What is continuous compounding?
Continuous compounding is a mathematical model of compound interest that assumes interest is being added to the principal an infinite number of times per period. In effect, interest is calculated and reinvested continuously, so the balance grows at every instant. It is an idealized limit rather than a practical banking convention, but it is useful as a theoretical benchmark and a component of many financial models.
Key definitions
– Compound interest: interest paid on both the original principal and on interest that has already been added to the account.
– Discrete compounding: interest is added at specific intervals (yearly, quarterly, monthly, daily).
– Continuous compounding: the limit of discrete compounding as the compounding frequency becomes infinite.
– e: the base of natural logarithms, a mathematical constant ≈ 2.718281828.
Main formula and variables
– Future value under continuous compounding:
FV = PV × e^(r × t)
where
PV = present value (initial principal),
r = nominal annual interest rate (expressed as a decimal),
t = time in years,
e = 2.71828…
– Relation to the discrete-compounding formula:
Discrete: FV = PV × (1 + r/n)^(n×t)
Continuous = limit of the discrete expression as n → ∞, which yields PV × e^(r t).
– Converting APY (annual percentage yield) and continuous rates:
If a nominal annual rate is compounded continuously, APY = e^r − 1.
Conversely, the continuous nominal rate that produces a given APY is r = ln(1 + APY).
Why it matters
– Continuous compounding gives the theoretical maximum growth for a given nominal rate r over a fixed time.
– It simplifies algebra in many models (exponential functions have convenient calculus properties).
– It is embedded in several financial formulas and models (for example, some option-pricing and discounting formulas use continuous-time assumptions).
Practical checklist — when and how to use continuous compounding
1. Check the model requirements: If a formula or textbook explicitly assumes continuous compounding, use FV = PV × e^(r t).
2. Make units consistent: r must be the annual rate (decimal) and t measured in years. Convert months or days to fraction of a year as needed.
3. If comparing products, compute their APYs first; use APY or convert discrete rates to an equivalent continuous rate using r = ln(1 + APY).
4. For numeric computation, use a calculator or spreadsheet exponent function (e.g., EXP in Excel).
5. Note limitations: verify whether the financial product actually compounds continuously (most consumer accounts do not).
Worked numeric example
Scenario: You invest $10,000 at a nominal annual rate r = 15% (0.15) for t = 1 year. Compare several compounding conventions.
Step 1 — Annual compounding:
FV_annual = 10,000 × (1 + 0.15) = 10,000 × 1.15 = 11,500
Interest earned = $1,500
Step 2 — Monthly compounding (n = 12):
FV_monthly = 10,000 × (1 + 0.15/12)^(12) ≈ 10,000 × (1.0125)^(12)