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Kelly criterion: the fraction that maximises growth

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Start with the idea

Kelly maximises expected log wealth, so it penalises deep losses more heavily than arithmetic expectancy does. Its apparent precision depends on knowing probabilities and payoffs that a trader can only estimate.

Symbols, units & horizon

p: win probability
q=1−p: loss probability
b: net payoff per unit staked on a win
f: fraction of bankroll staked
g(f): expected log wealth growth per bet
f*: unconstrained optimum
ln: natural log
g′ and g″: first and second derivatives
Domain: 1+bf>0 and 1−f>0

When and why to use this

Use it as a sensitivity exercise for growth and overbetting. A practical risk policy also needs leverage, concentration, drawdown and liquidity limits.

Betting a fixed fraction $f$ of equity on a repeated favourable bet, the long-run growth rate of wealth is the expected log return. Kelly is the $f$ that maximises it. For a bet that wins $b$ per unit risked with probability $p$ :

g (f) = p \ln (1 + b f) + q \ln (1 - f), f^{*} = \frac{b p - q}{b}

Differentiation + small-return approximation

Differentiate log growth and isolate the stake

Wealth multiplies by 1+bf on a win and 1−f on a loss. Expected log growth is $g = p \ln (1 + b f) + q \ln (1 - f)$ , with q=1−p.
Set the derivative to zero: $\frac{p b}{(1 + b f)} - \frac{q}{(1 - f)} = 0$ . Cross-multiply: $p b (1 - f) = q (1 + b f)$ .
Expand and collect f: $p b - q = b f (p + q) = b f$ , so $f^{*} = \frac{(b p - q)}{b}$ . The second derivative is negative in the feasible domain, confirming a maximum.
For small continuous returns, expand $\ln (1 + f R) \approx f R - \frac{f^{2} R^{2}}{2}$ . Maximising the expectation gives approximately $f^{*} = \frac{μ}{E [R^{2}]} \approx \frac{μ}{σ^{2}}$ when the squared mean is negligible.

Work it by hand

p=.55, b=1 gives f*=.10. At half that fraction, exact g=.55 ln1.05+.45 ln.95≈.003752 per bet.

With $p = 0.55$ , $b = 1$ : $f^{*} = \frac{(0.55 - 0.45)}{1} = 10 %$ of equity per bet. Bet less and you grow slower; bet more and you also grow slower — and near twice Kelly the small-return quadratic approximation predicts zero growth. The exact break-even depends on the payoff distribution. Sufficiently excessive leverage can make expected log growth negative.

For continuous outcomes (returns rather than win/lose) the analogue is $f^{*} \approx \frac{μ}{σ^{2}}$ — the Sharpe ratio divided by volatility. Same message: size scales with edge and inversely with variance.

Strategy: 40% win rate, wins are 2.5× losses. Kelly fraction of equity per trade?

$f^{*} = \frac{(2.5 \times 0.4 - 0.6)}{2.5} = \frac{(1.0 - 0.6)}{2.5} = 0.16$ . Half Kelly is 8%. Note that "risk 2% per trade" — the retail default — is roughly ⅛ Kelly here, which is conservative but not crazy given estimation error.

Why does betting more than Kelly reduce long-run growth even though each bet has positive expected value?

Growth compounds multiplicatively, so what matters is the expected log return, not the expected return. Log is concave: a −50% then +100% sequence has positive arithmetic mean and zero geometric growth. Larger fractions increase variance faster than they increase mean, and variance drags geometric growth by roughly $\frac{σ^{2}}{2}$ . Past Kelly the drag outruns the edge.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from math import log1p

def log_growth(fraction, win_probability, net_win_multiple):
    p, b, f = win_probability, net_win_multiple, fraction
    if not 0 <= p <= 1 or b <= 0 or not -1/b < f < 1:
        raise ValueError("Invalid probability, payoff, or wealth-factor domain")
    return p*log1p(b*f)+(1-p)*log1p(-f)

def binary_kelly(p, b):
    if not 0 <= p <= 1 or b <= 0:
        raise ValueError("Require probability in [0,1] and b > 0")
    return (b*p-(1-p))/b  # before constraints or fractional Kelly

print(binary_kelly(.55, 1), log_growth(.1, .55, 1))

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