Let's solve each part step by step. --- ### **Cash Flows** | Year | Project X | Project Y | |------|-----------|-----------| | 0 | -16,400 | -16,400 | | 1 | 6,660 | 7,190 | | 2 | 7,240 | 7,780 | | 3 | 4,760 | 3,530 | --- ## **1. IRR of Project X** Set NPV = 0 and solve for IRR (\( r \)): \[ 0 = -16,400 + \frac{6,660}{(1+r)^1} + \frac{7,240}{(1+r)^2} + \frac{4,760}{(1+r)^3} \] This equation is best solved numerically (trial-and-error or using financial calculator/Excel). ### **Let's estimate:** #### Try 10% \[ NPV = -16,400 + \frac{6,660}{1.1} + \frac{7,240}{1.21} + \frac{4,760}{1.331} \] \[ = -16,400 + 6,054.55 + 5,991.74 + 3,577.38 = -16,400 + 15,623.67 = -776.33 \] #### Try 12% \[ NPV = -16,400 + \frac{6,660}{1.12} + \frac{7,240}{1.2544} + \frac{4,760}{1.404928} \] \[ = -16,400 + 5,946.43 + 5,776.16 + 3,389.05 = -16,400 + 15,111.64 = -1,288.36 \] #### Try 8% \[ NPV = -16,400 + \frac{6,660}{1.08} + \frac{7,240}{1.1664} + \frac{4,760}{1.259712} \] \[ = -16,400 + 6,166.67 + 6,207.37 + 3,779.73 = -16,400 + 16,153.77 = -246.23 \] #### Try 6% \[ NPV = -16,400 + \frac{6,660}{1.06} + \frac{7,240}{1.1236} + \frac{4,760}{1.191016} \] \[ = -16,400 + 6,283.02 + 6,444.86 + 3,998.99 = -16,400 + 16,726.87 = +326.87 \] So, at 6%: +326.87, at 8%: -246.23 Interpolate: \[ IRR \approx 6\% + \frac{326.87}{326.87 + 246.23} \times (8\% - 6\%) = 6\% + \frac{326.87}{573.1} \times 2\% \] \[ = 6\% + 0.5706 \times 2\% = 6\% + 1.14\% = 7.14\% \] #### Check at 7.14% \[ (1.0714)^1 = 1.0714, \quad (1.0714)^2 = 1.148, \quad (1.0714)^3 = 1.230 \] \[ NPV = -16,400 + \frac{6,660}{1.0714} + \frac{7,240}{1.148} + \frac{4,760}{1.230} \] \[ = -16,400 + 6,216.93 + 6,308.36 + 3,869.92 = -16,400 + 16,395.21 = -4.79 \] Very close to zero! --- ### **IRR for Project X:** \[ \boxed{7.14\%} \] --- ## **2. IRR of Project Y** Set NPV = 0: \[ 0 = -16,400 + \frac{7,190}{(1+r)} + \frac{7,780}{(1+r)^2} + \frac{3,530}{(1+r)^3} \] #### Try 10% \[ NPV = -16,400 + \frac{7,190}{1.1} + \frac{7,780}{1.21} + \frac{3,530}{1.331} \] \[ = -16,400 + 6,536.36 + 6,429.75 + 2,653.16 = -16,400 + 15,619.27 = -780.73 \] #### Try 12% \[ NPV = -16,400 + \frac{7,190}{1.12} + \frac{7,780}{1.2544} + \frac{3,530}{1.404928} \] \[ = -16,400 + 6,419.64 + 6,205.22 + 2,513.89 = -16,400 + 15,138.75 = -1,261.25 \] #### Try 8% \[ NPV = -16,400 + \frac{7,190}{1.08} + \frac{7,780}{1.1664} + \frac{3,530}{1.259712} \] \[ = -16,400 + 6,657.41 + 6,672.04 + 2,803.72 = -16,400 + 16,133.17 = -266.83 \] #### Try 6% \[ NPV = -16,400 + \frac{7,190}{1.06} + \frac{7,780}{1.1236} + \frac{3,530}{1.191016} \] \[ = -16,400 + 6,783.02 + 6,922.54 + 2,965.40 = -16,400 + 16,670.96 = +270.96 \] So, at 6%: +270.96, at 8%: -266.83 Interpolate: \[ IRR \approx 6\% + \frac{270.96}{270.96 + 266.83} \times (8\% - 6\%) = 6\% + \frac{270.96}{537.79} \times 2\% \] \[ = 6\% + 0.504 \times 2\% = 6\% + 1.01\% = 7.01\% \] #### Check at 7.01% \[ (1.0701)^1 = 1.0701, \quad (1.0701)^2 = 1.144, \quad (1.0701)^3 = 1.224 \] \[ NPV = -16,400 + \frac{7,190}{1.0701} + \frac{7,780}{1.144} + \frac{3,530}{1.224} \] \[ = -16,400 + 6,719.51 + 6,801.40 + 2,884.64 = -16,400 + 16,405.55 = +5.55 \] Very close to zero! --- ### **IRR for Project Y:** \[ \boxed{7.01\%} \] --- ## **3. Crossover Rate** The **crossover rate** is the IRR of the differences in cash flows between the two projects. \[ \text{Diff} = X - Y \] | Year | X | Y | Diff (X - Y) | |------|------|------|--------------| | 0 | -16,400 | -16,400 | 0 | | 1 | 6,660 | 7,190 | -530 | | 2 | 7,240 | 7,780 | -540 | | 3 | 4,760 | 3,530 | 1,230 | So, cash flows for the "difference project" are: | Year | Cash Flow | |------|-----------| | 0 | 0 | | 1 | -530 | | 2 | -540 | | 3 | 1,230 | Set NPV = 0: \[ 0 = \frac{-530}{(1+r)} + \frac{-540}{(1+r)^2} + \frac{1230}{(1+r)^3} \] Let’s solve for \( r \): Multiply both sides by \( (1+r)^3 \): \[ 0 = -530(1+r)^2 - 540(1+r) + 1230 \] Let’s expand: \[ -530(1 + 2r + r^2) - 540(1 + r) + 1230 = 0 \] \[ -530 - 1060r - 530r^2 - 540 - 540r + 1230 = 0 \] \[ (-530 - 540 + 1230) + (-1060r - 540r) - 530r^2 = 0 \] \[ 160 - 1600r - 530r^2 = 0 \] \[ -530r^2 - 1600r + 160 = 0 \] Multiply both sides by -1: \[ 530r^2 + 1600r - 160 = 0 \] Now, use the quadratic formula: \[ r = \frac{-1600 \pm \sqrt{1600^2 - 4 \cdot 530 \cdot (-160)}}{2 \cdot 530} \] \[ = \frac{-1600 \pm \sqrt{2,560,000 + 339,200}}{1,060} \] \[ = \frac{-1600 \pm \sqrt{2,899,200}}{1,060} \] \[ \sqrt{2,899,200} \approx 1,703.24 \] So, \[ r_1 = \frac{-1600 + 1703.24}{1060} = \frac{103.24}{1060} \approx 0.0975 = 9.75\% \] \[ r_2 = \frac{-1600 - 1703.24}{1060} = \frac{-3303.24}{1060} \approx -3.117 \] Negative IRR is not relevant, so: --- ### **Crossover Rate:** \[ \boxed{9.75\%} \] --- ## **Final Answers** - **IRR Project X:** 7.14% - **IRR Project Y:** 7.01% - **Crossover Rate:** 9.75% --- **If you need the calculations done in Excel format or any clarification on steps, let me know!**