Continuous Compounding
Continuous compounding is a mathematical idealization of compound interest where interest accrues and is reinvested at every infinitesimally small interval, resulting in exponential growth described by e^(rT). It is the limiting case of discrete compounding as the number of compounding periods per year approaches infinity and is ubiquitously used in derivatives pricing, fixed income mathematics, and stochastic calculus.
Key takeaways
- The continuously compounded future value formula is FV = PV × e^(rT), where e ≈ 2.71828 is Euler's number.
- The continuously compounded rate r_c corresponding to a discretely compounded rate r_m (compounded m times per year) is: r_c = m × ln(1 + r_m/m).
- Black-Scholes, the Heath-Jarrow-Morton interest rate model, and virtually all stochastic calculus-based finance models assume continuous compounding for mathematical tractability.
- Log returns (continuously compounded returns) are additive over time: the 3-year log return = sum of 3 annual log returns, unlike discrete returns which are multiplicative.
- Continuously compounded returns are approximately normally distributed even when gross returns are lognormally distributed, simplifying statistical analysis.
Explanation
Continuous compounding emerges from taking the limit of discrete compounding. The discrete formula FV = PV × (1 + r/n)^(nT) as n → ∞ converges to FV = PV × e^(rT) by the definition of Euler's number: e = lim_{n→∞} (1 + 1/n)^n. This mathematical elegance makes continuous compounding the natural language of derivatives pricing and stochastic finance.
The key conversion between discrete and continuous rates: - Discrete rate r_m (compounded m times per year) → Continuous: r_c = m × ln(1 + r_m/m) - Continuous rate r_c → Discrete: r_m = m × (e^(r_c/m) − 1)
For example, a 5% annual rate compounded semiannually converts to r_c = 2 × ln(1 + 0.05/2) = 2 × ln(1.025) = 2 × 0.02469 = 4.938% continuously compounded.
In derivatives pricing, continuous compounding appears in the discounting of future payoffs. The present value of a cash flow C received at time T is PV = C × e^(−rT), where r is the continuously compounded risk-free rate. In the Black-Scholes option pricing model, the expected stock price at time T is S × e^((μ−σ²/2)T + σ√T·Z) where the lognormal distribution naturally arises from assuming continuously compounded returns are normally distributed with mean (μ−σ²/2) and variance σ²T.
Log returns — defined as r_log = ln(P_t/P_{t-1}) — are continuously compounded period returns. They have the crucial property of time additivity: ln(P_T/P_0) = Σ ln(P_t/P_{t-1}), making portfolio performance measurement straightforward. The geometric mean annual return of a fund is expressed as the average of log returns. This contrasts with arithmetic (discrete) returns, which are not time-additive — you cannot sum annual discrete returns to get a multi-year total return.
Formula
FV = PV × e^(rT) | r_continuous = m × ln(1 + r_m/m) | Log Return: r_log = ln(P_t/P_{t-1})
Example
A fixed-income portfolio manager values a zero-coupon bond paying $1,000 in 5 years using continuously compounded rates. The 5-year Treasury spot rate is 4.25% continuously compounded. Bond price = $1,000 × e^(−0.0425 × 5) = $1,000 × e^(−0.2125) = $1,000 × 0.8083 = $808.30. For comparison, if the 4.25% rate were compounded semiannually, the price would be $1,000 / (1 + 0.0425/2)^10 = $1,000 / (1.02125)^10 = $1,000 / 1.2342 = $810.30. The small but real difference ($2.00 per $1,000 face) illustrates why specifying the compounding convention matters in fixed income calculations.
Related terms
Bond Compound Interest Convexity Adjustment Discount Rate Eigenvalue Decomposition Internal Rate Of Return Jensens Inequality Option Option Pricing Model Present Value Risk Free Rate Spot Rate