Expected Value (EV), Kelly Criterion & 1,000-Run Monte Carlo Simulator (2026)

Published by John L. Kelly Jr. at Bell Labs in 1956, the formula maximizes the expected logarithm of wealth E[ln(W_n)] over repeated independent trials. Here, 'b' is the net decimal odds won per unit staked (Decimal Odds − 1), 'p' is the true probability of winning, and 'q = 1 − p' is the probability of losing. Equivalently, f* = Edge / Net Odds.

Full Kelly assumes you know your exact true win probability 'p' with zero estimation error and still carries a 1-in-3 (33.3%) chance of halving your bankroll before doubling it! Using Half-Kelly (0.5 × f*) sacrifices only 25% of the theoretical compound growth rate while cutting portfolio variance in half and drastically reducing drawdown risk from model error.

Because compound returns multiply rather than add, volatility drag (approximately ½ f² σ²) grows quadratically with bet size while expected return grows only linearly. At exactly 2× Full Kelly, your long-run geometric growth rate drops to 0%, and above 2× Full Kelly your median bankroll converges toward zero despite having a positive mathematical edge!

For Decimal odds (D), Implied Probability = 1 / D (so 2.50 odds = 1 / 2.50 = 40.0%). For negative American odds (−X), Implied Probability = X / (X + 100) (so −110 = 110 / 210 = 52.38%). For positive American odds (+Y), Implied Probability = 100 / (Y + 100) (so +150 = 100 / 250 = 40.0%).

Expected Value is the probability-weighted average profit per unit risked: EV = (p × NetProfit) − ((1 − p) × Stake). A +5.0% EV wager on a $100 stake means that across thousands of identical trials, the position yields an average net profit of +$5.00 per trial.