Continuous Compound Interest Calculator
Calculate growth with continuous compounding using Euler's number (e)
Calculate Continuous Compound Interest
Starting amount of your investment
Nominal annual interest rate
Number of years to invest
Additional months (0-11)
Investment Results
Interest Breakdown
Principal Breakdown
Formula used: FV = P × e^(r×t)
Where: P = Principal ($1,000.00), e = Euler's number (≈2.71828), r = Annual rate (4.00%), t = Time (20 years)
Calculation: $1,000.00 × 2.71828^(0.0400 × 20.00) = $2,225.54
Continuous vs. Discrete Compounding
💡 Continuous compounding provides the theoretical maximum return
Investment Analysis
Example Calculation
Investment Example
Initial Investment: $1,000
Annual Interest Rate: 4%
Investment Period: 20 years
Compounding: Continuous
Calculation Steps
1. Identify values: P = $1,000, r = 0.04, t = 20 years
2. Apply formula: FV = P × e^(r×t)
3. Calculate exponent: r × t = 0.04 × 20 = 0.8
4. Compute e^0.8: e^0.8 ≈ 2.22554
5. Final calculation: FV = $1,000 × 2.22554
Result: Final Balance = $2,225.54
Total Interest = $1,225.54
Comparison
• Daily Compounding: $2,225.34 (difference: $0.20)
• Monthly Compounding: $2,219.64 (difference: $5.90)
• Quarterly Compounding: $2,208.04 (difference: $17.50)
• Annual Compounding: $2,191.12 (difference: $34.42)
✓ Continuous gives the highest return!
Year-by-Year Growth
| Year | Balance | Interest |
|---|---|---|
| 1 | $1,040.81 | $40.81 |
| 2 | $1,083.29 | $83.29 |
| 3 | $1,127.50 | $127.50 |
| 4 | $1,173.51 | $173.51 |
| 5 | $1,221.40 | $221.40 |
| 6 | $1,271.25 | $271.25 |
| 7 | $1,323.13 | $323.13 |
| 8 | $1,377.13 | $377.13 |
| 9 | $1,433.33 | $433.33 |
| 10 | $1,491.82 | $491.82 |
| 11 | $1,552.71 | $552.71 |
| 12 | $1,616.07 | $616.07 |
| 13 | $1,682.03 | $682.03 |
| 14 | $1,750.67 | $750.67 |
| 15 | $1,822.12 | $822.12 |
| 16 | $1,896.48 | $896.48 |
| 17 | $1,973.88 | $973.88 |
| 18 | $2,054.43 | $1,054.43 |
| 19 | $2,138.28 | $1,138.28 |
| 20 | $2,225.54 | $1,225.54 |
Euler's Number (e)
Euler's number (e) is the base of natural logarithms and represents continuous exponential growth. It's approximately 2.71828 and appears in continuous compounding calculations.
Investment Tips
Continuous compounding gives the theoretical maximum return
The difference from daily compounding is minimal in practice
Use ln(2)/r to calculate doubling time
Effective Annual Rate (EAR) = e^r - 1
Higher rates show bigger differences from discrete compounding
Understanding Continuous Compound Interest
What is Continuous Compounding?
Continuous compounding is the theoretical limit of the compounding frequency. Instead of compounding monthly, daily, or even hourly, continuous compounding assumes interest is calculated and added to the principal infinitely often—at every possible moment.
Why Use Euler's Number?
As compounding frequency increases, the formula (1 + r/n)^(n×t) approaches e^(r×t). Euler's number (e ≈ 2.71828) naturally emerges as the limit of (1 + 1/n)^n as n approaches infinity, making it perfect for modeling continuous exponential growth.
Continuous Compound Interest Formula
FV = PV × e^(r×t)
- FV: Future Value (final amount)
- PV: Present Value (initial amount)
- e: Euler's number (≈2.71828)
- r: Annual interest rate (as decimal)
- t: Time in years
Key Insight: Continuous compounding provides the upper bound for compound interest. No matter how frequently you compound, you cannot exceed the return from continuous compounding.
Effective Annual Rate (EAR)
The Effective Annual Rate converts a continuously compounded rate to its equivalent annual rate:
EAR = e^r - 1
For example, a 4% continuously compounded rate is equivalent to an EAR of approximately 4.08%. This means that compounding once per year at 4.08% would give you the same result as continuous compounding at 4%.
Practical Applications
Finance & Banking
- • Theoretical maximum returns
- • Pricing derivatives and options
- • Present value calculations
- • Risk-free rate modeling
Real World
- • Population growth models
- • Radioactive decay calculations
- • Bacterial growth rates
- • Investment performance benchmarks
Doubling Time
With continuous compounding, you can calculate how long it takes for an investment to double using:
t = ln(2) / r ≈ 0.693147 / r
This is sometimes called the "Rule of 69.3" for continuous compounding (compared to the "Rule of 72" for annual compounding). For a 4% rate, doubling time = 0.693147 / 0.04 ≈ 17.33 years.