MTT Variance Calculator

One tournament, as a distribution

Take the field the calculator starts with: 1,000 entries, 150 paid, a standard ladder, and the 9.1% fee that a $10 + $1 entry actually costs you. At an ROI of 20.0%, the model says you walk away with nothing 80.8% of the time. The top prize is 163 buy-ins. A min-cash is 1.08 buy-ins, which is barely more than the entry cost and still counts as a good day.

That is the whole problem in one line: the top 10 finishes out of 1,000 carry 53.1% of everything you will ever collect. Your yearly result is decided by a handful of tournaments you have not played yet.

Where the payout curve comes from

The structure is generated rather than copied: each paid place gets a share of the pool proportional to one over its finishing position, raised to a steepness you can move. That is an approximation, so here is how far off it is. The left-hand figures are what our bankroll maths derives from real published structures; the right-hand ones are what the generated curve produces for the same field and the same number of paid places, through the same function.

The worst error across those structures is 10.8% of the standard deviation, and the steepness that fits them best runs from 0.80 to 1.07. So a standard ladder is a fair stand-in for a real one. The calculator above has no field for a full prize list, so the steepness is the control you get: its buttons run from 0.80 to 1.25, against the 0.80 to 1.07 those published ladders fit, so pick the button nearest your own ladder rather than reading one curve as the truth.

The standard deviation nobody quotes

The figure usually given for a large-field tournament is 3.0 buy-ins, with a range of 2.5 to 3.5. Worked out from the structure instead, this field at this ROI has a standard deviation of 7.5 buy-ins per tournament, which is about 2.5 times the quoted figure. That quoted range is an estimate rather than a measurement, and the note our own bankroll maths keeps beside it says which way it errs: commonly quoted range for large-field MTTs; a figure derived from a top-heavy structure comes out higher, so treat this as a floor. So if you enter one big structure over and over, the derived figure is the one that matches what happens to you, and every sample-size answer further down uses it.

One figure to reconcile before you go on. The published ladder of the same size in the table above, paying 15.0% of the field, comes out at 8.0 buy-ins, which is what the bankroll guide quotes for a field that size. This section says 7.5. Three things separate them and none of them is a disagreement: this one is tilted to a 20.0% ROI rather than no edge, it carries the 9.1% fee rather than none, and it uses the generated ladder, which sits -10.8% from the published one.

What a career looks like at four field sizes

Each row is 1,000 tournaments at the ROI named, simulated. The band is the middle ninety per cent of careers: one in twenty finishes below the low figure and one in twenty above the high one.

Seeded Monte Carlo: 1,500 careers of 1,000 tournaments, seed 20260917, drawn with mulberry32 from downswing-math. Every simulated figure below carries a 95% margin of error and is printed to the number of decimals that margin supports. The same inputs and the same seed reproduce these figures exactly. Profit is in buy-ins, and the cash rate is an output of the model rather than something entered: it is what an ROI that size implies about how often you reach the money in a field that shape. The widest margin on any band in that table is 180 buy-ins, on the 5,000-runner, 30% ROI row, which is why none of the figures is printed to more decimals than it is: a percentile from a simulation of this size is a range rather than a value, and the high end of a career is always the least certain part of it.

The deepest hole, and how long you are in it

The number most players actually want is not the spread of the final result, it is how bad it gets on the way. For the 1,000-runner at 20.0% ROI over 1,000 tournaments, half of all careers include a peak-to-trough fall of at least 135 buy-ins, and one in twenty includes one of at least 264 buy-ins. That is a winning player. The middle of the final results is 187 buy-ins, and the middle ninety per cent runs from -170 to 650.

The worst drawdown is the deepest peak-to-trough fall in cumulative profit inside the career, measured in buy-ins and counting from the first tournament. It has no useful closed form, which is why it is simulated, and it is the figure most players mean when they ask how bad it gets.

Read the last column as tournaments, not as buy-ins: it is the dry stretch that only one career in twenty exceeds. The finish-down column is a percentage of careers and its margin is in percentage points, so read the margins before ranking any two rows. The shortest and longest careers here differ by 44.1 points on how often you finish down, against a combined margin of 2.9 points. The gap is wider than the two margins of error together, so these two can be ranked. Re-running the 1,000-tournament row with nothing changed but the seed moves that figure by 0.9 points, against a combined margin of 2.9 points. The gap is inside the two margins of error together, so this sample cannot rank these two. Treat them as the same figure.

Dry stretches are exact arithmetic

Everything above is simulated and carries a margin of error. The gap between cashes does not need simulating at all. If every tournament is an independent draw with the same chance of cashing, the number of losing entries before the next cash is a geometric distribution, and that has closed-form answers.

The average gap is one minus the cash rate, over the cash rate. The one-in-twenty figure inverts the tail: the chance of a dry stretch reaching a given length is the chance of missing, raised to that length. Neither figure has a margin of error, and both of them are what the simulation is checked against rather than the other way round. One caveat on the last column, since a gap is a whole number of tournaments: it is the shortest gap that ninety-five per cent of gaps come in under, so the chance of actually reaching the figure on each row is 5.0% to 6.3% rather than a round five.

Paying more places

Inside one tournament, places paid is the lever with the largest effect on variance and the one players think about least: across the range below, with the field, the ROI and the steepness held still, it lowers the standard deviation by 25.7%. Which tournament you enter moves it further again: the four fields in the table above span a factor of 15.0 on the same figure. Same field, same ROI, same steepness, only the number of paid places moving:

All figures in buy-ins. Every extra paid place moves money out of the top of the ladder, which lowers the standard deviation and shortens the dry stretches at the same ROI. It is the same reason a single-table sit-and-go is the steadiest tournament format there is, and the reason satellites paying flat seats are steadier still.

Why a break-even player is down more often than half the time

Over 1,000 tournaments at zero ROI in this field, the simulation has you finishing down 56% of the time rather than half, give or take 3 percentage points. The reason is the shape, not bad luck. Most entries lose the buy-in and a rare deep run pays for them, so the middle of the distribution sits below the average, and a coin toss on being up or down is not the right picture until the sample is large enough for the sum to straighten out.

There is a closed-form reason for it: a single correction term for skewness turns the plain normal answer of 50.0% into 53.6%. The skewness of the sum is still too large for a single correction term, so this figure understates how often a break-even player is down. The simulated probability is the one to read; this one only shows which way the asymmetry pushes. The published figure is always the simulated one, with its margin attached.

When your ROI becomes evidence

A career-long result is not the same thing as proof of an edge. The sample below is where a ninety-five per cent confidence interval on the ROI stops touching zero, which is the weakest claim worth making, and it comes from the same ROI calculator maths rather than from a second formula.

The interval narrows with the square root of the sample, so halving the margin needs four times the tournaments. There is no shortcut: a smaller sample hides the uncertainty rather than removing it. Notice that it is not the biggest ROI that is easiest to prove, nor the smallest field: the fewest entries here belong to 180-runner, 15% ROI and the most to 5,000-runner, 30% ROI, a factor of 7.2 between them. What decides it is the standard deviation measured against the ROI, not either figure on its own. It also assumes the standard deviation is known and the interval is a normal approximation on the mean. Tournament results are heavily one-sided, so at small samples that interval is lop-sided rather than merely wide, and nearly every interval that misses the truth sits above it. Treat the figure as a floor on the sample you need.

What this model assumes, and where it stops being true

Every figure here treats your tournaments as independent draws from one fixed distribution at one fixed ROI. Three things that are true of real careers are therefore missing. Your ROI is not a constant: you get better, the games change, and a soft field gets found out. Your ROI is not known either, it is an estimate with its own error, and if the true figure is half what you entered then every band on this page is drawn in the wrong place. And results are not quite independent: the same tilt, the same fatigue and the same leak run through a session, which widens real swings beyond anything a model of independent draws produces. Read the output as the variance of the model you described, not as a forecast of your year.

• One tournament is a single draw from the payout structure, not a normal distribution. That is the whole reason this tool exists separately from the cash-game variance calculator.
• Finishing positions are tilted towards the top of the field by one parameter, chosen so the mean result equals the ROI you entered. It is the smallest change to an equal chance of every finish that reaches that ROI, which means the model assumes your edge shows up in how deep you run rather than in how often you scrape a min-cash.
• ROI is fixed and known. It is neither, and it is the single largest source of error in the answer: an ROI measured over a few hundred tournaments could be half or double what you typed.
• Every tournament is an independent draw. Re-entries, multi-day events, satellites into bigger fields, bounty money paid outside the prize pool, staking and rakeback are all outside the model.
• One buy-in means the total cost of entering, fee included, because that is the money you had to put up. Rake is entered as the share of that total which never reaches the prize pool.
• Field size and places paid are fixed for the whole career being simulated, and every tournament uses the same structure. A real schedule mixes fields, and mixing them changes the variance.
• The generated payout curve is a smooth approximation to a real structure. Real ladders round their prizes, flatten the top few places and pay a fixed min-cash, and the guide shows how far the approximation sits from published structures rather than claiming it is one.

For cash games, none of this applies and the normal approximation is the right tool: use the variance simulator for sample paths in big blinds and the downswing calculator for how deep a normal cash downswing goes. To size the roll that survives the drawdowns above, the bankroll calculator takes an ROI and a per-tournament standard deviation, and the figure to feed it is the one this page derives rather than a preset. If you are trying to work out your ROI in the first place, that is the ROI calculator.