External Rate Of Return method numerical||ERR Numerical||Unit-4||engineering economics lectures

External Rate of Return (ERR) Concept
📌 ERR is the interest rate that makes the Present Worth (PW) of all costs equal to the Future Worth (FW) of all revenues (or vice-versa when comparing cash flows).
💰 Costs (negative values like investment/maintenance) are discounted to the Present Worth (PW) at the given MARR (Minimum Attractive Rate of Return).
💸 Revenues (positive values/benefits) are compounded to the Future Worth (FW) at the MARR.
🔍 The resulting ERR is the rate (i%) that equates the accumulated cost (PW at year 0) to the accumulated benefits (FW at the project end year).

Numerical Example 1: Mutually Exclusive Alternatives (Projects A1, A2, A3)
🎯 The MARR is given as 5%. The analysis requires calculating the ERR for each project individually.
📉 Project A1 Calculation: Initial investment (PW at year 0) was $2,000. The compounded Future Worth (FW) at year 3, using MARR=5%, resulted in $39,338. Equating these values yielded an $\text{ERR}_{\text{A}1}$ of 25.29%.
📊 Project A2 Calculation: The steps involved discounting costs to PW at year 0 and compounding revenues to FW at year 3 (at 5% MARR), yielding an $\text{ERR}_{\text{A}2}$ of 28.71%.
🏆 Decision Rule: Since all calculated ERRs (A1: 25.29%, A2: 28.71%, A3: 22.76%) are greater than the MARR (5%), all projects are financially acceptable individually. For mutually exclusive choices, the project with the highest ERR (Project A2 at 28.71%) is selected.

Numerical Example 2: Single Project Analysis (IRR vs. ERR)
⚙️ This example requires calculating both the Internal Rate of Return (IRR) and the External Rate of Return (ERR) for a single project with negative cash flows at Year 0 ($5) and Year 2 ($10). The MARR is 5%.
💰 ERR Calculation Steps:
1. Discount all cash out-flows (costs) to PW at year 0 at the MARR (5%). This resulted in a PW of $7.56. (Note: The calculation involved discounting the Year 2 cost of $10 back 2 years).
2. Compound all cash inflows (revenues) to the project's end year (Year 5) at the MARR (5%). This resulted in an FW of $126.71.
3. Equate the PW of costs to the FW of revenues: $7.56(1+i)^5 = 126.71$.
4. Solving for $i$ yielded an $\text{ERR}$ of 7.09%.
✅ Comparison: Since $\text{ERR}$ (7.09%) is greater than MARR (5%), the project is acceptable. The calculated IRR was also higher than the ERR, as expected in this specific scenario.

Key Points & Insights
➡️ The External Rate of Return (ERR) method requires two distinct interest rates: the given MARR (used for initial PW/FW calculations) and the unknown ERR (used for the final equivalence calculation).
➡️ In mutually exclusive comparisons, the project with the highest calculated ERR is preferred, provided that $\text{ERR} > \text{MARR}$.
➡️ Cash Flow Diagrams are essential for visualizing which cash flows require discounting (moving back to PW) and which require compounding (moving forward to FW) at the MARR.
➡️ The next topic scheduled for numerical practice is the Payback Period Method.

📸 Video summarized with SummaryTube.com on Mar 15, 2026, 16:39 UTC