Bounded crash risk with capital units and optimal f

A before-entry bound on stake size

Risk of ruin is the chance that a run of independent losses exhausts the account, used as a before-entry bound on stake size. An independent event is an outcome whose chance does not change because of the prior trade, so a string of results is the product of the single-trade chances.

Capital units are how many equal slices the account is cut into before one loss can finish the game. Trading advantage is long-run winning share minus losing share, used as the edge input when estimating wipeout chance.

Independent wipeout paths

When successive outcomes are independent, the chance of a four-event even-odds string is 0.0625.

A 0.60-win and 0.40-loss process that risks the entire stake has a first-trade wipeout chance of 0.40, a win-then-two-loss path of 0.096, and a three-or-fewer-trade ruin chance of 0.496. Adding the distinct wipeout paths through five trades on that same all-in 0.60 and 0.40 process produces a ruin chance of 0.542.

Capital units and risk of ruin

With a trading advantage of 0.20 and one capital unit, long-run ruin chance is 67 percent. Two equal units lower it to 44 percent.

At a 20 percent trading advantage, ten capital units reduce wipeout chance to about 3 percent. At a 5 percent advantage the same ten units still leave roughly 35 percent ruin.

Fixed-percentage sizing

Fixed-percentage sizing resets the dollar stake as a constant share of the current balance after each result so a losing streak automatically shrinks the next risk. After a loss, a fixed fraction of remaining equity reduces the next dollar stake, so a losing streak drains the account more slowly than a constant dollar unit.

Kelly criterion and optimal f

Profit-loss ratio is average winning-trade dollars divided by average losing-trade dollars. Kelly criterion is a growth-rate fraction of equity set from win probability and payoff ratio so the next loss stays a bounded slice of the account. Optimal f is the single fraction of current capital that maximizes long-run growth for a stated win probability and profit-to-loss ratio.

With success probability 0.60 and an average profit-to-loss ratio of 3, the growth-maximizing fraction of capital is 0.47. A 0.30 hit rate with a profit-to-loss ratio of 4 still has a defined optimal fraction of 12.5 percent.

Editorial. TradersWeek applies that fraction only to the unit that remains after wipeout paths have been counted and the account has been split. The growth rule is not treated as a substitute for the ruin bound.