Article — Continuous Compound Interest
Continuous compound interest calculator
- What is continuous compound interest?
- The continuous compound formula
- Why e shows up in the formula
- Continuous vs monthly vs daily compounding
- Effective annual rate under continuous compounding
- Solving for time and rate
- Where continuous compounding is actually used
- Common continuous-compound mistakes
Continuous compound interest applies the formula A = Pe^(rt), where money grows by the factor e^(rt) and e is Euler's number, 2.71828. On $10,000 at 5% for 10 years the final amount is $16,487.21, just $0.56 above daily compounding and $198.26 above annual.
The continuous case is the theoretical limit of compounding, what happens as the number of compounding periods per year approaches infinity. Real bank accounts never compound this often; the math nonetheless underlies bond pricing, options models, and exponential growth in biology and physics.
What is continuous compound interest?
Continuous compound interest is interest credited at every instant rather than at fixed intervals like a year, quarter, or day. Each tiny moment of time earns a tiny fraction of interest, which immediately begins earning its own interest. The closed-form expression is A = Pe^(rt).
Annual compounding adds interest once a year. Monthly compounding adds it twelve times, daily compounding 365. As the period count grows, results converge on the continuous value but never exceed it. The continuous result sits at the top of every comparison table, by a hair.
Jacob Bernoulli discovered the number e in 1683 while studying compound interest. He asked what would happen if a 100% rate were split into ever-smaller pieces and proved the limit was a finite irrational number, today written 2.71828182...
The continuous compound formula
The continuous compound interest formula is A = P times e^(r times t). P is the starting principal, r is the annual rate as a decimal, and t is time in years. Interest earned alone is I = P(e^(rt) - 1).
A worked example: deposit $5,000 at 4% for 8 years. The exponent is 0.04 times 8 = 0.32. The growth factor e^0.32 is 1.37713. Final amount = 5,000 times 1.37713 = $6,885.64. Interest earned = $1,885.64.
A = P e^(rt) final amountI = P(e^(rt) - 1) interest earnedEAR = e^r - 1 effective annual ratet = ln(A/P) / r solve for timeWhy e shows up in the formula
Euler's number e appears because it is the limit of (1 + 1/n)^n as n grows without bound. Plug r/n into the discrete compound formula P(1 + r/n)^(nt), let n run to infinity, and the expression collapses to Pe^(rt). Continuous compounding is the natural endpoint of making the period smaller and smaller.
That same constant governs every exponential process: radioactive decay, population growth, charge on a capacitor. The shared math is why physicists, biologists, and traders all keep e on speed dial.
Continuous vs monthly vs daily compounding
The practical difference between continuous, daily, and monthly compounding is tiny. On $10,000 at 5% over 10 years, daily compounding gives $16,486.65 and continuous gives $16,487.21. The continuous edge is 56 cents. Compare that to the $198 gap between annual and continuous, and it is clear most of the benefit is captured by the time you reach daily.
This is why banks do not bother going past daily. The marginal customer benefit is rounded away, while the marketing line, "compounded daily", already implies near-maximum growth.
Effective annual rate under continuous compounding
The effective annual rate (EAR) under continuous compounding is e^r minus 1. At a nominal 6%, EAR = e^0.06 - 1 = 6.184%. At 10% nominal, EAR climbs to 10.517%. The EAR for continuous compounding is always slightly higher than the EAR of any finite compounding scheme using the same nominal rate.
EAR matters when comparing two products that quote rates at different frequencies. A 5.95% rate compounded monthly is roughly 6.12% effective; a 6.00% rate compounded continuously is 6.18%. Always compare EAR, never nominal.
Want a quick doubling-time estimate? Use t = 0.693 / r, where r is the decimal rate. At 5%, money doubles in 13.86 years under continuous compounding. The Rule of 72 (72 / rate-in-percent) approximates this to within a fraction of a year.
Solving for time and rate
To solve for time under continuous compounding, rearrange A = Pe^(rt) into t = ln(A/P) / r. To grow $5,000 into $10,000 at 4%, t = ln(2) / 0.04 = 17.33 years. For the required rate, use r = ln(A/P) / t. To double $5,000 in 10 years, r = ln(2) / 10 = 6.93%.
Both rearrangements use the natural logarithm. Spreadsheets expose it as LN(). Most financial calculators have a dedicated NAT-LOG key.
Where continuous compounding is actually used
Continuous compounding shows up everywhere quantitative finance lives: the Black-Scholes options model, bond pricing, the term structure of interest rates, the risk-free discount factor e^(-rt). Banks rarely use it in deposit products, but trading desks, actuaries, and economists work in it daily.
Outside finance, the same factor e^(rt) describes population growth in ecology, radioactive decay (with negative r), heat loss, and pharmacokinetic clearance. The continuous compounding formula is just one face of a universal exponential law.
A bank quoting "5% APR compounded continuously" delivers an effective rate of 5.127%. A bank quoting "5% APY" already includes compounding. Mixing these labels overstates returns or understates loan costs by a hundred basis points or more.
Common continuous-compound mistakes
Three errors recur. First, plugging the rate as a percent rather than a decimal: e^5 is 148.4, not 1.05. Always divide percent by 100 before applying the exponent. Second, forgetting to take the natural log when solving for t or r. Log base 10 will give the wrong answer. Third, assuming the continuous result is meaningfully higher than monthly. It is not; the difference is usually a few cents on $10,000.
A fourth, subtler error is mixing time units. The exponent must use the same unit for r and t. If the rate is annual, time is in years. A 5% annual rate over 18 months means r = 0.05 and t = 1.5, not t = 18. Convert before applying the formula.
One last note for spreadsheet users: Excel and Google Sheets both expose the exponential via EXP() and the natural log via LN(). The continuous compound formula in a cell is simply =P*EXP(r*t), where the cells P, r, and t hold principal, decimal rate, and years respectively. The same syntax works in Numbers, LibreOffice Calc, and most financial calculators.
- e = 2.71828 (Euler's number)
- A = Pe^(rt) = continuous compound formula
- 5% nominal = 5.127% EAR continuous
- 10% nominal = 10.517% EAR continuous
- Doubling at 5% = 13.86 years
- Doubling at 7% = 9.90 years
- $10k at 5% / 10yr = $16,487.21
- Daily-to-continuous gap = ~$0.56 per $10,000