Kelly Criterion: Advanced and Practical Limits

The Kelly criterion, derived by John Kelly in 1956, gives the bet size that maximizes the long-run geometric growth rate of capital. It is mathematically elegant and practically dangerous — which is why almost no one trades at full Kelly.

The basic formula

For a bet with win probability $p$, loss probability $q = 1-p$, and net odds $b$ (profit per unit risked on a win), the optimal fraction is:

$$f^* = \frac{bp - q}{b} = \frac{p(b+1) - 1}{b}$$

At $f^*$, the expected logarithm of wealth is maximized, and capital grows at the highest possible long-run rate.

Multi-asset Kelly and the growth rate

For multiple simultaneous opportunities, Kelly becomes a portfolio optimization problem maximizing:

$$g = \sum_i p_i \ln(1 + \sum_j f_j b_{ij})$$

subject to $\sum f_j \leq 1$. Solutions require numerical methods and the covariance structure of outcomes. The connection to Markowitz is deep: log-optimal portfolios lie on the efficient frontier in log-return space.

Why full Kelly is dangerous

Parameter uncertainty: $f^*$ depends on $p$ and $b$, which are estimated with error. Overestimating edge leads to over-betting, which under geometric growth guarantees ruin. The asymmetry is brutal — betting double Kelly produces zero growth rate and large drawdowns.

Drawdown severity: At full Kelly, the probability of a drawdown to fraction $x$ of peak equity is approximately $x$. A 50% drawdown has roughly a 50% probability over a long horizon. Most traders cannot endure that.

Variance drag: Geometric growth is always less than arithmetic mean return by approximately half the variance:

$$g \approx \mu - \frac{\sigma^2}{2}$$

Full Kelly maximizes the variance penalty by accepting high volatility for marginal growth gains.

Non-stationarity: Real edges are not stationary. A backtested $p$ decays, regimes shift, and the edge that justified Kelly yesterday may be gone today.

Fractional Kelly

The standard practical response is to bet a fraction $c \cdot f^*$ with $c \in [0.25, 0.5]$. This gives up a modest amount of expected growth for a large reduction in variance and drawdown depth. Half Kelly retains about 75% of the growth rate with half the variance.

Practical limits and refinements

• Cap individual position sizes regardless of Kelly output to avoid concentration.
• Use Bayesian shrinkage on edge estimates to pull them toward conservative values.
• Re-estimate edge continuously and reduce size when recent realized edge falls below estimate.
• Treat Kelly as a ceiling, not a target. It tells you when you are definitely betting too much, not exactly how much to bet.

The Kelly criterion's real value is conceptual: growth is geometric, edge and risk interact multiplicatively, and over-betting is catastrophic while under-betting is merely suboptimal. Conservative application of an honest edge beats aggressive application of a wishful one.