Let's break down the problem step by step. --- ## 1. **CFFA in Year 1** (Cash Flow From Assets) ### **Key Data** - **Initial Investment:** $50,000 (ignore for this question) - **Annual Savings (Year 1):** $40,000 - **Savings Growth Rate:** 3% per year - **Project Life:** 10 years - **Depreciation:** Straight-line over 10 years, zero salvage value \(\text{Annual Depreciation} = \frac{\$50,000}{10} = \$5,000\) per year - **Tax Rate:** 21% ### **Step 1: Calculate EBIT (Earnings Before Interest & Taxes)** - Savings = $40,000 (acts as a reduction in expense, i.e., positive EBIT impact) - Depreciation = $5,000 (non-cash expense) \[ \text{EBIT} = \text{Savings} - \text{Depreciation} = \$40,000 - \$5,000 = \$35,000 \] ### **Step 2: Calculate Taxes** \[ \text{Tax} = \text{EBIT} \times \text{Tax Rate} = \$35,000 \times 0.21 = \$7,350 \] ### **Step 3: Net Income** \[ \text{Net Income} = \text{EBIT} - \text{Tax} = \$35,000 - \$7,350 = \$27,650 \] ### **Step 4: Add Back Depreciation (Non-cash)** \[ \text{CFFA Year 1} = \text{Net Income} + \text{Depreciation} = \$27,650 + \$5,000 = \boxed{\$32,650} \] --- ## 2. **NPV of the Project** ### **Restate All Cash Flows** - **Year 0:** Initial investment = **-$50,000** - **Years 1–10:** - Annual savings in Year 1 = $40,000, grows at 3% per year - Depreciation = $5,000 per year - Tax rate = 21% - Discount rate (cost of capital) = 13% ### **General Formula for CFFA in Year t:** 1. Calculate savings in year t: \( S_t = \$40,000 \times (1.03)^{t-1} \) 2. EBIT\(_t\) = \( S_t - 5,000 \) 3. Taxes\(_t\) = \( \text{EBIT}_t \times 0.21 \) 4. Net Income\(_t\) = EBIT\(_t\) - Taxes\(_t\) 5. CFFA\(_t\) = Net Income\(_t\) + \$5,000 #### **Let's Generalize:** \[ \text{CFFA}_t = \left[ (S_t - 5,000) \times (1 - 0.21) \right] + 5,000 \] \[ \text{CFFA}_t = (S_t - 5,000) \times 0.79 + 5,000 \] \[ \text{CFFA}_t = 0.79 S_t - 0.79 \times 5,000 + 5,000 \] \[ \text{CFFA}_t = 0.79 S_t - 3,950 + 5,000 \] \[ \text{CFFA}_t = 0.79 S_t + 1,050 \] --- ### **Present Value Calculation** Let: - \( S_t = 40,000 \times (1.03)^{t-1} \) So, \[ \text{CFFA}_t = 0.79 \times 40,000 \times (1.03)^{t-1} + 1,050 \] \[ = 31,600 \times (1.03)^{t-1} + 1,050 \] #### **NPV Formula:** \[ \text{NPV} = -50,000 + \sum_{t=1}^{10} \frac{\text{CFFA}_t}{(1.13)^t} \] --- #### **Let's Break It Into Two Streams:** 1. **Growing Annuity:** \( 31,600 \) growing at 3% for 10 years at 13% 2. **Constant Annuity:** $1,050 for 10 years at 13% --- ### 2.1 **PV of Growing Annuity:** \[ PV_{GA} = C \times \frac{1 - \left(\frac{1+g}{1+r}\right)^N}{r-g} \] Where: - \( C = 31,600 \) - \( g = 0.03 \) - \( r = 0.13 \) - \( N = 10 \) Plug values in: \[ PV_{GA} = 31,600 \times \frac{1 - \left(\frac{1.03}{1.13}\right)^{10}}{0.13 - 0.03} \] Calculate \( \left(\frac{1.03}{1.13}\right)^{10} \): \[ \frac{1.03}{1.13} = 0.9115 \] \[ 0.9115^{10} = e^{10 \times \ln(0.9115)} = e^{10 \times -0.0924} = e^{-0.924} \approx 0.3966 \] So, \[ PV_{GA} = 31,600 \times \frac{1 - 0.3966}{0.10} \] \[ = 31,600 \times \frac{0.6034}{0.10} \] \[ = 31,600 \times 6.034 = \$190,674.40 \] --- ### 2.2 **PV of Level Annuity:** \[ PV_{A} = C \times \frac{1 - \frac{1}{(1 + r)^N}}{r} \] Where: - \( C = 1,050 \) - \( r = 0.13 \) - \( N = 10 \) Calculate \( (1.13)^{10} \): \[ (1.13)^{10} = e^{10 \times \ln(1.13)} = e^{10 \times 0.1220} = e^{1.220} \approx 3.387 \] \[ \frac{1}{3.387} = 0.2952 \] \[ PV_{A} = 1,050 \times \frac{1 - 0.2952}{0.13} \] \[ = 1,050 \times \frac{0.7048}{0.13} \] \[ = 1,050 \times 5.422 = \$5,693.10 \] --- ### 2.3 **Total PV of CFs:** \[ PV_{Total} = 190,674.40 + 5,693.10 = \$196,367.50 \] --- ### 2.4 **Subtract Initial Investment:** \[ NPV = PV_{Total} - 50,000 = \$146,367.50 \] --- ## **Final Answers** ### **1. Year 1 CFFA:** \[ \boxed{ \$32,650 } \] ### **2. NPV of the project:** \[ \boxed{ \$146,368 } \quad \text{(rounded to nearest dollar)} \] --- **Let me know if you need the cash flows for each year or an Excel-compatible table!**

Certainly! Here's a detailed explanation of each step involved in calculating both the Year 1 CFFA and the overall NPV of the project. --- ## **Part 1: Calculating the Year 1 CFFA** **Step 1: Understanding the cash flow components** - The project saves $40,000 in raw materials leaks in Year 1. - Savings grow by 3% annually. - The project costs $50,000 upfront but we're focusing on Year 1 cash flow, so we ignore the initial cost here. - The savings impact the company's EBIT (Earnings Before Interest & Taxes). - The savings are considered as a reduction in expenses, so they increase EBIT. **Step 2: Calculate EBIT** - EBIT = Savings - Depreciation - Since depreciation is a non-cash expense, it reduces taxable income but doesn't affect cash flow directly. \[ \text{EBIT} = \$40,000 - \$5,000 = \$35,000 \] **Step 3: Calculate taxes** - Taxes are paid on EBIT. \[ \text{Taxes} = \$35,000 \times 21\% = \$7,350 \] **Step 4: Determine Net Income** - Net Income = EBIT - Taxes \[ \$35,000 - \$7,350 = \$27,650 \] **Step 5: Add back Depreciation** - Since depreciation is a non-cash expense, it is added back to net income to determine cash flow. \[ \text{CFFA Year 1} = \$27,650 + \$5,000 = \boxed{\$32,650} \] --- ## **Part 2: Calculating the NPV of the entire project** **Step 1: Understand the cash flow structure over 10 years** - Savings start at $40,000 in Year 1, growing at 3% annually. - The annual savings in Year \( t \): \[ S_t = \$40,000 \times (1.03)^{t-1} \] - The savings impact EBIT, and after taxes and depreciation, we derive the cash flow for each year. **Step 2: Derive the formula for annual cash flows** - EBIT in year \( t \): \[ \text{EBIT}_t = S_t - \$5,000 \] - Taxes in year \( t \): \[ \text{Tax}_t = \text{EBIT}_t \times 21\% \] - Net income: \[ \text{Net Income}_t = \text{EBIT}_t - \text{Tax}_t = \text{EBIT}_t \times (1 - 0.21) = \text{EBIT}_t \times 0.79 \] - Add back depreciation to find total cash flow: \[ \text{CFFA}_t = \text{Net Income}_t + \text{Depreciation} = 0.79 \times (S_t - 5,000) + 5,000 \] Simplify: \[ \text{CFFA}_t = 0.79 \times S_t - 0.79 \times 5,000 + 5,000 = 0.79 \times S_t + 1,050 \] This formula allows us to compute the cash flow for each year directly. **Step 3: Calculate the present value of all cash flows** - Discount rate \( r = 13\% \). - The cash flows form a growing annuity due to increasing savings. - The cash flow in Year \( t \): \[ \text{CFFA}_t = 31,600 \times (1.03)^{t-1} + 1,050 \] - The present value (PV) of these cash flows over 10 years: \[ \text{NPV} = -\$50,000 + \sum_{t=1}^{10} \frac{\text{CFFA}_t}{(1.13)^t} \] **Step 4: Break down the sum** The total PV includes: - The PV of the growing component: \[ PV_{growth} = 31,600 \times \frac{1 - \left(\frac{1.03}{1.13}\right)^{10}}{0.13 - 0.03} \] - The PV of the constant component: \[ PV_{constant} = 1,050 \times \frac{1 - \frac{1}{(1.13)^{10}}}{0.13} \] Calculating these: - \(\left(\frac{1.03}{1.13}\right)^{10}\): converts the geometric growth into a factor for discounting. - Use natural logs and exponentials to compute powers efficiently. **Step 5: Final calculation** - Sum the two PV components: \[ PV_{total} = PV_{growth} + PV_{constant} \] - Subtract initial investment: \[ NPV = PV_{total} - 50,000 \] **Result:** \[ NPV \approx \$146,368 \] --- **Summary:** - **Year 1 CFFA** is about **\$32,650**, reflecting the net cash benefit from savings after taxes, plus non-cash depreciation. - **NPV** over 10 years, considering the growth of savings, discounting at 13%, and initial investment, is approximately **\$146,368**, indicating a highly profitable project. --- Would you like a step-by-step Excel formula setup or further clarification on any part?