Continuous Compound Interest Formula
The continuous compounding formula A = Pe^rt calculates growth when interest compounds infinitely often.
Uses Euler's number e.
The Formula
Continuous compounding represents the mathematical limit of compounding more and more frequently — yearly, monthly, daily, hourly, every second, and beyond. It uses Euler's number e ≈ 2.71828.
Variables
| Symbol | Meaning |
|---|---|
| A | Final amount (principal + interest, measured in currency units) |
| P | Principal (initial investment amount, measured in currency units) |
| e | Euler's number ≈ 2.71828 (mathematical constant, the base of natural logarithms) |
| r | Annual interest rate (expressed as a decimal — 5% = 0.05) |
| t | Time in years |
Comparison with Discrete Compounding
The standard compound interest formula is A = P(1 + r/n)^(nt), where n is the number of times compounded per year.
| Compounding | n value | $1,000 at 10% for 1 year |
|---|---|---|
| Annual | 1 | $1,100.00 |
| Quarterly | 4 | $1,103.81 |
| Monthly | 12 | $1,104.71 |
| Daily | 365 | $1,105.16 |
| Continuous | ∞ | $1,105.17 |
As n approaches infinity, (1 + r/n)^(nt) approaches e^(rt). The difference between daily and continuous compounding is very small, but the formula is mathematically elegant and important in finance and science.
Example 1
$5,000 is invested at 6% annual interest, compounded continuously, for 10 years. What is the final amount?
Apply the formula: A = 5000 × e^(0.06 × 10)
A = 5000 × e^0.6 = 5000 × 1.8221
A = $9,110.59
Example 2
How long does it take $1,000 to double at 5% continuous compounding?
Set up: 2000 = 1000 × e^(0.05t)
Simplify: 2 = e^(0.05t)
Take natural log: ln(2) = 0.05t
t = ln(2) / 0.05 = 0.6931 / 0.05
t = 13.86 years
When to Use It
Continuous compounding is used in finance, science, and mathematics.
- Theoretical interest calculations in banking and finance
- Bond pricing and yield calculations
- Population growth modeling (biology)
- Radioactive decay (with negative r)
- Any process where growth or decay happens continuously rather than in discrete steps
- The "Rule of 72" is an approximation: time to double ≈ 72 / (rate in percent)