The growth rate peak

Having a positive expected value is not enough to justify an arbitrary bet size. Overbetting a favorable bet reduces expected log growth and increases drawdown risk; sufficiently large stakes can make expected log growth negative. For repeated independent bets with fixed 0 < p < 1, net payout b > 0 and no borrowing, the Kelly criterion identifies the fraction f = p - q/b that maximizes expected log growth (subject to 0 <= f < 1), where p is win probability, q = 1 - p is loss probability, and b is the net payout per dollar staked. Because full Kelly assumes exact probabilities and carries high drawdown risk, quantitative traders routinely use fractional Kelly (such as half Kelly) to reduce exposure to model error and drawdowns; fractional Kelly does not guarantee safety.

Words you need

Expected value (EV): The average result if you could repeat the bet many times: each outcome times its probability, added up.e.g. A die pays its face in dollars: EV = 3.5.
Kelly criterion: A rule for what fraction of your money to bet on a favourable bet so your wealth grows fastest without going broke.
Bankroll: The total money you have available to bet or trade with.

Why it works

For this repeated binary-bet model, maximizing expected log wealth maximizes asymptotic compound growth. A stake fraction below 1 avoids exact bankruptcy in the ideal model, but can still produce severe drawdowns.

f* = (bp - q) / b = p - (1 - p) / b

Spot it: You have identified a positive-EV trade or bet and need to allocate capital.

Technique reference and sources

Picture compound wealth growth plotted against bet fraction. At f = 0, growth is zero. As bet fraction rises, compound growth climbs to a peak at the full Kelly fraction f*. Past f*, risk outpaces reward and growth declines: overbetting first lowers growth while it stays positive, and only further out does growth turn negative. Twice Kelly is an approximate zero-growth boundary for small edges, not an exact crossing. Calculate expected log growth for the actual probabilities and payouts: g(f) = p ln(1 + bf) + (1 - p) ln(1 - f). A fixed fraction below 1 never hits exactly zero in this idealized model; wealth after m losses is W0(1 - f)^m > 0, while a practical minimum-bankroll threshold can still be crossed. Betting everything (f = 1) does bankrupt you on a loss.

When this method fits

You have identified a positive-EV trade or bet and need to allocate capital.
An interview question asks how to size positions given win rate and odds.
You need to explain why practitioners trade fractional Kelly instead of full Kelly.
You want to evaluate the mathematical risk of overbetting and drawdown.

Identify probabilities. Find win probability p and loss probability q = 1 - p.
Determine net payout ratio. Find net profit b per dollar staked on a win (b = DecimalOdds - 1).
Calculate Kelly fraction. Compute f = p - q/b. If f <= 0, do not bet.
Scale to fractional Kelly. Multiply f by a fraction like 0.5 to buffer against estimation error and lower drawdown.

Even-money wager (b = 1): When payout is 1 to 1, Kelly fraction simplifies to edge: f = p - q = 2p - 1.

p = 0.55: 2(0.55) - 1 = 0.10 (10%)
p = 0.60: 2(0.60) - 1 = 0.20 (20%)

Expected edge divided by net odds: Compute expected profit per dollar staked (p x b - q), then divide by b. Exact: p = 0.5, b = 2 gives edge 0.5 x 2 - 0.5 = 0.5, so Kelly is 0.5 / 2 = 0.25.

p = 0.5, b = 2: edge 0.5, Kelly 0.25 (25%)
p = 0.6, b = 1: edge 0.2, Kelly 0.2 (20%)

Asymmetric payout (b = 2): When net payout is 2 to 1, f = p - q/2.

p = 0.50, b = 2: 0.50 - 0.50 / 2 = 0.25 (25%)
p = 0.40, b = 3: 0.40 - 0.60 / 3 = 0.20 (20%)

Fractional Kelly safety: For small edges, half Kelly retains approximately 75% of optimal expected log growth. Halving the fraction exactly halves one-period simple-return standard deviation and quarters its variance under the same payoff distribution.

Full 0.20 scaled by 0.5 gives 0.10
Full 0.24 scaled by 0.5 gives 0.12

Common traps

Betting an arbitrary large fraction (such as 50% or 100%) simply because a bet has positive EV. Positive EV describes average profit per stake, not a safe stake size. With p = 0.6 and b = 1, Kelly is 0.2, but f = 0.3 still has positive expected log growth, about 0.01475, below the optimum. Larger fractions can have negative growth. Fractions below 1 avoid exact bankruptcy in the ideal model, but not severe drawdowns or practical ruin.

Using gross decimal odds instead of net payout b in the Kelly formula. The parameter b is net profit per dollar staked: b = DecimalOdds - 1. For decimal odds 3.0, b is 2.0.

Betting more than twice the Kelly fraction (2f). Twice Kelly is an approximate zero-growth boundary for small edges, not an exact crossing. Overbetting first lowers growth while it stays positive; further out, negative expected log growth grinds wealth toward zero. Always calculate g(f) for the actual probabilities and payouts.

Another example: Half-Kelly sizing to manage estimation risk

Full Kelly fraction: f = 0.2
Half-Kelly multiplier: 0.5
Scaled fraction: 0.5 x 0.2 = 0.1
Wager 10% of bankroll to cut drawdown volatility in half

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

J. L. Kelly, Jr., A New Interpretation of Information Rate (1956)
Foundational paper for the Kelly criterion optimizing compound wealth growth. Lesson framing and calculations are original.

Edward O. Thorp, The Kelly Criterion in Blackjack Sports Betting and the Stock Market (2006)
Discussion of full vs fractional Kelly, parameter uncertainty, and drawdown risk. Lesson examples and questions are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.