Growth by a fixed percentage
A quantity grows exponentially when it increases by the same percentage each period, rather than by the same amount. Savings with compound interest, a population, the spread of an early epidemic and inflation all follow this pattern. When the percentage is negative, the quantity shrinks the same way: exponential decay describes depreciation, radioactive decay or a drug leaving the body.
The formulas
- Discrete growth: P(t) = P₀ × (1 + r)ᵗ, with r the rate per period as a decimal
- Continuous growth: P(t) = P₀ × e^(r × t)
- Doubling time: ln(2) ÷ ln(1 + r) for discrete growth, ln(2) ÷ r for continuous growth
- Half-life for decay: ln(2) ÷ |ln(1 + r)|
- Rule of 70: doubling time ≈ 70 ÷ rate in percent, a quick mental shortcut
Worked example: 1,000 growing 5% a period
- After 10 periods: 1,000 × 1.05¹⁰ = 1,628.89, a gain of 62.89%.
- Doubling time: ln 2 ÷ ln 1.05 = 14.21 periods.
- The rule of 70 gives 70 ÷ 5 = 14, close enough for a quick estimate.
- With continuous growth at 5%, the value after 10 periods would be 1,000 × e^0.5 = 1,648.72.
Growth over time at 5%
| Periods | Value | Total change |
|---|---|---|
| 1 | 1,050.00 | 5.00% |
| 5 | 1,276.28 | 27.63% |
| 10 | 1,628.89 | 62.89% |
| 20 | 2,653.30 | 165.33% |
| 30 | 4,321.94 | 332.19% |
| 50 | 11,467.40 | 1,046.74% |
The gain in the second ten periods (1,024) is far larger than in the first ten (629): that acceleration is what makes exponential growth so powerful, and so easy to underestimate.
Decay and half-life
Enter a negative rate for decay. A car losing 15% of its value a year keeps 0.85 each year: after 5 years, 0.85⁵ = 44% of the price remains, and its value halves every ln 2 ÷ |ln 0.85| = 4.3 years. Radioactive isotopes are usually described the other way round, by their half-life: carbon-14 halves every 5,730 years.
Discrete or continuous?
Discrete growth applies the percentage once per period, like interest paid yearly. Continuous growth compounds at every instant and gives a slightly higher result for the same nominal rate. The calculator shows the equivalent continuous rate: 5% per period equals 4.879% compounded continuously.
Limits of the model
Real growth rarely stays exponential for long. Populations run out of resources and markets saturate, which leads to S-shaped, logistic curves. Use exponential projections for short horizons, or as an upper bound.