Ruin odds, the Kelly fraction and optimal f as size filters

Protect remaining equity first

Protecting remaining equity is treated as the first system objective because a depleted account cannot continue. Seeking profit is framed as a secondary business objective.

Loss potential is named before entry. The capital at risk on the next trade, and the maximum drawdown that would force the system to be replaced, are decided in advance rather than left to chance.

Bound the next loss before entry

Two pre-trade controls bound loss: a position size that does not consume a large share of equity or margin, and a stop-loss whose dollar risk is only a small share of account value. The stop-loss is a pre-set exit that caps adverse move size so the dollar loss on one trade stays a small share of equity.

Convert the next stake into ruin odds

Risk of ruin is the chance that a chosen stake, relative to remaining equity, will exhaust the account after a losing streak. It is the link between the fraction of capital risked and the chance of losing the entire stake. With a 10-unit bank and 5 units risked, or 50 percent of capital, a favorable two-to-one game can still exhaust the bank after a short losing streak.

In the coin-toss illustration, ruin probability equals 1 divided by (1 plus expectancy) raised to the number of risk units. Expectancy is the average edge per trial used in the ruin and optimal-fraction formulas, and it is 0.5 in the even coin-toss case. The risk-unit count is starting capital divided by the amount risked on the next trial. A 50 percent stake maps to a 44 percent chance of failure.

Read the peak fraction as a ceiling

The illustrated optimal f, written as optimal risk percentage equal to expectancy divided by the risk-to-reward ratio, is 25 percent when expectancy is 0.5 and the payoff ratio is 2. That same fraction is the Kelly-criterion size for those inputs.

Optimal f is the risk percentage at which illustrated growth peaks for a given payoff. Larger fractions then cut profit and raise ruin odds. The Kelly criterion is a pre-trade equity fraction implied by expectancy and payoff, used as an upper bound on exposure rather than a size to chase.

That 25 percent fraction is shown producing 18 units of profit from 10 units of starting capital over 10 tosses, with a 20 percent ruin probability.

Sit below the peak

Sitting below that peak, at 15 percent risk instead of 25 percent, is shown cutting illustrated profit to 16 units while lowering ruin probability to 7 percent. That is how the same edge can be taken at a lower wipeout chance.

A payoff edge is not enough on its own

Positive expectancy can exist when wins are larger than losses even if winning frequency is about 40 percent or lower. An even-money coin-toss stake reverts toward zero, while a two-to-one payoff on the same sequence is illustrated as about 500 currency units after 100 tosses.

A risk-to-reward ratio above 1.0 is necessary but not sufficient. That ratio is planned gain divided by planned loss on one trade. A planned ten-to-one payoff still loses if winning frequency is below 10 percent, and costs such as slippage and commissions shrink the gap between wins and losses.

Increase size only after gains exist

An anti-martingale stance increases size only after profits have already been earned, so added exposure is funded by gains rather than by the original stake.