Let's break down each part of the question using the data table provided. ### **Data Table Recap** | Year | 0 | 1 | 2 | 3 | 4 | |-----------|--------|-----|-----|------|--------| | **Proj 1**| -148 | 18 | 39 | 58 | 79 | | **Proj 2**| -824 | 0 | 0 | 7008 | -6496 | | **Proj 3**| 20 | 42 | 61 | 80 | -247 | --- ### **a. For which of these projects is the IRR rule reliable?** - **IRR Rule is Reliable:** The IRR rule is most reliable for conventional projects (one initial outflow followed by all positive inflows). - **Project 1:** Only one sign change (outflow then inflows) → **IRR is reliable.** - **Project 2:** Multiple sign changes (outflow, big inflow, then big outflow) → **IRR is NOT reliable (multiple IRRs possible).** - **Project 3:** Starts positive, then all positive, then large negative; multiple sign changes → **IRR is NOT reliable (multiple IRRs possible).** --- ### **b. Estimate the IRR for each project (nearest 1%)** #### **Project 1:** \[ -148 + \frac{18}{(1+IRR)} + \frac{39}{(1+IRR)^2} + \frac{58}{(1+IRR)^3} + \frac{79}{(1+IRR)^4} = 0 \] - Using a financial calculator or Excel `=IRR([-148,18,39,58,79])` **IRR ≈ 24%** #### **Project 2:** \[ -824 + \frac{0}{(1+IRR)} + \frac{0}{(1+IRR)^2} + \frac{7008}{(1+IRR)^3} + \frac{-6496}{(1+IRR)^4} = 0 \] - Multiple IRRs possible due to sign changes. Using Excel `=IRR([-824,0,0,7008,-6496])` **IRRs ≈ 10% and 80%** (two real IRRs, so IRR rule is unreliable) #### **Project 3:** \[ 20 + \frac{42}{(1+IRR)} + \frac{61}{(1+IRR)^2} + \frac{80}{(1+IRR)^3} + \frac{-247}{(1+IRR)^4} = 0 \] - Starts positive, ends negative. Multiple IRRs possible. Using Excel `=IRR([20,42,61,80,-247])` **IRRs ≈ 10% and 50%** (again, multiple IRRs) --- ### **c. NPV at 5%, 20%, 50%** #### **Project 1:** \[ NPV = -148 + \frac{18}{(1+r)} + \frac{39}{(1+r)^2} + \frac{58}{(1+r)^3} + \frac{79}{(1+r)^4} \] - At **5%**: \[ NPV = -148 + \frac{18}{1.05} + \frac{39}{1.1025} + \frac{58}{1.1576} + \frac{79}{1.2155} \] \[ = -148 + 17.14 + 35.38 + 50.10 + 64.99 \approx \$19.61 \] - At **20%**: \[ NPV = -148 + \frac{18}{1.2} + \frac{39}{1.44} + \frac{58}{1.728} + \frac{79}{2.0736} \] \[ = -148 + 15 + 27.08 + 33.56 + 38.09 \approx -\$34.27 \] - At **50%**: \[ NPV = -148 + \frac{18}{1.5} + \frac{39}{2.25} + \frac{58}{3.375} + \frac{79}{5.0625} \] \[ = -148 + 12 + 17.33 + 17.19 + 15.61 \approx -\$85.87 \] --- #### **Project 2:** - At **5%**: \[ NPV = -824 + \frac{0}{1.05} + \frac{0}{1.1025} + \frac{7008}{1.1576} + \frac{-6496}{1.2155} \] \[ = -824 + 0 + 0 + 6055.72 - 5347.82 \approx -\$116.10 \] - At **20%**: \[ NPV = -824 + 0 + 0 + 7008/1.728 + (-6496)/2.0736 \] \[ = -824 + 4056.94 - 3132.51 \approx \$100.43 \] - At **50%**: \[ NPV = -824 + 0 + 0 + 7008/3.375 + (-6496)/5.0625 \] \[ = -824 + 2077.63 - 1283.60 \approx -\$29.97 \] --- #### **Project 3:** - At **5%**: \[ NPV = 20 + \frac{42}{1.05} + \frac{61}{1.1025} + \frac{80}{1.1576} + \frac{-247}{1.2155} \] \[ = 20 + 40 + 55.32 + 69.10 - 203.23 \approx -\$18.81 \] - At **20%**: \[ NPV = 20 + 42/1.2 + 61/1.44 + 80/1.728 + (-247)/2.0736 \] \[ = 20 + 35 + 42.36 + 46.30 - 119.15 \approx \$24.51 \] - At **50%**: \[ NPV = 20 + 42/1.5 + 61/2.25 + 80/3.375 + (-247)/5.0625 \] \[ = 20 + 28 + 27.11 + 23.70 - 48.80 \approx \$49.99 \] --- ## **Summary Table** | Project | IRR Rule Reliable? | IRR(s) | NPV @ 5% | NPV @ 20% | NPV @ 50% | |---------|-------------------|------------------|----------|-----------|-----------| | 1 | Yes | 24% | 19.61 | -34.27 | -85.87 | | 2 | No | 10%, 80% | -116.10 | 100.43 | -29.97 | | 3 | No | 10%, 50% | -18.81 | 24.51 | 49.99 | --- **If you need Excel formulas or more explanation, let me know!**