Maximizing Long-Term Growth with Optimal Risk Sizing
When it comes to investing and betting, few models are as powerful—or as misunderstood—as the Kelly Criterion. Originally developed by Bell Labs researcher John L. Kelly Jr. in 1956, this formula offers a mathematically sound method to maximize the long-term growth rate of capital by optimizing bet size or investment allocation relative to risk and reward.
Let’s break down why this model matters, how it works, and how it’s used in professional investing.
📌 What is the Kelly Criterion?
The Kelly Criterion is a formula used to determine the optimal size of a series of bets or investments to maximize logarithmic wealth growth over the long run, while minimizing the risk of ruin. It helps answer a key question: How much should I allocate to a single opportunity given the odds and potential returns?
At its core, the model balances risk and reward to avoid two extremes:
- Betting too little (missing out on compounding opportunities)
- Betting too much (increasing the risk of losing your bankroll)
🔢 The Formula
For binary outcomes (like a win/loss scenario in gambling), the classic Kelly formula is: f∗=bp−qbf^* = \frac{bp – q}{b}f∗=bbp−q
Where:
- f∗f^*f∗ = Fraction of capital to wager/invest
- bbb = Net odds received (e.g. 1:1 odds → b = 1)
- ppp = Probability of winning
- qqq = Probability of losing (1 – p)
In investment applications, this is often generalized using expected return and variance: f∗=E[R]σ2f^* = \frac{E[R]}{\sigma^2}f∗=σ2E[R]
Where:
- E[R]E[R]E[R] = Expected excess return of the asset
- σ2\sigma^2σ2 = Variance of returns
🧠 Why It Works: The Power of Logarithmic Utility
Unlike linear models that aim to maximize expected value, the Kelly Criterion maximizes the expected logarithmic growth rate of wealth. This reflects a more realistic utility function where the pain of losses grows faster than the joy of gains.
This makes Kelly a risk-aware optimizer—you’re not just shooting for high returns, you’re optimizing for sustainable, compounding returns over time.
📈 Practical Applications in Investing
Many legendary investors—Warren Buffett, Ed Thorp, Bill Gross, and James Simons—have referenced or used versions of the Kelly Criterion in their portfolio construction or position sizing strategies.
Use cases include:
- Position Sizing in high-conviction trades
- Capital Allocation across multiple asymmetric bets
- Risk-adjusted Sizing of venture capital or early-stage investments
- Hedging strategies in derivatives and options
“If you’re not following the Kelly Criterion (or a derivative of it), you’re probably either leaving money on the table or risking too much.” — Ed Thorp
⚠️ Caveats & Modifications
Despite its elegance, the Kelly Criterion comes with practical limitations:
- It requires accurate inputs (expected returns, probabilities, variance)—which are rarely known with certainty.
- Full Kelly allocations can be volatile, so many investors apply “fractional Kelly” (e.g., 50% Kelly) to reduce drawdowns and improve risk-adjusted returns.
Additionally, correlations between investments must be taken into account when applying Kelly across a portfolio.
✅ Summary: Why the Kelly Criterion Matters
| Benefit | Description |
|---|---|
| Optimized Growth | Maximizes wealth over time through compounding |
| Risk Aware | Avoids overbetting and the risk of financial ruin |
| Quantitative Discipline | Encourages precision in sizing positions |
| Tailored to Conviction | Higher conviction = higher allocation |
The Kelly Criterion provides a mathematical foundation for smart risk-taking. In a world where emotions, noise, and volatility can derail rational decisions, Kelly offers a North Star: only risk what you can afford to lose, and only when the odds are clearly in your favor.
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Murray Slatter
Strategy, Growth, and Transformation Consultant: Book time to meet with me here!
