Risk of ruin: the arithmetic nobody wants to do

Most traders spend their attention on the entry. The entry is the least important decision they make. What determines whether an account survives long enough for a genuine edge to express itself is the size of each position relative to capital — and the relationship between that size and survival is not linear. It is exponential, and it turns against you far earlier than intuition suggests.

Risk of ruin is the probability that a sequence of bets drives capital below a level from which the strategy can no longer function. "Ruin" rarely means zero. It means the point at which you are forced to stop: the margin call, the mandate breach, the drawdown at which you stop trusting the system and abandon it. That last one is the most common form of ruin and the one no formula captures. But the arithmetic of the first two is exact, and it is worth doing once, carefully, because it reorders every other priority in trading.

The recovery asymmetry

Start with the simplest fact and the one most often waved away. A loss of x% requires a gain of x/(1−x) to get back to even. The two numbers diverge violently.

• Lose 10%, you need +11.1% to recover.
• Lose 20%, you need +25%.
• Lose 33%, you need +50%.
• Lose 50%, you need +100%.
• Lose 75%, you need +300%.
• Lose 90%, you need +900%.

Nothing here is behavioural or psychological. It is the mechanical consequence of compounding on a smaller base. A trader who is down 50% and produces an excellent year of +40% is still down 30% from the high-water mark.

This is why drawdown control is not a defensive afterthought bolted onto an offensive system. It is the return engine. The compounding rate of a strategy is governed by its geometric mean return, and the geometric mean is dragged below the arithmetic mean by variance. A useful approximation: geometric return ≈ arithmetic return − σ²/2, where σ is the standard deviation of returns. Double the position size and you double arithmetic return but quadruple the variance term. There is a size past which more leverage produces less wealth. Not less risk-adjusted wealth — less wealth, full stop.

The clean demonstration: a position that gains 50% then loses 33.3% has an arithmetic mean return of +8.35% per period and a geometric return of exactly zero. 1.50 × 0.667 = 1.00. The arithmetic average is a description of the average trade. The geometric average is a description of your bank balance. Only one of them buys anything.

The ruin formula, with real numbers

Take a strategy with a genuine edge: it wins 55% of the time and each win and each loss is the same size. Risk a fixed fraction of starting capital per trade. For fixed-unit betting, the classical gambler's-ruin result gives the probability of eventually losing everything as (q/p)^N, where p is the win probability, q = 1−p, and N is the number of units of capital you hold.

With p = 0.55, q = 0.45, so q/p = 0.45/0.55 = 0.818.

Risking 2% per trade gives you N = 50 units. 0.818^50 ≈ 0.000044. That is roughly one chance in 23,000. Effectively safe.

Risking 5% per trade gives N = 20. 0.818^20 ≈ 0.0182 — about 1.8%. Still small, but you have moved from "will not happen" to "will happen to roughly one trader in fifty running this exact system".

Risking 10% per trade gives N = 10. 0.818^10 ≈ 0.134. A 13% chance of total ruin, with a real, positive, 55% edge. One trader in seven running a genuinely profitable system into the ground because of sizing alone.

Risking 20% per trade gives N = 5. 0.818^5 ≈ 0.367. More than one in three.

Read that progression again. The edge never changed. Only the position size did, and ruin probability went from negligible to a coin-flip-adjacent risk across a range of sizes that traders casually describe as "aggressive but fine".

And note what happens when the edge is absent. If p = 0.50, q/p = 1, and the ruin probability is 1 for any finite capital given unlimited trials. With no edge, ruin is not a risk, it is a certainty deferred. If p = 0.45 — you are paying spread and commission on a coin flip, which describes a great deal of retail activity — ruin is likewise certain, and arrives quickly.

Losing streaks are not rare, they are scheduled

Traders underestimate streaks because they think about the probability of the next loss rather than the probability of a run appearing somewhere in a long sequence.

At a 55% win rate, the chance of six consecutive losses starting at any particular trade is 0.45^6 = 0.0083, roughly one in 120. Over 500 trades there are about 495 places such a run could start, so the expected number of six-loss runs is around four. A six-loss streak is not a tail event in that sample. It is the base case, and you should expect several.

A ten-loss run has probability 0.45^10 = 0.00034 per starting point, around one in 2,900. Across 500 trades the expected count is about 0.17, which corresponds to roughly a one-in-seven chance of seeing at least one such run in that stretch. Over a multi-year career of several thousand trades, you will see it.

Now price those streaks at different sizings, using compounding on remaining equity:

• At 2% risk: six losses → 0.98^6 = 0.886, an 11.4% drawdown. Ten losses → 0.98^10 = 0.817, an 18.3% drawdown. Unpleasant, survivable, recoverable with +22%.
• At 10% risk: six losses → 0.90^6 = 0.531, a 47% drawdown. Ten losses → 0.90^10 = 0.349, a 65% drawdown, requiring +187% to return to the high-water mark.

The 10% trader has not been unlucky. They have experienced exactly the streak the model predicts, and their system is now mathematically unrecoverable on any reasonable horizon. This is the mechanism by which competent traders with real edges blow up: the edge was real, the streak was ordinary, the sizing was not.

Kelly, and why nobody should trade full Kelly

The Kelly criterion, from John Kelly's 1956 paper at Bell Labs and developed for markets by Ed Thorp, gives the fraction of capital that maximises the long-run geometric growth rate. For a bet paying b to 1 with win probability p:

f* = p − q/b

For our 55%, even-money strategy: f* = 0.55 − 0.45/1 = 0.10. Ten percent per trade.

Compute the growth rate per trade, g = p·ln(1+f) + q·ln(1−f):

• At f = 10% (full Kelly): 0.55·ln(1.10) + 0.45·ln(0.90) = 0.0524 − 0.0474 = +0.50% per trade.
• At f = 5% (half Kelly): 0.55·ln(1.05) + 0.45·ln(0.95) = 0.0268 − 0.0231 = +0.375% per trade.
• At f = 20% (double Kelly): 0.55·ln(1.20) + 0.45·ln(0.80) = 0.1003 − 0.1004 = −0.01% per trade.
• At f = 25%: 0.55·ln(1.25) + 0.45·ln(0.75) = 0.1227 − 0.1295 = −0.67% per trade.

Three things fall out of those four lines.

Half Kelly delivers 75% of the maximum growth rate (0.375/0.50) while roughly halving the volatility and dramatically reducing drawdown depth. That is one of the best trades available anywhere in finance.

At double Kelly, growth is zero. You are taking twice the risk of the optimal bettor for none of the return. Past that point, expected wealth declines toward zero with probability one, no matter how good the edge.

And the whole curve is flat near the top and cliff-like on the right. Being half the optimal size costs you a quarter of your growth. Being twice the optimal size costs you everything. When you are uncertain about your true win rate — and you always are, because it was estimated from a finite sample — the asymmetry says to err small. Practitioners typically run a quarter to a half of Kelly for exactly this reason.

One further caution: full Kelly is optimal only if p and b are known exactly. If you estimated a 55% win rate from 200 trades, the standard error on that estimate is about 3.5 percentage points, so the true rate might be 51%. At p = 0.51 the Kelly fraction is 2%, and your "optimal" 10% is five times oversized. Parameter uncertainty is not a rounding error in this calculation; it is the dominant term.

Where Switzerland changes the arithmetic

Three Swiss specifics belong in this calculation.

Leverage is not capped for you the way it is in the EU. ESMA's 2018 product-intervention measures limited retail leverage for EU clients to 30:1 on major currency pairs and far lower for volatile instruments. Switzerland is not in the EU, and FINMA has not imposed an equivalent retail leverage cap. A Swiss-resident retail client can access leverage that an EU-resident retail client cannot. That is a freedom, and freedom here means the ruin arithmetic above is entirely your responsibility.

Wealth tax is a fixed leak on the compounding path. Cantons and communes levy an annual tax on net wealth, assessed on the value of your assets at the end of the tax period, and it falls due whether the year was good or bad. Rates, allowances and the communal multiplier differ substantially between cantons, and within a canton between communes, so the same portfolio faces a different drag depending on where its owner is registered. The effect on the arithmetic above is small but structural: the compound return you need in order to stand still is your gross edge, minus trading costs, minus that levy, and the levy is charged on the reduced balance during a drawdown as well as on the peak. It will not ruin you on its own. It does mean the break-even growth rate in the ruin model is not zero, and a strategy whose expectancy is a whisker above zero before tax is below it after.

The professional answer to ruin risk is a hard allocation ceiling, written down in advance. Swiss occupational pension funds invest under the BVV2/OPP2 investment rules, which set maximum shares per asset category — broadly half the assets in equities, less in real estate, and a much smaller share in alternative investments — alongside limits on how much may sit with a single debtor or in a single company, and a ceiling on foreign-currency exposure left unhedged. A fund may go beyond a category limit only if it documents in its accounts why doing so is still consistent with its overall duty of prudent investment. Argue with the individual numbers if you like; the design is the point. The institutions that cannot afford ruin do not rely on judgement in the moment. They bind themselves ahead of time with a written ceiling, and require an explicit, recorded justification from anyone who wants to breach it. A private trader's position-size limit is the same instrument on a smaller balance sheet.

What to take away

• Compute your own ruin probability before you size a trade: with an even-money system and a 55% win rate, risking 2% per trade gives roughly a 1-in-23,000 chance of ruin, while risking 10% gives roughly 1-in-7 — the edge is identical in both cases and only the sizing differs.
• Expect the streaks the model predicts: at a 55% win rate you should anticipate several six-loss runs in every 500 trades, so choose a position size at which a six-to-ten loss run is an inconvenience rather than a career event.
• Size at a fraction of Kelly, typically a quarter to a half, because half Kelly captures about 75% of the growth rate for far less variance while double Kelly reduces long-run growth to zero.
• Treat your estimated win rate as uncertain: a 55% rate measured over 200 trades carries a standard error near 3.5 points, and since the Kelly curve is gentle to the left and catastrophic to the right, uncertainty should always push size down.
• Borrow the institutional discipline: Swiss pension funds operate under written per-category and per-issuer investment ceilings that may be exceeded only with a documented justification, which is exactly the function a stated maximum position size performs for a private account.

Sources

• J. L. Kelly Jr., "A New Interpretation of Information Rate", Bell System Technical Journal, 1956.
• Edward O. Thorp, Beat the Market and his subsequent writing on the Kelly criterion in portfolio choice; also his memoir A Man for All Markets.
• William Feller, An Introduction to Probability Theory and Its Applications, Volume I — the standard treatment of the gambler's-ruin problem.
• Leo Breiman, "Optimal Gambling Systems for Favorable Games", Berkeley Symposium on Mathematical Statistics and Probability, 1961.
• Ordinance on Occupational Retirement, Survivors' and Disability Pension Plans (BVV2/OPP2), the articles governing permitted investments, debtor and company limits, and category limits.
• European Securities and Markets Authority (ESMA), product intervention decisions on contracts for differences for retail clients, 2018 — the comparison point for Swiss retail leverage.
• Swiss Federal Tax Administration, cantonal tax information on the taxation of net wealth of individuals, including the variation in rates and allowances across cantons and communes.
• Nassim Nicholas Taleb, Dynamic Hedging and The Black Swan — on gap risk and the failure of stop-based loss models under discontinuous price moves.