← back to writing · December 2025

How I Think About the Kelly Criterion

A useful answer to a narrow question: if you have an edge, how much risk should you take without overbetting it?

I first encountered Kelly as a position-sizing rule: estimate your edge, divide by the payoff, and the formula tells you what fraction of capital to risk. That version is useful, but it hides both the interesting part and the reason I would rarely use the full answer in practice.

The original result came from John L. Kelly Jr. at Bell Labs. His 1956 paper, "A New Interpretation of Information Rate," studied a gambler receiving imperfect information through a noisy channel. The question was not simply whether the gambler had an edge, but how aggressively to use it when wealth compounds over repeated bets.

That distinction matters in trading. A positive expected return does not make every position size sensible. Kelly gives a clean benchmark for the size that maximizes long-run logarithmic growth under a particular model. The assumptions in that sentence do a lot of work, so I want to keep them visible.

The Setup: A Gambler with a Private Wire

Kelly's paper posed a specific problem. Imagine a gambler who receives information about horse races through a noisy private channel — think of an insider tip that's correct, say, 70% of the time. The gambler knows the channel is noisy. The question: how much of their bankroll should they bet on each race to maximize long-term wealth?

A few intuitive policies illustrate the trade-off:

Kelly derived that rule and connected the resulting growth rate to Shannon's information theory under the betting setup in his paper.

The Formula

For a simple bet where you either win or lose, the Kelly criterion says:

The Kelly Fraction
f* = (bp − q) / b
f* = fraction of your bankroll to bet
b = net odds (e.g., b = 1 for even money, b = 2 for 2-to-1)
p = probability of winning
q = 1 − p = probability of losing

For even-money bets (b = 1), this simplifies to:

Even-money Kelly
f* = 2p − 1
If you have a 60% chance of winning an even-money bet, Kelly says bet 20% of your bankroll.
If you have a 55% edge, bet 10%. If you have a 51% edge, bet 2%.
If you have no edge (p = 0.5), bet zero.

This can be stated even more intuitively: f* = edge / odds. Your bet size should be proportional to your edge and inversely proportional to the odds. The bigger your informational advantage, the more you bet. The larger the potential loss, the less you bet.

Why Logarithms? The Math of Compounding

The derivation reveals why Kelly works. The key insight is that wealth grows multiplicatively, not additively.

If you start with bankroll \(W\) and make \(n\) bets, each wagering fraction \(f\), your wealth after \(n\) bets is:

Multiplicative wealth growth
Wn = W0 × (1 + bf)wins × (1 − f)losses

Taking the logarithm converts multiplication to addition:

Log growth rate
G(f) = p · log(1 + bf) + q · log(1 − f)
By the law of large numbers, this converges to your long-run growth rate.
Kelly maximizes G(f) by taking the derivative, setting it to zero, and solving for f.

Setting G'(f) = 0:

Derivation
pb / (1 + bf) − q / (1 − f) = 0
Rearranging: pb(1 − f) = q(1 + bf)
→ pb − pbf = q + qbf → pb − q = fb(p + q) = fb
→ f* = (bp − q) / b

The useful feature is that the growth-rate function G(f) is concave. There is a single maximum. Betting less than f* sacrifices some modeled growth; betting more also lowers growth while increasing drawdowns. In the even-money example below, 2× Kelly brings expected log growth back to zero. The exact boundary depends on the bet.

Having an edge and sizing it well are separate problems. Beyond the growth-optimal size, taking more risk can reduce long-run compound growth.

The Information Theory Connection

The information-theory connection is what makes the original paper more than a position-sizing exercise.

In Kelly's idealized setup, with fair odds matched to the underlying outcomes, the advantage in optimal log-growth from side information is tied to the information carried by the channel. This is a result about that model, not a general identity between bits and investment returns.

The value of side information

Under fair odds and Kelly's repeated-betting assumptions, the increase in optimal expected log growth from observing a signal Y about outcome X is:

ΔG* = I(X;Y)

Here I(X;Y) is mutual information. Channel capacity is the largest value this information rate can attain over the allowed input distributions.

Within that setup:

I like this connection because it separates confidence from information. The formula only responds to the edge supplied to it; it has no way to reward conviction. In markets, of course, estimating that edge is the difficult part.

A Concrete Example

The biased coin

You're offered a game. A coin comes up heads 60% of the time. It pays even money. You start with $1,000. How should you bet?

Kelly says: f* = 2(0.6) − 1 = 0.20. Bet 20% of your current bankroll each round.

If you bet 20% (Kelly): Expected log growth is G = 0.6·log(1.2) + 0.4·log(0.8) ≈ 0.020 per bet. The corresponding long-run typical-growth projection doubles roughly every 35 bets and reaches about $150,000 after 250 bets. This is not the arithmetic expected bankroll.

If you bet 40% (2× Kelly): G = 0.6·log(1.4) + 0.4·log(0.6) ≈ 0.000. Growth rate is zero. You go nowhere on average despite having a 60% edge. Your bankroll just random-walks.

If you bet 10% (half Kelly): G ≈ 0.75 × G(f*). The modeled log-growth rate is lower, but so is the variability. The same typical-growth projection reaches roughly $43,000 after 250 bets.

If you bet 100% (all-in): First tails and you have $0. Game over. Probability of surviving 100 rounds: 0.6100 ≈ 6.5 × 10−23. Effectively zero.

In this particular even-money example, 2× Kelly is a useful warning. The bet still has positive expected profit in a one-period arithmetic sense, but betting 40% of the bankroll produces zero expected log growth. Repetition and compounding change the answer.

This is the difference between additive and multiplicative thinking. A stake that looks acceptable in isolation can be too large for a repeated compounding process.

Kelly in the Continuous World: The Portfolio Connection

Under a simplified continuous-return model, the multi-asset Kelly solution takes a matrix form familiar from mean-variance portfolio theory:

Continuous Kelly (multi-asset)
f* = Σ−1(μ − r·𝟏)
Σ−1 = inverse covariance matrix of asset returns
μ = vector of expected returns
r = risk-free rate
Under these assumptions, this has the same direction as the Markowitz tangency portfolio, with leverage determined by log-growth.

Under the model assumptions, three related formulations line up:

  1. Kelly: "Maximize my long-run geometric growth rate"
  2. Markowitz: "Maximize expected return for a given level of risk"
  3. Log utility: "Maximize my expected logarithmic utility of wealth"

At the corresponding parameterization, they produce the same portfolio weights. That equivalence is useful, but it depends on the return model and does not hold unchanged for arbitrary distributions or constraints.

The Fractional Kelly Tradeoff

Full Kelly maximizes modeled long-run log growth, but it can produce drawdowns that many investors cannot or should not tolerate. The exact drawdown distribution depends on the strategy; Kelly optimality does not make the path comfortable.

This is why Kelly is often used fractionally: take some fraction α of the model's full-Kelly amount. In the simple approximation used here, the trade-off is:

A fractional Kelly approximation

In the continuous-return quadratic model, betting fraction α of the Kelly amount gives:

G(αf*) = α(2 − α) · G(f*)

Half Kelly (α = 0.5): The model retains 75% of full-Kelly growth while return variance scales to roughly one quarter.

Quarter Kelly (α = 0.25): The model retains 43.75% of full-Kelly growth with still less variability.

The practical reasons for fractional Kelly are compelling:

What I Take From It

The formula is compact. The useful habits around it are not:

  1. An edge estimate comes before a sizing rule. Kelly cannot rescue an inaccurate probability model.
  2. Selection and sizing are different skills. A good trade can still be a bad portfolio decision at the wrong size.
  3. Overbetting is costly. Near the optimum, giving up some modeled growth can buy a meaningful reduction in risk; going above it moves in the opposite direction.
  4. Compounding is multiplicative. Arithmetic averages can obscure the effect of volatility and drawdowns on terminal wealth.

I still like Kelly because it turns a vague instruction—"bet more when the edge is better"—into a precise optimization problem. I trust the full numerical answer much less than I trust the questions it forces me to ask.

Further Reading