Position sizing: Kelly, fractional Kelly, and why most students get it wrong

Two traders with the same investment ideas can have wildly different returns based on how they size their positions. Position sizing isn't the boring administrative layer underneath the "real" work of picking ideas. It's at least as important as the ideas themselves.

The math foundation here is the Kelly criterion, formulated in 1956 by John Kelly Jr. at Bell Labs. It tells you the position size that maximizes long-run wealth growth, given known probabilities and known payoffs. It's exact, it's mathematically correct, and it will blow up your account if you use it as written.

The Kelly formula

For a single trade with known edge:

f* = (bp - q) / b

where:

• f* = fraction of wealth to bet
• b = net odds (if you win, you win b times your stake)
• p = probability of winning
• q = probability of losing (1 - p)

Concrete example: a trade has 60% chance of paying +30% and 40% chance of paying −20%. Then b = 1.5 (you win 1.5× your stake on a win, since 30% / 20% = 1.5), p = 0.6, q = 0.4.

f* = (1.5 × 0.6 − 0.4) / 1.5 = (0.9 − 0.4) / 1.5 = 0.333

Kelly says: bet 33% of your wealth on this trade.

Sounds aggressive. It IS aggressive. And the math is correct — IF you know p, q, and b precisely. Which you don't.

Why pure Kelly destroys most accounts

Three failure modes:

Failure 1: You overestimate p. Humans systematically overestimate their probability of being right. If you think you have 60% edge and actually have 55%, the Kelly formula tells you to bet 33% but optimal is 17%. The penalty for betting too large is asymmetric — over-betting reduces compound returns much more than under-betting does.

Failure 2: Returns are not binary. The Kelly formula above assumes win/lose outcomes. Real trades have continuous outcome distributions. The "Kelly fraction" for continuous returns involves the ratio of expected return to variance, not just probabilities. Most students who quote Kelly are using the wrong version.

Failure 3: Drawdowns are intolerable. Even at the correct full Kelly fraction, expected drawdowns are huge. With f* = 0.33 on a 60/40 trade, your expected maximum drawdown over a 30-trade sequence is approximately 50%. Most traders can't psychologically tolerate a 50% drawdown — they capitulate at 30% and lock in a permanent loss.

Fractional Kelly: the actual recommendation

Industry practice: bet half-Kelly or quarter-Kelly.

Half-Kelly gives you 75% of the long-run growth rate of full Kelly with roughly 50% of the volatility. Quarter-Kelly gives you 50% of the growth with 25% of the volatility.

For the 60/40 trade above:

• Full Kelly: 33% of wealth
• Half-Kelly: 17%
• Quarter-Kelly: 8%

Most retail traders should be at quarter-Kelly or smaller, because:

• They systematically overestimate their edge (failure 1)
• They underestimate drawdown tolerance (failure 3)
• They have concentrated portfolios, so the binary-Kelly approximation overstates allowable bet size

A simpler heuristic that works

If the math is too much, use this rule of thumb:

No single position should be more than 5% of your portfolio for high-conviction trades, 2% for medium-conviction trades, 0.5% for speculative trades.

This is roughly equivalent to quarter-Kelly for typical retail edge levels and assumes 15-30 positions in the portfolio. It's intentionally conservative because real-world deviation from your modeling assumptions is always larger than you expect.

Position sizing for the Zunko Competition

The Spring 2026 portfolio competition starts with \$10,000 per student. You'll be tempted to size large to win the leaderboard. Don't. Here's the math:

Strategy A (large bets): Six 17% positions. To win the leaderboard, you need most to be right. But one wrong call is a 17% portfolio hit, hard to recover from in 14 weeks.

Strategy B (medium bets): Fifteen 5% positions. Diversification cushion. To win, you don't need spectacular accuracy — 60% hit rate with average +12% on winners and average −10% on losers gives you a 6% portfolio return per cycle, top-quartile in most cohorts.

Strategy B wins more competitions than strategy A historically. The "lottery ticket" approach (5 huge positions, hope for one to triple) can produce occasional spectacular winners but loses much more frequently. The Sharpe ratio of A is about half of B.

Position sizing across time

Your sizing should NOT be constant across trades. Three adjustments make a real difference:

1. Larger when conviction is higher. Not by 2× — by 1.4×. (Kelly is roughly proportional to edge squared, but capping at 1.4× prevents over-sizing on overconfident calls.) High-conviction = 5% of portfolio. Medium = 3.5%. Low-conviction speculative = 2%.

2. Smaller during regime uncertainty. When market volatility is unusually high (VIX above 25-30), cut all position sizes by 30-50%. The mean-reversion of volatility is real but uncertain in timing; in the meantime your model's assumptions about correlation and tail-risk are likely wrong.

3. Smaller when you're on a losing streak. Counterintuitive but mathematically correct. After three losing trades in a row, your prior on "my edge is real" should update downward. The honest response is smaller sizing until you've recalibrated. Most traders do the opposite — they size up to "make back" losses, which compounds the problem.

What you should NOT do

• Don't use leverage to size up to "full Kelly." That just amplifies the failure modes.
• Don't size based on conviction alone without checking your historical accuracy on similar trades.
• Don't ignore correlation. Five 5% positions in different tech stocks is one 25% position in tech.
• Don't average down on losing positions without first asking: "if I had cash and saw this stock fresh today, would I still buy it?" If not, the right action is to sell, not to average down.

Try this in the simulator

For the next 10 trades you place:

1. Write down your conviction level (high/medium/low/speculative).
2. Apply the position sizing rule (5% / 3.5% / 2% / 0.5%).
3. Track outcomes. After 10 trades, compute your actual hit rate and average return.
4. Compare to your conviction labels: were your "high-conviction" trades actually more accurate? By how much?

Most students discover their conviction labels are weakly predictive of outcomes — meaning their sizing should be flatter than they thought. The students who realize this and adjust have a measurable performance edge after one term.

Reading

• John Kelly, "A New Interpretation of Information Rate" (1956, original paper) — free, technical
• Edward Thorp, Beat the Dealer (1962) — first popular treatment, gambling-focused but the framework is identical to investing
• William Poundstone, Fortune's Formula (2005) — Kelly's history applied to markets

This concludes the Beginner / Intermediate / Pro core. Continue to the Advanced track for: options pricing intuitions, factor exposure analysis, and constructing your own risk model.