The Risk-of-Ruin Formula

We have an edge (expectancy) and we know bet size matters. Now we put a number on survival. The risk-of-ruin formula turns three ingredients — your edge, your payoff odds, and how many units of risk your bankroll holds — into a single probability of going broke. Its most important lesson is geometric: ruin doesn’t fall linearly as you build capital, it falls exponentially, which is why a little extra cushion buys a huge amount of safety. We’ll derive it, work several examples, and meet the units-of-risk idea that makes it usable.

Units of risk: the right way to count a bankroll

Analogy. Don’t count your bankroll in dollars — count it in lives, like a video game. If each bet risks one life and you have 20 lives, you can survive a long unlucky streak before “game over.” A bigger dollar account that bets huge per trade might have only 3 lives; a smaller account betting tiny might have 50. Survival is about lives, not dollars.

Definition. A unit of risk is the amount you put at risk on one bet (one R, from last lesson). Your bankroll’s units of capital is $N = \frac{\text{bankroll}}{\text{risk per bet}}.$ This single number — how many losing bets in a row it takes to wipe you out — is what the risk-of-ruin formula actually depends on. Edge sets the lean of the walk; $N$ sets how far the walls are from the start.

Worked example. A $50,000 account risking$1,000 per trade has $N = 50\,000 / 1\,000 = 50$ units. The same account risking $5,000 per trade has$N = 10$units. Same dollars, same edge — five times fewer "lives," and (we'll see) a vastly higher risk of ruin. Position sizing *is* the choice of$N$.

The classic formula

The cleanest version is for an even-money bet (win or lose one unit each time) with win probability $p$ and loss probability $q = 1 - p$.

Definition. For a bankroll of $N$ units, the probability of eventual ruin (touching 0 before growing without bound) is $\text{RoR} = \left(\frac{q}{p}\right)^{N} = \left(\frac{1-p}{p}\right)^{N},$ valid when you have an edge ($p > 0.5$, so $q/p < 1$). If $p \le 0.5$ the ratio $q/p \ge 1$ and ruin is certain (RoR = 1) — no amount of capital saves a game with no edge against an unbounded horizon, the gambler’s-ruin result in disguise.

The geometry. $q/p$ is a number less than 1 (because you have an edge), and you’re raising it to the $N$-th power. Raising a fraction to higher powers crushes it toward zero fast. That’s the exponential decay: each extra unit of capital multiplies your ruin probability by $q/p$ again.

Worked example 1. Even-money bet, win probability $p = 0.55$, so $q = 0.45$ and $q/p = 0.45/0.55 = 0.818$.

• With $N = 5$ units: $\text{RoR} = 0.818^5 \approx 0.366$ — a 37% chance of ruin. Scary, for a real edge.
• With $N = 10$ units: $\text{RoR} = 0.818^{10} \approx 0.134$ — 13%.
• With $N = 20$ units: $\text{RoR} = 0.818^{20} \approx 0.018$ — 1.8%.
• With $N = 40$ units: $\text{RoR} = 0.818^{40} \approx 0.0003$ — 0.03%.

Look at the pattern: each doubling of $N$ roughly squares the ruin probability (0.37 → 0.13 → 0.018 → 0.0003). Doubling your capital-in-units doesn’t halve ruin — it squares it. Capital is the exponential lever; edge only sets the base.

Adding a payoff ratio: the general formula

Real systems don’t win and lose the same amount. The general fixed-fraction risk-of-ruin formula folds in the payoff ratio $b$ (average win ÷ average loss, in R) alongside win probability $p$.

Definition (Kelly-based approximation). A widely used closed form expresses ruin in terms of the edge per unit risked and the fraction of bankroll risked per trade $f$. With per-bet expectancy (in R) $a = E_R$ and per-bet variance $v$, and risking fraction $f$ of the bankroll per unit, the continuous approximation for ruin starting with the full bankroll is $\text{RoR} \approx \exp\!\left(-\frac{2\,a\,N}{v / E_R}\right)\quad\text{(schematically: ruin} \downarrow \text{ as edge} \uparrow \text{and as } N \uparrow).$ The exact algebra varies by model, but every version shares the same two-handle structure: ruin falls as the edge grows and as the units of capital grow, and it falls exponentially in their product. You don’t need to memorize a specific closed form; you need the shape.

A cleaner practical handle — the Kelly link. If you bet a fixed fraction $f$ of your bankroll and your full-Kelly fraction is $f^*$, then betting fraction-of-Kelly $k = f/f^*$ has an elegant long-run ruin (drawdown-to-a-fraction) relationship: the probability of your equity ever dropping to a fraction $x$ of its start is approximately $P(\text{ever drop to } x) \approx x^{\,(2/k) - 1}.$ So at full Kelly ($k=1$) the chance of ever halving is $\approx (0.5)^1 = 50\%$ — full Kelly is brutal. At half Kelly ($k=0.5$) it’s $\approx (0.5)^3 = 12.5\%$. At quarter Kelly ($k=0.25$) it’s $\approx (0.5)^7 \approx 0.8\%$. Betting smaller fractions of Kelly slashes deep-drawdown (and ruin) probability super-linearly — the same exponential lever, viewed through Kelly.

Worked example. You bet half Kelly ($k = 0.5$). Probability of ever suffering a 75% drawdown (dropping to $x = 0.25$ of start): $\approx 0.25^{(2/0.5)-1} = 0.25^{3} = 0.0156$, about 1.6%. At full Kelly the same 75%-drawdown probability is $\approx 0.25^{(2/1)-1} = 0.25^1 = 25\%$ — sixteen times worse. Halving your Kelly fraction barely dents your growth rate (it costs about 25% of the growth) but cuts catastrophic-drawdown risk by an order of magnitude. This is why nobody sane bets full Kelly.

Reading the formula like a risk manager

What it tells you to do

• Add units, not just edge. A modest edge with deep capital (high $N$) beats a strong edge run on a thin account. Survival is bought primarily with $N$ — i.e. by betting smaller fractions.
• Bet a fraction of Kelly. Half or quarter Kelly costs little growth and slashes drawdown/ruin risk super-linearly. The standard professional default is half Kelly or less.
• Stress the inputs. Recompute RoR with a pessimistic edge (say, 70% of your estimate) to see how fragile your survival is to being wrong.

When the simple formula breaks down

• Non-stationary edge. If $p$ drifts (regime change, alpha decay), a static RoR is fiction. Real edges fade; the formula assumes they don’t.
• Correlated bets. The formula assumes independent trades. If your positions move together (everything’s long tech), a “losing streak” is really one big correlated loss, and effective $N$ collapses.
• Fat tails / gaps. A jump that blows through your stop means a single “loss” can be several R, shrinking $N$ unexpectedly — the next lessons (drawdowns, sequencing, Monte Carlo) handle exactly these real-world wrinkles the clean formula glosses over.

Putting it together

The risk-of-ruin formula turns edge, odds, and capital into a survival probability. Count your bankroll in units of risk ($N =$ bankroll ÷ risk-per-bet) — the number of consecutive losses you can absorb — because that, not raw dollars, is what the formula keys off. For an even-money edge the clean form is $\text{RoR} = (q/p)^N$: a fraction below 1 raised to the power of your units, which means ruin decays exponentially — each doubling of $N$ roughly squares (shrinks) the ruin probability. The general version folds in the payoff ratio but keeps the same shape: ruin falls as edge grows and as units grow, exponentially in their product. The cleanest practical handle is the Kelly link — deep-drawdown risk scales like $x^{(2/k)-1}$ — which is why half or quarter Kelly slashes catastrophe risk while barely denting growth. And every number assumes you know your edge, so bet smaller than the formula says, because the real wrinkles (drifting edge, correlation, fat tails) all make true ruin worse than the clean math admits.

Next up — drawdown distributions — we stop treating the worst loss as a single number. Depth, duration, and the dreaded time under water are all random variables with their own distributions, and the same system run twice can hand you two very different worst-cases.