Let's solve step-by-step: **Given:** - **Machine X:** - Initial cost = $25 million - Life = 10 years - After-tax inflow = $6 million/year (end of each year, for 10 years) - Needs to be replaced after 10 years (for a 20-year project, need to buy it twice) - Replacement cost increases by 3% per year due to inflation - **Machine Y:** - Initial cost = $55 million - Life = 20 years - After-tax inflow = $8 million/year (end of each year, for 20 years) - **Cost of capital:** 11% - **Inflation for machine price:** 3% per year (for replacement price only) --- ### **Step 1: Calculate NPV for Machine X (over 20 years)** #### **A. First Machine X: Years 0-9** - Cost at time 0 = $25M - Cash inflows: $6M/year for years 1-10 #### **B. Second Machine X: Years 10-19** - Cost at year 10 = $25M × (1.03)¹⁰ - Cash inflows: $6M/year for years 11-20 --- #### **A1. PV of first 10 years' inflows with Machine X** PV = \( \sum_{t=1}^{10} \frac{6}{(1.11)^t} \) This is a present value of an ordinary annuity: PV = \( 6 \times \left[\frac{1 - (1+0.11)^{-10}}{0.11}\right] \) - \( (1 + 0.11)^{-10} = (1.11)^{-10} ≈ 0.3522 \) - \( 1 - 0.3522 = 0.6478 \) - \( 0.6478 / 0.11 = 5.889 \) - PV = \( 6 \times 5.889 = 35.33 \) million #### **A2. PV of first outflow (cost)** - At year 0: -$25M --- #### **B1. Cost of replacing Machine X in year 10** Replacement cost = $25M × (1.03)¹⁰ Calculate \( (1.03)^{10} \): - \( (1.03)^{10} = 1.3439 \) - Replacement cost = $25M × 1.3439 = $33.60M PV of this outflow at year 0: PV = \( \frac{33.60}{(1.11)^{10}} \) - \( (1.11)^{10} ≈ 2.8394 \) - PV = \( 33.60 / 2.8394 = 11.84 \) million #### **B2. PV of 2nd 10-years' inflows (years 11-20)** Find PV at year 10, then discount to present: PV at year 10: Same formula as above: - PV10 = \( 6 \times 5.889 = 35.33 \) million Discount to present: PV0 = \( \frac{35.33}{(1.11)^{10}} = 35.33 / 2.8394 = 12.45 \) million --- #### **A3. Total NPV for Machine X** Sum all: \[ NPV_X = \text{(PV of 1st 10 years inflows)} - \text{(Initial cost)} + \text{(PV of 2nd 10 years inflows)} - \text{(PV of replacement cost)} \] \[ NPV_X = 35.33 - 25 + 12.45 - 11.84 = 10.94 \text{ million} \] --- ### **Step 2: Calculate NPV for Machine Y (over 20 years)** - Initial cost = $55M at Year 0 - After-tax inflow = $8M/year for 20 years PV of inflows: PV = \( 8 \times \left[\frac{1-(1.11)^{-20}}{0.11}\right] \) - \( (1.11)^{-20} ≈ 0.1240 \) - \( 1 - 0.1240 = 0.8760 \) - \( 0.8760 / 0.11 = 7.964 \) - PV = \( 8 \times 7.964 = 63.71 \) million NPV = PV of inflows - PV of cost NPV_Y = 63.71 - 55 = **8.71 million** --- ### **Step 3: Compare NPVs and Find Value Added** - NPV of Machine X: **$10.94 million** - NPV of Machine Y: **$8.71 million** **Machine X** is the better machine for the project. **Difference in value to company:** \[ 10.94 - 8.71 = 2.23 \text{ million} \] --- ## **Final Answers** ### **1. Which is the better machine?** **Machine X** is the better machine for this 20-year project. ### **2. By how much would the value of the company increase if it accepted the better machine (X) instead of the worse (Y)?** **By $2.23 million.** --- ## **Summary Table** | Machine | NPV ($ million) | Better? | |-----------|-----------------|---------| | X | 10.94 | Yes | | Y | 8.71 | No | **Difference in value if choosing X over Y:** $2.23 million --- If you need to see the calculations in a table or with more decimal precision, let me know!