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Kelly Criterion: History, Formula and Fractional Kelly

By Tally Calculators · Published Oct 27, 2026 · Updated Oct 27, 2026 · 3 min read

The Kelly criterion is a formula for deciding what fraction of your bankroll to stake on a bet or investment when you know, or can estimate, your edge. It maximises the long-run growth rate of your capital and, uniquely, avoids ruin in theory, because you never bet everything.

The formula

f* = (b × p − q) ÷ b, where f* is the fraction of your bankroll to stake, p is the probability of winning, q = 1 − p, and b is the net odds, the profit per unit staked. With decimal odds, b = decimal odds − 1.

If the formula gives a negative number, the bet has negative expected value and Kelly says to stay out.

Origins at Bell Labs

John L. Kelly Jr., a physicist at Bell Labs, published “A New Interpretation of Information Rate” in the Bell System Technical Journal in 1956. He was working on Claude Shannon’s information theory and showed how a gambler with a private signal about a race could grow capital at the highest possible rate. The idea was largely academic until Edward Thorp, a mathematician, used it for blackjack card counting, described in his 1962 book Beat the Dealer, and then for investing through his hedge fund Princeton Newport Partners, which operated from 1969 to 1988. Later research collected in The Kelly Capital Growth Investment Criterion (2011) extended it to portfolios.

Why people use fractional Kelly

Full Kelly is aggressive. It assumes you know your probabilities exactly and accepts large drawdowns: drops of 50% in a bankroll are not unusual even when you are right. Because probability estimates are almost always too optimistic, many practitioners use half Kelly or quarter Kelly. Under standard approximations, half Kelly gives roughly 75% of the maximum growth rate with much lower volatility, and it protects you if your edge is smaller than you think.

Criticisms

The economist Paul Samuelson criticised the approach in a 1971 paper, arguing that maximising long-run growth is not the right goal for everyone, because investors differ in their tolerance for risk. In practice:

Using the calculator

  1. Enter your estimated win probability and the decimal odds.
  2. Choose full, half or quarter Kelly.
  3. If the result is zero, you have no edge according to your own inputs.
  4. Test how sensitive the answer is: if lowering your probability by 3 points wipes out the stake, the edge is thin.

This is a mathematical tool, not betting or investment advice. If gambling is causing you problems, local helplines can provide free, confidential support.

Frequently asked questions

What does a negative Kelly result mean?

You have no positive edge, so the formula says not to bet.

Why use half Kelly?

It sharply reduces volatility and protects you if your probability estimate is too high, at a modest cost in expected growth.

Can Kelly be used for investing?

Yes, in modified forms, but expected returns and risks are much harder to estimate than in a simple bet.

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This article is for general information only and is not financial, medical or professional advice.