Forced-close odds from monthly edge and leverage

The close line as ruin

A pre-set account-close loss can be treated as ruin. Its chance over a chosen horizon can be computed from three empirical inputs: the chance a month is profitable, the fraction of starting capital added in a winning month, and the fraction lost in a losing month.

Risk of ruin is the chance that equity reaches that pre-set close-out loss within the stated horizon, counted by stopping every path that hits the loss.

What the calculation assumes

The calculation assumes process characteristics stay constant, months have no breakeven outcome so win and loss probabilities sum to one, and successive months form no usable sequence.

Why a stop ends the usual count

If a losing month has probability 0.4 and removes 0.25 of original equity, two opening losses occur with probability 0.16 and stop the account at half of original capital when the close line is a 50 percent loss. Ordinary binomial formulas do not apply after that stop.

An exact path solution

The solver enumerates every surviving and closed branch so open plus closed probabilities sum to one. Under the stated assumptions it is an exact path solution rather than a simulation sample, and it can look jumpy versus daily data.

Leverage along a locked curve

With the win-month fraction and the gain-to-loss size ratio held fixed, expected annualized return and ruin probability are locked together. Expected value is the annualized mix of win-month probability times gain size minus loss-month probability times loss size, plotted as the return coordinate against ruin.

Raising leverage is the remaining way to move up the return axis. Leverage control is changing exposure relative to starting equity so the same monthly win rate and gain-to-loss ratio move to a different expected-return and close-out-probability pair. Many of those curves bend at a knee where extra return coincides with a sharp rise in close-out risk.

Shared maps for comparison

Processes that share the same descriptive ratios share the same return-versus-ruin map, so those coordinates can be used to compare relative efficiency the way an efficient-frontier plot compares portfolios.

Where the constant-size reading fades

Cumulative close-out probability generally flattens because surviving accounts have grown. Scaling exposure up with equity would invalidate the constant-size assumption beyond about two or three years.

Allowing month-to-month variation around the average gain and loss is expected to widen dispersion and raise ruin probability relative to the constant-average model.