Kelly Criterion

The formula and its derivation

Suppose you have a repeated, independent bet. Each round, you stake a fraction f of your current bankroll. On a win (probability p), you receive net odds b — so a $1 stake returns $b net. On a loss (probability q = 1 − p), you lose the stake. After one round, wealth multiplies by (1 + bf) with probability p or (1 − f) with probability q. After N rounds:

W_N / W_0 = ∏_{i=1}^N M_i ,  where M_i ∈ {1 + bf, 1 − f}

The long-run growth rate is (1/N)·log(W_N/W₀) = (1/N)·Σ log(M_i), which by the law of large numbers converges to E[log(M)] = p·log(1 + bf) + q·log(1 − f) — the Kelly objective. Differentiate with respect to f and set to zero:

d/df [p·log(1+bf) + q·log(1−f)] = pb/(1+bf) − q/(1−f) = 0

Solving gives f* = (bp − q)/b. Equivalently, f* = (p(b+1) − 1)/b — the edge per dollar bet, divided by the loss per dollar bet.

A worked example: 60/40 even-money

You have a bet where p = 0.60 (win probability), q = 0.40, and b = 1 (even-money payoff). Kelly says:

f* = (1 × 0.60 − 0.40) / 1 = 0.20

Bet 20% of bankroll each round. Starting at $1,000, after one round you're at $1,200 (win) or $800 (loss). Expected log-growth per round:

g(0.20) = 0.6·log(1.2) + 0.4·log(0.8) = 0.6·0.182 + 0.4·(−0.223) = 0.020

Two percent per round, compounding. Over 100 rounds, expected bankroll growth ≈ e² ≈ 7.4×. Now compare with betting 10% (under-Kelly) or 40% (over-Kelly, 2× full):

f	Round outcome (win)	Round outcome (loss)	E[log-growth]/round	100-round factor
0.05 (¼ Kelly)	+5.0%	−5.0%	0.0094	≈ 2.55×
0.10 (½ Kelly)	+10.0%	−10.0%	0.0152	≈ 4.59×
0.20 (Kelly)	+20.0%	−20.0%	0.0204	≈ 7.71×
0.30	+30.0%	−30.0%	0.0166	≈ 5.27×
0.40 (2× Kelly)	+40.0%	−40.0%	0.0000	≈ 1.00× (no growth)
0.50 (over)	+50.0%	−50.0%	−0.0203	≈ 0.13× (loses money)
0.80	+80.0%	−80.0%	−0.0959	≈ 6.8 × 10−5 (ruin)

Two things to notice. First, the growth curve is concave with a peak at f = 0.20 exactly. Second, betting more than 2× Kelly turns expected growth negative — you lose money on average despite having a 60% win rate. The 80% bet leads to near-certain ruin.

Kelly vs other position-sizing rules

	Fixed dollar	Fixed fraction (non-Kelly)	Kelly	Half-Kelly	Mean-variance / Merton
Sizing rule	Same $ each bet	Same % each bet	(bp−q)/b each bet	0.5 × Kelly	μ/(γσ²) for power utility γ
Account for edge?	No	Implicit	Yes — edge in numerator	Yes — half-strength	Yes
Long-run growth	Linear (eventually)	Below Kelly	Maximum	~75% of Kelly	Maximum at γ=1
Variance / drawdown	Low	Moderate	High	Moderate	Adjustable via γ
Sensitivity to edge mis-estimation	Low	Low	High — exponential damage	Moderate	Linear in γ
Used by	Conservative retail	Index funds (rebalance to fixed weight)	Pure quant — theoretical	Thorp, professionals	Pension funds, robo-advisors
First-principles foundation	None	Heuristic	E[log W] max (Kelly 1956)	Same, dampened	E[u(W)] for general u

Bell Labs to Las Vegas to Wall Street

John Larry Kelly Jr. was a 33-year-old Bell Labs physicist working on information theory when he derived the formula. The 1956 paper "A New Interpretation of Information Rate" was written as an information-theoretic exercise — Kelly observed that the growth rate of a horse-racing bankroll under optimal sizing equals the mutual information between race odds and inside knowledge, in Shannon's sense. Claude Shannon, his Bell Labs colleague and the founder of information theory, suggested the gambling framing. The paper has 1,800+ citations and seeded both quantitative finance and information theory's gambling-theoretic interpretation.

Kelly died of a stroke in 1965 at age 41 without seeing his formula go mainstream. Edward O. Thorp, an MIT mathematician working on blackjack card-counting, found Kelly's paper and applied it to casino play in his 1962 best-seller Beat the Dealer. Thorp later co-founded Princeton-Newport Partners (1969–1988), the first true statistical-arbitrage hedge fund, returning roughly 19% net annually for two decades using Kelly-sized positions on small-edge trades.

Kelly's formula re-entered academic finance through Mossin (1968), Markowitz (1976), and the consumption-CAPM literature. In continuous time it becomes Merton's optimal-portfolio share (1969): f* = μ/(γσ²) for power utility u(W) = W^1−γ/(1−γ), reducing to Kelly when γ = 1 (log utility). Modern quant funds — Renaissance, Two Sigma, D.E. Shaw — implicitly or explicitly use Kelly-derived sizing on each independent signal, then aggregate.

Continuous-time Kelly (the Merton share)

Suppose returns follow a Brownian motion with drift μ and volatility σ. The continuous-time analogue of Kelly is the Merton share:

f* = μ / σ²    =  Sharpe ratio / σ

where the Sharpe ratio is (μ − r)/σ and r is the risk-free rate. A strategy with Sharpe 1.0 and 20% vol calls for 25% leverage; Sharpe 2.0 same vol calls for 50%. The Kelly objective in continuous time becomes the geometric growth rate μ·f − (σ·f)²/2, maximized at f = μ/σ². The half-variance term is why over-betting hurts — volatility drag eats arithmetic returns.

Variants and extensions

• Fractional Kelly (Thorp 1969). Bet kf* for k ∈ (0,1). Half-Kelly (k=0.5) captures ~75% of growth with half the variance and is far more robust to errors in estimating p and b. The professional default.
• Multi-bet Kelly. When N independent bets are available, solve for the joint optimal allocation. Closed-form when bets are independent (sum of individual Kellys scaled by total available wealth).
• Continuous Kelly with stochastic volatility. When σ is itself random, the Merton share becomes f* = E[μ]/Var[μ·dt + σ·dW], picking up additional precautionary terms.
• Tail-risk Kelly (Vince 1990, MacLean-Thorp-Ziemba 2010). Adjustments for non-Gaussian return distributions; explicit drawdown constraints add risk-off triggers.
• Generalized Kelly under uncertainty. Robust Kelly bets the lower bound of a confidence interval for p — protects against edge mis-estimation explicitly.
• Pure information-theoretic Kelly (Algoet-Cover 1988). The fundamental limit of any betting strategy is the mutual information between side information and outcomes — Kelly bets achieve it.

Real-world applications

• Quant trading. Renaissance Medallion is widely believed to size positions via Kelly-like rules on each independent alpha signal. Reported gross returns of ~66% annually since 1988.
• Sports betting. Bill Benter's Hong Kong horse-racing fund used fractional Kelly to extract an estimated $1 billion over 1984–2006. Haralabos Voulgaris built one of NBA gambling's biggest bankrolls in the 2000s using similar logic.
• Blackjack card-counting. Ed Thorp's original 1962 application. Counters increase bet size proportional to the running count, approximating Kelly on each hand's edge.
• Venture capital portfolio sizing. Some VCs explicitly use Kelly-derived allocation across deals; concentrated portfolios at high-conviction stages reflect the formula's tendency to load up when edge is large.
• Insurance underwriting. Lloyd's syndicates implicitly Kelly-size individual underwriting decisions — when the loss ratio implies an edge, larger exposure is acceptable.
• Cryptocurrency leverage. Perpetual futures traders on Binance, dYdX, GMX use Kelly-style sizing on signal strength; over-leveraging is what kills retail traders during volatility spikes.
• Personal finance heuristic. Stock-allocation rules of thumb ("100 minus age") implicitly approximate fractional Kelly given long-run equity Sharpe ratios and life-cycle risk aversion.

Common pitfalls and critiques

• Estimating p too aggressively. Kelly is brutally sensitive to overestimating edge — a 60% perceived edge that's really 55% leads to 2× Kelly betting and zero long-run growth. Use fractional Kelly, always.
• Treating one big bet as repeated. Kelly's optimality is asymptotic; in finite samples with one bet, expected-utility maximization (with a non-log u) gives different sizing.
• Forgetting correlations across bets. If your N "independent" trades are actually correlated, summed Kelly allocations exceed the joint optimum. De-correlate before summing.
• Ignoring transaction costs. Continuous rebalancing to maintain a fixed Kelly fraction incurs costs that eat the edge for small-edge / high-volatility strategies.
• Confusing Kelly with risk-neutral profit maximization. Kelly grows wealth fastest in the long run, but in any short window expected dollar return is higher under more aggressive sizing — at the cost of much higher variance and ruin probability.
• Ignoring the drawdown distribution. Full Kelly has expected maximum drawdown of roughly 50% over a 100-bet horizon; many funds cannot survive that. Half-Kelly cuts the expected drawdown to ~25% with only ~25% loss of growth.