Risk of Ruin: The Formula Behind Not Going Broke

Most people who allocate capital agree that position sizing matters more than selection. Very few can write down the equation that tells them how much is too much.

Poker players can, and they learned it first. Not because they are better mathematicians, but because the feedback loop is short enough to be unignorable. A trader who is overleveraged might find out in a decade. A poker player finds out this month.

The formula transfers directly, and it is worth knowing whether or not you ever sit at a table.

Why the Game Got There First

Poker produces thousands of independent trials with a measurable edge and measurable variance, which makes it one of the cleanest laboratories for capital allocation that exists outside a textbook.

The culture that grew around that reflects it. According to Pokerology, which has published free strategy lessons since 2004, the object of studying the game is to help players make better decisions at the table. Decisions, not results. That distinction sounds like a platitude until you notice how few investing frameworks are built on it, and how many are built on last year’s returns.

The practical consequence is that serious players size their bankroll before they think about strategy, because a correct strategy applied at the wrong stake is still a losing proposition.

The Equation

For a player or portfolio with a positive expected return, the probability of eventually losing everything is approximately:

RoR = e^(−2μB / σ²)

Three inputs. μ is your expected return per unit of time. σ is your standard deviation over that same unit. B is your bankroll, expressed in the same currency as the other two.

Notice what is absent. There is no term for skill, confidence, experience or conviction. There is your edge, your volatility, and your capital.

Notice also what the shape tells you. The relationship is exponential, not linear, and almost every intuition people carry about bankroll sizing is a linear intuition.

Running the Numbers

Take a competent online cash player: a win rate of 5 big blinds per 100 hands against a standard deviation of 90 big blinds per 100, which is typical for six-handed no-limit. One buy-in is 100 big blinds.

That is the same player with the same skill throughout. Only the capital changes. Going broke at ten buy-ins is not bad luck, it is the expected outcome nearly a third of the time.

Now hold the bankroll at 30 buy-ins and vary the edge instead:

A player with a genuine, positive, measurable edge of 1 BB/100 and a thirty buy-in roll busts roughly half the time. They are not doing anything wrong. Their edge is simply too thin for their capital.

Two Properties Nobody Expects

The exponential form produces two results that are worth committing to memory, and I checked both numerically rather than trusting the intuition.

Doubling your bankroll squares your safety. At 30 buy-ins the risk of ruin is 2.463%. At 60 buy-ins it is 0.0607%, which is 2.463% squared. Not halved. Squared. Capital is the single most efficient lever available.

Halving your edge square-roots it. Drop from 5 BB/100 to 2.5 and the risk of ruin moves from 2.463% to 15.695%, which is precisely the square root of 0.02463. A modest deterioration in edge produces a violent deterioration in survival odds.

Put together: overestimating your edge is far more dangerous than underestimating your capital requirement, because the penalty compounds in the wrong direction.

Kelly, and Why Nobody Sane Bets Full Kelly

The companion result is the Kelly criterion, which gives the fraction of capital that maximises long-run growth. In continuous form it is simply f* = μ/σ².

Apply it honestly to an 8% expected excess return with 16% volatility and it recommends roughly 3.1 times your capital. Which is the first clue that something needs qualifying.

Full Kelly maximises growth and produces drawdowns almost nobody can hold through. The standard response is to bet a fraction of it, and the arithmetic of that trade is unusually favourable:

Half Kelly gives up a quarter of the theoretical growth rate and roughly halves the volatility. That is among the better trades available in finance, and it is why professional practice in both fields clusters well below the theoretical optimum. The portfolio construction literature reaches the same conclusion from a different direction.

The Assumption That Breaks Everything

Every number above depends on knowing μ. You do not know μ. You have an estimate of it, drawn from a finite sample, and that estimate is itself uncertain.

This is the failure mode, and it is the same one in both disciplines. A player calculates their required bankroll using a win rate measured over 20,000 hands, which is nowhere near enough to distinguish a 5 BB/100 winner from a 1 BB/100 winner. A trader sizes positions off a two-year track record and treats the resulting Sharpe ratio as a property of their strategy rather than a noisy sample statistic.

Then look at the second table again. The gap between believing you are a 5 BB/100 player and actually being a 1 BB/100 player is the gap between a 2.5% risk of ruin and a 48% one. Nothing about the strategy changed. Only the accuracy of one input.

Which is why fractional Kelly is not timidity. It is the correct response to parameter uncertainty, and if you are honest about how little you know your own edge, the fraction should be smaller than feels comfortable.

What to Take From It

Three things, none of which require the formula in front of you.

Capital is the most efficient lever you have, because its effect on survival is quadratic rather than linear.

Your edge estimate is the weakest link in the calculation, and errors there punish you disproportionately.

And the growth you surrender by sizing conservatively is much smaller than the growth you surrender by not being in the game at all. That is the whole argument, in poker and in finance alike, and it is a great deal easier to accept before the drawdown than during it.