TradingCalcKelly Growth-Security Frontier Calculator
RISK/STATS · KELLY FRONTIER · ADVANCED
MacLean-Ziemba-Blazenko fractional-Kelly frontier: probability a strategy compounding at a fraction of full Kelly ever loses a given fraction of its starting capital.
e.g. 50 means "ever falls to half my starting bankroll"
Continuous-compounding (Brownian-motion/lognormal) approximation. Valid for λ in (0, 2] - beyond λ=2 the drift turns negative and ruin becomes certain.
The formula
MacLean, Ziemba & Blazenko (1992): for a strategy compounding at a fraction λ of full Kelly, the probability wealth ever falls to a fraction α of its starting value:
- P = α(2/λ − 1)
Invertible to solve for the largest λ that keeps that probability at or below a target: λ = 2 / (1 + ln(P) / ln(α)). Valid for λ in (0, 2] - at λ=2 the strategy's drift hits exactly zero, and beyond it ruin becomes certain (P=1), not a formula breakdown.
Worked example
Half Kelly (λ=0.5), worried about ever losing half your capital (α=0.5):
- P = 0.5(2/0.5 − 1) = 0.53 = 12.5%
At full Kelly (λ=1) the same α=0.5 gives P = 0.51 = 50%: half-Kelly cuts the chance of ever losing half your bankroll from 1-in-2 to 1-in-8.
Where to go next
This is the continuous-compounding growth-vs-ruin frontier for a strategy sized as a fraction of its own full Kelly, not a one-shot bet size. For sizing a single prediction- market bet directly from your edge, see the prediction market edge sizer, or get the full risk/return picture of your actual return series with the portfolio tearsheet.
Use via API or MCP
This calculation is available as a deterministic API call for bots and AI agents.
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What assumptions and edge cases does this calculator use?
Alpha, the fraction of starting capital you must never fall to, must be strictly between 0 and 1. Give exactly one of the fraction of full Kelly you bet (above 0) or the largest probability you accept (strictly between 0 and 1), never both and never neither. The probability is of ever falling to alpha, not at a fixed date. Not covered: the real edge of a strategy and its estimation error, transaction costs and discrete bets.
What is the Kelly growth-security frontier?
A closed-form result (MacLean, Ziemba & Blazenko, 1992) answering: if a strategy compounds at a fraction lambda of its own full-Kelly bet size, what's the probability it ever falls to a fraction alpha of its starting capital? It quantifies the classic growth-vs-security tradeoff: full Kelly maximizes long-run growth but with genuinely painful drawdown risk along the way.
What does "fraction of full Kelly" mean?
Full Kelly (lambda=1) is the bet size that maximizes long-run expected growth rate. Half Kelly (lambda=0.5) means betting half that size: still growth-positive, but with a much smaller chance of a deep drawdown, at the cost of somewhat slower long-run growth.
Why does the probability not depend on my actual edge or growth rate?
The formula is scale-free: the underlying growth rate cancels out algebraically, leaving only lambda and alpha. This makes the frontier a universal statement about the relationship between fractional-Kelly sizing and drawdown risk, regardless of how big your specific edge is.
What happens at lambda > 1 or lambda >= 2?
Lambda > 1 ("super-Kelly") means betting more aggressively than growth-optimal: still valid, but always a strictly worse point on the frontier than lambda=1. At lambda=2 the strategy's drift hits exactly zero, and beyond that the expected growth rate turns negative, so ruin (falling to any fraction alpha) becomes certain: the calculator reports 100%, not the raw formula's value.
Is this the same as the prediction-market edge sizer?
No. The edge sizer answers "how much should I stake on this one bet, given my edge?" (the standard discrete Kelly formula). This frontier answers a different question: "if I repeatedly compound at some fraction of full Kelly over time, what's my risk of a deep drawdown along the way?"