# Step-by-Step Solution: Equivalent Rate of Return (ERR) ## 1. **List the Cash Flows** | Year | 0 | 1 | 2 | 3 | 4 | |------|----------------|---------------|---------------|---------------|---------------| | CF | $500,000 (in) | -$400,000 | -$900,000 | $0 | $1,200,000 | - Receipts (+): $500,000 at start (Year 0), $1,200,000 at Year 4 - Expenditures (−): $400,000 at Year 1, $900,000 at Year 2 - Net CF at Year 3: $0 ## 2. **Understanding ERR (External Rate of Return)** ERR is found by: - Financing cash flows (outflows) are compounded forward at the MARR. - Reinvestment (inflow) cash flows are discounted back at the MARR. - Solve for the rate \( i^* \) that equates present worth of outflows and inflows. ### Step-by-Step for ERR Calculation **Step 1: Identify Outflows and Inflows** - **Outflows (Financing):** Years 1 ($400,000), 2 ($900,000) - **Inflows (Reinvestment):** Year 0 ($500,000), Year 4 ($1,200,000) **Step 2: Compound Outflows to End at MARR** - MARR = 12% (0.12) - Compound all outflows to Year 4 (project end): \[ \text{Future worth of outflows at Year 4} = 400{,}000 \cdot (1.12)^{3} + 900{,}000 \cdot (1.12)^{2} \] \[ 400,000 \cdot 1.404928 = 561,971.2 \] \[ 900,000 \cdot 1.2544 = 1,129,000 \] \[ \text{Total outflows at Year 4} = 561,971.2 + 1,129,000 = 1,690,971.2 \] **Step 3: Discount Inflows to Present at MARR** - Year 0 inflow: $500,000 (already at present) - Year 4 inflow: $1,200,000 discounted to present \[ 1,200,000 \cdot (P/F, 12\%, 4) = 1,200,000 / (1.12)^4 = 1,200,000 / 1.5748 = 762,356 \] \[ \text{Total inflows at Year 0} = 500,000 + 762,356 = 1,262,356 \] **Step 4: Find ERR (\( i^* \))** ERR is the interest rate that makes the compounded outflows at Year 4 equal to the compounded inflows at Year 4. Let \( i^* \) be the ERR. The inflows at Year 4 (compounded from present) are: \[ 500,000 \cdot (1 + i^*)^{4} + 0 \ (\text{from Year 3}) + 1,200,000 \] But normally, you set: \[ \text{Future worth of inflows at Year 4 (compounded at ERR)} = \text{Future worth of outflows at Year 4 (compounded at MARR)} \] So: \[ 500,000 \cdot (1 + i^*)^{4} + 1,200,000 = 400,000 \cdot (1.12)^{3} + 900,000 \cdot (1.12)^{2} \] But inflow of $1,200,000 is already at year 4, so only $500,000 needs to be compounded: \[ 500,000 \cdot (1 + i^*)^{4} + 1,200,000 = 1,690,971.2 \] \[ 500,000 \cdot (1 + i^*)^{4} = 1,690,971.2 - 1,200,000 = 490,971.2 \] \[ (1 + i^*)^{4} = \frac{490,971.2}{500,000} = 0.9819424 \] \[ 1 + i^* = (0.9819424)^{1/4} \] \[ 1 + i^* = 0.995448 \] \[ i^* = 0.995448 - 1 = -0.004552 \text{ or } -0.4552\% \] ## 3. **Check the Calculation** A negative ERR indicates the project does **not** meet the MARR of 12%. ## 4. **Excel Goal Seek Output** **How to do in Excel:** 1. Set up cash flows in a row: `500,000`, `-400,000`, `-900,000`, `0`, `1,200,000` 2. Use Excel's `=IRR()` function for regular IRR, but for ERR, you must manually set up the compounding as above. 3. Set up a cell for ERR guess (say, cell B7). 4. In another cell, compute: `=500000*(1+B7)^4 + 1200000 - (400000*(1.12)^3 + 900000*(1.12)^2)` 5. Use Data > What-If Analysis > Goal Seek to set that cell to zero by changing B7. ### Example Excel Output (copy this into your assignment): ``` Cash Flows: 500,000 -400,000 -900,000 0 1,200,000 MARR: 12% ERR formula: =500000*(1+ERR)^4 + 1200000 - (400000*1.12^3 + 900000*1.12^2) Goal Seek: Set result cell to 0 by changing ERR Result: ERR ≈ -0.46% ``` ## 5. **Decision** - **Should Enigma Engineering pursue the project at MARR = 12%?** - **No.** ERR is negative and far below the required MARR. The project is **not financially acceptable**. --- ## **Summary Table** | Parameter | Value | |-------------------|-------------| | ERR (approximate) | -0.46% | | MARR | 12% | | Decision | **Reject** | --- ## **Conclusion** The project yields an ERR of approximately **-0.46%**, which is less than the MARR of 12%. **Enigma Engineering should not pursue this project.**