Let's break down the problem step by step and **compare the break-even analysis using the Accounting Approach and the Present Value (Financial) Approach**. --- ## **Given Data** - **Number of cars per restaurant:** 2 - **Cost per car:** \$23,000 - **Total investment:** 2 × \$23,000 = \$46,000 - **Depreciation period:** 3 years (straight line) - **Salvage value (at end of year 5):** \$5,000 - **Annual operating cost:** \$18,000 - **Variable cost per delivery:** \$1.50 - **Revenue per delivery:** \$3.00 - **Tax rate:** 21% - **Discount rate:** 8% - **Days per year:** Assume 365 (unless otherwise specified) Let **N** = number of deliveries per day. --- ## **Step 1: Annual Fixed and Variable Costs** ### **Annual Depreciation (Straight Line):** - Depreciable base: = (Total cost – Salvage value) = \$46,000 – \$5,000 = \$41,000 - **Annual depreciation:** = \$41,000 / 3 = **\$13,666.67 per year** (for 3 years) ### **Annual Operating Cost:** = \$18,000 --- ## **Step 2: Break-Even Calculation (Accounting Approach)** ### **Annual Revenue and Cost Function** Let **D** = deliveries per year = N × 365 - **Annual Revenue:** = \$3.00 × D - **Annual Variable Cost:** = \$1.50 × D - **Annual Fixed Cost:** = Operating cost + Depreciation = \$18,000 + \$13,666.67 = **\$31,666.67** - **Annual Profit (before tax):** = Revenue – Variable cost – Fixed cost = \$3.00 × D – \$1.50 × D – \$31,666.67 = \$1.50 × D – \$31,666.67 - **Taxable profit:** = (\$1.50 × D – \$31,666.67) - **Net income (after tax):** = (Profit before tax) × (1 – tax rate) = (\$1.50 × D – \$31,666.67) × 0.79 **Set Net Income = 0 for break-even:** \[ (\$1.50 \times D - \$31,\!666.67) \times 0.79 = 0 \\ \$1.50 \times D - \$31,\!666.67 = 0 \\ \$1.50 \times D = \$31,\!666.67 \\ D = \frac{\$31,\!666.67}{\$1.50} = 21,\!111.11 \text{ deliveries per year} \] \[ \text{Deliveries per day} = \frac{21,\!111.11}{365} \approx 57.84 \] **Accounting Break-Even:** \[ \boxed{58 \text{ deliveries per day}} \] --- ## **Step 3: Present Value Approach (Financial Break-Even)** Now, let's find the number of deliveries per day so that the **Net Present Value (NPV) = 0**. ### **Cash Flows:** - **Initial investment (Year 0):** –\$46,000 - **Annual cash flow (Years 1-3):** - Revenue: \$3.00 × D - Variable costs: \$1.50 × D - Operating costs: \$18,000 - Depreciation: \$13,666.67 (non-cash, affects taxes) - Tax shield from depreciation: Depreciation × 21% = \$2,870 - Taxable income = (Revenue – Variable cost – Operating cost – Depreciation) - Taxes paid = Taxable income × 21% - **Net operating cash flow = (Revenue – Variable cost – Operating cost – Taxes paid) + Depreciation × tax rate** - **Year 4 and 5:** No depreciation (depreciation ends after year 3), but still possible operating and variable costs unless specified otherwise (assume delivery service continues for 5 years). At Year 5, add after-tax salvage value. ### **Let's assume the service runs for 5 years**: #### **Annual Cash Flow Calculation (Years 1-3):** - Revenue: \$3 × D - Variable cost: \$1.5 × D - Operating cost: \$18,000 - Depreciation: \$13,666.67 - EBIT = Revenue – Variable cost – Operating cost – Depreciation = \$3D – \$1.5D – \$18,000 – \$13,666.67 = \$1.5D – \$31,666.67 - Taxes = EBIT × 21% - Net income = EBIT × 0.79 - **Operating cash flow = Net income + Depreciation** \[ \text{Net income} = (\$1.5D – \$31,666.67) \times 0.79 \] \[ \text{Operating cash flow} = \text{Net income} + \$13,666.67 \] #### **Years 4-5:** - No depreciation, so: - EBIT = \$1.5D – \$18,000 - Taxes = EBIT × 21% - Net income = EBIT × 0.79 - Operating cash flow = Net income #### **Year 5: Salvage Value** - Salvage value = \$5,000 (tax-free, as stated) --- ### **Set Up NPV Equation:** Let’s denote annual deliveries as **D**. \[ NPV = -\$46,000 + \sum_{t=1}^{3} \frac{CF_{1-3}}{(1.08)^t} + \frac{CF_4}{(1.08)^4} + \frac{CF_5 + \$5,000}{(1.08)^5} \] - \(CF_{1-3} = (\$1.5D – \$31,666.67) \times 0.79 + \$13,666.67\) - \(CF_{4-5} = (\$1.5D – \$18,000) \times 0.79\) #### **Calculate \(CF_{1-3}\):** \[ CF_{1-3} = (\$1.5D – \$31,666.67) \times 0.79 + \$13,666.67 \] \[ = 0.79 \times 1.5D – 0.79 \times 31,666.67 + 13,666.67 \] \[ = 1.185D – 25,016.67 + 13,666.67 \] \[ = 1.185D – 11,350 \] #### **Calculate \(CF_{4-5}\):** \[ CF_{4-5} = (\$1.5D – \$18,000) \times 0.79 \] \[ = 1.185D – 14,220 \] --- ### **Plug into NPV Equation:** \[ NPV = -46,000 + \frac{CF_{1-3}}{1.08} + \frac{CF_{1-3}}{(1.08)^2} + \frac{CF_{1-3}}{(1.08)^3} + \frac{CF_{4}}{(1.08)^4} + \frac{CF_{5} + 5,000}{(1.08)^5} \] \[ NPV = -46,000 + (1.185D – 11,350) \left(\frac{1}{1.08} + \frac{1}{1.08^2} + \frac{1}{1.08^3}\right) + \frac{1.185D – 14,220}{1.08^4} + \frac{1.185D – 14,220 + 5,000}{1.08^5} \] #### **Calculate the Discount Factors:** - 1 / 1.08 = 0.92593 - 1 / 1.08^2 = 0.85734 - 1 / 1.08^3 = 0.79383 - 1 / 1.08^4 = 0.73503 - 1 / 1.08^5 = 0.68058 Sum for years 1-3: 0.92593 + 0.85734 + 0.79383 = **2.5771** --- #### **Now, substitute:** \[ NPV = -46,000 + (1.185D – 11,350) \times 2.5771 + (1.185D – 14,220) \times 0.73503 + (1.185D – 14,220 + 5,000) \times 0.68058 \] \[ NPV = -46,000 + [1.185D \times 2.5771 – 11,350 \times 2.5771] + [1.185D \times 0.73503 – 14,220 \times 0.73503] + [1.185D \times 0.68058 – 14,220 \times 0.68058 + 5,000 \times 0.68058] \] \[ NPV = -46,000 + [3.052D – 29,242.585] + [0.870D – 10,444.127] + [0.806D – 9,684.648 + 3,402.9] \] \[ NPV = -46,000 + (3.052D + 0.870D + 0.806D) – (29,242.585 + 10,444.127 + 9,684.648) + 3,402.9 \] \[ NPV = -46,000 + 4.728D – 49,371.36 + 3,402.9 \] \[ NPV = (4.728D) – 46,000 – 49,371.36 + 3,402.9 \] \[ NPV = (4.728D) – 91,968.46 \] **Set NPV = 0:** \[ 4.728D = 91,968.46 \] \[ D = \frac{91,968.46}{4.728} = 19,454.6 \text{ deliveries per year} \] \[ \text{Deliveries per day} = \frac{19,454.6}{365} = 53.32 \] --- ## **Final Answers and Comparison** ### **Accounting Approach Break-Even:** \[ \boxed{58 \text{ deliveries per day}} \] ### **Present Value (Financial) Approach Break-Even:** \[ \boxed{54 \text{ deliveries per day (rounded up)}} \] --- ## **Summary Table:** | Approach | Break-Even Deliveries/Day | |-----------------------|--------------------------| | Accounting | 58 | | Present Value (NPV=0) | 54 | - **Accounting break-even** is slightly higher than the **present value break-even**, as the latter accounts for the time value of money and the tax shield more accurately. --- ### **If you have follow-up questions or need an Excel template, let me know!**