--- title: "Payback, ERR and BCR for four projects" subject: "Engineering Economics" chapter: 4 key: "semester_6/engineering_economics:payback-err-bcr-four-projects" source: https://www.bctengineer.com/q/semester_6/engineering_economics/payback-err-bcr-four-projects site: BCT Engineering --- # Payback, ERR and BCR for four projects Engineering Economics · Chapter 4: Comparative Analysis of Alternatives - Asked in: 2065 Chaitra, 16 marks - Video: [Rate of return [IRR ,ERR ] | Benefit cost ratio](https://www.youtube.com/watch?v=t2MpiusaZhA) — OPengineer - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/engineering_economics/payback-err-bcr-four-projects - Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-4 ## Question From the following four mutually exclusive projects recommend the best one using Payback Period, ERR and BCR methods. The study period is 5 years and MARR = ε = 15%. | Project | A | B | C | D | |---|---|---|---|---| | Initial investment | 500000 | 400000 | 700000 | 600000 | | Net annual revenue | 125000 | 110000 | 170000 | 135000 | Salvage value is 20% of the initial investment. ## Short answer Simple payback: B has shortest at 3.64 years. ERR: only B (15.49%) exceeds 15%. BCR: only B (1.021) exceeds 1. All methods select project B. ## Answer Data: N = 5 years; MARR = reinvestment rate ε = 15%; salvage value S = 20% of the initial investment (A: 100,000; B: 80,000; C: 140,000; D: 120,000). Net annual revenue is the annual benefit. Factors: (F/A,15%,5) = 6.7424; (P/A,15%,5) = 3.3522; (P/F,15%,5) = 0.4972. #### 1. Payback period (simple) Payback = initial investment / net annual revenue (salvage ignored). | Project | Investment | Net annual revenue | Payback = I / annual | |---|---|---|---| | A | 500,000 | 125,000 | 4.00 yr | | B | 400,000 | 110,000 | 3.64 yr | | C | 700,000 | 170,000 | 4.12 yr | | D | 600,000 | 135,000 | 4.44 yr | Shortest payback: **B (3.64 years)**. Payback ignores the cash flows after recovery and the time value of money, so it is only a screening measure. #### 2. External rate of return (ERR) All receipts (annual revenue and salvage) are compounded to year 5 at ε = 15%, and ERR is the rate i' with I(1 + i')⁵ = FW. ```text FW = A(F/A,15%,5) + S Project B: FW = 110,000(6.7424) + 80,000 = 821,662 ERR = (821,662 / 400,000)^(1/5) − 1 = 15.49% ``` | Project | FW of receipts at yr 5 | ERR = (FW / I)^(1/5) − 1 | Acceptable (≥ 15%)? | |---|---|---|---| | A | 942,798 | 13.52% | No | | B | 821,662 | 15.49% | Yes | | C | 1,286,205 | 12.94% | No | | D | 1,030,221 | 11.42% | No | Only B has ERR above MARR = 15%. #### 3. Benefit–cost ratio (conventional) BCR = PW(annual revenue + salvage) / initial investment. ```text Project B: PW(B) = 110,000(3.3522) + 80,000(0.4972) = 408,511 BCR = 408,511 / 400,000 = 1.021 ``` | Project | PW of benefits | BCR = PW(B) / I | Acceptable (≥ 1)? | |---|---|---|---| | A | 468,737 | 0.937 | No | | B | 408,511 | 1.021 | Yes | | C | 639,471 | 0.914 | No | | D | 512,202 | 0.854 | No | Only B has BCR > 1. #### Decision | Method | Best project | |---|---| | Payback | B | | ERR | B (15.49%) | | BCR | B (1.021) | A, C and D fail ERR and BCR tests at 15% (their PWs are negative), so no incremental comparison is needed. **Answer: Select project B.** It has the shortest payback (3.64 years), ERR = 15.49% > 15% and BCR = 1.021 > 1. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is simple payback computed here and what is its weakness? A. Investment / net annual revenue; it ignores post-payback flows and time value of money B. Investment / salvage; it ignores revenue C. Investment x revenue; it ignores risk D. PW of revenue / investment; it ignores salvage Answer: A. Payback = I / annual revenue (B: 400,000/110,000 = 3.64 yr) and it ignores later cash flows and the time value of money. (Key point: Simple payback) 2. Which equation gives ERR for these projects? A. I = FW (1 + i')^5 B. I(1 + i')^5 = FW of annual revenue and salvage compounded at 15% C. I(1 + i')^5 = PW of benefits D. I = FW of revenues discounted at i' Answer: B. Receipts including salvage are compounded to year 5 at E = 15%, and I(1+i')^5 = FW. (Key point: ERR calculation) 3. In the conventional BCR used here, what is the ratio? A. Initial investment / PW of benefits B. FW of benefits / PW of costs C. PW of (annual revenue + salvage) / initial investment D. Net annual revenue / salvage Answer: C. BCR = PW(benefits)/I; accept when BCR > 1. B: 408,511/400,000 = 1.021. (Key point: BCR definition) 4. Why is project B recommended? A. It has the largest investment B. It has the largest FW C. Only A, C, D pass the MARR test D. It has the shortest payback and is the only one with ERR >= 15% and BCR > 1 Answer: D. B satisfies all three criteria (3.64 yr, 15.49%, 1.021) while A, C and D fail ERR and BCR tests. 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