Let's solve the problem step-by-step: --- ## **Step 1: Data Summary** | Item | Old Machine | New Machine | |-----------------------|-------------------|-------------------------------| | Purchase Cost (Year 0)| $0 | $250,000 | | Depreciation Method | - | MACRS 5-year, half-year conv. | | Depreciation Years | - | 6 years (years 0-5) | | Annual Op. Cost | $120,000 | $45,000 | | Life | 5 years | 5 years | | Salvage Value | 0 | 0 | | Tax Rate | 35% | 35% | | MARR | 16% | 16% | --- ## **Step 2: MACRS Depreciation Schedule** For a 5-year property, the MACRS percentages (half-year convention) are: | Year | MACRS % | |------|---------| | 1 | 20.00% | | 2 | 32.00% | | 3 | 19.20% | | 4 | 11.52% | | 5 | 11.52% | | 6 | 5.76% | So, depreciation for each year: \[ \text{Depreciation} = \$250,000 \times \text{MACRS \%} \] | Year | Depreciation | |------|-----------------| | 1 | $50,000 | | 2 | $80,000 | | 3 | $48,000 | | 4 | $28,800 | | 5 | $28,800 | | 6* | $14,400 | *Year 6 depreciation is not relevant for our 5-year analysis since the machine is disposed at the end of year 5 (assume no recapture since salvage value is zero). --- ## **Step 3: Calculate After-Tax Cash Flows** ### **A. Old Machine** - No purchase cost. - Op. cost: $120,000/year (years 1-5) - No depreciation. - No salvage. - No tax shield. \[ \text{After-tax Op. Cost} = \text{Op. Cost} \times (1 - \text{Tax Rate}) = \$120,000 \times (1 - 0.35) = \$78,000 \] But since there is no revenue effect, the cash outflow per year is $120,000, but we can also say, for cash flow analysis, it remains $120,000 outflow each year. But for incremental analysis, let's keep all numbers. ### **B. New Machine** - Year 0: $250,000 outflow - Op. cost: $45,000/year (years 1-5) - Depreciation (as above) - Tax shield from depreciation: Depreciation × Tax Rate #### For each year (1-5): \[ \text{Taxable Income Impact} = \text{Op. Cost} + \text{Depreciation} \] \[ \text{Tax Savings} = (\text{Depreciation}) \times 0.35 \] \[ \text{After-tax Op. Cost} = \text{Op. Cost} \times (1 - 0.35) = \$29,250 \] But we need to add back depreciation for cash flow (since it's a non-cash expense): \[ \text{After-tax Cash Flow} = -\text{Op. Cost} + (\text{Depreciation} \times \text{Tax Rate}) \] So, for each year: | Year | Op. Cost | Depreciation | Depreciation Tax Shield | Total Cash Outflow | |------|----------|--------------|------------------------|--------------------| | 0 | - | - | - | $250,000 | | 1 | $45,000 | $50,000 | $17,500 | $45,000 - $17,500 = $27,500 | | 2 | $45,000 | $80,000 | $28,000 | $45,000 - $28,000 = $17,000 | | 3 | $45,000 | $48,000 | $16,800 | $45,000 - $16,800 = $28,200 | | 4 | $45,000 | $28,800 | $10,080 | $45,000 - $10,080 = $34,920 | | 5 | $45,000 | $28,800 | $10,080 | $45,000 - $10,080 = $34,920 | --- ## **Step 4: Incremental Analysis** Subtract old machine cash flows from new machine cash flows (New - Old): - Year 0: -$250,000 - Years 1-5: (New outflow) - (Old outflow) = (from above) - $120,000 | Year | New OCF | Old OCF | Incremental CF (New - Old) | |------|---------------|------------|----------------------------| | 0 | -$250,000 | $0 | -$250,000 | | 1 | -$27,500 | -$120,000 | $92,500 | | 2 | -$17,000 | -$120,000 | $103,000 | | 3 | -$28,200 | -$120,000 | $91,800 | | 4 | -$34,920 | -$120,000 | $85,080 | | 5 | -$34,920 | -$120,000 | $85,080 | --- ## **Step 5: Net Present Value (NPV) Calculation** Discount rate = 16% NPV = Sum of \(\frac{\text{Cash Flow}_t}{(1 + i)^t}\) \[ \text{NPV} = -250,000 + \frac{92,500}{1.16^1} + \frac{103,000}{1.16^2} + \frac{91,800}{1.16^3} + \frac{85,080}{1.16^4} + \frac{85,080}{1.16^5} \] Let's compute each term: - Year 1: 92,500 / 1.16 = **$79,741** - Year 2: 103,000 / (1.16^2) = 103,000 / 1.3456 = **$76,532** - Year 3: 91,800 / (1.16^3) = 91,800 / 1.5609 = **$58,813** - Year 4: 85,080 / (1.16^4) = 85,080 / 1.8106 = **$47,002** - Year 5: 85,080 / (1.16^5) = 85,080 / 2.1003 = **$40,504** Total NPV: \[ \text{NPV} = -250,000 + 79,741 + 76,532 + 58,813 + 47,002 + 40,504 \] \[ = -250,000 + 302,592 = \boxed{52,592} \] --- ## **Step 6: Conclusion** - **NPV of incremental cash flows (replacing old with new): \$52,592** - Since NPV > 0 at the MARR of 16%, **replacing the old machine with the new machine is economically justified**. --- ### **Final Answer** **The incremental NPV of replacing the old machine with the new one is \$52,592 (at 16% discount rate). Since NPV is positive, you should REPLACE the old machine with the new one.**

Let's analyze the problem carefully and compare the two options: keep the old machine or replace it with a new one, using incremental analysis. --- ## Step 1: Data Summary | Item | Old Machine | New Machine | |---------------------------|--------------------------|-------------------------------------| | Purchase/Capital Cost | $0 | $250,000 at Year 0 | | Operating Cost (annual) | $120,000 | $45,000 | | Life | 5 years | 5 years | | Salvage Value | $0 | $0 | | Depreciation Method | N/A | MACRS 5-year (half-year convention) | | Tax Rate | 35% | 35% | | MARR (discount rate) | 16% | 16% | --- ## Step 2: Depreciation Schedule (MACRS 5-year property) | Year | MACRS % | Depreciation ($250,000) | |-------|---------|-------------------------| | 1 | 20.00% | $50,000 | | 2 | 32.00% | $80,000 | | 3 | 19.20% | $48,000 | | 4 | 11.52% | $28,800 | | 5 | 11.52% | $28,800 | | 6 | 5.76% | $14,400 (not relevant for 5-year analysis) | --- ## Step 3: Operating Cost Savings & Tax Implications - Operating cost difference: Old = $120,000/year, New = $45,000/year - Annual savings: $120,000 - $45,000 = **$75,000** - Tax savings from depreciation (depreciation × tax rate): - Year 1: $50,000 × 0.35 = $17,500 - Year 2: $80,000 × 0.35 = $28,000 - Year 3: $48,000 × 0.35 = $16,800 - Year 4: $28,800 × 0.35 = $10,080 - Year 5: $28,800 × 0.35 = $10,080 - Terminal tax shield at end of Year 5 due to remaining book value: - Book value after 5 years: $14,400 (remaining depreciation not taken) - Tax shield from this loss: $14,400 × 0.35 = **$5,040** --- ## Step 4: Calculate Incremental Cash Flows (New – Old) ### Year 0: - Outflow: $250,000 (cost of new machine) ### Years 1-5: - **Annual operating cash savings:** $75,000 - **Tax shield from depreciation:** as above - **After-tax operating savings:** \[ \text{Tax savings (from depreciation)} + \text{Cost savings after tax} = \text{(Operating savings)} \times (1 - t) + (\text{Depreciation} \times t) \] Calculating the after-tax cash flow for each year: | Year | Operating Savings | Depreciation | Tax shield from depreciation | After-tax cash flow (incremental) | |-------|---------------------|--------------|------------------------------|----------------------------------| | 1 | $75,000 | $50,000 | $17,500 | $(75,000 \times 0.65) + 17,500 = 48,750 + 17,500 = \$66,250$ | | 2 | $75,000 | $80,000 | $28,000 | $48,750 + 28,000 = \$76,750$ | | 3 | $75,000 | $48,000 | $16,800 | $48,750 + 16,800 = \$65,550$ | | 4 | $75,000 | $28,800 | $10,080 | $48,750 + 10,080 = \$58,830$ | | 5 | $75,000 | $28,800 | $10,080 | $48,750 + 10,080 = \$58,830$ plus terminal tax shield of $5,040$ (since remaining book value is $14,400) | **Note:** The terminal tax shield applies at the end of Year 5 due to sale/disposal of the asset, which reduces taxable gain. It adds $5,040 in Year 5 cash flow. --- ## Step 5: Discounted Cash Flows and NPV Calculation Calculate present value of each year’s incremental cash flow using 16% discount rate: \[ PV = \frac{\text{Cash Flow}_t}{(1 + 0.16)^t} \] | Year | Cash Flow | Discount Factor | Present Value | |-------|--------------|------------------|------------------------------| | 0 | -$250,000 | 1.00 | -$250,000 | | 1 | $66,250 | 1/1.16 = 0.8621 | $66,250 × 0.8621 ≈ $57,168 | | 2 | $76,750 | 1/1.16²=0.7432 | $76,750 × 0.7432 ≈ $57,000 | | 3 | $65,550 | 1/1.16³=0.6410 | $65,550 × 0.6410 ≈ $42,055 | | 4 | $58,830 | 1/1.16⁴=0.5537 | $58,830 × 0.5537 ≈ $32,558 | | 5 | $58,830 + $5,040 = $63,870 | 1/1.16⁵=0.4781 | $63,870 × 0.4781 ≈ $30,558 | **Total NPV:** \[ NPV = -250,000 + 57,168 + 57,000 + 42,055 + 32,558 + 30,558 \approx \$ -30,661 \] --- ## **Final Conclusion:** - The **NPV is approximately -$30,661**, which is **negative**. - The **internal rate of return (IRR)** is less than 16%, around 10–11%, indicating the investment does **not** meet the minimum required return. **Therefore, it is NOT economically justified to replace the old machine with the new one.** The best decision is to **keep the old machine**. --- ## **Summary:** | **Result** | **Conclusion** | |--------------|-----------------------------------| | Incremental NPV | Approximately **-$30,661** (negative) | | IRR | Less than 16% (~10–11%) | | **Decision** | **Do not replace** (keep the old machine) | --- **This aligns with the second analysis you pasted, which indicates that replacing is not economically justified at a 16% MARR.**

Let's carefully analyze whether to keep the old machine or replace it with a new one, using incremental analysis and considering taxes, depreciation, and a minimum rate of return (MARR) of 16%. --- ## **Step 1: Data Summary** | Item | Old Machine | New Machine | |-----------------------------------|----------------------------------|----------------------------------------------| | Purchase / Capital Cost | $0 | $250,000 (Year 0) | | Operating Cost (per year) | $120,000 | $45,000 | | Life | 5 years | 5 years | | Salvage Value | $0 | $0 | | Depreciation Method | N/A | MACRS 5-year (half-year convention) | | Tax Rate | 35% | 35% | | Discount Rate (MARR) | 16% | 16% | --- ## **Step 2: MACRS 5-Year Depreciation Schedule** For MACRS 5-year property (half-year convention), the percentage depreciation per year is: | Year | MACRS % | Depreciation ($250,000) | |-------|---------|-------------------------| | 1 | 20.00% | $50,000 | | 2 | 32.00% | $80,000 | | 3 | 19.20% | $48,000 | | 4 | 11.52% | $28,800 | | 5 | 11.52% | $28,800 | | 6 | 5.76% | $14,400 (not relevant for 5-year analysis) | **Note:** The remaining book value after 5 years is $14,400, which is a loss if disposed, providing a terminal tax shield. --- ## **Step 3: Operating Cost Savings and Tax Effects** - **Annual operating cost savings** when switching to the new machine: \[ \$120,000 - \$45,000 = \$75,000 \] - **Tax savings from depreciation**: \[ \text{Tax shield}_t = D_t \times 0.35 \] where \( D_t \) is depreciation in year \( t \). | Year | Depreciation \( D_t \) | Tax shield \( D_t \times 0.35 \) | |-------|------------------------|----------------------------------| | 1 | $50,000 | $17,500 | | 2 | $80,000 | $28,000 | | 3 | $48,000 | $16,800 | | 4 | $28,800 | $10,080 | | 5 | $28,800 | $10,080 | **Terminal tax shield:** At the end of Year 5, the remaining book value ($14,400) leads to a loss that results in a tax shield: \[ \$14,400 \times 0.35 = \$5,040 \] --- ## **Step 4: Calculate Incremental After-Tax Cash Flows (New – Old)** ### Year 0: - **Cost of new machine:** \(\$250,000\) outflow. ### Years 1–5: - **Pre-tax operating savings:** \$75,000 annually. - **Tax shield from depreciation** each year, adding cash flow. - **Net incremental cash flow**: \[ \text{CF}_t = (\text{Savings} \times (1 - t)) + (\text{Depreciation}_t \times t) \] which simplifies to: \[ \text{CF}_t = 75,000 \times 0.65 + D_t \times 0.35 \] Calculations per year: | Year | Operating Savings after tax | Depreciation \( D_t \) | Tax shield \( D_t \times 0.35 \) | Total Cash Flow \( \text{CF}_t \) | |-------|------------------------------|------------------------|----------------------------------|----------------------------------| | 1 | \$75,000 \times 0.65 = \$48,750 | $50,000 | $17,500 | \$48,750 + \$17,500 = \$66,250 | | 2 | \$48,750 | $80,000 | $28,000 | \$48,750 + \$28,000 = \$76,750 | | 3 | \$48,750 | $48,000 | $16,800 | \$48,750 + \$16,800 = \$65,550 | | 4 | \$48,750 | $28,800 | $10,080 | \$48,750 + \$10,080 = \$58,830 | | 5 | \$48,750 | $28,800 | $10,080 | \$48,750 + \$10,080 = \$58,830 | **Add terminal tax shield at Year 5**: \[ \$14,400 \times 0.35 = \$5,040 \] Total Year 5 cash flow: \[ \$58,830 + \$5,040 = \$63,870 \] --- ## **Step 5: Discounted Cash Flows and NPV Calculation** Using a 16% discount rate: \[ PV = \frac{\text{Cash Flow}_t}{(1 + 0.16)^t} \] Calculate each year's present value: | Year | Cash Flow | Discount Factor \( (1.16)^t \) | PV of Cash Flow | |-------|--------------|------------------------------|----------------| | 0 | -\$250,000 | 1.00 | -\$250,000 | | 1 | \$66,250 | 0.8621 | \$66,250 \times 0.8621 \approx \$57,168 \) | | 2 | \$76,750 | 0.7432 | \$76,750 \times 0.7432 \approx \$57,000 \) | | 3 | \$65,550 | 0.6410 | \$65,550 \times 0.6410 \approx \$42,055 \) | | 4 | \$58,830 | 0.5537 | \$58,830 \times 0.5537 \approx \$32,558 \) | | 5 | \$63,870 | 0.4781 | \$63,870 \times 0.4781 \approx \$30,558 \) | **Total NPV:** \[ NPV \approx -\$250,000 + \$57,168 + \$57,000 + \$42,055 + \$32,558 + \$30,558 \approx -\$30,661 \] --- ## **Final Result & Conclusion** - **The NPV of replacing the old machine with the new one is approximately \(-\$30,661\).** - Since the NPV is negative at the 16% MARR, **replacing is not economically justified**. ### **Final Recommendation:** **Do not replace the old machine; keep it.** --- ## **Summary:** | **Decision Metric** | **Result** | |----------------------|----------------------------------------| | NPV at 16% | \(-\$30,661\) | | IRR (approximate) | Around 10–11%, less than 16% | | **Conclusion** | **Reject replacement; keep the old machine** | This comprehensive analysis confirms that replacing the old machine is not economically beneficial given the data and assumptions.