Let's solve each part step by step, with explanations after each step. --- ### **Step 1: Calculate IRR for Project X** **Cash flows for Project X:** - Year 0: -$16,400 - Year 1: $6,660 - Year 2: $7,240 - Year 3: $4,760 The IRR is the rate \( r \) that makes NPV = 0: \[ 0 = -16,400 + \frac{6,660}{(1+r)^1} + \frac{7,240}{(1+r)^2} + \frac{4,760}{(1+r)^3} \] Let's estimate IRR using trial and error or a financial calculator. I'll show two rates and interpolate: #### Try r = 12% \[ NPV = -16,400 + \frac{6,660}{1.12} + \frac{7,240}{1.12^2} + \frac{4,760}{1.12^3} \] \[ = -16,400 + 5,946.43 + 5,779.71 + 3,393.96 = -16,400 + 15,120.10 = -1,279.90 \] #### Try r = 9% \[ NPV = -16,400 + \frac{6,660}{1.09} + \frac{7,240}{1.09^2} + \frac{4,760}{1.09^3} \] \[ = -16,400 + 6,110.09 + 6,098.35 + 3,660.63 = -16,400 + 15,869.07 = -530.93 \] #### Try r = 7% \[ NPV = -16,400 + \frac{6,660}{1.07} + \frac{7,240}{1.07^2} + \frac{4,760}{1.07^3} \] \[ = -16,400 + 6,224.30 + 6,321.27 + 3,887.19 = -16,400 + 16,432.76 = +32.76 \] So, at 7%, NPV is positive; at 9%, NPV is negative. Let's interpolate between 7% and 9%. \[ IRR \approx 7\% + \left(\frac{32.76}{32.76+530.93}\right) \times (9\%-7\%) \] \[ = 7\% + \left(\frac{32.76}{563.69}\right) \times 2\% \] \[ = 7\% + (0.0581) \times 2\% = 7\% + 0.116\% = 7.12\% \] But the NPV at 9% is only -530.93, so let's try 8%: #### Try r = 8% \[ NPV = -16,400 + \frac{6,660}{1.08} + \frac{7,240}{1.08^2} + \frac{4,760}{1.08^3} \] \[ = -16,400 + 6,166.67 + 6,207.50 + 3,784.36 = -16,400 + 16,158.53 = -241.47 \] So, between 7% (NPV = +32.76) and 8% (NPV = -241.47): \[ IRR \approx 7\% + \left(\frac{32.76}{32.76+241.47}\right) \times 1\% \] \[ = 7\% + \left(\frac{32.76}{274.23}\right) \times 1\% = 7\% + 0.1195\% = 7.12\% \] So, **IRR for Project X ≈ 7.12%** **Explanation:** We used the NPV formula at several rates and interpolated to find the rate at which NPV = 0. --- ### **Step 2: Calculate IRR for Project Y** **Cash flows for Project Y:** - Year 0: -$16,400 - Year 1: $7,190 - Year 2: $7,780 - Year 3: $3,530 \[ 0 = -16,400 + \frac{7,190}{(1+r)^1} + \frac{7,780}{(1+r)^2} + \frac{3,530}{(1+r)^3} \] #### Try r = 10% \[ NPV = -16,400 + \frac{7,190}{1.1} + \frac{7,780}{1.1^2} + \frac{3,530}{1.1^3} \] \[ = -16,400 + 6,536.36 + 6,421.49 + 2,654.37 = -16,400 + 15,612.22 = -787.78 \] #### Try r = 8% \[ NPV = -16,400 + \frac{7,190}{1.08} + \frac{7,780}{1.08^2} + \frac{3,530}{1.08^3} \] \[ = -16,400 + 6,648.15 + 6,681.95 + 2,804.40 = -16,400 + 16,134.50 = -265.50 \] #### Try r = 6% \[ NPV = -16,400 + \frac{7,190}{1.06} + \frac{7,780}{1.06^2} + \frac{3,530}{1.06^3} \] \[ = -16,400 + 6,783.02 + 6,920.37 + 2,963.47 = -16,400 + 16,666.86 = +266.86 \] Now interpolate between 6% (+266.86) and 8% (-265.50): \[ IRR \approx 6\% + \left(\frac{266.86}{266.86+265.50}\right) \times 2\% \] \[ = 6\% + \left(\frac{266.86}{532.36}\right) \times 2\% = 6\% + 0.5015 \times 2\% = 6\% + 1.003\% = 7.00\% \] So, **IRR for Project Y ≈ 7.00%** **Explanation:** We tested NPV at several discount rates and interpolated to find the IRR where NPV = 0. --- ### **Step 3: Find the Crossover Rate** The crossover rate is the IRR of the difference in cash flows between Projects X and Y. **Difference (X-Y):** - Year 0: 0 - Year 1: 6,660 - 7,190 = -530 - Year 2: 7,240 - 7,780 = -540 - Year 3: 4,760 - 3,530 = 1,230 Set NPV of differences to zero and solve for r: \[ 0 = \frac{-530}{(1+r)^1} + \frac{-540}{(1+r)^2} + \frac{1,230}{(1+r)^3} \] Let's try r = 10% \[ NPV = \frac{-530}{1.1} + \frac{-540}{1.1^2} + \frac{1,230}{1.1^3} \] \[ = -481.82 + (-446.28) + 922.73 = -481.82 - 446.28 + 922.73 = -1,010.1 + 922.73 = -5.37 \] Try r = 11% \[ NPV = \frac{-530}{1.11} + \frac{-540}{1.11^2} + \frac{1,230}{1.11^3} \] \[ = -477.48 + (-438.70) + 894.30 = -477.48 - 438.70 + 894.30 = -916.18 + 894.30 = -21.88 \] Try r = 13% \[ NPV = \frac{-530}{1.13} + \frac{-540}{1.13^2} + \frac{1,230}{1.13^3} \] \[ = -469.03 + (-422.83) + 840.93 = -469.03 - 422.83 + 840.93 = -891.86 + 840.93 = -50.93 \] Wait, the NPV is negative and getting more negative as r increases. Let's try r = 8% \[ NPV = \frac{-530}{1.08} + \frac{-540}{1.08^2} + \frac{1,230}{1.08^3} \] \[ = -490.74 + (-463.07) + 972.91 = -490.74 - 463.07 + 972.91 = -953.81 + 972.91 = +19.10 \] So, at 8%: +19.10; at 10%: -5.37. Now interpolate between 8% and 10%: \[ Crossover = 8\% + \frac{19.10}{19.10+5.37} \times 2\% = 8\% + \frac{19.10}{24.47} \times 2\% \] \[ = 8\% + 0.7809 \times 2\% = 8\% + 1.56\% = 9.56\% \] **Explanation:** The crossover rate is the IRR of the difference in cash flows. We set up the difference, calculated NPV at various rates, and interpolated. --- ## **Final Answers** - **IRR for Project X:** **7.12%** - **IRR for Project Y:** **7.00%** - **Crossover Rate:** **9.56%**