Let's break the problem into two parts: --- ## **1. Simple Payback Period** The **simple payback period** is the time required to recover the initial investment from the annual net benefits. ### **Given:** - Initial investment (cost) = \$13.5 million - Annual benefits (savings/income) = \$2.7 million - Residual value at end of 40 years = negligible (effectively zero) ### **Formula:** \[ \text{Payback Period} = \frac{\text{Initial Investment}}{\text{Annual Benefit}} \] ### **Calculation:** \[ \text{Payback Period} = \frac{13.5}{2.7} = 5 \text{ years} \] --- ## **2. Internal Rate of Return (IRR)** The **IRR** is the discount rate (\( r \)) that makes the net present value (NPV) of all cash flows equal to zero. ### **Cash Flows:** - Year 0: \(-\$13.5\) million (outflow) - Years 1–40: \(+\$2.7\) million per year (inflow) - Year 40 residual: \$0 (negligible) ### **NPV Equation:** \[ 0 = -13.5 + 2.7 \times \left[ \frac{1-(1+r)^{-40}}{r} \right] \] We need to solve for \( r \). ### **Step-by-Step Calculation:** #### **Step 1: Set up the equation** \[ 13.5 = 2.7 \times \left[ \frac{1-(1+r)^{-40}}{r} \right] \] #### **Step 2: Isolate the annuity factor** \[ \frac{13.5}{2.7} = \frac{1-(1+r)^{-40}}{r} \] \[ 5 = \frac{1-(1+r)^{-40}}{r} \] #### **Step 3: Rearranging for \( r \)** \[ 1-(1+r)^{-40} = 5r \] \[ (1+r)^{-40} = 1-5r \] \[ -40 \ln(1+r) = \ln(1-5r) \] \[ \ln(1+r) = -\frac{1}{40} \ln(1-5r) \] But it's easier to just **trial values** for \( r \): --- ### **Step 4: Trial and Error (Guess and Check)** #### Try \( r = 10\% = 0.10 \): \[ (1+r)^{-40} = (1.10)^{-40} \approx 0.0221 \] \[ 1 - (1+r)^{-40} = 1 - 0.0221 = 0.9779 \] \[ \frac{0.9779}{0.10} = 9.779 \] But we want this to be **5**. #### Try \( r = 20\% = 0.20 \): \[ (1.20)^{-40} \approx 0.000103 \] \[ 1 - 0.000103 = 0.999897 \] \[ \frac{0.999897}{0.20} = 4.999 \] Perfect! This is almost exactly 5. --- ## **Final Answers:** ### **1. Simple Payback Period:** \[ \boxed{5\ \text{years}} \] ### **2. Internal Rate of Return (IRR):** \[ \boxed{20\%} \] --- ### **Summary Table** | Metric | Value | |-------------------------|-----------| | Payback Period | 5 years | | IRR | 20% | --- ### **Explanation** - The payback period is found by dividing the total investment by the annual benefit. - For IRR, we set NPV = 0 and solve for \( r \) using the present value of an annuity formula, then use trial and error or a financial calculator. Here, the IRR is 20%, as shown by our calculation. Let me know if you want to see the calculation with more decimal accuracy or using formulas!