Let's walk through each MIRR calculation step by step using the data: **Cash Flows:** | Year | Cash Flow | |------|-----------| | 0 | -53,000 | | 1 | 16,700 | | 2 | 21,900 | | 3 | 27,300 | | 4 | 20,400 | | 5 | 8,600 | **Discount Rate:** 11% **Reinvestment Rate:** 8% --- ## 1. **MIRR using the Discounting Approach** *All positive cash flows are discounted to present at the finance rate (11%). MIRR is the rate that equates the sum of discounted positive CFs with the initial outflow.* ### Step 1: Discount all positive cash flows to year 0 at 11% \[ \text{PV} = \sum_{t=1}^n \frac{CF_t}{(1+0.11)^t} \] - Year 1: \( \frac{16,700}{(1.11)^1} = 15,045.05 \) - Year 2: \( \frac{21,900}{(1.11)^2} = 17,786.24 \) - Year 3: \( \frac{27,300}{(1.11)^3} = 19,753.94 \) - Year 4: \( \frac{20,400}{(1.11)^4} = 13,396.56 \) - Year 5: \( \frac{8,600}{(1.11)^5} = 5,114.61 \) \[ \text{Total PV of positive CFs} = 15,045.05 + 17,786.24 + 19,753.94 + 13,396.56 + 5,114.61 = 71,096.40 \] ### Step 2: Find MIRR \[ \text{MIRR} = \left(\frac{\text{PV of positive CFs}}{|\text{PV of negative CFs}|}\right)^{1/n} - 1 \] \[ \text{MIRR} = \left(\frac{71,096.40}{53,000}\right)^{1/5} - 1 \] \[ = (1.3414)^{0.2} - 1 \] \[ = 1.0602 - 1 = 0.0602 = 6.02\% \] **Answer (Discounting Approach):** \[ \boxed{6.02\%} \] --- ## 2. **MIRR using the Reinvestment Approach** *All positive cash flows are compounded to the end of the project (year 5) at the reinvestment rate (8%). MIRR is the rate that equates the FV of positive CFs at year 5 to the initial outflow.* ### Step 1: Compound all positive cash flows to year 5 at 8% \[ \text{FV} = \sum_{t=1}^n CF_t \times (1+0.08)^{n-t} \] - Year 1: \( 16,700 \times (1.08)^{4} = 16,700 \times 1.3605 = 22,724.35 \) - Year 2: \( 21,900 \times (1.08)^{3} = 21,900 \times 1.2597 = 27,599.43 \) - Year 3: \( 27,300 \times (1.08)^{2} = 27,300 \times 1.1664 = 31,852.48 \) - Year 4: \( 20,400 \times (1.08)^{1} = 20,400 \times 1.08 = 22,032.00 \) - Year 5: \( 8,600 \times (1.08)^{0} = 8,600 \) \[ \text{Total FV of positive CFs (year 5)} = 22,724.35 + 27,599.43 + 31,852.48 + 22,032.00 + 8,600 = 112,808.26 \] ### Step 2: Find MIRR \[ \text{MIRR} = \left(\frac{\text{FV of positive CFs}}{|\text{PV of negative CFs}|}\right)^{1/n} - 1 \] \[ \text{MIRR} = \left(\frac{112,808.26}{53,000}\right)^{1/5} - 1 \] \[ = (2.1294)^{0.2} - 1 \] \[ = 1.1639 - 1 = 0.1639 = 16.39\% \] **Answer (Reinvestment Approach):** \[ \boxed{16.39\%} \] --- ## 3. **MIRR using the Combination Approach** *Discount all negative cash flows to present at finance rate (11%), compound all positive cash flows to the end at reinvestment rate (8%), then calculate the rate that equates them.* - Only one negative cash flow: -53,000 at year 0, so PV is -53,000. - FV of positive cash flows at year 5 (already calculated): 112,808.26 \[ \text{MIRR} = \left(\frac{\text{FV of positive CFs}}{|\text{PV of negative CFs}|}\right)^{1/n} - 1 \] \[ \text{MIRR} = \left(\frac{112,808.26}{53,000}\right)^{1/5} - 1 \] \[ = (2.1294)^{0.2} - 1 \] \[ = 1.1639 - 1 = 0.1639 = 16.39\% \] **Answer (Combination Approach):** \[ \boxed{16.39\%} \] --- ## **Summary Table** | Approach | MIRR | |------------------|---------| | Discounting | 6.02% | | Reinvestment | 16.39% | | Combination | 16.39% | --- ### **Final Answers (rounded to 2 decimals):** - **Discounting approach:** 6.02% - **Reinvestment approach:** 16.39% - **Combination approach:** 16.39%

Let's carefully re-calculate the MIRR using each approach, ensuring accuracy and clarity. --- ### Given Data: - **Initial Investment (Year 0):** -$53,000 - **Cash Flows (Years 1-5):** $16,700, $21,900, $27,300, $20,400, $8,600 - **Discount Rate (Finance rate):** 11% - **Reinvestment Rate:** 8% - **Project Duration:** 5 years --- ## 1. MIRR Using the Discounting Approach **Step 1:** Discount all positive cash flows to Year 0 at the **finance rate (11%)** (though here, only positive cash flows are compounded forward). For the discounting approach, the positive cash flows are compounded at the reinvestment rate, and negative cash flows are discounted at the finance rate, but since only the initial investment is negative, the calculation simplifies. **Step 2:** Calculate the **present value (PV)** of positive cash flows discounted at 11%: \[ PV_{positive} = \frac{16,700}{(1.11)^1} + \frac{21,900}{(1.11)^2} + \frac{27,300}{(1.11)^3} + \frac{20,400}{(1.11)^4} + \frac{8,600}{(1.11)^5} \] Calculating each term: - Year 1: \( \frac{16,700}{1.11} = 15,045.05 \) - Year 2: \( \frac{21,900}{1.2321} = 17,786.24 \) - Year 3: \( \frac{27,300}{1.3676} = 19,953.94 \) - Year 4: \( \frac{20,400}{1.5164} = 13,436.56 \) - Year 5: \( \frac{8,600}{1.6841} = 5,114.61 \) Sum: \[ PV_{positive} = 15,045.05 + 17,786.24 + 19,953.94 + 13,436.56 + 5,114.61 = 71,336.40 \] **Step 3:** Calculate the **MIRR**: \[ MIRR = \left( \frac{FV_{positive}}{|PV_{negative}|} \right)^{1/n} - 1 \] In the discounting approach, positive cash flows are compounded to the end of the project at the reinvestment rate (8%). **Step 4:** Compound positive cash flows to Year 5: \[ FV_{positive} = 16,700 \times (1.08)^4 + 21,900 \times (1.08)^3 + 27,300 \times (1.08)^2 + 20,400 \times (1.08)^1 + 8,600 \] Calculations: - Year 1: \( 16,700 \times 1.3605 = 22,724.35 \) - Year 2: \( 21,900 \times 1.2597 = 27,599.43 \) - Year 3: \( 27,300 \times 1.1664 = 31,852.32 \) - Year 4: \( 20,400 \times 1.08 = 22,032 \) - Year 5: \( 8,600 \) Sum: \[ FV_{positive} = 22,724.35 + 27,599.43 + 31,852.32 + 22,032 + 8,600 = 112,808.43 \] **Step 5:** Calculate MIRR: \[ MIRR = \left( \frac{112,808.43}{53,000} \right)^{1/5} - 1 = (2.1313)^{0.2} - 1 \] \[ = 1.1639 - 1 = 0.1639 \text{ or } \boxed{16.39\%} \] --- ## 2. MIRR Using the Reinvestment Approach **Step 1:** Calculate the **future value (FV)** of positive cash flows at the **reinvestment rate (8%)**: \[ FV = 16,700 \times (1.08)^4 + 21,900 \times (1.08)^3 + 27,300 \times (1.08)^2 + 20,400 \times (1.08)^1 + 8,600 \] Same as above, sum is: \[ FV = 22,724.35 + 27,599.43 + 31,852.32 + 22,032 + 8,600 = 112,808.43 \] **Step 2:** Compute MIRR: \[ MIRR = \left( \frac{FV}{|PV_{initial}|} \right)^{1/n} - 1 = \left( \frac{112,808.43}{53,000} \right)^{0.2} - 1 \] \[ = 2.1313^{0.2} - 1 = 1.1639 - 1 = \boxed{16.39\%} \] --- ## 3. MIRR Using the Combination Approach In this approach, negative cash flows are discounted to Year 0 (already at Year 0), and positive cash flows are compounded to Year 5 at the reinvestment rate. **Since the only negative cash flow is at Year 0,** the calculation is identical to the reinvestment approach: \[ \boxed{16.39\%} \] --- ## **Final Results:** | Approach | MIRR (%) | |---------------------------|-----------| | Discounting Approach | **16.39%** | | Reinvestment Approach | **16.39%** | | Combination Approach | **16.39%** | --- ### **Summary:** - **All methods yield approximately 16.39% MIRR in this case.** - The slight differences in intermediate calculations are due to rounding, but the core conclusion remains consistent. **Note:** The initial calculations in your previous attempt underestimated the positive cash flows' future value and present value calculations, so the accurate MIRR is approximately **16.39%**.

The final MIRR for the project, calculated using all three approaches, is approximately **16.39%**. **Final Answers:** - **MIRR (Discounting Approach):** **16.39%** - **MIRR (Reinvestment Approach):** **16.39%** - **MIRR (Combination Approach):** **16.39%**

Introduction: The Modified Internal Rate of Return (MIRR) is a comprehensive investment appraisal metric that addresses certain limitations inherent in the traditional IRR. Unlike IRR, which can produce multiple values when cash flows change signs more than once, MIRR provides a unique solution by incorporating reinvestment and financing assumptions explicitly. It considers the cost of capital for negative cash flows (financing rate) and the reinvestment rate for positive cash flows, thus providing a more realistic measure of a project's profitability and efficiency. The MIRR helps in making better investment decisions by evaluating the efficiency of cash flows over the project's lifespan, accounting for the cost of borrowing and reinvestment opportunities. The calculation involves two key processes: discounting all negative cash flows to the present (or the project start) at the financing rate and compounding all positive cash flows to the end of the project at the reinvestment rate. The MIRR is then the discount rate that equates the present value of negative cash flows with the future value of positive cash flows, effectively capturing the project's profitability under these assumptions. Understanding these concepts allows for a more accurate assessment of the project's viability compared to traditional IRR, especially in scenarios where cash flows are irregular or multiple sign changes occur. **Explanation:** This introduction emphasizes the importance of MIRR as a more realistic evaluation tool that overcomes IRR's limitations by explicitly incorporating financing and reinvestment assumptions. Recognizing these concepts is essential for understanding the subsequent calculations, as they underpin the formulas and steps used in deriving the MIRR. This foundational knowledge ensures clarity in interpreting the results and their implications for investment decisions. --- Presentation of Relevant Formulas Required To Solve The Question: 1. **Present Value of Negative Cash Flows (PV_neg):** \[ PV_{neg} = \sum_{t=0}^{n} \frac{CF_t}{(1 + r_{f})^{t}} \] **Description:** This formula discounts all negative cash flows to the present (or year 0) at the financing rate \( r_{f} \). It reflects the current value of outflows, considering the cost of financing. 2. **Future Value of Positive Cash Flows (FV_pos):** \[ FV_{pos} = \sum_{t=1}^{n} CF_t \times (1 + r_{r})^{n - t} \] **Description:** This formula compounds all positive cash flows to the end of the project (year \( n \)) at the reinvestment rate \( r_{r} \). It accounts for the opportunity to reinvest positive inflows at the reinvestment rate. 3. **MIRR Calculation Formula:** \[ MIRR = \left( \frac{FV_{pos}}{|PV_{neg}|} \right)^{1/n} - 1 \] **Description:** This formula computes the rate of return that equates the future value of positive cash flows with the present value of negative cash flows over \( n \) periods, incorporating the reinvestment and financing assumptions. **Explanation:** These formulas are essential tools in the MIRR calculation, enabling the incorporation of differing rates for positive and negative cash flows. They provide a structured method to evaluate the project's profitability under realistic assumptions about reinvestment and financing costs, thereby offering a superior measure for decision-making. --- Detailed Step-by-Step Solution: **Step 1: Identify Cash Flows and Rates** - Initial investment at Year 0: \(-\$53,000\) - Cash flows over Years 1-5: \$16,700, \$21,900, \$27,300, \$20,400, \$8,600 - Discount (financing) rate \( r_{f} = 11\%\) - Reinvestment rate \( r_{r} = 8\%\) - Project duration \( n = 5 \) years *Explanation:* Accurate identification of cash flows and applicable rates is fundamental. The negative cash flow occurs only at Year 0, while positive inflows are spread over subsequent years. The rates for discounting and reinvestment are given and critical for the calculations. **Step 2: Calculate the Present Value of Negative Cash Flows (PV_neg)** \[ PV_{neg} = \frac{-53,000}{(1 + 0.11)^0} = -53,000 \] *Explanation:* The initial outflow occurs at Year 0, so its present value is simply \(-\$53,000\). This forms the denominator in the MIRR formula, representing the initial investment adjusted for financing costs. **Step 3: Calculate the Future Value of Positive Cash Flows (FV_pos)** \[ FV_{pos} = 16,700 \times (1.08)^4 + 21,900 \times (1.08)^3 + 27,300 \times (1.08)^2 + 20,400 \times (1.08)^1 + 8,600 \] Calculations step-by-step: - Year 1: \( 16,700 \times 1.3605 = 22,722.35 \) - Year 2: \( 21,900 \times 1.2597 = 27,599.43 \) - Year 3: \( 27,300 \times 1.1664 = 31,852.32 \) - Year 4: \( 20,400 \times 1.08 = 22,032 \) - Year 5: \( 8,600 \) Sum: \[ FV_{pos} = 22,722.35 + 27,599.43 + 31,852.32 + 22,032 + 8,600 = 112,806.43 \] *Explanation:* This step compounds all positive cash flows to the project's end at the reinvestment rate, reflecting the best-case scenario for reinvested inflows. **Step 4: Compute MIRR** \[ MIRR = \left( \frac{FV_{pos}}{|PV_{neg}|} \right)^{1/n} - 1 \] \[ = \left( \frac{112,806.43}{53,000} \right)^{1/5} - 1 \] \[ = (2.1313)^{0.2} - 1 \] \[ = 1.1639 - 1 = 0.1639 \] \[ \boxed{16.39\%} \] *Explanation:* This calculation yields the annualized rate of return that equates the compounded positive cash flows with the initial outflow, considering reinvestment and financing assumptions. **Step 5: Repeat for Reinvestment and Combination Approaches** - Since the only negative cash flow is at Year 0, the calculations for the reinvestment and combination approaches are identical to the above, leading to the same MIRR. *Explanation:* In this scenario, with a single initial outflow, the different approaches converge, simplifying the calculation and interpretation. --- ## **Conclusion:** The detailed calculations demonstrate that the project's Modified Internal Rate of Return (MIRR) is approximately **16.39%** across all approaches. This rate reflects the project's profitability considering both the cost of capital for negative cash flows and the reinvestment opportunities for positive inflows, providing a realistic measure for decision-making. **Final Answer:** **The MIRR of the project, calculated using the discounting, reinvestment, and combination approaches, is approximately 16.39%.**

Introduction: The Modified Internal Rate of Return (MIRR) is an advanced investment appraisal metric designed to address limitations inherent in the traditional IRR. Unlike IRR, which can produce multiple solutions when cash flow signs change multiple times, MIRR guarantees a unique result by explicitly incorporating assumptions about reinvestment and financing costs. It considers the cost of capital for negative cash flows (financing rate) and the reinvestment rate for positive cash flows, thus providing a more realistic evaluation of a project's profitability and efficiency over its lifespan. Calculating MIRR involves two key steps: discounting all negative cash flows to the present (or optionally to the end of the project) at the financing rate, and compounding all positive cash flows to the end of the project at the reinvestment rate. The MIRR is then the discount rate that equates the present value of negative cash flows with the future value of positive cash flows, considering these assumptions. Understanding these concepts is essential for interpreting the project’s financial viability accurately, especially when cash flows are irregular or multiple sign changes occur. **Explanation:** This introduction underscores the importance of MIRR as a more comprehensive measure that incorporates realistic assumptions about the costs of financing and reinvestment. Recognizing these concepts provides clarity in subsequent calculations, ensuring that the computed MIRR reflects true project profitability under specified rates. --- Presentation of Relevant Formulas Required To Solve The Question: 1. **Present Value of Negative Cash Flows (PV_neg):** \[ PV_{neg} = \sum_{t=0}^{n} \frac{CF_t}{(1 + r_{f})^{t}} \] *Rationale:* This formula discounts all negative cash flows to the present (or project start) at the financing rate \( r_{f} \). Since in this problem the only negative cash flow occurs at Year 0, it simplifies to the initial investment amount. This step ensures the negative cash flows are valued consistently with the financing cost assumptions. 2. **Future Value of Positive Cash Flows (FV_pos):** \[ FV_{pos} = \sum_{t=1}^{n} CF_t \times (1 + r_{r})^{n - t} \] *Rationale:* This formula compounds all positive cash flows to the end of the project at the reinvestment rate \( r_{r} \). It captures the growth potential of inflows assuming reinvestment at the specified rate, thus reflecting the maximum future value of positive cash flows. 3. **MIRR Calculation Formula:** \[ MIRR = \left( \frac{FV_{pos}}{|PV_{neg}|} \right)^{1/n} - 1 \] *Rationale:* This formula calculates the constant rate of return that equates the future value of positive cash flows to the present value of negative cash flows over the project’s duration. It incorporates the assumptions about reinvestment and financing, providing a more realistic measure than traditional IRR. **Explanation:** These formulas are fundamental tools in the MIRR calculation, allowing separation of positive and negative cash flows, valuation at appropriate rates, and synthesis into a single rate of return that accurately reflects project profitability under specified assumptions. --- Detailed Step-by-Step Solution: **Step 1: Identify Cash Flows and Rates** - Initial investment at Year 0: \(-\$53,000\) - Cash inflows: Year 1: \$16,700; Year 2: \$21,900; Year 3: \$27,300; Year 4: \$20,400; Year 5: \$8,600 - Discount (financing) rate \( r_{f} = 11\% \) - Reinvestment rate \( r_{r} = 8\% \) - Duration \( n=5 \) years *Explanation:* Accurate identification ensures correct application of formulas. The initial outflow is at Year 0; positive inflows occur in subsequent years. --- **Step 2: Calculate the Present Value of Negative Cash Flows (PV_neg)** Since the only negative cash flow is at Year 0: \[ PV_{neg} = -53,000 \] *Explanation:* At Year 0, the cash flow equals the initial investment. No discounting is necessary here because the cash flow occurs at Year 0, which is the base point. --- **Step 3: Calculate the Future Value of Positive Cash Flows (FV_pos)** Reinvest positive cash flows at the reinvestment rate (8%) to Year 5: \[ FV_{pos} = \sum_{t=1}^{5} CF_t \times (1 + r_{r})^{n - t} \] Calculations: - Year 1 (\( t=1 \)): \[ 16,700 \times (1.08)^4 = 16,700 \times 1.3605 = 22,722.35 \] - Year 2 (\( t=2 \)): \[ 21,900 \times (1.08)^3 = 21,900 \times 1.2597 = 27,599.43 \] - Year 3 (\( t=3 \)): \[ 27,300 \times (1.08)^2 = 27,300 \times 1.1664 = 31,852.32 \] - Year 4 (\( t=4 \)): \[ 20,400 \times (1.08)^1 = 20,400 \times 1.08= 22,032.00 \] - Year 5 (\( t=5 \)): \[ 8,600 \times (1.08)^0 = 8,600 \times 1= 8,600 \] Sum: \[ FV_{pos} = 22,722.35 + 27,599.43 + 31,852.32 + 22,032 + 8,600 = 112,806.43 \] *Explanation:* This step models the growth of all positive cash inflows assuming they are reinvested at the reinvestment rate until Year 5, representing the maximum accumulated value of inflows. --- **Step 4: Calculate the MIRR** Using the formula: \[ MIRR = \left( \frac{FV_{pos}}{|PV_{neg}|} \right)^{1/n} - 1 \] Plugging in values: \[ MIRR = \left( \frac{112,806.43}{53,000} \right)^{1/5} - 1 = (2.132)^{0.2} - 1 \] Calculating: \[ (2.132)^{0.2} \approx e^{0.2 \times \ln(2.132)} \approx e^{0.2 \times 0.757} \approx e^{0.1514} \approx 1.1639 \] Therefore: \[ MIRR = 1.1639 - 1 = 0.1639 \text{ or } 16.39\% \] *Explanation:* This rate indicates the annualized return that equates the future value of inflows to the initial outflow under the specified reinvestment and financing assumptions. --- **Step 5: Finalize the Results** Since the only negative cash flow occurs at Year 0, the calculations for the reinvestment and combination approaches are identical in this case and yield the same MIRR: \[ \boxed{\text{MIRR} \approx 16.39\%} \] --- ### **Summary:** - **Reinvestment Approach MIRR:** 16.39% - **Combination Approach MIRR:** 16.39% Both approaches produce the same result here due to the absence of negative cash flows in intermediate periods, simplifying the calculations. **This comprehensive step-by-step calculation confirms that the project's MIRR, considering the specified rates, is approximately 16.39%.**