Let's break down the problem step-by-step: **Given Data:** - Initial outlay, \( C_0 = -\$5,000 \) - Cash flows for Years 1 through 10 are the same - IRR = 13% - WACC = 10% --- ### Step 1: Find the annual cash flow \( CF \) Since cash flows are the same each year and IRR is 13%, we can use the IRR formula: \[ 0 = -C_0 + \sum_{t=1}^{10} \frac{CF}{(1 + IRR)^t} \] This is equivalent to: \[ C_0 = \sum_{t=1}^{10} \frac{CF}{(1 + IRR)^t} \] which is the present value (PV) of an annuity: \[ C_0 = CF \times \text{PVIFA}_{IRR,10} \] where: \[ \text{PVIFA}_{r,n} = \frac{1 - (1 + r)^{-n}}{r} \] Calculating PVIFA at IRR = 13%, n=10: \[ \text{PVIFA}_{13\%,10} = \frac{1 - (1 + 0.13)^{-10}}{0.13} \] Calculate denominator: \[ (1 + 0.13)^{10} = (1.13)^{10} \] Using a calculator: \[ (1.13)^{10} \approx e^{10 \times \ln(1.13)} \approx e^{10 \times 0.121} \approx e^{1.21} \approx 3.354 \] Thus, \[ (1.13)^{-10} = \frac{1}{3.354} \approx 0.298 \] Now, \[ \text{PVIFA} = \frac{1 - 0.298}{0.13} = \frac{0.702}{0.13} \approx 5.404 \] Now, solve for \( CF \): \[ 5,000 = CF \times 5.404 \implies CF = \frac{5,000}{5.404} \approx 924.86 \] --- ### Step 2: Calculate the future value (FV) of cash inflows at WACC (10%) Since the project’s cash flows are the same each year, the future value at year 10 (FV at year 10): \[ FV_{10} = CF \times \text{FVIFA}_{WACC,10} \] Where: \[ \text{FVIFA}_{r,n} = \frac{(1 + r)^n - 1}{r} \] At \( r = 10\% \), \( n=10 \): \[ (1 + 0.10)^{10} = (1.10)^{10} \approx e^{10 \times \ln 1.10} \approx e^{10 \times 0.0953} \approx e^{0.953} \approx 2.593 \] So, \[ \text{FVIFA}_{10\%,10} = \frac{2.593 - 1}{0.10} = \frac{1.593}{0.10} = 15.93 \] Now, compute FV at year 10: \[ FV_{10} = 924.86 \times 15.93 \approx 924.86 \times 15.93 \approx 14,744.76 \] --- ### Step 3: Calculate the MIRR **MIRR formula:** \[ \text{MIRR} = \left( \frac{FV_{\text{positive cash flows at WACC}}}{|C_0|} \right)^{1/n} - 1 \] Where: - \( FV_{\text{positive cash flows at WACC}} = FV_{10} \) - \( C_0 = -\$5,000 \) - \( n = 10 \) Plug in: \[ \text{MIRR} = \left( \frac{14,744.76}{5,000} \right)^{1/10} - 1 \] Calculate ratio: \[ \frac{14,744.76}{5,000} = 2.94895 \] Now, take the 10th root: \[ (2.94895)^{1/10} = e^{\frac{1}{10} \times \ln(2.94895)} \approx e^{0.1 \times 1.080} = e^{0.108} \approx 1.114 \] Finally: \[ \text{MIRR} \approx 1.114 - 1 = 0.114 \text{ or } 11.40\% \] --- ### **Final Answer:** \(\boxed{11.40\%}\) **The project's MIRR is approximately \(\boxed{11.40\%}\).**

To calculate the project's Modified Internal Rate of Return (MIRR), we first determine the annual cash flow using the IRR. With a $5,000 initial outlay and constant annual cash flows for 10 years, and knowing the IRR is 13%, we reverse the annuity formula to find the annual cash flow is approximately $924.86 per year. Next, we compound all positive cash flows forward at the project's WACC of 10% to the end of year 10, giving a future value of about $14,744.76. The MIRR is then the rate that equates the initial investment ($5,000) to the future value of the cash inflows over 10 years: \(\text{MIRR} = \left(\frac{14,744.76}{5,000}\right)^{1/10} - 1\), which gives approximately 11.40%. This MIRR reflects the project's true return, assuming interim cash flows are reinvested at the WACC, and is a better indicator of profitability than IRR when the reinvestment rate assumption matters. **Final answer: The project's MIRR is approximately 11.40%.**