Compound Interest
Compound interest is the process by which interest is earned not only on the original principal but also on previously accumulated interest, causing wealth to grow at an exponential rather than linear rate over time. It is the mathematical foundation of all time value of money calculations and underlies asset pricing, bond valuation, option theory, and long-term investment return compounding.
Key takeaways
- The future value formula FV = PV × (1 + r/n)^(nT) captures compounding with periodic reinvestment at rate r with n compounding periods per year over T years.
- More frequent compounding (monthly vs. annual) generates higher returns; the limiting case as n → ∞ is continuous compounding: FV = PV × e^(rT).
- The effective annual rate (EAR) standardizes different compounding frequencies: EAR = (1 + r/n)^n − 1.
- The Rule of 72 provides a quick approximation: an investment doubles in approximately 72/r% years (e.g., 9 years at 8% annual return).
- Compounding is the core mechanism behind long-term wealth accumulation; the power of compounding explains why return consistency over decades dominates return magnitude in any single year.
Explanation
Compound interest transforms a finite principal into an exponentially growing quantity through the continuous reinvestment of returns. This stands in contrast to simple interest, where interest is calculated only on the original principal and thus grows linearly. The difference between simple and compound interest becomes dramatic over long time horizons — a crucial insight for both investment management and debt management.
The fundamental formula for discrete compounding is:
FV = PV × (1 + r/n)^(n×T)
where PV is present value, r is the annual nominal interest rate, n is the number of compounding periods per year, and T is the number of years. For annual compounding (n=1), this reduces to FV = PV × (1+r)^T. For continuous compounding, n → ∞ and FV = PV × e^(rT), where e ≈ 2.71828 is Euler's number.
The concept of effective annual rate (EAR) is critical for comparing investment vehicles with different compounding conventions: EAR = (1 + r/n)^n − 1. For example, a money market fund quoting 5.20% compounded daily has an EAR = (1 + 0.052/365)^365 − 1 = 5.337%. Continuous compounding is used throughout derivatives pricing, fixed income analytics, and risk management because it simplifies mathematical derivations. The Black-Scholes model, for example, assumes continuous compounding of the risk-free rate, expressed as e^(rT).
From a practitioner standpoint, understanding compounding is essential for return attribution. A hedge fund that earns 10% in year 1, −5% in year 2, and 8% in year 3 has a geometric mean return of [(1.10)(0.95)(1.08)]^(1/3) − 1 = (1.1286)^(0.333) − 1 = 4.12% per year — significantly below the arithmetic mean of (10 − 5 + 8)/3 = 4.33%. The difference is the 'variance drain' — a concept formalized by Jensen's Inequality. Minimizing volatility is thus mathematically equivalent to maximizing long-run compound growth at a given arithmetic average return.
Formula
FV = PV × (1 + r/n)^(n×T) | Continuous: FV = PV × e^(rT) | EAR = (1 + r/n)^n − 1
Example
An investor places $100,000 in a diversified equity portfolio earning 8% per year, compounded annually. After 10 years: $100,000 × (1.08)^10 = $215,892. After 30 years: $100,000 × (1.08)^30 = $1,006,266. The same investor's twin places $100,000 in a savings account earning 8% simple interest. After 30 years, simple interest yields: $100,000 + ($100,000 × 0.08 × 30) = $340,000. The compounding investor ends up with $1,006,266 versus $340,000 — nearly three times as much — illustrating Einstein's reputed description of compound interest as the 'eighth wonder of the world.'
Related terms
Black Scholes Model Bond Continuous Compounding Copula Equity Finite Difference Method Hedge Fund Interest Rate Jensens Inequality Nominal Interest Rate Option Perpetuity