Kelly & Geometric Growth

You have an edge — and a question that decides whether it makes you rich or broke: how much do you bet? Bet too little and your edge crawls; bet too much and the same winning edge still drags you to zero, because volatility eats geometric growth alive. There is exactly one bet size that compounds your money fastest, and the math behind it is one of finance’s most beautiful — and most abused — results.

Here’s what we’ll build, brick by brick:

• Arithmetic vs geometric mean — why +50% then −50% averages to 0% yet loses you 25%, and why only the geometric mean actually compounds.
• CAGR & volatility drag — the variance penalty $g \approx \mu - \tfrac{1}{2}\sigma^2$, made visual, and why identical average returns can compound at wildly different rates.
• The Kelly criterion — bet $f^* = p - q/b$ of your bankroll, the fraction that maximizes the expected log of your wealth (and thus its long-run growth).
• Continuous Kelly — $f^* = \mu/\sigma^2$ for real markets with smooth returns.
• Fractional Kelly — why half-Kelly keeps three-quarters of the growth for half the volatility, and why almost every professional bets a fraction.
• Multi-asset Kelly & the edge of ruin — vector Kelly across correlated bets, what leverage truly costs, and the iron law that betting past twice Kelly turns growth negative: you go broke with probability one, edge or no edge.

This is where probability, compounding, and risk management fuse into one discipline. By the end you’ll size positions so your edge actually compounds — and understand in your bones why the trading graveyard is full of people who were right about the market and wrong about the bet.