Expected value versus leverage, drawdown, and Kelly sizing

The same expected value, three different paths

Three betting styles with odds of 35-to-1, 1-to-1, and 1-to-35 can share the same negative expected value of -5.26% while producing very different equity spreads and drawdowns. Expected value is the shared statistic across those styles, not a shared equity path.

Drawdown and distribution at a shared expected value

In a 1,000-trade, 1,000-iteration comparison, maximum drawdown was about -68.6% at 35-to-1 odds, -15.4% at even odds, and -5.6% at 1-to-35 odds.

As payoff odds fell from 35-to-1 to 1-to-35, input series standard deviation dropped from 5.77 to 0.998 to 0.277, matching a narrower output distribution.

To keep maximum drawdown small at a fixed expected value, high payoff odds and low win bias should be avoided.

Win-rate lift needed to reach breakeven

The high-odds case needed only a small win-rate lift, from about 0.026 to 0.028, to reach breakeven, while the 1-to-35 case needed the win rate to rise from 0.921 to 0.972.

At 1-to-35 payoff odds, even a 100% win rate produced only about +2.86% because the payout could not offset the compressed reward.

Expected value does not forecast a fixed-fraction path

Expected value can forecast the linear path of fixed-dollar betting, but it cannot by itself forecast the exponential path of fixed-fraction betting.

For fractional equity betting, a positive expected value is only a first filter; the stake size remains the binding control on survival and growth.

Kelly criterion and risk of ruin

Betting at the full Kelly criterion fraction was judged too aggressive on Risk of ruin grounds. Half Kelly or less was treated as the more conservative bound.

Raising the stake to twice the Kelly fraction approached breakeven, and raising it further toward about 35% of equity drove annualized return toward zero or ruin.