--- title: "Nominal vs effective; FW with monthly compounding" subject: "Engineering Economics" chapter: 2 key: "semester_6/engineering_economics:fw-monthly-compounding-annual-begin" source: https://www.bctengineer.com/q/semester_6/engineering_economics/fw-monthly-compounding-annual-begin site: BCT Engineering --- # Nominal vs effective; FW with monthly compounding Engineering Economics · Chapter 2: Interest and Time Value of Money - Asked in: 2071 Bhadra, 3+5 marks - Video: [Compounding More Frequent than Payments - Engineering Economics Lightboard](https://www.youtube.com/watch?v=kEcGEXlK5oY) — Engineering Economics Guy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/engineering_economics/fw-monthly-compounding-annual-begin - Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-2 ## Question What is nominal and effective interest rate? Evaluate FW at the end of 15 years with 10% interest compounded monthly of a cash flow of Rs. 50,000 at the beginning of each year for first 10 years. ## Short answer Effective annual rate = (1+0.10/12)^12 -1 = 10.4713%. Annuity-due of 10 payments gives FV at year 9, then compound 6 years: FW = 50,000 × 16.302 × 1.8175 = Rs 14,81,529. ## Answer #### Nominal and effective interest rate The nominal rate is the stated yearly rate, ignoring the effect of compounding within the year (here 10% per year). The effective rate is the actual yearly rate after compounding m times a year: i = (1 + r/m)ᵐ − 1. #### Effective annual rate i = (1 + 0.10/12)¹² − 1 = **10.4713% per year** #### Cash flow diagram ```text Rs 50,000 at t = 0, 1, ..., 9 (beginning of years 1-10) ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ | | | | | | | | | | +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--> year 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 FW = ? ``` #### FW at the end of year 15 The payments at t = 0 to 9 form an annuity-due. The (F/A, i, 10) factor gives their value at t = 9, so this is then moved to t = 15 (6 years): FW = 50,000 (F/A, 10.4713%, 10) (F/P, 10.4713%, 6) - (F/A, 10.4713%, 10) = [(1.104713)¹⁰ − 1] / 0.104713 = 16.302 - (F/P, 10.4713%, 6) = (1.104713)⁶ = 1.8175 FW = 50,000 × 16.302 × 1.8175 = **Rs 1,481,529** **Answer: FW at the end of 15 years = about Rs 14.8 lakh.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Effective rate for 10% compounded monthly? A. (1 + 0.10/12)^12 - 1 = 10.4713% B. 10% C. 0.8333% D. 10.25% Answer: A. Compounding monthly lifts 10% nominal to 10.47% effective. (Key point: Effective annual rate) 2. Payments at the beginning of years 1-10: where does (F/A, i, 10) place their value? A. At t = 10 B. At t = 9, so it must be moved 6 more years to t = 15 C. At t = 15 D. At t = 0 Answer: B. First payment at t=0 means last at t=9. (Key point: Annuity due) 3. Which expression gives FW at 15? A. 50,000 (F/A, 10.4713%, 15) B. 50,000 (P/A, 10.4713%, 10)(F/P, 10.4713%, 15) C. 50,000 (F/A, 10.4713%, 10)(F/P, 10.4713%, 6) D. 50,000 (F/A, 10%, 10)(F/P, 10%, 5) Answer: C. Annuity to t=9 then 6 years of growth. (Key point: Setup) ## Close variants set in other exams - [Nominal vs effective; FW with continuous compounding](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Afw-continuous-compounding-500) — Engineering Economics, chapter 2, asked 2× ## Related in Engineering Economics - [Compounding semi-annually and monthly, 10% nominal](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Adeposit-10000-semiannual-monthly) — Engineering Economics, chapter 2 - [Nominal vs effective rate; quarterly rate from monthly](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Anominal-effective-monthly-1-quarterly) — Engineering Economics, chapter 2 - [Monthly deposits, quarterly compounding, balance at year 8](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Aequal-monthly-deposits-balance-8-years) — Engineering Economics, chapter 2 - [Nominal vs effective rate; simple vs compound deposit](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Anominal-effective-simple-compound-1000) — Engineering Economics, chapter 2 - [Car loan interest rate and lump sum at year 3](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Acar-loan-installments-lump-sum-year3) — Engineering Economics, chapter 2 - [Monthly deposits with quarterly compounding, future amount](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Adeposit-quarterly-monthly-five-years) — Engineering Economics, chapter 2 ## Related in other subjects - [Capitalised value from Rs 20,000 monthly rent](https://www.bctengineer.com/question.md?key=semester_8%2Fconstruction_management%3Avalue-building-20000-rent-2078) — Construction Management, chapter 13 - [EPS; total capital Rs 15,00,000](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Aeps-capital-1500000-pref-loan-a74) — Project Engineering, chapter 6 - [EPS and book value; Rs 10,00,000, 3500 shares](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Aeps-bv-capital-1000000-3500-shares) — Project Engineering, chapter 6