---
title: "Short note: continuous compounding"
subject: "Engineering Economics"
chapter: 2
key: "semester_6/engineering_economics:short-note-continuous-compounding"
source: https://www.bctengineer.com/q/semester_6/engineering_economics/short-note-continuous-compounding
site: BCT Engineering
---

# Short note: continuous compounding

Engineering Economics · Chapter 2: Interest and Time Value of Money

- Asked in: 2068 Jestha, 4 marks
- Video: [Continuous Compounding - Engineering Economics Lightboard](https://www.youtube.com/watch?v=LxvCFTslVz4) — 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/short-note-continuous-compounding
- Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-2

## Question

Write a short note on continuous compounding.

## Short answer

Continuous compounding adds interest at every instant (m→∞). Future worth is P e^(r n), effective rate i = e^r − 1. It yields the highest possible return for a given nominal rate and is used in theoretical analysis and continuous cash flow models.

## Answer

Continuous compounding is the limiting case of compounding in which the number of compounding periods per year (m) tends to infinity, so interest is added at every instant.

#### Formulas
Let r = nominal annual rate and n = number of years.

- Effective annual rate: i = e^r − 1, because lim (1 + r/m)^m = e^r as m → ∞
- Future worth: F = P·e^(rn)
- Present worth: P = F·e^(−rn)
- Single payment factors: (F/P, r, n) = e^(rn), (P/F, r, n) = e^(−rn)
- Uniform series: (F/A, r, n) = (e^(rn) − 1)/(e^r − 1), (A/F, r, n) = (e^r − 1)/(e^(rn) − 1), (P/A, r, n) = (e^(rn) − 1)/[e^(rn)(e^r − 1)]

#### Example
Rs 10,000 at 12% nominal for 5 years.

- Continuous: F = 10,000 × e^(0.12 × 5) = 10,000 × 1.8221 = Rs 18,221; effective rate = e^0.12 − 1 = 12.75%
- Monthly compounding: F = 10,000 × (1.01)^60 = Rs 18,167; effective rate = 12.68%
- Annual compounding: F = 10,000 × (1.12)^5 = Rs 17,623

#### Remarks
- It gives the highest return for a given nominal rate, but only slightly more than daily compounding.
- Used for theoretical analysis, flows that occur steadily (such as oil production or continuous revenue) and some bank products.
- Most real banks compound monthly, quarterly or yearly, so it is mainly a limiting model.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is continuous compounding?
   A. Compounding once a year
   B. Compounding daily
   C. Simple interest
   D. Limit as number of compounding periods per year tends to infinity
   Answer: D. Interest is added at every instant. (Key point: Definition)

2. Effective rate for nominal r?
   A. i = e^r - 1
   B. i = r
   C. i = e^r
   D. i = 1 + r
   Answer: A. lim (1+r/m)^m = e^r. (Key point: Effective rate)

3. F in terms of P?
   A. F = P(1+r)^n
   B. F = P e^(rn)
   C. F = P e^(-rn)
   D. F = P r n
   Answer: B. Present worth uses e^(-rn). (Key point: Future worth)

4. How does it compare with daily compounding?
   A. Much higher than daily
   B. Lower than annual
   C. Highest return but only slightly more than daily
   D. Equal to simple
   Answer: C. It is mainly a limiting model. (Key point: Comparison)

## Related in Engineering Economics

- [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×