A positive expected value says the bet is favorable once. It does not say what fraction of wealth to stake. When the same gamble is repeated, wealth is a product, and the quantity that compounds is expected log-wealth. Sizing is the choice of that fraction.
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Lesson close · Special: Quantitative Decision & Markets
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when the edge is positive and . For even money (), .
After a stake , wealth is multiplied by on a win and by on a loss. Expected log-growth, in the natural logarithm, is
Differentiate:
Set the derivative to zero: . Expand, , then collect terms in :
using . Therefore . The second derivative is negative wherever the logs are defined, so the critical point is a maximum.
The numerator is the edge. If it is zero, . Expected profit of staking equals , which rises with whenever the edge is positive. That sum never penalizes losing most of the bankroll, because it adds wealth instead of multiplying it. One round still loses with probability . Over rounds, converges to . A stake with shrinks wealth geometrically even though every round has positive expected profit. Fractional Kelly uses with . It does not maximize ; it shrinks the stake when is only an estimate.
Even money. Let , so and . The edge is and . Then , and over rounds the growth factor is . Half-Kelly, , gives and a factor near . Staking keeps the edge at per unit, but and .
Longer odds. Let , , . The edge per unit is , yet
with . A stake instead gives . Positive edge, negative growth. The denominator stops a large payoff from being read as a large fraction of wealth.
The same stake appears in a card counter who bets more only when the shoe is rich, in a systematic book that sets exposure from an estimated edge and variance, and in a growth-optimal portfolio that maximizes . A quoted "fraction of Kelly" is the constant in .
Interactive · Kelly fraction f*=p−(1−p)/b
f*=0.100 (edge)
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