Scope and assumptions
Continuous growth assumes change happens at every instant; A = P·e^(rt) with r as a decimal rate. Illustrative.
Compound interest gets more frequent until, in the limit, it never stops — and that limit has a name: e. The formula A = P·e^(rt) is what you get when growth is happening at every instant, and at the same nominal rate it always edges out annual compounding.
The instant-growth formula
A = P · e^(rt)
P = principal/start, r = rate as a decimal, t = time, e ≈ 2.71828.
| P | r | t | A = P·e^(rt) |
|---|---|---|---|
| 100 | 5% | 10 | 100·e^0.5 ≈ 164.87 |
| 1,000 | 3% | 5 | 1,000·e^0.15 ≈ 1,161.83 |
| 500 | 8% | 2 | 500·e^0.16 ≈ 586.19 |
Continuous vs the annual cap
Annual: A = P(1 + r)^t. Continuous is the limit as compounding frequency → ∞. At 5%/10yr:
- Annual: 100 × 1.05^10 ≈ 162.89
- Continuous: 100·e^0.5 ≈ 164.87
Continuous always yields slightly more because interest never waits for a period boundary. The gap (164.87 vs 162.89) is the value of compounding between the discrete ticks — small at normal rates, but it grows with r and t. Daily compounding already lands almost on the continuous ceiling, which is why daily and continuous look identical in practice.
Decay is the same formula
Flip the sign: A = P·e^(−rt) shrinks the value. Radioactive decay, cooling, and drug clearance all use a negative rate. The same formula covers growth and decay, only the sign of r changes.
When to reach for it
- Population/biology models (cells, bacteria)
- Radioactive decay (with negative r)
- Finance theory and some loans
For normal savings, discrete compounding is the right call.
When Continuous Growth Doesn't Fit
- Forgetting r is a decimal — 5% means r = 0.05, not 5.
- Mixing with (1+r)^t — that's discrete; e^rt is continuous.
- Dropping e — the constant 2.71828 is what makes it continuous.
- Wrong sign on decay — negative r shrinks the value.
- Expecting a big edge over daily — continuous only slightly beats daily compounding.
- Using it for simple savings — discrete compounding is the correct tool for bank accounts.
Related advanced operations
For the rest of the advanced percentage operations, start at the hub: Percentage Difference Formula. Adjacent: Successive Percentage Change and Reverse Percentage Increase.
FAQ
Why does e appear at all? e is the limit of (1 + 1/n)^n as n → ∞ — the natural constant of compounding that never stops. Continuous growth is discrete compounding with infinitely many ticks, and e is what falls out.
How do I find the continuous rate from a known annual rate? Invert the limit: if annual compounding gives factor (1 + r), the continuous rate is ln(1 + r). A 5% annual rate ≈ 4.88% continuous (ln 1.05). Useful when a model expects the r in e^(rt).
Is continuous growth realistic for money? Barely — most accounts compound daily at most, which is already within a hair of continuous. Continuous is mainly a clean math model for biology, physics, and theory; real bank products use discrete terms.
Can time t be in anything other than years? Yes, as long as r matches the period. r = 0.05 per month with t in months is fine; just keep r and t on the same clock or the answer is wrong.
What happens if r is negative and t is huge? e^(−rt) approaches 0 — the value decays toward nothing but never quite reaches it, the continuous mirror of discrete halving.
Sources: Khan Academy, Continuously compounding interest; Investopedia, Continuous compounding.
Disclaimer: Illustrative. Real financial products use specific compounding terms; verify with the issuer. Informational only.
Sources and references
- Khan Academy — Continuously compounding interesthttps://www.khanacademy.org/math
- Investopedia — Continuous compoundinghttps://www.investopedia.com/terms/c/continuouscompounding.asp
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