Fractional Kelly & Risk of Ruin

Hard4–5 hrsFractional KellyRisk of RuinDrawdownUncertainty

Overview

Full Kelly is theoretically optimal but practically dangerous. The reason: you never know your edge μ precisely, and overestimating μ leads to overbetting. If your true edge is 0 and you bet as if it's positive, Kelly says bet a lot — and you go bankrupt. The solution is fractional Kelly: bet a fraction λ of f*. At λ=0.5 (half Kelly), you give up 25% of growth rate but cut variance by 75%. The practical rule: if you're very confident in your edge, use 50% Kelly. If uncertain, use 25% or less.

How to Recognize

→You want Kelly but are uncertain about your edge estimate
→'What is P(bankroll falls below X% of initial?)'
→Risk management: how much cushion do you need?
→Comparing full Kelly vs fractional Kelly over real horizons
→Any question about ruin probability with positive edge

Step-by-Step Approach

1.Full Kelly maximizes long-run growth but has high volatility (50% drawdowns expected)
2.Fractional Kelly (fraction f of Kelly): reduces volatility at cost of slower growth
3.At half Kelly: growth rate = 3/4 of full Kelly, variance = 1/4 of full Kelly
4.Risk of ruin formula: P(ruin before N × bankroll) for a continuous Kelly process
5.In practice: use 25–50% Kelly to account for parameter estimation error

Key Formulas

G(\lambda f^*) = \lambda G(f^*) - \frac{\lambda^2 - \lambda}{2}\sigma^2

— growth at fraction

\lambda

of Kelly

\text{Half Kelly}: G = \frac{3}{4}G(f^*),\; \sigma = \frac{1}{2}\sigma(f^*)

P(\text{ruin to level }\epsilon) \approx \epsilon^{2\mu/\sigma^2}

— continuous Kelly ruin probability

\text{Max expected drawdown} \approx \frac{1}{2} \cdot \ln(1/\alpha)

at level

\alpha

Worked Examples

0Quick Example

Full Kelly: expected growth 2%/round but σ=10%/round. Half Kelly: growth 1.5%/round, σ=5%/round. After 100 rounds: full Kelly wealth ~7.5×, half Kelly ~4.5× — but half Kelly has far lower probability of 50%+ drawdown.

Problem

A strategy has μ = 0.5%/day and σ = 2%/day. Compare full Kelly, half Kelly, and quarter Kelly over 500 days.

Solution

Full Kelly: f* = μ/σ² = 0.005/0.0004 = 12.5× leverage. G(f*) = μ²/(2σ²) = (0.005)²/(2×0.0004) = 0.03125/day.

After 500 days: W = W₀ × e^{500 × 0.03125} = W₀ × e^{15.6} ≈ W₀ × 6×10^6. Enormous growth — but enormous variance too.

Half Kelly: λ=0.5. f = 6.25×. G = (3/4)G* ≈ 0.0234/day. After 500 days: W ≈ W₀ × e^{11.7} ≈ W₀ × 120,000.

Quarter Kelly: λ=0.25. f = 3.125×. G = 0.75×(1-0.5)×G* ≈ 0.0117/day... Actually: G(λf*) = λ(2-λ)/2 × μ²/σ² = 0.25×1.75/2 × G* ≈ 0.219G*.

In practice: the 6×10^6 full Kelly gain assumes perfect μ and σ knowledge. Any estimation error makes half/quarter Kelly far more robust.

Answer

Full Kelly: astronomical in theory, fragile in practice. Half Kelly: 75% of growth, 50% of leverage, far better tail risk. Use half Kelly as default when edge is well-estimated, quarter Kelly when uncertain.

Common Mistakes

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Thinking fractional Kelly is 'playing it safe' in a bad way. Half Kelly still outperforms any fixed-fractional strategy that doesn't use Kelly in the long run.
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Not accounting for correlation between bets. If you run 10 correlated strategies and Kelly-size each independently, total leverage is 10× too high.
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Ignoring costs (spreads, commissions, funding rates). These reduce μ and therefore reduce f* significantly. Always use net-of-costs edge.

Practice Problems

Click "Show Answer" to reveal

A portfolio has Sharpe ratio = 1.0 (annualized). Daily expected return = μ, daily vol = σ. What is the Kelly leverage? What leverage does half-Kelly imply?

Hint: Sharpe = μ√252/σ = 1 → μ/σ = 1/√252. Kelly = μ/σ² = (μ/σ)/σ.

Back to Study Plan

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