC}{OCF} = 1 + \frac{\$183,000}{\$70,221.64} = 1 + 2.606 = \textbf{3.606} $$

Solution:

Degree of Operating Leverage (DOL):

$$ DOL = 1 + \frac{FC}{OCF} = 1 + \frac{\$76,000}{\$108,700} = 1 + 0.699 = \textbf{1.699} $$

Increase in Operating Cash Flow:

First, find the percentage change in quantity sold:

$$ \% \Delta Q = \frac{10,000 - 9,200}{9,200} = \frac{800}{9,200} = 0.08696 \text{ or } 8.696\% $$

Now, use the DOL to find the percentage change in OCF:

$$ \% \Delta OCF = DOL \times \% \Delta Q = 1.699 \times 0.08696 = 0.1477 \text{ or } 14.77\% $$

The dollar increase is:

$$ \text{Increase in OCF} = OCF_{current} \times \% \Delta OCF = \$108,700 \times 0.1477 = \textbf{\$16,056.49} $$

New OCF and DOL:

$$ OCF_{new} = OCF_{current} + \text{Increase} = \$108,700 + \$16,056.49 = \$124,756.49 $$

$$ DOL_{new} = 1 + \frac{FC}{OCF_{new}} = 1 + \frac{\$76,000}{\$124,756.49} = 1 + 0.609 = \textbf{1.609} $$

Solution:

Fixed Costs (FC):

$$ DOL = 1 + \frac{FC}{OCF} $$

$$ 3.41 = 1 + \frac{FC}{\$81,000} \Rightarrow 2.41 = \frac{FC}{\$81,000} \Rightarrow FC = 2.41 \times \$81,000 = \textbf{\$195,210} $$

Operating Cash Flow at 15,500 units:

First, find the contribution margin ($P-v$) which is the change in OCF per unit of output. $$ \text{OCF} = (P-v)Q - FC $$

$$ \$81,000 = (P-v)(14,500) - \$195,210 $$

$$ (P-v)(14,500) = \$276,210 \Rightarrow P-v = \frac{\$276,210}{14,500} = \$19.049 $$

Now, find the OCF at 15,500 units:

$$ OCF = \$19.049 \times 15,500 - \$195,210 = \$295,259.50 - \$195,210 = \textbf{\$100,049.50} $$

Operating Cash Flow at 13,500 units:

$$ OCF = \$19.049 \times 13,500 - \$195,210 = \$257,161.50 - \$195,210 = \textbf{\$61,951.50} $$

Solution:

DOL at 15,500 units:

$$ DOL = 1 + \frac{FC}{OCF} = 1 + \frac{\$195,210}{\$100,049.50} = 1 + 1.951 = \textbf{2.951} $$

DOL at 13,500 units:

$$ DOL = 1 + \frac{FC}{OCF} = 1 + \frac{\$195,210}{\$61,951.50} = 1 + 3.151 = \textbf{4.151} $$