Kelly Criterion Derivation

Medium4–5 hrsKelly CriterionExpected LogGeometric GrowthOptimization

Overview

The Kelly Criterion answers the question: given a bet with positive edge, what fraction of your bankroll maximizes long-run growth? The answer — maximize the expected log of wealth — leads to a clean formula: f* = (pb - q)/b. Kelly is universally used at prop trading firms and hedge funds for position sizing. The critical insight is that over many bets, bankroll growth is MULTIPLICATIVE — so you maximize the geometric mean, not the arithmetic mean. Maximizing E[wealth] (not log) leads to overbetting and eventual ruin.

How to Recognize

→'What fraction of your bankroll should you bet?'
→You have a positive-edge bet and need to size it
→'How do you avoid ruin while maximizing growth?'
→Multi-round betting/investment with reinvestment
→Any question about balancing risk and return over time

Step-by-Step Approach

1.Write bankroll after n rounds as W₀ × (1+bf)^{wins} × (1-f)^{losses}
2.Maximize log-growth rate: G(f) = p·log(1+bf) + q·log(1-f)
3.Differentiate with respect to f, set to zero
4.Solve: f* = (pb - q)/b = p - q/b
5.Check: G(f*) > 0 iff pb > q (positive edge condition)

Key Formulas

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

— Kelly fraction

f^* = \frac{\text{edge}}{\text{odds}} = \frac{pb-q}{b}

G(f^*) = p\ln(1+bf^*) + q\ln(1-f^*)

— growth rate at Kelly

\text{Even odds }(b=1): f^* = 2p-1 = p-q

(net edge)

Worked Examples

0Quick Example

Even odds, p=0.55: f*=2(0.55)−1=0.10. Bet 10%. Growth rate: 0.55ln(1.1)+0.45ln(0.9)≈0.55(0.095)+0.45(−0.105)≈0.0051 per round (0.51%/bet).

Problem

You have a biased coin: P(heads) = 0.6. You bet on heads at even odds (win what you wager). What fraction of your bankroll should you bet each round? What is the long-run growth rate?

Solution

p = 0.6, q = 0.4, b = 1 (win $b for every$ 1 wagered).

f* = p - q/b = 0.6 - 0.4/1 = 0.2. Bet 20% of bankroll each round.

Growth rate per round: G(0.2) = 0.6·ln(1.2) + 0.4·ln(0.8).

= 0.6(0.1823) + 0.4(−0.2231) = 0.1094 − 0.0892 = 0.0202.

≈ 2% growth per round. After 100 rounds: W₀ × e^{100×0.0202} ≈ W₀ × e^2.02 ≈ 7.5× initial wealth.

Compare: betting 40% (double Kelly) → growth rate ≈ 0 (breakeven). Betting 60%: growth NEGATIVE → ruin.

Answer

f* = 20%. Growth rate ≈ 2%/round. After 1000 rounds: wealth grows ~400×. At f=40% (double Kelly): no long-run growth. At f>50%: bankrupt despite edge.

Common Mistakes

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Maximizing E[wealth] instead of E[log(wealth)]. Maximizing E[W] suggests betting 100% (if positive edge), but this leads to ruin almost surely. You must maximize log for long-run growth.
!
Forgetting that f* = 0 when edge = 0. If pb = q, the expected log-growth at any f > 0 is negative. Never bet on a fair game.
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Using Kelly on correlated bets. The formula assumes independent bets. With correlated positions, the multi-dimensional Kelly (mean-variance optimization) applies.
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Ignoring estimation error. If your edge estimate is wrong, full Kelly can destroy wealth. Most practitioners use half-Kelly or quarter-Kelly as a safety margin.

Practice Problems

Click "Show Answer" to reveal

You have a bet: win $3 for every$ 1 risked with P(win) = 0.35. (a) Is the edge positive? (b) What is f*? (c) At f = 2f*, what happens to long-run growth?

Hint: Edge = pb - q = 0.35(3) - 0.65. For (c): at double Kelly, log-growth rate ≈ 0 (breakeven). Beyond double Kelly: negative growth.

A market maker quotes a 2-tick wide spread on an asset. Their model says 60% of trades they facilitate are 'uninformed' (return to mid within 10 min) and 40% are 'informed' (price moves adversely). If uninformed trades profit $0.01 and informed trades lose$ 0.02, what fraction of the market maker's capital should they deploy per trade?

Hint: Frame as a Kelly bet: p=0.6, b = 0.01 (win), q = 0.4, loss = 0.02. Adjust the formula for asymmetric payoffs.

Back to Study Plan

Sanity Checking Calculations Fractional Kelly & Risk of Ruin