Continuous Growth Rate Calculator

Convert period growth rate to its continuous counterpart and visualize exponential growth.

Enter Period Growth Rate

Period Growth Rate (e.g., 0.05 for 5%)

Continuous Growth Rate

Understanding Continuous Growth Rate

The continuous growth rate is the constant rate at which a quantity would grow if it were compounding continuously. Unlike period growth rates (e.g., annual growth), continuous growth provides an instantaneous rate of change. It's particularly useful in financial modeling, population growth, and other areas where growth is happening constantly rather than at discrete intervals.

Formula

The continuous growth rate (r) is calculated from the period growth rate (R) using the natural logarithm:

$$ r = ln(1 + R) $$

r is the continuous growth rate.
ln is the natural logarithm function.
R is the period growth rate (expressed as a decimal, e.g., 0.05 for 5%).

Example

If a population grows by 5% per year (period growth rate R = 0.05), the continuous growth rate is calculated as:

$$ r = ln(1 + 0.05) ≈ 0.0488 $$

This means the population is growing at an instantaneous rate of approximately 4.88% continuously.

Use Cases

Finance: Calculating continuously compounded interest rates.
Population Growth: Modeling population increases over time.
Radioactive Decay: Describing the continuous decay of radioactive substances (in this case, a negative growth rate).
Ecology: Analyzing continuous changes in ecological systems.