Continuous compounding: what it means and how to use it
Model growth when interest is compounded continuously.
Continuous compounding is one of 3 finance formulas in the arithmetic section of this library, and it is used at high school level.
Why continuous compounding works
Compounding more often adds a little more money each time, but the gains get smaller and smaller. As the number of periods per year grows without limit, the factor (1 + r/n) raised to the n settles down on e raised to the r. So e is not chosen for style; it is the ceiling that all the faster schedules climb toward.
What each symbol means
$P$ is principal, $r$ the annual decimal rate, $t$ time in years, and $A$ the balance.
Continuous compounding: when it holds
The rate and time units must agree; this is an ideal continuous-compounding model.
When it stops applying
No real bank pays interest every instant, so this is an upper bound rather than a description. It also cannot handle money added along the way, and for a fixed schedule the ordinary compound formula is the accurate one.
Continuous compounding: a worked example
$1000$ at $5\%$ for $2$ years gives $1000e^{0.1}\approx1105.17$.
The mistake to avoid
What people do: Students enter the rate as a percent, computing 1000 times e raised to the 5 times 2.
Why it goes wrong: The exponent then means 1000 percent, and the balance explodes to over 22 million dollars from a 1000 dollar deposit.
Do this instead: Convert the rate to a decimal before it meets the exponent: 5 percent for 2 years gives an exponent of 0.05 times 2 = 0.1, and 1000 times e raised to the 0.1 is 1105.17.
Continuous compounding: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $P$ is principal, $r$ the annual decimal rate, $t$ time in years, and $A$ the balance.
- Check the conditions before substituting. The rate and time units must agree; this is an ideal continuous-compounding model.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most arithmetic slips.
Where this formula fits
- Subject
- Arithmetic formulas — 13 entries in this library
- Topic
- Finance
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where continuous compounding comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms lesson — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Arithmetic Calculator — check your substitution and the value it produces.
- Study arithmetic — the subject guide that explains the ideas these formulas compress.
- Arithmetic Practice — questions that make you retrieve the formula instead of recognising it.
- All 13 arithmetic formulas — the full grouped reference, or the complete formula library.
Questions about continuous compounding
How much more does continuous compounding actually pay?
Very little. On 1000 dollars at 5 percent for 2 years, monthly compounding gives 1104.94, daily gives 1105.16, and continuous gives 1105.17, so the whole race is decided by a few cents.
Can I use it for something that shrinks?
Yes, with a negative r. Radioactive decay, drug levels in the bloodstream, and a cooling cup of coffee are all modelled by the same expression with the exponent turned negative.
Is Pe^(rt) the same as P(1 + r)^t?
No, though they are close for small rates. At 5 percent for one year the first gives 1051.27 and the second gives 1050, and the gap widens as the rate or the time grows.
How do I solve for the time?
Take the natural logarithm of both sides. Dividing A by P and then dividing ln(A/P) by r gives t, so doubling your money at 5 percent takes ln 2 divided by 0.05, about 13.86 years.