--- title: "Nominal vs effective; FW with continuous compounding" subject: "Engineering Economics" chapter: 2 key: "semester_6/engineering_economics:fw-continuous-compounding-500" source: https://www.bctengineer.com/q/semester_6/engineering_economics/fw-continuous-compounding-500 site: BCT Engineering --- # Nominal vs effective; FW with continuous compounding Engineering Economics · Chapter 2: Interest and Time Value of Money - Asked in: 2070 Bhadra, 2+4 marks - Video: [Nominal and Effective Interest Rates - Engineering Economics Lightboard](https://www.youtube.com/watch?v=aT1n_bbQQbM) — 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-continuous-compounding-500 - Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-2 ## Question What is effective and nominal interest rate? Evaluate FW at the end of 10 years with 8% interest rate compounded continuously of a cash flow of Rs. 500 at the beginning of each year for first 5 years. ## Short answer Continuous compounding effective rate = e^0.08 -1 = 8.3287%. Annuity-due of 5 payments yields value at year4; compound 6 years using e^(0.08×6)=1.6161: FW = 500 × 5.9052 × 1.6161 = Rs 4,772. ## Answer #### Effective and nominal rate Nominal rate r is the stated annual rate (8%). Effective rate is the actual annual growth after compounding within the year. For continuous compounding, m → ∞: i = e^r − 1 i = e^0.08 − 1 = **8.3287% per year** #### Cash flow diagram ```text Rs 500 at t = 0,1,2,3,4 (beginning of years 1-5) ^ ^ ^ ^ ^ 0 1 2 3 4 5 6 7 8 9 10 FW = ? ``` #### FW at the end of year 10 The five payments (t = 0 to 4) form an annuity-due; (F/A, i, 5) gives the value at t = 4, which is then moved 6 years to t = 10. FW = 500 (F/A, 8.3287%, 5) (F/P, 8.3287%, 6) - (F/A, 8.3287%, 5) = [(1.083287)⁵ − 1] / 0.083287 = 5.9052 - (F/P, 8.3287%, 6) = e^(0.08×6) = e^0.48 = 1.6161 FW = 500 × 5.9052 × 1.6161 = **Rs 4,772** **Answer: FW at the end of 10 years = about Rs 4,772.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Effective rate for 8% compounded continuously? A. 8% B. 8.24% C. e^8 - 1 D. e^0.08 - 1 = 8.3287% Answer: D. Limit of (1 + r/m)^m is e^r. (Key point: Effective rate) 2. (F/P, r, n) under continuous compounding? A. e^(rn) B. (1 + r)^n C. e^(-rn) D. r e^n Answer: A. F = P e^(rn); 6 years gives e^0.48 = 1.6161. (Key point: Single payment factor) 3. Five payments at t=0..4: how is the FW at t=10 obtained? A. Move 5 years B. (F/A, i, 5) gives value at t=4, then move 6 years C. Move 10 years D. No shift Answer: B. Value at t=4 is compounded 6 years to t=10. (Key point: Annuity due shift) ## Close variants set in other exams - [Nominal vs effective; FW with monthly compounding](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Afw-monthly-compounding-annual-begin) — Engineering Economics, chapter 2, asked 2× ## Related in Engineering Economics - [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 - [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 - [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, 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 and effective rate for perpetual withdrawals](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Anominal-effective-perpetual-withdrawal-1000) — Engineering Economics, chapter 2