Two investors identify the same mispriced stock. Both are right—it doubles over the next two years. The first investor allocated 2% of their portfolio to the position. The second allocated 25%. Same analysis, same stock, same outcome. The first made a modest contribution to their returns. The second transformed their portfolio.
This thought experiment reveals the hidden variable in investing: position sizing.
The investment industry obsesses over stock selection. Analysts produce detailed reports on what to buy. Investment committees debate which opportunities are most attractive. Academic research focuses on factors that predict returns. An entire edifice of effort centers on the question: what should I own?
The question that receives almost no attention: how much?
Yet the math is unambiguous. Over a lifetime of investing, position sizing has at least as much impact on wealth accumulation as security selection—probably more. Two investors with identical analytical skill but different sizing strategies will end up in dramatically different places. The one who sizes well will compound wealth. The one who sizes poorly—either too aggressively or too timidly—will underperform despite picking the same winners.
This isn’t intuition. It’s mathematics, formalized over sixty years ago by a Bell Labs researcher named John Kelly.
The Kelly Story
In 1956, John Kelly Jr. published a paper titled “A New Interpretation of Information Rate” in the Bell System Technical Journal. The paper was ostensibly about information theory—Kelly’s colleague Claude Shannon had recently developed the mathematical framework for communication, and Kelly was extending it. But buried in the mathematics was something unexpected: a formula for optimal gambling.
Kelly’s insight was elegant. If you have an edge—if you know the odds are in your favor—there exists a mathematically optimal bet size that maximizes long-term wealth growth. Bet more than this optimal amount and you risk ruin despite having an edge. Bet less and you leave money on the table. The Kelly formula finds the precise point that maximizes geometric growth of capital.
The formula itself is simple. For a bet with probability p of winning and payoff odds of b-to-1:
Kelly % = (bp - q) / b
Where q = (1-p), the probability of losing.
Or in a more intuitive form:
Kelly % = Edge / Odds
Where Edge = (Probability of Win × Payoff if Win) - (Probability of Loss × Loss if Lose)
Consider a coin flip where heads pays 2:1 (you win $2 for every $1 bet) and tails loses your stake, but the coin is biased—60% heads, 40% tails. Your edge is (0.60 × 2) - (0.40 × 1) = 1.20 - 0.40 = 0.80, or 80 cents per dollar bet. Kelly says bet: 0.80 / 2 = 40% of your bankroll on each flip.
That sounds aggressive. Betting 40% on a coin flip? But this is the exact amount that maximizes long-term growth when you have such a massive edge.
Ed Thorp: From Blackjack to Wall Street
The Kelly Criterion might have remained an academic curiosity if not for Ed Thorp, a mathematics professor who saw its practical power.
Thorp first achieved fame by solving blackjack. In 1962, he published Beat the Dealer, demonstrating that card counting could give players a mathematical edge over the casino. But having an edge wasn’t enough—you also needed to size bets correctly. Bet too small and your edge barely matters. Bet too large and a run of bad luck wipes you out before the edge materializes.
Thorp used Kelly to size his blackjack bets. When the count was favorable—more high cards remaining in the deck—Kelly said bet big. When unfavorable, Kelly said bet small or not at all. The combination of edge identification (card counting) and optimal sizing (Kelly) made Thorp so successful that casinos banned him and eventually changed their rules.
Then Thorp turned to Wall Street. In 1969, he founded Princeton Newport Partners, one of the first quantitative hedge funds. The fund used sophisticated models to identify mispriced warrants and options, then sized positions using Kelly principles. Over 19 years, Princeton Newport compounded at 19.1% annually with remarkable consistency—only three down months over two decades. His later fund, Ridgeline Partners, continued the approach.
Thorp’s memoir, A Man for All Markets, details this journey from academic mathematics to blackjack tables to Wall Street. The through-line is Kelly: having an edge matters, but sizing that edge correctly is what transforms edge into wealth.
The Mathematics of Ruin
Why does bet sizing matter so much? Because of a mathematical asymmetry that most investors underappreciate: the asymmetry between gains and losses.
If you lose 50% of your capital, you need a 100% gain to recover. Lose 75% and you need 300%. Lose 90% and you need 900%. Losses create holes that get progressively harder to climb out of.
This asymmetry makes over-betting catastrophic. Consider an investor with a genuine edge who bets too aggressively. They win often—the edge is real—but occasional losses dig deep holes. A few bad outcomes in sequence can destroy years of gains. The edge never had time to compound because ruin came first.
Kelly avoids this. The formula automatically sizes smaller when the edge is smaller or the odds are worse. It never suggests bet sizes that risk total ruin. The worst case under Kelly is gradual decline during unlucky streaks, followed by recovery as the edge reasserts—not permanent destruction of capital.
Conversely, under-betting leaves wealth on the table. An investor who bets far below Kelly grows capital much more slowly. They’re protected against volatility they don’t actually face, paying an opportunity cost measured in foregone compounding.
Kelly finds the knife’s edge: maximum growth without ruin risk.
Kelly in Practice
The Kelly formula is elegant, but applying it to real investing requires adaptation. Investing isn’t blackjack—the probabilities are uncertain, the payoffs are complex, and the opportunities aren’t repeated coin flips.
The Problem of Unknown Edges
In blackjack, Thorp knew his edge precisely. Card counting produced quantifiable advantages in specific situations. The Kelly formula could be applied directly.
In investing, edges are estimated, not known. You believe a stock is worth $100 and trades at $70—a potential 43% gain. But that belief is uncertain. Your valuation might be wrong. The market might stay irrational longer than you can stay solvent. The company’s fundamentals might change.
When you don’t know your edge precisely, full Kelly is dangerous. If you overestimate your edge, Kelly tells you to bet more than is actually optimal. And overestimating edges is exactly what overconfident humans do.
This is why practitioners use fractional Kelly—betting a fraction of what the full formula suggests.
Half Kelly bets 50% of the Kelly-optimal amount. This reduces expected growth to about 75% of full Kelly growth, but reduces variance dramatically and provides significant protection against estimation errors.
Quarter Kelly bets 25% of optimal. Growth drops further, but the strategy becomes extremely robust to estimation errors. You could be wrong about your edge by a factor of two and still be sizing appropriately.
The math: if you’re betting half Kelly and your true edge is half what you estimated, you’re actually betting at full Kelly for your true edge—optimal, not catastrophic. If you’re betting full Kelly and your true edge is half estimated, you’re betting double Kelly—which leads to eventual ruin.
Fractional Kelly builds margin of safety into position sizing. You sacrifice some expected growth for protection against being wrong about how good your opportunities are.
The Problem of Correlated Bets
Kelly assumes independent bets. Each coin flip is unrelated to the last. But investments are correlated. If you own five bank stocks, their outcomes are linked—a banking crisis hits all of them simultaneously.
Correlated bets must be sized smaller than Kelly suggests for individual positions. The portfolio’s overall exposure to any single risk factor should follow Kelly logic, not the individual position sizes.
Practically, this means two things:
First, consider your effective bet. Five 10% positions in highly correlated stocks isn’t five bets—it’s closer to one 50% bet on the shared factor. Size accordingly.
Second, diversify exposures. If Kelly suggests 30% in your highest-conviction idea, don’t put another 30% in a highly correlated second idea. The combined risk exceeds what Kelly would recommend for the shared exposure.
The Problem of Uncertainty About Uncertainty
Kelly requires probability estimates. “60% chance this stock doubles, 40% chance it halves.” But how confident are you in those probabilities?
This is uncertainty about uncertainty—second-order uncertainty—and it matters. If you’re confident in your probabilities, Kelly sizing can be more aggressive. If your probabilities themselves are uncertain, you need additional margin.
One approach: estimate a range of probabilities and size based on the pessimistic end. If you think the probability is “somewhere between 50% and 70%,” use 50% for Kelly calculations. This builds in conservatism for estimation uncertainty.
Another approach: weight confidence into fractional Kelly. When probability estimates are highly uncertain, use quarter Kelly. When you have high confidence in your estimates, move toward half Kelly. Never use full Kelly because you’re never certain enough to warrant it.
Practical Kelly Implementation
Given these complications, here’s a practical framework:
Step 1: Estimate your edge. What’s the expected value of this investment? This requires estimating both probability of various outcomes and payoff in each outcome. Be conservative—assume you’re overestimating your edge because you probably are.
Step 2: Estimate odds. What’s the ratio of potential gain to potential loss? If you expect to make 50% if right and lose 25% if wrong, your odds are 2:1.
Step 3: Calculate theoretical Kelly. Use the formula: (bp - q) / b. This gives you a theoretical optimal bet size.
Step 4: Apply fractional Kelly. Multiply by 0.25 to 0.5 depending on confidence in your estimates. Less confidence = smaller fraction.
Step 5: Check against portfolio constraints. Does this position create correlated exposure that exceeds your Kelly-implied risk for any factor? Adjust down if so.
Step 6: Reality check. Would this position size keep you awake at night? Kelly optimizes mathematics, not psychology. Reduce further if the emotional cost exceeds what the math justifies.
What Kelly Reveals About Conventional Wisdom
Kelly mathematics expose several flaws in conventional investment thinking.
The Diversification Delusion
Modern portfolio theory emphasizes diversification: spread your bets across many assets to reduce risk. The result is portfolios with dozens or hundreds of positions, none large enough to matter much.
Kelly suggests something different. If you have genuine edge in a position—if your analysis is right—Kelly says bet meaningfully. A diversified portfolio of small bets is suboptimal if some of those bets represent better opportunities than others.
Warren Buffett, who runs one of the least diversified portfolios in professional investing, puts it bluntly: “Diversification is protection against ignorance. It makes little sense if you know what you’re doing.”
This doesn’t mean concentration is always right. If you don’t have edge—if you can’t reliably identify mispriced securities—diversification makes sense because you’re not leaving Kelly edge on the table. But if you do have edge, Kelly says use it. Size your best ideas appropriately rather than diluting them across a portfolio that doesn’t reflect your conviction hierarchy.
Charlie Munger makes the same point differently: “The idea of excessive diversification is madness.” For investors with genuine analytical edge, Kelly-informed concentration outperforms conventional diversification.
The Cost of Timidity
Most professional investors are under-bet. They have genuine insights but size positions timidly—2-3% positions that barely move the portfolio even when they’re right.
This timidity has rational explanations. Career risk punishes volatility more than it rewards returns. A fund manager who outperforms steadily will be rewarded less than they’ll be punished for a volatile year that underperforms. The incentives favor smooth returns over maximum wealth creation.
But for individual investors managing their own capital, these incentives don’t apply. Career risk is irrelevant. Only long-term wealth accumulation matters. And Kelly shows that timid sizing leaves enormous returns on the table.
If you have an investment with 60% probability of gaining 50% and 40% probability of losing 25%, your Kelly allocation is 20-25%. Most investors would put 3-5% in such a position. They’d be betting at roughly one-fifth of Kelly—surrendering most of the mathematical edge they’d identified.
The Courage to Size
Stanley Druckenmiller, with one of the best track records in macro investing, explains his approach: “When you have tremendous conviction on a trade, you have to go for the jugular.”
This isn’t recklessness. It’s Kelly logic translated to investor language. When edge is large and conviction is high, Kelly says bet big. Betting small in those situations is mathematically suboptimal.
George Soros, Druckenmiller’s mentor at the Quantum Fund, operated the same way. His famous trade against the British pound in 1992—betting $10 billion, effectively the entire fund—was sizing proportional to perceived edge. The conviction was high, the payoff asymmetry was favorable, and Kelly mathematics justified the concentration.
Most investors lack the psychological fortitude to size this way. When the math says bet 25%, they bet 5% because 25% feels scary. This gap between mathematical optimality and psychological tolerance is where returns get left on the table.
Kelly and Risk of Ruin
One of Kelly’s most valuable properties is that it protects against ruin while maximizing growth. But “ruin” in Kelly terms requires definition.
In the strict formula, ruin is losing everything—going to zero. Kelly-optimal sizing makes this virtually impossible because the formula never suggests betting your entire stake, and losses reduce subsequent bet sizes proportionally.
But in practice, ruin has softer forms. A 50% drawdown might be technically recoverable but psychologically devastating. A 30% loss might trigger margin calls or career consequences that end the game before recovery.
This is why fractional Kelly matters beyond just edge uncertainty. It also manages behavioral and practical forms of ruin that the mathematics don’t capture.
Consider: Full Kelly produces maximum long-term growth but also maximum volatility. Half Kelly produces 75% of the growth with dramatically less volatility. For investors who need to stay in the game—who can’t psychologically or practically tolerate deep drawdowns—half Kelly trades some growth for stability. That’s often a worthwhile exchange.
The deep insight: Kelly tells you the mathematically optimal sizing, but wisdom involves adjusting for non-mathematical constraints. What matters isn’t maximum possible wealth but maximum wealth you can actually achieve given your specific situation, psychology, and time horizon.
Position Sizing Hierarchy
Kelly principles suggest a hierarchy for position sizing that differs from conventional practice:
Tier 1: High-conviction positions (10-25% each) These are your best ideas—positions where analysis is deepest, edge is largest, and Kelly suggests significant allocation. Most portfolios have only 2-4 such positions at any time. Size them meaningfully.
Tier 2: Medium-conviction positions (3-10% each) Good opportunities with solid edge but less certainty. Kelly suggests moderate sizing. A portfolio might hold 3-6 such positions.
Tier 3: Low-conviction positions (1-3% each) Opportunities with some edge but high uncertainty. Kelly suggests small sizing. These positions matter only in aggregate.
Tier 4: Tracking positions (<1%) Positions held more for learning than returns. Kelly would say “why bother?” but real-world learning has value that mathematics doesn’t capture.
The conventional approach inverts this: many small positions, few large ones. Kelly inverts back: concentrate where edge is greatest, minimize allocation where edge is smallest.
Portfolio Construction Implications
Kelly-informed portfolio construction looks different from conventional diversification:
Fewer positions. You can’t meaningfully size dozens of positions. If everything is 2%, nothing matters. Kelly-informed portfolios typically hold 10-15 positions, not 50-100.
Explicit conviction ranking. Which positions are Tier 1? Why? The ranking should reflect Kelly inputs: edge, odds, confidence. Most portfolios lack explicit ranking—everything just gets roughly equal weight.
Concentration at the top. The top 3-5 positions should constitute the majority of the portfolio. If your best idea represents 5% and your worst idea represents 4%, your sizing doesn’t reflect your analysis.
Correlation awareness. Positions must be sized considering correlations. Two 15% positions in highly correlated stocks might constitute a single 30% effective bet—above what Kelly suggests for any single opportunity.
Common Mistakes
Kelly is powerful but commonly misapplied:
Ignoring Estimation Uncertainty
The most common mistake: treating probability estimates as precise when they’re rough guesses. If you estimate 60% probability, your true confidence interval might be 45-75%. Using point estimates in Kelly produces sizing that doesn’t reflect your actual uncertainty.
Solution: use fractional Kelly and size based on conservative probability estimates.
Ignoring Correlation
Sizing each position independently ignores portfolio-level risk. Five 15% positions in technology stocks isn’t Kelly-optimal for five opportunities—it’s massively over-bet on the technology factor.
Solution: think about effective bets, not position counts. What factors drive your portfolio? Size total exposure to each factor using Kelly logic.
Ignoring Psychology
Full Kelly produces drawdowns that most humans can’t tolerate. A 30% drawdown is mathematically temporary; psychologically it might trigger panic selling that locks in losses.
Solution: adjust Kelly for your actual risk tolerance, not theoretical optimal. The best sizing is that which you can maintain through volatility.
Sizing for Wins, Not Edge
Some investors size based on how much they’ll make if right, ignoring the probability of being wrong. “This could double, so I’m going big.” That’s not Kelly logic—that’s gambling disguised as strategy.
Solution: always size based on expected value (probability × payoff), not maximum payoff.
Overconfidence in Edge
Most investors overestimate their edge. Careful analysis produces confident conclusions, but confidence doesn’t equal accuracy. Kelly applied to an overestimated edge produces over-betting and eventual disaster.
Solution: assume your edge is roughly half what you think it is. Then apply fractional Kelly. Double-margin against overconfidence.
The Practice
Building Kelly-informed sizing into your process requires deliberate habits:
Exercise 1: The Conviction Ranking
For your current portfolio, rank positions by conviction. Which are your best ideas? Which would you add to if you could? Which would you reduce? Does your sizing reflect this ranking? If your second-best idea has a larger position than your best idea, something is wrong.
Exercise 2: Calculate Theoretical Kelly
For your highest-conviction position, estimate:
- Probability it achieves your target return
- Magnitude of return if successful
- Probability it fails
- Magnitude of loss if it fails
Calculate Kelly: (p × win - q × loss) / win. Compare to your actual position size. Are you sized at full Kelly, half, quarter? Is that appropriate given your confidence in the estimates?
Exercise 3: The Correlation Audit
Map your portfolio’s factor exposures. What percentage is effectively a bet on technology? On interest rates? On China? On the economy? Are these effective bets sized appropriately using Kelly logic, or have they accidentally grown through position accumulation?
Exercise 4: The Opportunity Cost Analysis
Calculate: If you had sized your winners at Kelly-optimal levels (or half Kelly) over the past five years, how would your returns differ? Most investors discover they’ve dramatically under-bet their best ideas.
This retrospective creates visceral understanding of what timid sizing costs.
Exercise 5: The Pre-Commitment
Before your next investment, write down your Kelly-implied sizing and commit to it. “Based on my estimates, half Kelly suggests a 12% position. I commit to 12%, not the 5% that feels comfortable.”
Pre-commitment overcomes the psychological pull toward timidity at the moment of execution.
Kelly and Other Frameworks
Kelly deepens understanding of related frameworks:
Margin of Safety as Sizing Input
Margin of safety affects Kelly calculations through two channels. First, margin creates asymmetric payoffs—if you’re buying below intrinsic value, your upside exceeds your downside, improving Kelly-suggested sizing. Second, margin affects probability estimates—larger discounts mean margin for error, increasing confidence in positive outcomes.
The connection: deep margin of safety both justifies and enables larger positions.
Circle of Competence as Edge Validation
Kelly requires knowing your edge. But edge only exists inside your circle of competence. Outside your circle, you don’t have edge—you have guesses. And Kelly-sizing based on guesses is just optimized gambling.
This creates a sizing hierarchy: size largest inside your circle, where edge is real. Size smallest (or not at all) outside your circle, where edge is imaginary.
Base Rates as Probability Foundation
Base rates provide the outside-view foundation for probability estimates that Kelly requires. Before estimating “60% chance this turnaround succeeds,” ask: “What percentage of turnarounds succeed historically?” That base rate anchors your estimate and prevents overconfidence that leads to over-sizing.
Asymmetric Returns as Kelly’s Goal
Asymmetric returns—positions with limited downside and substantial upside—produce favorable Kelly calculations. When upside significantly exceeds downside, Kelly suggests larger positions than for symmetric bets. Seeking asymmetry isn’t just about expected value; it’s about creating situations where Kelly-optimal sizing is meaningful rather than trivial.
The Deep Insight
Kelly reveals a profound truth about wealth creation: it’s not just about being right—it’s about being sized right when you’re right.
Most investors focus entirely on security selection. They search for mispriced assets, analyze competitive advantages, evaluate management teams, study industry dynamics. All of this effort goes toward answering: “What should I buy?”
Kelly asks the prior question that most never consider: “Given what I should buy, how much should I own?” And the answer to this question—not the answer to stock selection—often determines long-term wealth.
Consider two investors with identical analytical skill. Both identify the same mispriced stocks. Both are right the same percentage of the time. But one sizes according to conviction; the other sizes timidly and evenly. Over a 30-year career, the Kelly-informed investor will compound at dramatically higher rates. Same analysis, different outcomes—driven entirely by sizing.
This is the hidden edge. It’s hidden because no one talks about it. Finance education focuses on asset pricing, not position sizing. Analyst reports recommend “buy” or “sell” without specifying how much. The infrastructure of investing ignores the dimension that matters most for wealth creation.
Ed Thorp understood this in 1962 at blackjack tables. He understood it in 1969 on Wall Street. The mathematics haven’t changed. Only the widespread ignorance persists.
Kelly isn’t just a formula. It’s a reframing: position sizing isn’t a secondary consideration after security selection. It’s at least equally important—the variable that turns analytical edge into actual wealth.
Position sizing through Kelly principles transforms how you construct portfolios—concentrating capital in highest-conviction ideas rather than spreading it thin across many mediocre positions. For building conviction that justifies concentration, see circle of competence. For understanding why deep discounts enable larger positions, explore margin of safety. For the probability estimation that Kelly requires, see base rates. And for structuring positions where Kelly suggests meaningful sizing, explore asymmetric returns.