Let's solve the problem in **3 steps**: **Step 1:** Use the Present Worth (PW) method at MARR = 15% to identify the best alternative. **Step 2:** Use the Incremental IRR method to confirm the same decision. **Step 3:** Compare the results and explain. --- ### **Step 1: Present Worth (PW) Method at 15%** #### **Calculate PW for Each Alternative** The formula for Present Worth is: \[ PW = \sum_{k=0}^{n} \frac{CF_k}{(1+i)^k} \] #### **Alternative A:** \[ PW_A = -200{,}000 + \sum_{k=1}^6 \frac{90{,}000}{(1.15)^k} \] \[ PW_A = -200{,}000 + 90{,}000 \times \left(\frac{1-(1.15)^{-6}}{0.15}\right) \] \[ = -200{,}000 + 90{,}000 \times 3.784 = -200{,}000 + 340,560 = \boxed{140,560} \] --- #### **Alternative B:** \[ PW_B = -230{,}000 + \sum_{k=1}^6 \frac{108{,}000}{(1.15)^k} \] \[ PW_B = -230{,}000 + 108{,}000 \times 3.784 = -230{,}000 + 408,672 = \boxed{178,672} \] --- #### **Alternative C:** \[ PW_C = -212{,}500 + \frac{-15{,}000}{1.15} + \sum_{k=2}^6 \frac{122{,}500}{(1.15)^k} \] First, find the present worth for years 2–6: \[ \text{PW of 122,500 for 5 years (years 2-6):} \] \[ PW_{2-6} = 122,500 \times \left(\frac{1-(1.15)^{-5}}{0.15}\right) \times (1.15)^{-1} \] But easier: Discount each cash flow back. - Year 0: -212,500 - Year 1: -15,000/(1.15) = -13,043 - Year 2: 122,500/(1.15^2) = 92,615 - Year 3: 122,500/(1.15^3) = 80,535 - Year 4: 122,500/(1.15^4) = 70,030 - Year 5: 122,500/(1.15^5) = 60,887 - Year 6: 122,500/(1.15^6) = 52,944 Sum: \[ PW_C = -212,500 - 13,043 + 92,615 + 80,535 + 70,030 + 60,887 + 52,944 = 131,468 \] **PW Comparison:** - A: 140,560 - B: 178,672 (**Best**) - C: 131,468 **Explanation:** Alternative **B** has the highest PW at 15%. So, B is preferred by the PW method. --- ### **Step 2: Incremental IRR Method** #### **Rank by Initial Cost:** A ($200k), C ($212.5k), B ($230k) #### **Compare C vs A:** Incremental cash flows (C-A): | Year | C-A | |------|----------| | 0 | -12,500 | | 1 | -105,000 | | 2-6 | 32,500 | Find IRR for these cash flows: \[ 0 = -12,500 - 105,000/(1+IRR) + 32,500 \times \left[\frac{1-(1+IRR)^{-5}}{IRR}\right] \times (1+IRR)^{-1} \] You can use a financial calculator or Excel for IRR, but since all incremental cash flows are positive except year 1, it's likely IRR is negative or much less than 15%. So, **A is preferred over C** (since incremental IRR < 15%). #### **Compare B vs A:** Incremental cash flows (B-A): | Year | B-A | |------|----------| | 0 | -30,000 | | 1-6 | 18,000 | Find IRR for these cash flows: \[ 0 = -30,000 + 18,000 \times \left( \frac{1-(1+IRR)^{-6}}{IRR} \right) \] Try IRR = 15%: \[ = -30,000 + 18,000 \times 3.784 = -30,000 + 68,112 = 38,112 > 0 \] Try IRR = 25%: \[ = -30,000 + 18,000 \times 3.326 = -30,000 + 59,868 = 29,868 > 0 \] Try IRR = 50%: \[ = -30,000 + 18,000 \times 2.139 = -30,000 + 38,502 = 8,502 > 0 \] Try IRR = 100%: \[ = -30,000 + 18,000 \times 1.0 = -30,000 + 18,000 = -12,000 < 0 \] So, IRR is between 50% and 100%. Since IRR (B-A) > 15%, **B is preferred over A**. **Explanation:** Incremental IRR between B and A is much greater than 15%, so B is preferred. --- ### **Step 3: Comparison & Conclusion** Both methods (PW and Incremental IRR) select **Alternative B**. **Final Answer:** > **Both the Present Worth (PW) method at 15% and the Incremental IRR method select Alternative B as the best design. This demonstrates that the same capital investment decision results from both methods when using the incremental investment analysis procedure.**