The Kelly Criterion

Kelly sizing maximizes the long-run growth rate of wealth by betting a fraction proportional to your edge.

1. The idea

The Kelly criterion answers 'what fraction of my bankroll should I bet' for a repeated favorable bet, choosing the fraction that maximizes the expected long-run exponential growth rate of wealth, $E [lo g (wealth)]$ , rather than expected wealth itself.

For a simple even-money bet won with probability $p$ and lost with probability $q = 1 - p$ , the optimal fraction is $f^{*} = 2 p - 1$ , the 'edge'. More generally, for a bet that pays $b$ to 1 on a win and loses the full stake on a loss, $f^{*} = (b p - q) / b$ .

Betting more than Kelly increases risk of ruin and lowers long-run growth; betting less than Kelly ('fractional Kelly') sacrifices some growth for lower variance — a common practical compromise.

2. Why it matters

Kelly sizing is a favorite interview topic because it combines expected-value reasoning with a non-obvious log-utility argument, and testing the even-odds formula $2 p - 1$ is extremely common.

3. Core mathematics

Even-odds Kelly fraction

f^{*} = 2 p - 1

Bet doubles on win, loses stake on loss (b=1)

General Kelly fraction

f^{*} = \frac{b p - q}{b} = p - \frac{q}{b}

$b$ = net odds received on a win (b to 1), $q = 1 - p$

Growth rate maximized

g (f) = p ln (1 + b f) + q ln (1 - f)

4. How to recognize it

Think of this when…

Asks for the optimal fraction of bankroll to bet given a win probability and payout
Mentions maximizing long-run growth rate or avoiding ruin
Gives even-money or b-to-1 odds with a favorable edge

5. Worked example

Worked example

A bet wins with probability 0.6, paying even money (win $1 per $1 bet, lose $1 per $1 bet on a loss). What fraction of bankroll should be bet under Kelly, and what if odds are 2-to-1 instead (b=2)?

1
Even money means b=1, so use $f^{*} = 2 p - 1$ .
$f^{*} = 2 (0.6) - 1 = 0.2$
2
For b=2, use the general formula with q=0.4.
$f^{*} = 0.6 - \frac{0.4}{2} = 0.6 - 0.2 = 0.4$
Answer: Bet 20% of bankroll at even odds; bet 40% of bankroll at 2-to-1 odds.

6. Common mistakes

Applying $f^{*} = 2 p - 1$ to bets that aren't even-money — this formula only holds for b=1
Confusing Kelly (maximizes long-run log growth) with maximizing expected value (which implies betting everything if edge is positive)
Forgetting Kelly can be negative, meaning don't take the bet (or bet the other side) when there's no edge

7. Quick check

At even odds with p = 0.5, the Kelly fraction is:

Betting more than the Kelly fraction:

8. Practice

6 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Jensen's Inequality

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