Continuous Compounding

Business Calculus · Axiom Academy

What happens when interest compounds infinitely often? What If We Compound More Often? We've seen that monthly compounding beats annual compounding. What if we compound even more frequently? Daily? Hourly? Every second? Let's investigate what happens to \ 1 invested at 100\% interest for 1 year as we increase the number of compounding periods: As we compound more and more often, the value approaches a special number that appears throughout mathematics and finance: This is an irrational number (like π) that never repeats The number e is the natural base for exponential growth. It appears in biology, physics, economics, and anywhere continuous growth or decay occurs. It's the base of the natural logarithm . The Continuous Compounding Formula When interest compounds continuously, we use a beautifully simple formula: Notice there's no n (compounding periods) to worry about. The formula A = Pe^ rt is elegant precisely because continuous compounding is the limiting case—it's mathematically cleaner than dealing with discrete periods. Comparing Monthly vs. Continuous Let's compare monthly compounding with continuous compounding: Continuous compounding yields 13.61 more than monthly compounding over 10 years. The difference is small but real. For very large amounts or very long time periods, this difference becomes more significant. Let's see how different compounding frequencies compare:

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