Kelly Criterion & Bet Sizing

Medium2–3 hrsKellyBet SizingLog UtilityEdge

Overview

Kelly answers the question: given an edge, how much should you bet? The naive answer (bet everything!) is wrong — if you ever bet 100% and lose, you're ruined. Kelly maximizes the long-run compound growth rate. The elegant insight: maximize E[log(wealth)] rather than E[wealth]. Over-betting beyond Kelly actually reduces long-run growth. This is directly used in trading for position sizing.

How to Recognize

→'What fraction of your bankroll should you bet?'
→Repeated betting with known edge and fixed odds
→Questions about long-run growth vs. short-run expected value
→Sizing a position in a strategy with known win rate

Step-by-Step Approach

1.Kelly maximizes E[log(wealth)], NOT E[wealth]
2.Binary bet (win b:1 with prob p, lose stake with prob q=1−p): f* = p − q/b
3.Equivalently: f* = edge / odds = (pb − q) / b
4.Half-Kelly: bet f*/2 for the same expected growth rate with much less variance
5.Over-betting is catastrophically worse than under-betting (ruin risk)

Key Formulas

01Kelly fraction:

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

02For 1:1 bets:

f^* = 2p - 1

(e.g., 60% win → bet 20%)

E[\log(W_n)/n] \to p\log(1+f\cdot b) + q\log(1-f)

— maximize this

04Growth rate =

p \ln(1+fb) + q \ln(1-f)

, maximized at

f = f^*

Worked Examples

0Quick Example

Biased coin 60% heads, pays 1:1. Kelly: f* = 0.6 − 0.4 = 20% of bankroll per flip.

Problem

You have a biased coin: heads 55%, tails 45%. You can bet any fraction of your bankroll. The bet pays 1:1. What is the Kelly-optimal fraction?

Solution

p = 0.55, q = 0.45, b = 1 (win 1 unit for every 1 unit bet).

Kelly formula: f* = p − q/b = 0.55 − 0.45/1 = 0.55 − 0.45 = 0.10.

Bet 10% of your bankroll each flip.

Intuition check: if p = 0.5 (fair coin), f* = 0.5 − 0.5 = 0. Don't bet on a fair game!

If p = 1 (certain win), f* = 1 − 0 = 1. Bet everything on a sure thing.

Answer

$f^* = 10\%$ . With 55% edge on a 1:1 game, bet 10% of bankroll each time.

Common Mistakes

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Applying Kelly to maximize single-bet EV. Kelly maximizes GROWTH RATE (log utility), not EV.
!
Betting full Kelly with estimation uncertainty in p. In practice, use half-Kelly to account for errors in probability estimates.
!
Forgetting that Kelly applies per-bet given current bankroll. Your bet size changes after each outcome.
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Not checking that EV > 0 before applying Kelly. f* < 0 means don't bet.

Practice Problems

Click "Show Answer" to reveal

A strategy wins 70% of the time, pays 1:1. What is the Kelly fraction?

Hint: f* = p − q/b with p=0.7, q=0.3, b=1.

A bet pays 10:1. What is the minimum probability of winning needed to have Kelly > 0?

Hint: Set f* = p − q/10 > 0, with q = 1−p.

You're a market maker quoting a security. Your edge per trade is $0.02 on a$ 10 security (0.2%). How does Kelly apply to position sizing?

Hint: Think of each trade as a bet with edge = 0.2%. What limits your position size?

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