Let's break down the problem and solve it step by step: ### **Given Data:** - **Initial cost of solar panels (\(C_0\)):** \$3,000 (outflow, at time 0) - **Monthly savings:** \$30 per month (inflow, for 10 years) - **Lifetime:** 10 years - **Residual value:** Negligible (so, zero) - **We need to find:** The **annual effective IRR** (Internal Rate of Return), rounded to two decimal places. --- ## **Step 1: Calculate the Cash Flows** - **Initial cash flow (time 0):** \(-\$3,000\) - **Monthly cash inflow:** \(+ \$30\) for each month, for 10 years (\(10 \times 12 = 120\) months). --- ## **Step 2: Convert Monthly Cash Flows to Annual Cash Flows** - **Annual savings:** \(30 \times 12 = \$360\) per year - **Total number of years:** 10 So, the cash flow stream is: - Year 0: \(-\$3,000\) - Year 1 to Year 10: \(+\$360\) each year --- ## **Step 3: Set Up the IRR Equation** The IRR is the rate \(r\) that makes the NPV zero: \[ -3,000 + 360 \times \left(\frac{1 - (1 + r)^{-10}}{r}\right) = 0 \] --- ## **Step 4: Solve for IRR** \[ 360 \times \left(\frac{1 - (1 + r)^{-10}}{r}\right) = 3,000 \] \[ \frac{1 - (1 + r)^{-10}}{r} = \frac{3,000}{360} = 8.3333 \] So, \[ 1 - (1 + r)^{-10} = 8.3333r \] \[ 1 - 8.3333r = (1 + r)^{-10} \] Take the reciprocal: \[ \frac{1}{1 - 8.3333r} = (1 + r)^{10} \] Now, take the 10th root: \[ 1 + r = \left(\frac{1}{1 - 8.3333r}\right)^{1/10} \] This equation is transcendental and must be solved numerically. --- ## **Step 5: Numerical Solution (Trial & Error or Use Financial Calculator or Excel)** Alternatively, let's estimate using trial and error: ### **Try \(r = 8\%\) (\(0.08\))** \[ (1 + 0.08)^{-10} = (1.08)^{-10} \approx 0.4632 \] \[ 1 - 0.4632 = 0.5368 \] \[ \text{NPV} = -3,000 + 360 \times \frac{0.5368}{0.08} = -3,000 + 360 \times 6.71 = -3,000 + 2,415.6 = -584.4 \] Still negative. Need a lower rate. ### **Try \(r = 5\%\) (\(0.05\))** \[ (1.05)^{-10} = 0.6139 \] \[ 1 - 0.6139 = 0.3861 \] \[ \text{NPV} = -3,000 + 360 \times \frac{0.3861}{0.05} = -3,000 + 360 \times 7.722 = -3,000 + 2,779.92 = -220.08 \] Still negative, but closer. ### **Try \(r = 3\%\) (\(0.03\))** \[ (1.03)^{-10} = 0.7441 \] \[ 1 - 0.7441 = 0.2559 \] \[ \frac{0.2559}{0.03} = 8.63 \] \[ 360 \times 8.63 = 3,106.8 \] \[ \text{NPV} = -3,000 + 3,106.8 = +106.8 \] Now positive! So IRR is between 3% and 5%. --- ## **Step 6: Interpolate** Let’s use linear interpolation between 3% and 5%: - At 3%: NPV = +106.8 - At 5%: NPV = -220.08 \[ \text{IRR} \approx 3\% + \frac{106.8}{106.8 + 220.08} \times (5\% - 3\%) = 3\% + \frac{106.8}{326.88} \times 2\% \] \[ \frac{106.8}{326.88} \approx 0.3266 \] \[ 0.3266 \times 2\% = 0.6532\% \] \[ \text{IRR} \approx 3\% + 0.65\% = 3.65\% \] --- ## **Step 7: Refine the Estimate** Let’s try \(r = 4\%\): \[ (1.04)^{-10} = 0.6756 \] \[ 1 - 0.6756 = 0.3244 \] \[ 0.3244 / 0.04 = 8.11 \] \[ 360 \times 8.11 = 2,919.6 \] \[ \text{NPV} = -3,000 + 2,919.6 = -80.4 \] Now between 3% (NPV = +106.8) and 4% (NPV = -80.4). \[ \text{IRR} = 3\% + \frac{106.8}{106.8 + 80.4} \times (4\% - 3\%) = 3\% + \frac{106.8}{187.2} \times 1\% \] \[ \frac{106.8}{187.2} = 0.571 \] \[ 0.571 \times 1\% = 0.571\% \] \[ \text{IRR} \approx 3.57\% \] --- ## **Step 8: Final Answer (Rounded to Two Decimal Places)** \[ \boxed{3.57\%} \] --- ## **Summary Table** | Rate | NPV | |------|----------| | 3% | +106.8 | | 4% | -80.4 | So the **annual effective IRR** of this investment is: \[ \boxed{3.57\%} \] --- ### **Explanation** - We set up the cash flows: \(-\$3,000\) at time 0, then \$360 per year for 10 years. - We used the IRR formula for an annuity and solved numerically. - By interpolating between 3% and 4%, we found that the IRR is approximately \(3.57\%\). --- **Final Answer:** **The annual effective IRR of this investment is** \(\boxed{3.57\%}\) **(rounded to two decimal places).**