Monte Carlo Ruin Curves & Stops

The clean risk-of-ruin formula assumed independent, even-money bets with a known edge. Real life has fat tails, correlated positions, drifting edges, and lumpy payoffs the formula can’t digest. So practitioners reach for the tool from the Monte Carlo topic: simulate the strategy thousands of times and just count how often it blows up. This lesson uses simulation to estimate ruin probability, plots the ruin curve that maps risk-per-trade to blow-up probability, and closes the loop with the stop-loss arithmetic that converts a risk budget into the exact position size to trade — the practical machinery every risk desk runs before sizing anything.

Estimating ruin by simulation

Analogy. You can’t know your real risk of ruin from a single life — you only get to live one. Monte Carlo gives you a thousand parallel lives: run the strategy through a thousand plausible futures, see how many end in ruin, and the fraction that blow up is your estimated probability. It’s the law of large numbers turned into a risk gauge.

The procedure.

1. Model the per-trade outcome — from your edge: win probability $p$, payoff distribution (or resample your actual historical trade R-multiples).
2. Set a bet-sizing rule — e.g. risk a fixed fraction $f$ of equity per trade.
3. Define the ruin barrier — the equity level at which you’re out (50% drawdown, redemption trigger, zero).
4. Simulate one path — walk the equity through $T$ trades, flagging the path as ruined if it ever touches the barrier.
5. Repeat thousands of times and compute $\text{RoR} \approx \dfrac{\text{ruined paths}}{\text{total paths}}$.

Worked example. Simulate 10,000 paths of a system with $p = 0.55$, even money, risking 4% per trade over 120 trades, ruin barrier at 50% of starting equity. Suppose 1,820 paths breach the barrier. Then $\text{RoR} \approx \frac{1\,820}{10\,000} = 0.182 = 18.2\%.$ An 18% chance of a ruinous drawdown — from a winning system — purely because 4% per trade is too aggressive. Crucially, simulation also hands you the whole distribution for free: the median drawdown, the time-under-water, the spread of final equities — not just the single ruin number.

The accuracy caveat. A simulated probability has its own sampling error of roughly $\sqrt{\text{RoR}(1-\text{RoR})/M}$ for $M$ paths. To pin down a 1% ruin probability to a meaningful precision you need many thousands of paths — and, more importantly, the result is only as good as the return model you feed it. Garbage assumptions (too-rosy edge, ignored correlations, thin tails) produce a confidently wrong number. Simulation removes the formula’s structural assumptions but not the need for honest inputs.

The ruin curve: ruin vs risk-per-trade

The single most useful output of all this is a curve, not a number: plot risk of ruin (y) against the fraction of equity you risk per trade (x). It turns position sizing into a picture.

The shape. Starting from tiny bets, ruin is near zero. As risk-per-trade rises, ruin stays low for a while, then climbs — and past a point it shoots up toward certainty. There’s typically a ‘knee’: a region where a small increase in bet size causes a large jump in ruin. Sane position sizing lives to the left of the knee, where ruin is comfortably small and edge can do its work.

The Kelly connection. Recall from the formula lesson: long-run growth is maximized at the full-Kelly fraction, but ruin/drawdown risk keeps climbing past it, and beyond twice Kelly the long-run growth rate goes negative and ruin becomes certain. So the ruin curve and the growth curve tell a joint story: there’s a band — roughly fractional Kelly — where growth is near-optimal and ruin is acceptably low. The whole art of sizing is finding that band, and the ruin curve is how you see it.

Worked reading. Suppose the curve shows: 1% risk → 0.2% ruin; 2% → 1.5%; 4% → 18%; 6% → 45%; 8% → 78%. The knee is around 4–6%. A trader wanting ruin under ~2% must size at or below 2% per trade — and notice that going from 4% to 6% (a 1.5× bigger bet) nearly triples ruin while adding little to growth. The curve makes the bad trade-off visible: past the knee you’re buying a sliver of extra growth with a mountain of extra ruin.

Stop-loss math: from risk budget to position size

Now the most practical arithmetic in the whole topic — turning a risk decision into an order size. This is what makes all the abstract ruin math actionable.

The identity. Decide two things independently: how much of your account you’re willing to lose on this trade (your risk-per-trade, e.g. 1% of equity), and where your stop-loss sits (how far, in %, the price can move against you before you exit). The position size that delivers exactly that risk is $\text{Position size} = \frac{\text{Account} \times \text{Risk-per-trade \%}}{\text{Stop distance \%}}.$ Equivalently, in shares: $\text{Shares} = \dfrac{\text{Account} \times \text{Risk\%}}{\text{Entry price} - \text{Stop price}}$.

The key insight: two independent dials. Risk-per-trade sets the dollars you can lose; the stop distance sets how big a position delivers that loss. A tighter stop lets you hold a bigger position for the same dollar risk — but a stop too tight gets you knocked out by ordinary noise (whipsaws). A wider stop means a smaller position for the same risk but more room to be right. The dollars-at-risk never change; only the share count does.

Worked example. Account $100,000, willing to risk 1% = $1,000 on a trade. Stock trades at $50, and you’ll exit if it falls to $46 — a stop distance of $4, which is $4/50 = 8\%$. $\text{Position size} = \frac{100\,000 \times 0.01}{0.08} = \frac{1\,000}{0.08} = 12\,500 \text{ dollars (i.e. 250 shares).}$ Check: 250 shares × $4 stop =$1,000 at risk = 1% of the account. ✓ Now suppose you tightened the stop to $48 (a 4$1,000 risk, but position size doubles to $25,000 (500 shares) — a tighter stop buys a bigger position for identical dollar risk. The risk budget controls survival; the stop placement controls how that budget is spent.

Bringing the whole topic together

The survival workflow

1. Confirm an edge (positive expectancy, net of costs) — without it, no sizing saves you.
2. Choose a per-trade risk budget small enough that the ruin curve (simulated for your return distribution) sits comfortably left of the knee — typically a fraction of Kelly.
3. Place stops where the trade thesis is wrong, then size from the identity so each trade risks exactly the budget.
4. Stress everything — pessimistic edge, correlated losing streaks, bad sequencing, fat-tailed gaps — because the real world is harsher than any clean model.

Pitfalls to retire

• Sizing on a single backtest instead of a distribution — the drawdown and ruin tails won’t be in one history.
• Treating a positive edge as safety — the central paradox: edge plus over-betting still equals ruin.
• Confusing tighter stops with lower risk — risk is the per-trade budget; the stop just sets position size.
• Ignoring correlation and gaps — they shrink your effective units of capital exactly when you most need them.

Putting it together

When the clean formula’s assumptions break, Monte Carlo estimates risk of ruin by brute force: model the per-trade outcome, set a sizing rule and a ruin barrier, simulate thousands of paths, and take the fraction that blow up — getting the whole drawdown and time-under-water distribution as a bonus, but only as honest as the return model you feed it. Plotting ruin against risk-per-trade gives the ruin curve, with its tell-tale knee where a small increase in bet size sends ruin soaring; sane sizing lives left of the knee, in the fractional-Kelly band where growth is near-optimal and ruin is low. Finally, the stop-loss identity — position size = (account × risk%) ÷ stop-distance% — turns a risk budget into an exact order, with risk-per-trade and stop placement as independent dials (a tighter stop buys a bigger position for the same dollar risk, not less risk). Mind the leaks: gaps, slippage, and correlation make real losses exceed −1R, shrinking your effective capital and pushing true ruin above the clean math — so size for the tails, because the tails are where ruin lives.

That completes the survival toolkit. You can define a ruin barrier, compute and stress an edge, read the risk-of-ruin formula and its exponential payoff to capital, think in drawdown distributions rather than single numbers, respect sequencing risk wherever cashflows live, and turn a simulated ruin curve plus a stop into an exact, survivable position size. Next is the final exam — a graded run across the whole topic.