How to model risk of ruin

Two questions decide whether a betting edge ever becomes money. The first is whether the edge is real, and closing line value answers that far better than profit and loss ever will. The second question is quieter and gets asked much less often. Even if the edge is real, will the bankroll survive long enough for it to arrive? A bettor can be right about almost every price and still go broke, because variance does not wait for the long run to be polite. Risk of ruin is the mathematics of that second question.

This is a maths article. The formulas below are simple enough to run in a spreadsheet, and every number states its assumptions so it can be checked. The conclusion, stated up front, is uncomfortable for most bettors moving to sharp venues: the correct stake is almost always smaller than it feels like it should be.

The short version

• Risk of ruin is the probability that a bankroll reaches zero before an edge plays out. In the continuous approximation it is R = exp(−2μB / σ²), where μ is the expected profit per bet, B is the bankroll measured in unit stakes, and σ² is the variance of a single bet.
• The relationship is exponential in bankroll and inverse in variance. Adding units to the bankroll drives ruin down fast. Raising variance, for example by betting longshots, undoes that protection just as fast.
• At sharp venues the edge is compressed to roughly one to three percent. Sizing that was safe at a soft book on a five percent edge can be close to reckless once the edge collapses to that range.
• Full Kelly is growth-optimal in theory and brutal in practice. A full-Kelly bettor has around a fifty percent chance of the bankroll halving at some point and a ten percent chance of it falling to a tenth. Fractional Kelly is how those drawdowns are made survivable.
• The rule “never risk more than two percent” has no mathematical basis. Ruin depends jointly on edge, variance, bankroll size, and horizon. A flat percentage ignores all four.

Risk of ruin calculator

The calculator below runs that formula on your own numbers. Every input it needs is explained in the sections underneath it, so a reader who wants the reasoning rather than the result can skip past it and come back.

What risk of ruin actually models

The idea comes from the classic gambler’s ruin problem, first set out rigorously by William Feller. Two players stake against each other until one is wiped out. For a bettor, the opponent is the market, which can be treated as having effectively unlimited capital. The question is then narrower: starting with some bankroll and a positive edge, what is the probability of ever hitting zero?

Feller’s discrete solution assumes fixed, symmetric payouts. Sports betting does not work that way. Odds vary from bet to bet, payouts are asymmetric, and stakes change as the bankroll moves. For that reason the useful tool is the continuous approximation, which treats the bankroll as a random walk with drift rather than a series of identical coin flips. The drift is the edge. The spread around it is the variance.

The approximation gives a single, compact formula for the probability of ruin: R = exp(−2μB / σ²). Three inputs drive it. μ is the expected profit per bet, expressed as a fraction of one unit stake. B is the current bankroll measured in unit stakes, so a €5,000 bankroll staking €50 a bet is 100 units. σ² is the variance of a single bet.

Two properties of that formula do most of the work in this article. Ruin falls exponentially as bankroll units rise, so small additions to B crush the risk rather than nudging it. And variance sits in the denominator of the exponent, so doubling it, which happens naturally when a bettor moves to longer odds, badly weakens the protection the bankroll was giving.

One caveat before the numbers. B is a snapshot, not a fixed constant. It is the bankroll in current stakes at this moment, and it has to be recomputed as the bankroll and the stake change. There is also a small gap between theory and practice. In a perfectly continuous model, stakes shrink smoothly toward zero and the mathematical ruin point is never quite reached. Real venues impose minimum stakes, so once the bankroll falls below the level needed to place the required bet, the account is finished in practice even if the formula says otherwise. Real risk of ruin is therefore always slightly higher than the continuous approximation suggests.

The three inputs, made concrete

Edge (μ). Everything downstream assumes the edge is real and positive. If it is negative, ruin is not a probability but a certainty, delayed only by luck. This is why the honest starting point is not this article but the earlier ones: a bettor should have verified a positive edge against the closing line over a meaningful sample before modelling ruin at all. That comparison only means something once the bookmaker’s margin has been stripped out of the reference price, which is what devigging does.

The size of the edge also shrinks with the venue. Steven Levitt’s study of a full NFL betting season in The Economic Journal found that bookmakers do not balance their books at all. They hold positions and price deliberately away from the market-clearing level, because they predict bettor behaviour more reliably than bettors predict results. That is the mechanism a soft-book edge exploits, and it is also the reason the edge thins out at venues that price to be right rather than to be popular. Pinnacle runs roughly two to two and a half percent margin on liquid top-tier soccer against six to ten percent at a typical retail book, which compresses the realistic edge to something like one to three percent rather than the five percent or more an alert bettor can sometimes find at a soft book.

Variance (σ²). The variance of a single unit bet at decimal odds O, for a roughly fair price, is approximately σ² ≈ p(1−p)O², which reduces to the clean relationship σ² ≈ O − 1. Variance rises with the odds, and it rises steeply. Buchdahl’s work on the favourite-longshot bias shows the same thing empirically: at long odds the standard deviation of results dwarfs the expected profit. The table below gives the approximate single-bet variance across common odds.

A bettor playing even-money markets carries roughly one unit of variance per bet. A bettor chasing 10.0 shots carries nine. That difference feeds straight into the ruin formula, and its effect is dramatic, as the next section shows.

Bankroll in units (B). Because B enters the exponent, it is the most powerful lever a bettor controls. Halving the stake doubles the number of units in the bankroll. Doubling B squares the ruin probability downward. If ruin is 13.5% at 100 units, it is about 1.8% at 200 units under the same edge and variance. This is also why bankroll is best thought of in units rather than currency. The formula does not care whether a unit is one euro or one thousand; it cares how many of them stand between the bettor and zero.

What the numbers actually say

The table below shows the probability of ruin for a flat 1-unit staking plan at even-money odds, which fixes the variance at σ² ≈ 1.0. Rows are the bankroll in units. Columns are the edge, expressed as expected profit per bet. These are the assumptions; change them and the numbers change.

Assumptions: flat 1-unit stakes, even-money odds (σ² ≈ 1.0), infinite horizon. Edge is expected profit per bet as a fraction of the unit stake. Variance here uses the fair-price approximation σ² ≈ O − 1. The calculator above uses the exact single-bet figure σ² = p(1−p)O², which runs marginally higher once the price carries an edge, so its outputs sit a fraction above these table values.

Read across a single row and the point about the two percent rule makes itself. At 100 units, a 1% edge carries a 13.5% chance of ruin, a 2% edge carries 1.8%, and a 3% edge carries a quarter of a percent. Same bankroll, same stake, wildly different survival, because ruin is a function of the edge and not just the stake size. A flat rule that ignores the edge is guessing.

Variance changes the picture just as sharply, and this is where longshot strategies get dangerous. Take a 2% edge bettor with 100 units. At even money, variance is about 1.0 and the ruin probability is 1.8%. Move the same bettor, same edge, same bankroll, onto 10.0 shots, where variance is about 9.0, and the exponent is divided by nine. The ruin probability jumps from 1.8% to roughly 64%. Nothing changed except the odds being played.

Why full Kelly is the wrong default

The Kelly criterion gives the stake that maximises long-run growth. It is correct about growth and misleading about survival, because the growth-optimal stake is far more volatile than most bettors can tolerate. Thorp’s work on the Kelly criterion in betting and investing makes the drawdown property precise: a full-Kelly bettor has a probability of exactly x of the bankroll ever falling to a fraction x of its starting value. That means a fifty percent chance of the bankroll halving at some point, and a ten percent chance of it collapsing to a tenth. Those are not tail scenarios. They are the expected behaviour of the growth-optimal strategy.

Buchdahl’s own simulation makes the same case with sports-betting numbers. In his walkthrough of the pitfalls of Kelly staking, a bettor with a genuine 2.5% edge at odds of 1.909 has a full-Kelly stake of about 2.75%, and even with that real edge spends a large share of the time below the starting bankroll, with roughly a one-in-three chance of finishing a 1,000-bet sample down. The more important result is what happens when the edge is misjudged. Overestimate a 2.5% edge as 5%, and full Kelly turns the system into a coin toss, with a better than eight percent chance of losing ninety percent of the bankroll.

Fractional Kelly fixes this by staking a fraction c of the full-Kelly amount. The trade is favourable because variance scales with the square of the stake while growth does not. At half Kelly the variance drops to a quarter of full Kelly while growth stays at three quarters of the maximum. At quarter Kelly the variance falls to about six percent of full Kelly while growth holds at roughly forty-four percent of the maximum. In Buchdahl’s simulation, moving to half Kelly cuts the chance of the bankroll halving from around ten percent to about one percent. The practical reading is that fractional Kelly is not timidity. It is the main defence against the one input every bettor gets wrong, which is the size of their own edge.

Drawdown: the pain that arrives before ruin

For most bettors the bankroll never actually reaches zero. They quit, or they tilt, long before that. The number that governs when they break is not the ruin probability but the maximum drawdown, the largest fall from a previous high-water mark to the low that follows it.

The reference result comes from Magdon-Ismail and colleagues, who worked out the expected maximum drawdown of a random walk with drift. Their analysis of maximum drawdown identifies three regimes, split by the sign of the edge. With a negative edge, the expected drawdown grows linearly with the number of bets, which is just ruin arriving on schedule. At exactly break-even, it grows with the square root of the number of bets. With a positive edge, it grows only logarithmically. That last regime is the one a verified bettor lives in, and it still means the bankroll spends most of its life below its own record, because percentage staking produces a lognormal path that sits under its high-water mark far more often than above it.

The practical takeaway does not need the full formula. Expected maximum drawdown scales roughly with variance divided by edge. It grows when the edge shrinks and when the variance rises, which is exactly the combination a bettor walks into at a sharp venue. And because both the drift and the spread scale with stake size, halving the stake roughly halves the expected drawdown. A bettor who models this in advance knows the size of the hole they should expect to sit in, which is the difference between riding out normal variance and abandoning a sound strategy at the bottom of it.

The sharp-venue squeeze

Put the pieces together and the move most readers are considering, from limited soft-book accounts to a broker or sharp book, looks different. The variance of the bets does not change. The edge collapses. That single change, feeding into the exponent of the ruin formula, blows up both the ruin probability and the expected drawdown at identical sizing.

Take two bettors with 100 units, flat even-money stakes. The first has a 5% edge, the kind of number a soft-book price lag can produce, and a ruin probability far below one hundredth of a percent. The second has a 1.5% edge, a realistic sharp-venue figure, and a ruin probability of about 5%. To pull the second bettor back to the first bettor’s level of safety, they need roughly 330 units instead of 100. In plain terms, the same person moving to sharper prices has to stake about three times smaller to stand as safe as they were before.

Broker mechanics make this harder, not easier. Minimum stakes create a floor below which the required fractional-Kelly bet cannot be placed, which is the discrete ruin point mentioned earlier. Capital is also locked in ways a soft-book account never locks it. Sportmarket, for instance, requires a deposit to be turned over five times at odds of 1.50 or higher before a fee-free withdrawal, so a bettor in a drawdown is forced to keep betting through the variance rather than pulling funds out or moving them elsewhere. What each route costs in minimum deposit, turnover requirement and withdrawal fee is set out in the broker comparison. The combination of a smaller edge, unchanged variance, and locked liquidity is why quarter Kelly or lower at these venues is not a matter of personal caution. It is a survival requirement, and it is the practical core of managing a bankroll at sharp venues.

Circuit breakers and the psychology

The mathematics is exact. The bettor executing it is not. Loss aversion means the pain of a drawdown is felt more sharply than the pleasure of an equivalent gain, and because a percentage-staked bankroll spends most of its time below its high-water mark, that pain is close to constant. Two failure modes follow. The first is abandonment, quitting a sound strategy at the bottom and forfeiting the recovery. The second, and the more expensive, is tilt, raising stakes to claw back losses quickly. Since the ruin formula reacts exponentially to a smaller B, and a bigger percentage stake is a smaller B, raising stakes inside a drawdown drives the ruin probability toward one hundred percent.

The defence is to decide the rules before the drawdown, not during it. A pre-set fractional-Kelly stake and a drawdown threshold that forces a pause both act as circuit breakers. When the pause triggers, the question is whether the fall is still inside the range the drawdown model predicted, in which case it is variance and the strategy holds, or whether it signals that the edge itself has eroded, which shows up first as decaying closing line value rather than as the losing run itself.

What this means

Risk of ruin is not a single number and cannot be reduced to a flat percentage rule. It is a joint function of the edge, the variance of the bets, the bankroll measured in units, and the horizon. The honest sequence is to verify the edge first, size for survival second, and expect drawdowns that have been modelled in advance rather than discovered in a panic. The tools for the first and last of those, a bet log and a closing-line tracker, sit in the wider infrastructure serious bettors use. The sizing sits with Kelly. This article is the part in between: the arithmetic that decides whether a real edge ever gets the chance to pay.