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Category Deposits & Savings

Principal Amount ($) Initial deposit or investment

Annual Interest Rate (%) Nominal rate, not effective

Time Period Duration of investment

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About

Continuous compounding represents the mathematical limit of the compounding process: interest accrues and reinvests at every infinitesimal instant rather than at discrete intervals. The governing formula A = P ⋅ e^rt uses Euler's number e ≈ 2.71828 as its base. Most financial institutions compound daily or monthly, but continuous compounding sets the theoretical upper bound on growth. The difference between daily and continuous compounding on a $10,000 deposit at 5% over 10 years is roughly $0.31. The practical gap is small, yet the model matters in derivatives pricing (Black-Scholes), bond mathematics, and population dynamics where the continuous assumption is standard.

Miscalculating effective annual rate or ignoring the compounding method can distort yield comparisons across financial products. This calculator applies the exact e^rt formula and also computes the equivalent periodic-compounding result so you can compare directly. Doubling time is derived analytically as ln(2) ÷ r, not approximated by the Rule of 72. Note: this tool assumes a fixed nominal rate for the entire term. Variable-rate instruments require period-by-period recalculation.

Formulas

The continuous compound interest formula derives from taking the limit of periodic compounding as the number of compounding periods n → ∞:

A = P ⋅ e^{r ⋅ t}

where A = final amount, P = principal (initial deposit), r = nominal annual interest rate (decimal), t = time in years, and e = Euler's number ≈ 2.71828.

Interest earned is the difference between the future value and principal:

I = A − P = P ⋅ (e^{r ⋅ t} − 1)

The effective annual rate under continuous compounding converts the nominal rate to an equivalent annual yield:

EAR = e^r − 1

Doubling time is derived by setting A = 2P and solving for t:

t_double = ln(2)r ≈ 0.6931r

For comparison, standard periodic compounding uses:

A_periodic = P ⋅ (1 + rn)^{n ⋅ t}

where n = number of compounding periods per year (1 = annually, 4 = quarterly, 12 = monthly, 365 = daily).

Reference Data

Nominal Rate	EAR (Continuous)	EAR (Monthly)	EAR (Daily)	Doubling Time
1%	1.00502%	1.00460%	1.00502%	69.31 yr
2%	2.02013%	2.01844%	2.02007%	34.66 yr
3%	3.04545%	3.04160%	3.04535%	23.10 yr
4%	4.08108%	4.07415%	4.08085%	17.33 yr
5%	5.12711%	5.11619%	5.12675%	13.86 yr
6%	6.18365%	6.16778%	6.18313%	11.55 yr
7%	7.25082%	7.22901%	7.25009%	9.90 yr
8%	8.32871%	8.29995%	8.32776%	8.66 yr
9%	9.41743%	9.38069%	9.41621%	7.70 yr
10%	10.51709%	10.47131%	10.51558%	6.93 yr
12%	12.74969%	12.68250%	12.74746%	5.78 yr
15%	16.18342%	16.07545%	16.17965%	4.62 yr
20%	22.14028%	21.93910%	22.13360%	3.47 yr
25%	28.40254%	28.07320%	28.39168%	2.77 yr

Frequently Asked Questions

The difference is marginal for typical consumer rates. On a $100,000 deposit at 5% over 10 years, continuous compounding yields $164,872.13 while daily compounding (365 days) yields $164,866.47 - a gap of about $5.66. The gap grows with higher rates and longer terms. At 20% over 30 years the spread exceeds $200 per $10,000 invested.

Black-Scholes assumes asset prices follow a geometric Brownian motion, a continuous-time stochastic process. Using discrete compounding would introduce artificial time-step artifacts. The continuous framework makes the mathematics tractable via Itô calculus and yields closed-form solutions. The risk-free rate r in the model is always expressed as a continuously compounded rate, so any quoted APR must be converted before substitution.

Doubling time t_double = ln(2) ÷ r approaches infinity as r → 0. At r = 0 the formula is undefined because the principal never grows. For very small rates like 0.1%, doubling takes 693.1 years. The Rule of 72 approximation (72 ÷ rate-as-percentage) becomes less accurate below 2% and above 20%.

Technically yes, but almost no lender uses continuous compounding. Mortgage and loan contracts specify discrete periods (monthly for most US mortgages, daily for some credit lines). Applying continuous compounding to a loan would slightly overstate the cost. If a lender quotes a continuously compounded rate, convert it to an effective periodic rate before building an amortization schedule: the monthly rate would be (e^r/12 − 1).

Subtract the inflation rate from the nominal rate to get the real rate: r_real ≈ r_nominal − r_inflation. For a more precise result, use the Fisher equation: 1 + r_real = (1 + r_nominal) ÷ (1 + r_inflation). At 5% nominal and 3% inflation, the real continuously compounded rate is approximately 1.94%, nearly doubling the real doubling time to about 35.7 years.

Each row in the year-by-year schedule is computed using the exact continuous formula A = P ⋅ e^{r ⋅ t} evaluated at integer year boundaries. This is not a recursive approximation - each value is independently calculated from the original principal P, ensuring no floating-point error accumulates across rows.