1. **Problem Statement:** You are evaluating a new product launch with the following data: - Initial cost: 2150000 - Life: 4 years - No salvage value - Straight-line depreciation to 0 - Sales: 150 units/year - Price/unit: 28000 - Variable cost/unit: 17000 - Fixed costs/year: 580000 - Required return: 12% - Tax rate: 35% You need to find upper/lower bounds for sales, variable cost, fixed costs (±10%), base-case NPV, best/worst-case NPVs, sensitivity of NPV to fixed costs, cash break-even output, accounting break-even output, and degree of operating leverage at accounting break-even. 2. **Formulas and Important Rules:** - Depreciation per year: $$\frac{\text{Initial Cost}}{\text{Life}} = \frac{2150000}{4} = 537500$$ - Earnings Before Interest and Taxes (EBIT): $$\text{EBIT} = (\text{Price} - \text{Variable Cost}) \times \text{Units} - \text{Fixed Costs} - \text{Depreciation}$$ - Net Income: $$\text{Net Income} = \text{EBIT} \times (1 - \text{Tax Rate})$$ - Operating Cash Flow (OCF): $$\text{OCF} = \text{Net Income} + \text{Depreciation}$$ - NPV: $$\text{NPV} = -\text{Initial Cost} + \sum_{t=1}^4 \frac{\text{OCF}}{(1 + 0.12)^t}$$ - Cash break-even: output where OCF = 0 ignoring taxes - Accounting break-even: output where Net Income = 0 - Degree of Operating Leverage (DOL) at output Q: $$\text{DOL} = \frac{Q(\text{Price} - \text{Variable Cost})}{Q(\text{Price} - \text{Variable Cost}) - \text{Fixed Costs} - \text{Depreciation}}$$ 3. **Calculate bounds for variables (±10%):** - Unit Sales: Lower = 150 × 0.9 = 135, Upper = 150 × 1.1 = 165 - Variable Cost/unit: Lower = 17000 × 0.9 = 15300, Upper = 17000 × 1.1 = 18700 - Fixed Costs: Lower = 580000 × 0.9 = 522000, Upper = 580000 × 1.1 = 638000 4. **Calculate Base-case NPV:** - EBIT = (28000 - 17000) × 150 - 580000 - 537500 = 11000 × 150 - 580000 - 537500 = 1650000 - 580000 - 537500 = 532500 - Net Income = 532500 × (1 - 0.35) = 532500 × 0.65 = 346125 - OCF = 346125 + 537500 = 883625 - NPV = -2150000 + 883625 × (PVIFA 12%,4) - PVIFA 12%,4 = $$\sum_{t=1}^4 \frac{1}{(1.12)^t} = 3.03735$$ - NPV = -2150000 + 883625 × 3.03735 = -2150000 + 2683457.34 = 533457.34 5. **Best-case scenario (max sales, min costs):** - Units = 165, Variable Cost = 15300, Fixed Costs = 522000 - EBIT = (28000 - 15300) × 165 - 522000 - 537500 = 12700 × 165 - 522000 - 537500 = 2095500 - 522000 - 537500 = 1038000 - Net Income = 1038000 × 0.65 = 674700 - OCF = 674700 + 537500 = 1212200 - NPV = -2150000 + 1212200 × 3.03735 = -2150000 + 3681636.87 = 1531636.87 6. **Worst-case scenario (min sales, max costs):** - Units = 135, Variable Cost = 18700, Fixed Costs = 638000 - EBIT = (28000 - 18700) × 135 - 638000 - 537500 = 9300 × 135 - 638000 - 537500 = 1255500 - 638000 - 537500 = 8000 - Net Income = 8000 × 0.65 = 5200 - OCF = 5200 + 537500 = 542700 - NPV = -2150000 + 542700 × 3.03735 = -2150000 + 1648257.15 = -501742.85 7. **Sensitivity of NPV to Fixed Costs:** - Change in Fixed Costs: $$\Delta FC = 1000$$ (example) - Change in EBIT = -1000 - Change in Net Income = -1000 × (1 - 0.35) = -650 - Change in OCF = -650 (Net Income) + 0 (Depreciation) = -650 - Change in NPV = -650 × 3.03735 = -1974.28 - Sensitivity: $$\frac{\Delta NPV}{\Delta FC} = \frac{-1974.28}{1000} = -1.974$$ 8. **Cash Break-even (OCF=0 ignoring taxes):** - OCF = EBIT + Depreciation = 0 - EBIT = -Depreciation = -537500 - EBIT = (Price - Variable Cost) × Q - Fixed Costs - Depreciation - Set EBIT = -537500: $$(28000 - 17000)Q - 580000 - 537500 = -537500$$ $$11000Q - 580000 = 0$$ $$11000Q = 580000$$ $$Q = \frac{580000}{11000} = 52.727... \approx 53 \text{ units}$$ 9. **Accounting Break-even (Net Income=0):** - Net Income = EBIT × (1 - Tax Rate) = 0 => EBIT = 0 - $$0 = (28000 - 17000)Q - 580000 - 537500$$ - $$11000Q = 1117500$$ - $$Q = \frac{1117500}{11000} = 101.59 \approx 102 \text{ units}$$ 10. **Degree of Operating Leverage at Accounting Break-even:** - At Q=102, - $$\text{DOL} = \frac{Q(Price - VC)}{Q(Price - VC) - Fixed Costs - Depreciation} = \frac{102 \times 11000}{102 \times 11000 - 580000 - 537500}$$ - Denominator = 1122000 - 1117500 = 4500 - Numerator = 1122000 - $$\text{DOL} = \frac{1122000}{4500} = 249.3333$$ **Final answers:** - Base-case NPV = 533457.34 - Best-case NPV = 1531636.87 - Worst-case NPV = -501742.85 - Sensitivity $$\frac{\Delta NPV}{\Delta FC} = -1.974$$ - Cash break-even = 53 units - Accounting break-even = 102 units - Degree of operating leverage = 249.3333