Let’s solve the Rate of Return (ROR) on the extra investment required for the EMT alternative using the **Annual Worth (AW) method**. ### **Step 1: Define the Differences** - **Initial Cost Difference:** \( \Delta P = 800,000 - 680,000 = \$120,000 \) - **Annual Cost Difference:** \( \Delta A = 152,000 - 200,000 = -\$48,000 \) - **Salvage Value Difference:** \( \Delta S = 130,000 - 70,000 = \$60,000 \) - **Project Life:** 5 years ### **Step 2: Set Up the AW Equation** The incremental cash flows are: - **Year 0:** \(-\$120,000\) (extra initial cost) - **Years 1-5:** \(+\$48,000\) (annual savings) - **Year 5:** \(+\$60,000\) (extra salvage value) The AW-based ROR equation is: \[ 0 = -\Delta P + \Delta A \cdot (P/A, i, 5) + \Delta S \cdot (P/F, i, 5) \] Substitute the values: \[ 0 = -120,000 + 48,000 \cdot (P/A, i, 5) + 60,000 \cdot (P/F, i, 5) \] ### **Step 3: Express in Terms of i** \[ 120,000 = 48,000 \cdot (P/A, i, 5) + 60,000 \cdot (P/F, i, 5) \] Where: - \( (P/A, i, 5) = \frac{(1+i)^5-1}{i(1+i)^5} \) - \( (P/F, i, 5) = \frac{1}{(1+i)^5} \) ### **Step 4: Guess and Check (or Use Financial Calculator/Excel)** Let’s try to estimate \( i \): #### **Approximation Using Excel or Calculator** Alternatively, you can use **Excel's IRR** or **NPV functions**, or solve iteratively. #### **Manual Estimate** - If you try \( i = 20\% \): \[ (P/A, 20\%, 5) = 2.9906 \] \[ (P/F, 20\%, 5) = 0.4019 \] \[ 48,000 \times 2.9906 = 143,548.8 \] \[ 60,000 \times 0.4019 = 24,114 \] \[ 143,548.8 + 24,114 = 167,662.8 \] So, \[ 120,000 / 167,662.8 \approx 0.716 \] which means the ROR is higher than 20%. Try \( i = 30\% \): \[ (P/A, 30\%, 5) = 2.644 \] \[ (P/F, 30\%, 5) = 0.2865 \] \[ 48,000 \times 2.644 = 126,912 \] \[ 60,000 \times 0.2865 = 17,190 \] \[ 126,912 + 17,190 = 144,102 \] \[ 120,000/144,102 \approx 0.833 \] Try \( i = 40\% \): \[ (P/A, 40\%, 5) = 2.379 \] \[ (P/F, 40\%, 5) = 0.2401 \] \[ 48,000 \times 2.379 = 114,192 \] \[ 60,000 \times 0.2401 = 14,406 \] \[ 114,192 + 14,406 = 128,598 \] \[ 120,000/128,598 \approx 0.934 \] Try \( i = 50\% \): \[ (P/A, 50\%, 5) = 2.1768 \] \[ (P/F, 50\%, 5) = 0.1317 \] \[ 48,000 \times 2.1768 = 104,486 \] \[ 60,000 \times 0.1317 = 7,902 \] \[ 104,486 + 7,902 = 112,388 \] \[ 120,000/112,388 \approx 1.068 \] So, at 40% the sum is slightly above 120,000, and at 50% it’s below 120,000. The answer is **slightly below 50%**, roughly **45%**. ### **Final Answer** **The ROR on the extra investment for the EMT alternative is approximately:** \[ \boxed{45\%} \] If you need a more precise answer, you can interpolate or use Excel's IRR or Goal Seek function. Let me know if you want to see how to do it in Excel!

**Step-by-Step Comprehensive Solution to Determine the ROR on the Extra Investment for EMT Using an AW-Based Equation** --- ### **Step 1: Identify and Quantify the Incremental Cash Flows** **Given Data:** | Parameter | Hydraulic Press (HP) | Electric Machine Technology (EMT) | Difference (Δ) | |----------------------------------|---------------------|------------------------------|------------------------------| | First Cost (Initial Investment) | \$680,000 | \$800,000 | ΔP = \$800,000 - \$680,000 = **\$120,000** | | Annual Operating Costs | \$200,000 | \$152,000 | ΔA = \$152,000 - \$200,000 = **-\$48,000** | | Salvage Value after 5 years | \$70,000 | \$130,000 | ΔS = \$130,000 - \$70,000 = **\$60,000** | | Project Life | 5 years | 5 years | - | **Explanation:** The incremental cash flows stem from choosing the EMT over the hydraulic press, representing additional initial investment, annual operating savings, and additional salvage value. These differences form the basis for the ROR calculation. --- ### **Step 2: Formulate the AW-Based Equation for the Incremental Cash Flows** **Methodology:** The ROR on the extra investment is the interest rate \(i\) that equates the present worth of incremental costs and benefits to zero, which can be expressed using an annual worth approach: \[ 0 = -\Delta P + \Delta A \times (P/A, i, 5) + \Delta S \times (P/F, i, 5) \] Where: - \( (P/A, i, 5) \) = Present worth factor of an annuity of 1 per year over 5 years at interest rate \(i\). - \( (P/F, i, 5) \) = Present worth factor of a single sum of 1 at the end of 5 years at interest rate \(i\). **Substituting the known differences:** \[ 0 = -120,000 + (-48,000) \times (P/A, i, 5) + 60,000 \times (P/F, i, 5) \] Rearranged: \[ 120,000 = 48,000 \times (P/A, i, 5) + 60,000 \times (P/F, i, 5) \] **Explanation:** This equation equates the incremental initial investment to the present worth of the incremental annual savings and salvage value, expressed as annual worth factors at interest rate \(i\). Solving for \(i\) yields the ROR on the extra investment. --- ### **Step 3: Express the Present Worth Factors** The two key factors are: \[ (P/A, i, 5) = \frac{(1 + i)^5 - 1}{i \times (1 + i)^5} \] \[ (P/F, i, 5) = \frac{1}{(1 + i)^5} \] These formulas depend on the interest rate \(i\). Since \(i\) is unknown, iterative estimation or interpolation is used to find the interest rate that satisfies the equation. --- ### **Step 4: Approximate the ROR Using Trial and Error** **Method:** - Use standard present worth factors for various \(i\) values. - Calculate the right side of the equation for each \(i\). - Find the interest rate \(i\) that makes the equation balance. **Calculations:** | \(i\) (%) | \((P/A, i, 5)\) | \((P/F, i, 5)\) | RHS Calculation | Sum of Terms | Difference from \$120,000 | |------------|------------------|------------------|-----------------|----------------|-------------------------| | 20% | 2.991 | 0.402 | \(48,000 \times 2.991 = 143,568\) | \(143,568 + 60,000 \times 0.402 = 143,568 + 24,120 = 167,688\) | \$120,000 - \$167,688 = -\$47,688 | | 30% | 2.626 | 0.376 | \(48,000 \times 2.626 = 125,088\) | \(125,088 + 60,000 \times 0.376 = 125,088 + 22,560 = 147,648\) | -\$27,648 | | 40% | 2.380 | 0.286 | \(48,000 \times 2.380 = 114,240\) | \(114,240 + 60,000 \times 0.286 = 114,240 + 17,160 = 131,400\) | -\$11,400 | | 50% | 2.325 | 0.188 | \(48,000 \times 2.325 = 111,600\) | \(111,600 + 60,000 \times 0.188 = 111,600 + 11,280 = 122,880\) | -\$2,880 | | 55% | 2.262 | 0.165 | \(48,000 \times 2.262 = 108,576\) | \(108,576 + 60,000 \times 0.165 = 108,576 + 9,900 = 118,476\) | +\$1,524 | Observation: - At 50%, the sum is \$122,880, slightly above \$120,000. - At 55%, the sum drops below \$120,000. - The interest rate \(i\) that balances the equation is approximately between 50% and 55%. Interpolating: \[ i \approx 50\% + \frac{120,000 - 122,880}{122,880 - 118,476} \times (55\% - 50\%) \approx 50\% + \frac{-2,880}{4,404} \times 5\% \] \[ i \approx 50\% - 0.654 \times 5\% \approx 50\% - 3.27\% = 46.7\% \] **Result:** The approximate ROR is **46.7%**, which can be rounded to **approximately 47%** for reporting purposes. --- ### **Final Answer:** \[ \boxed{ \text{The ROR on the extra investment for EMT is approximately } \mathbf{47\%} } \] This indicates that the additional investment in EMT yields an annual return of about 47%, based on the incremental cash flows and the AW method. --- ### **Summary:** - The analysis involved calculating incremental cash flows. - Formulating an AW-based equation relating these flows to the interest rate. - Estimating the interest rate through trial and interpolation. - The resulting ROR (~47%) helps in assessing whether the extra investment in EMT is justified compared to other investment opportunities or benchmarks. --- **Note:** For more precise results, financial calculator or software (e.g., Excel's IRR function) can be used to directly compute the exact interest rate satisfying the cash flow equation.