Let's break down the problem step by step. ## 1. **Given Data** - **Cars per restaurant:** 2 - **Cost per car:** $23,000 - **Total car cost:** $46,000 - **Depreciation period:** 3 years (straight line) - **Salvage value (in year 5):** $5,000 per car = $10,000 total - **Operation costs per year:** $18,000 - **Variable cost per delivery:** $1.50 - **Price per delivery:** $3.00 - **Tax rate:** 21% - **Discount rate:** 8% - **Time horizon:** We'll solve for break-even on an **annual basis** and then per day. --- ## 2. **Accounting Break-Even (Ignoring Time Value of Money)** ### **Step 1: Depreciation Calculation** - **Total depreciation per year** (straight-line, 3 years, ignoring salvage for depreciation): \[ \text{Annual Depreciation} = \frac{\$46,000 - \$10,000}{3} = \frac{\$36,000}{3} = \$12,000 \] ### **Step 2: Break-Even Equation (Before Tax)** Let \( Q \) = number of deliveries per year. \[ \text{Total Revenue} = 3Q \] \[ \text{Total Variable Cost} = 1.5Q \] \[ \text{Total Fixed Cost} = \$18,000 \text{ (operation)} + \$12,000 \text{ (depreciation)} = \$30,000 \] Net income before tax: \[ \text{Profit before tax} = (3Q - 1.5Q) - 30,000 = 1.5Q - 30,000 \] At break-even (after-tax profit = 0): \[ \text{Profit after tax} = (\text{Profit before tax}) \times (1 - 0.21) = 0 \] \[ 1.5Q - 30,000 = 0 \implies Q = 20,000 \text{ deliveries per year} \] **But:** This is *before* considering taxes. At accounting break-even, the pre-tax profit is zero, so after-tax is negative (tax shield from depreciation). The true *accounting* break-even requires after-tax profit = 0. Let’s set: \[ \text{EBT} = 1.5Q - 30,000 \] \[ \text{Net income} = (\text{EBT}) \times (1-0.21) \] Set Net income to 0: \[ 0 = (1.5Q - 30,000) \times (1 - 0.21) \] \[ 0 = 1.5Q - 30,000 \] So, per above, **accounting break-even is 20,000 deliveries per year**. **Check:** - Earnings before tax: $1.5Q - 30,000 = 0$ - No taxes due, so after-tax profit is zero. #### **Deliveries per day:** Assuming 365 days per year: \[ \text{Deliveries per day} = \frac{20,000}{365} \approx 54.8 \approx 55 \] --- ## 3. **Financial (NPV) Break-Even (Considering Time Value of Money)** Here, we look for \( Q \) such that **NPV = 0** over the project life. ### **Step 1: Cash Flows** #### a. **Initial Investment (Year 0):** - $46,000 (cars) #### b. **Operating Cash Flows (Years 1-3):** Each year: - Revenue: \( 3Q \) - Variable cost: \( 1.5Q \) - Fixed operating cost: $18,000 - Depreciation: $12,000 Taxable income per year: \[ \text{EBT} = 3Q - 1.5Q - 18,000 - 12,000 = 1.5Q - 30,000 \] Taxes: \[ \text{Taxes} = (\text{EBT}) \times 0.21 \] Net income: \[ \text{Net Income} = (\text{EBT}) \times (1 - 0.21) \] Operating cash flow (OCF): \[ \text{OCF} = \text{Net Income} + \text{Depreciation} \] So, \[ \text{OCF} = (1.5Q - 30,000) \times (1 - 0.21) + 12,000 \] \[ = (1.5Q - 30,000) \times 0.79 + 12,000 \] \[ = 1.185Q - 23,700 + 12,000 \] \[ = 1.185Q - 11,700 \] #### c. **Year 4 and 5:** - No depreciation after year 3 - Assume no operation or variable costs (delivery service ends after 3 years) - Salvage value realized in year 5: $10,000 (no tax on salvage according to problem) ### **Step 2: NPV Equation** \[ NPV = -46,000 + \sum_{t=1}^3 \frac{\text{OCF}}{(1.08)^t} + \frac{10,000}{(1.08)^5} \] Where OCF is as above. ### **Step 3: Solve for Q where NPV = 0** Let \( OCF = 1.185Q - 11,700 \). Plug in values: Calculate present value factors: - \( PV_1 = 1/1.08 = 0.9259 \) - \( PV_2 = 1/(1.08)^2 = 0.8573 \) - \( PV_3 = 1/(1.08)^3 = 0.7938 \) - \( PV_5 = 1/(1.08)^5 = 0.6806 \) Sum for years 1–3: \[ PV_{1-3} = 0.9259 + 0.8573 + 0.7938 = 2.577 \] So, \[ NPV = -46,000 + (1.185Q - 11,700) \times 2.577 + 10,000 \times 0.6806 \] \[ = -46,000 + (1.185Q - 11,700) \times 2.577 + 6,806 \] \[ = -39,194 + (1.185Q - 11,700) \times 2.577 \] Set NPV = 0: \[ 0 = -39,194 + (1.185Q - 11,700) \times 2.577 \] \[ (1.185Q - 11,700) \times 2.577 = 39,194 \] \[ 1.185Q - 11,700 = \frac{39,194}{2.577} = 15,206 \] \[ 1.185Q = 15,206 + 11,700 = 26,906 \] \[ Q = \frac{26,906}{1.185} = 22,705 \] ### **Deliveries per day:** \[ \frac{22,705}{365} = 62.2 \approx 63 \] --- ## **Summary Table** | Approach | Deliveries/year | Deliveries/day | |------------------|----------------|---------------| | Accounting | 20,000 | 55 | | Financial (NPV) | 22,705 | 63 | --- ## **Conclusion** - **Accounting break-even:** Chipotle needs to deliver about **55 meals per day**. - **Financial break-even (NPV=0):** Chipotle needs to deliver about **63 meals per day**. **The financial (NPV) break-even is higher because it accounts for the time value of money and the fact that depreciation is a non-cash expense, while the accounting break-even is lower because it only considers the matching of revenues and expenses on a yearly basis.** --- **If you want to see the full calculations shown step by step, let me know!**