Ruin Is a Function of Size

Streaks are the ordinary weather

The frequency of losing runs is the part intuition gets wrong, because people read a run of losses as evidence about the edge rather than as a property of any series with a losing side. The arithmetic is undramatic. At a 45% win rate the chance of at least one run of eight consecutive losses somewhere in five hundred trades is about **85%**; a run of ten is about **43%** and a run of twelve about **16%**. At the more comfortable-sounding 55% win rate, eight in a row still appears in about 37% of five-hundred-trade samples and ten in a row in about 9%. The number that matters for a plan is not the probability but the cost, and the cost is a function of the fraction rather than the streak. Eight losses at 1% per trade leave 92.3% of capital — a 7.7% drawdown that most people would describe as a bad month. The same eight losses at 5% leave 66.3%, and at 10% they leave 43.1%, needing **132%** to get back to even. The streak is identical; the account is not. Which means the statement "my system lost eight in a row" tells you almost nothing until you know the size, and the statement "I risk 10% a trade" tells you almost everything. This is also the cleanest answer to a question that comes up after every drawdown: is the system broken? A streak is not evidence, because a working system produces them at a computable rate. What would be evidence is a realised expectancy over a decent sample that has drifted outside the range the sample size can explain — which is a calculation, done at fifty trades and again at a few hundred, rather than a feeling formed over a weekend. How often, and what it costs — 8 losses in a row, 45% win rate, in 500 trades: about 85% · 10 in a row at the same win rate: about 43% · 8 in a row at 55%: about 37% · That streak at 1% risk: −7.7% ← · At 5% / at 10%: −33.7% / −56.9%, needing +132% Ruin is defined in this subject as reaching a fifth of starting capital, because a fixed-fractional account never quite reaches zero: it is the point at which a career is over rather than the point at which the arithmetic stops.

What the simulation adds to the arithmetic

The streak arithmetic is a single scenario. What a plan actually faces is a sequence of them, with wins interleaved and the drawdowns stacking, and that is what the simulator in this lesson runs: 400 accounts, 200 trades each, one shared sequence of wins and losses so that size is the only variable. Three numbers come out of it, and they tell the story in different ways. The first is the **ruin rate**, which goes 0% at 1% risk, 0.3% at 5%, 15.8% at 10% and 65% at 20%. The second is the **median outcome**, which rises before it collapses: 1.26× at 1%, 2.38× at 5%, 2.69× at 10%, and 0.19× at 20%. The third is the **bad-luck path** — the fifth percentile — which is 0.99× at 1% risk and 0.19× at 10%, meaning that one path in twenty ends flat at the small size and loses four fifths of the account at the larger one. The shape of those three curves is the lesson. Ruin is monotone in size, always. The median is not, because for a while the extra capital at work out-earns the extra variance — and then the drawdowns, which are larger than the returns by construction, take over. At 20% risk the growth rate itself turns negative, about −0.8% a trade in the simulation, on a strategy whose expectancy is positive: the account is being given a fair chance on every trade and destroyed by the size of the bet. That is the definition of a ruin problem, and it is why the fraction is the input a plan fixes in advance rather than the output it discovers in a bad month. The same 200 trades at four sizes — 1% risk: ruin 0% · median 1.26× · worst path 0.99× · 5% risk: ruin 0.3% · median 2.38× · 10% risk: ruin 15.8% · median 2.69× · drawdown 78% · 20% risk: ruin 65% · median 0.19× · growth negative ← The median rising to 2.69× at 10% risk is not an argument for 10%: the paths that survive look wonderful and the ones that do not are excluded from the median. The distribution, not the median, is the object of interest.

Ruin is not a big drawdown

A 50% drawdown and ruin are different in kind, not in degree. A drawdown is a temporary state you can recover from with a larger percentage gain — 50% down needs 100% up. Ruin is a **boundary you cross**, after which the process stops: margin called out, capital at zero, the game over. Risk-of-ruin mathematics cares about the probability of touching that boundary, which grows with the size of each bet and with the number of bets taken, and which is why the same system can have a comfortable expected value and an unacceptable chance of ending. The asymmetry is what makes ruin the right thing to manage and drawdown the wrong proxy. A strategy with a positive expectancy but a large fixed fraction of capital per trade will, given enough trades, eventually meet the run of losses that ends it — and the probability of that run rises with time even as the expected profit also rises. That is the trap in “the system works, I just need to survive long enough”: survival is not a side condition, it is the binding constraint. The practical implication is that position size should be set by the worst plausible streak, not by the average trade or by the largest loss in a backtest. The relevant question is not “what is my edge?” but “at this size, across hundreds of trades, what is the chance of a sequence that ends me?” A fractional bet that retains most of the growth and much less of the ruin probability is almost always the better choice, and the mathematics of why is the next lesson. Ruin is absorbing: you do not trade your way back from a zero balance. That single property is why the bound is treated differently from every other loss.

The de-risking ladder

Ruin is a size problem, and size is a decision. What turns that decision into something that can be held is a pre-committed ladder: a set of drawdown levels at which the risk taken per position drops, defined before the drawdown arrives, so that the reduction is an instruction rather than a judgement made by an account that has just lost a third of its value. The design is short. Pick a maximum drawdown that the plan intends never to exceed, divide the distance to it into steps, and attach a smaller risk fraction to each step. A book risking one percent per position might reduce to three-quarters of a percent at a ten percent drawdown, to half at fifteen, and stop opening new positions at the maximum. The specific numbers matter far less than the structure, because the structure is what converts a boundary that everyone intends to respect into one that the account has already agreed to. Why it works is arithmetic plus behaviour. The arithmetic is that a smaller risk fraction lengthens the time to a given drawdown and reduces the size of the next loss, so the ladder cannot prevent the loss but it slows the approach to the boundary. The behaviour is the more important part: at a deep drawdown, the trader’s own judgement is the least reliable input available. A ladder replaces that input with a rule set in a state that was not stressed. Two design cautions keep the ladder from doing harm. First, it must not be so tight that ordinary variance reaches it — the same point that governs position limits applies here, and a ladder keyed to noise will reduce size permanently as a result of trading normally. Second, the reversals need defining: at what point, on the way back up, does the risk fraction return to each level? Without an answer, the ladder is a ratchet that ratchets only downward, which is a slow way to stop trading entirely. The last piece is the language, because it decides whether the ladder survives a conversation with yourself. A de-risking step is not a punishment and not a judgement about the method; it is the mechanism by which the method gets to keep operating. An account that halves its risk in a drawdown and then recovers is a functioning account. An account that holds its size because reducing would “lock in the loss” is one bet away from the outcome the whole subject is about. And a note on the honest asymmetry: recovering from a deep drawdown requires a larger percentage gain than the loss that caused it, so the ladder is worth more than its arithmetic suggests. It keeps the account inside the range where recovery is a matter of ordinary returns rather than of exceptional ones. • Pre-commit drawdown levels and the smaller risk fraction each one triggers. • The ladder is the answer to “my judgement is least reliable exactly when I need it”. • Key it to something larger than ordinary variance, and define how it reverses on the way up. • Recovery is a division, so staying inside the recovery range is worth more than it looks. Write the ladder somewhere the account can see it — a dated note, a broker setting, a line in the plan — because the version of the trader who wrote it is not the one who will be reading it during the drawdown.

Sources

• Risk of ruin and the arithmetic of surviving a losing runRalph Vince, "The Mathematics of Money Management"
• Streak probabilities in independent trialsStandard probability; run-length distributions
• Monte Carlo evidence that sizing, not selection, drives survivalVan Tharp, "Trade Your Way to Financial Freedom"
• Why 20% risk turns a positive edge into a negative growth rateThe log-growth (Kelly) formulation of geometric growth