Kelly Criterion Sizing

Kelly Criterion

The Kelly Criterion is a formula for determining the optimal fraction of capital to wager on a bet (or trade) to maximize the long-term growth rate of wealth.

Formula

Given:

• p = probability of winning
• q = 1 - p = probability of losing
• b = win/loss ratio (how much you win per unit risked)

kelly_fraction = p - q / b

Interpretation

• Positive Kelly: bet this fraction of your capital
• Zero or negative: do not take the bet

Half-Kelly

In practice, many traders use "half-Kelly" — betting half the Kelly fraction — to reduce volatility while sacrificing some growth:

half_kelly = kelly_fraction / 2

Example

If you win 60% of the time (p = 0.6) with a 2:1 payout (b = 2):

kelly = 0.6 - 0.4 / 2 = 0.6 - 0.2 = 0.4

Bet 40% of your capital on each trade.

Drawdown Limits

A maximum drawdown limit caps how much the portfolio can fall from its peak before the strategy stops trading. The drawdown at time t is:

drawdown(t) = (peak - value(t)) / peak

If drawdown exceeds the limit, the position size is set to zero. This is a critical risk control that prevents catastrophic losses — even a strategy with a positive Kelly fraction can experience devastating drawdowns.

Volatility Targeting

Volatility targeting scales position size so the portfolio maintains a constant annualized volatility target. Given recent realized volatility σ_realized and a target σ_target:

vol_scale = sigma_target / sigma_realized

The Kelly fraction (or any position size) is then multiplied by vol_scale. When the market is calm, you trade larger; when volatile, you reduce exposure. This is the most common professional risk management technique and is used by virtually all systematic funds.

Task

Implement:

• kelly_fraction(win_prob, win_loss_ratio) and half_kelly(win_prob, win_loss_ratio)
• max_drawdown(portfolio_values) — returns the maximum drawdown (as a positive fraction) from a list of portfolio values
• vol_target_scale(recent_returns, target_vol) — returns the scaling factor given a list of recent daily returns and an annualized target volatility. Annualize daily std by multiplying by sqrt(252). If realized vol is 0, return 1.0.