Lesson 2 of 7
The Kelly criterion and its dangers
Compute the Kelly fraction for a simple bet, and see why investors who use it stake less than the formula allows.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
Who writes thisEditorial processLinkedIn (opens in a new tab)X (opens in a new tab)
In one minute
- The Kelly criterion is a formula for the fraction of a bankroll that grows it fastest over many repeated bets, given the true odds and the true probability of winning.
- For a 60 percent, even money bet, the Kelly fraction is 20 percent of the bankroll. Staking more grows the bankroll slower, and staking twice as much can make growth negative.
- Markets never hand you a probability this exact, which is why investors who use Kelly at all typically stake a fraction of what the formula names.
The Kelly criterion answers one narrow question: given a bet with a known edge, what fraction of your bankroll, the money you bet with, grows it fastest if you could repeat the bet many times. It comes from information theory, not from investing, and its answer is a ceiling, not a target.
Log growth per bet at five stakes, on a 60 percent even money bet
- Stake 5%0.88%
- Stake 10%1.50%
- Stake 20% · the Kelly fraction2.01%
- Stake 30%1.47%
- Stake 40%−0.24%
The formula, for one simple case
For a bet that pays even money, win or lose the same amount, the Kelly fraction is twice your probability of winning, minus one. A 60 percent chance of winning gives a Kelly fraction of 0.2, or 20 percent of the bankroll: (2 times 0.6) minus 1 is 0.2. Below even odds, the formula turns negative: it stakes nothing.
Why growth peaks, then turns negative
The five stakes above are staked fractions of the same bankroll on the same bet, each one repeated many times. Growth rises from a 5 percent stake to a 10 percent stake, peaks exactly at the 20 percent Kelly fraction, then falls, and by a 40 percent stake it has gone negative. A bet with a real edge can still shrink your bankroll over time if you stake enough of it on each repetition.
Why investors stake less than Kelly says
A coin that lands heads 60 percent of the time is a known, fixed probability. A stock's odds of a good outcome are not measured that precisely by anyone. Kelly's formula takes the probability as given, and a probability that is even slightly too high produces a fraction that overstates what should be staked. Staking a fraction of Kelly gives up little. In the chart, half Kelly (10 percent) keeps 1.50 of the 2.01 percent growth, three quarters of it, while risking half as much on each bet.
Check yourself
A bet pays even money and wins 60 percent of the time. What is its Kelly fraction?
- 10 percent
- 20 percent
- 60 percent
Show the answer
Answer: B.
(2 times 0.6) minus 1 is 0.2, or 20 percent of the bankroll. The formula uses the edge over even odds, not the win probability itself.In the example, what happens to growth if you stake twice the Kelly fraction?
- It doubles
- It can fall, and even turn negative, despite the same edge
- It stays the same
Show the answer
Answer: B.
Staking 40 percent instead of 20 percent, on the same 60 percent bet, turns growth from a positive 2.01 percent to a negative 0.24 percent per bet.Why do investors who use the Kelly criterion typically stake less than it names?
- Because the formula is mathematically wrong
- Because the probability it needs is never known as exactly in markets as it is in a coin bet
- Because smaller stakes are always more profitable
Show the answer
Answer: B.
The formula assumes a known, fixed probability. An investor's estimate of an edge is never that precise, so a fraction of the named stake protects against the estimate being too high.Sources
- Kelly, "A New Interpretation of Information Rate", Bell System Technical Journal, 1956.
- Poundstone, Fortune's Formula, 2005, on the history of the formula from Bell Labs to gambling and markets.
What this leaves out
- The bet, the 60 percent probability and every growth figure here are illustrative, chosen so the arithmetic can be checked by hand.
- The formula assumes an even money payout and a probability you already know. A stock has neither stated to you in advance.
- This is not a sizing rule for any position. It shows what over-betting a known edge does to growth over time.
- Nothing here is advice about any security or any bet.