Kelly Criterion Position Sizing Calculator

Purpose and model boundary

This calculator sizes a repeated bet with the Kelly criterion. You enter your chance of winning each bet and what a win pays, and it returns the Kelly bet: the share of your bankroll that makes the bankroll grow fastest over many bets. It then sizes your own bet as a share of the Kelly bet, and simulates 100 possible sequences of 200 bets at full, half and quarter Kelly and at your size, all on the same wins and losses, to show how far each one falls from a previous high and how its bankroll spreads.

It is the betting-and-sizing companion of the Monte Carlo Retirement Success Calculator, which uses the same seeded generator. Every figure on this page is computed by the spreadsheet; the page adds nothing to it. It is an educational estimate from the chance and odds you enter. It does not know whether you really have an edge, it does not recommend a bet, and it is not financial, investment, betting, tax or legal advice.

Inputs and units

• Chance of winning each bet, as a percentage from 1% to 99%: your estimate. The Kelly bet is only as good as this estimate; an overstated chance gives an oversized bet.
• Amount won per 1 lost, the net odds b, from 0.05 to 20: what a winning bet pays per 1 staked beyond the stake, when a losing bet loses the whole stake. Even money is 1; odds of 3 to 1 are 3; decimal odds of 2.5 are 1.5.
• Bankroll, in dollars: the money set aside for these bets.
• Share of the Kelly bet you place, from 5% to 100%: 50% is half Kelly.
• Largest fall from a peak you can accept, from 5% to 95% of that peak.
• Random seed: the starting number of the spreadsheet's random-number generator. The page does not show it: each press of Simulate first replaces it with the next seed in the stream, except the first press after an example is loaded or after Reset, which keeps that seed. Through the API or MCP the seed you pass is used as given; blank means 42, the seed of the examples.

Shares, chances and growth rates are percentages. Money is in dollars.

Governing relationships

A bet wins with probability p and pays b per 1 staked; a loss loses the stake. Betting a share f of the bankroll multiplies it by 1 + f b after a win and by 1 − f after a loss, so the expected growth of its logarithm per bet is:

g(f) = p × ln(1 + f × b) + (1 − p) × ln(1 − f)

The Kelly bet maximises g:

f* = p − (1 − p) ÷ b

When f* is zero or less the bet has no edge, and the bet is nothing. Half and quarter Kelly bet f*/2 and f*/4, and your size is f* times the share you enter. Over many bets the typical bankroll grows like e raised to the number of bets times g, so g is the growth rate per bet. For a small edge, half Kelly keeps about three quarters of the full-Kelly growth rate and quarter Kelly about seven sixteenths. Above the Kelly bet growth falls, and near twice the Kelly bet it turns negative.

Each future draws its bets from one seeded generator, the minimal standard generator with multiplier 48,271, x(next) = 48271 × x mod (2^31 − 1). Future r owns four consecutive stretches of 64 steps of the stream, and its 200 bets are the first 200 states of that run, so no two futures share a draw. A bet wins when its state is below INT(p × (2^31 − 1)) + 1, which is the same as its uniform draw x ÷ (2^31 − 1) being below p. Every bet size uses the same wins and losses.

The ratio R of the bankroll to its previous high starts at 1 and after each bet is R = min(1, R × the bet's multiplier). The largest fall from a peak is 1 minus the lowest R, and it counts as larger than you can accept when the lowest R is below 1 minus your tolerance; a fall exactly equal to the tolerance counts as not larger. After t bets with K wins the bankroll is:

B(t) = bankroll × (1 + f × b)^K × (1 − f)^(t − K)

Calculation sequence

The spreadsheet first computes the Kelly bet, the four bet sizes and their growth rates. It then draws 100 futures of 200 bets, counts the wins after every 10 bets, tracks each bet size's ratio to its previous high bet by bet, and computes each bankroll after 0, 10, …, 200 bets. From these it takes, for each bet size, the share of futures with a fall larger than you can accept, the median of each future's largest fall, the median, 10th and 90th percentile bankroll after 200 bets, the share ending below the starting bankroll, and the median and 10th percentile after every 10 bets. Beside the simulation it evaluates the growth rate at 61 bet sizes, from 0 to three times the Kelly bet in steps of a twentieth of it, stopping at 95% of the bankroll, so the Kelly bet and half of it are exact points of the curve.

Outputs and interpretation

The headline is the Kelly bet as a share of the bankroll. Beside it are your bet as a share and in dollars, and the chance of a fall from a peak larger than you can accept at your size, an estimate from 100 futures whose 95 percent sampling margin is about 9.8 percentage points, which the page reports with the seed used and the workbook's self-check. The sizing table compares the four bet sizes, and the bankroll table gives the median and 10th percentile of the three fixed sizes after every 10 bets. The growth rates, the share of the growth kept, each size's chance of a fall and its median and 10th-percentile bankroll, the next seed, the number of futures and the growth curve's table are returned to the API and MCP.

Under the default inputs (a 55% chance at even money, half Kelly, a 50% tolerance) the Kelly bet is 10% of the bankroll, so half Kelly bets $500 of $10,000 and keeps about 75% of the full-Kelly growth rate. Over 200 bets, 93 of the 100 futures fall more than 50% from a peak at full Kelly, 24 at half Kelly and none at quarter Kelly. Almost every full-Kelly path falls far from some peak over 200 bets; that is the usual case for betting a fraction of Kelly. The often-quoted result that the chance of ever falling to a share x of the starting bankroll is x concerns the start, not the running peak, and is not the chance shown here.

The growth chart draws the growth rate per bet against the bet size, in red past the Kelly bet, with dashed lines at the Kelly bet and half of it. The bankroll chart draws the median of each fixed size with its 10th percentile dashed beneath it and the starting bankroll. Strong edges give enormous simulated bankrolls after 200 bets; they follow from the model, and real limits on bets and an uncertain edge make them unreachable.

Validation and status logic

The workbook evaluates the conditions below from top to bottom and returns the first matching status.

NOT VALID is a refusal, not a numeric answer: every number reads zero and the tables are empty. CHECK accompanies a computed result but calls for the stated review; with no edge the simulation still runs, every bet size is nothing and every bankroll stays where it started. The self-check is reported separately as arithmetic-integrity evidence; it does not replace this ordered status chain.

Assumptions and limitations

The chance of winning you enter is taken as exact: the model has no uncertainty about the edge itself, which is the main real-world risk of Kelly betting. Every bet has the same chance and odds, bets are independent, a loss loses the whole stake and a win pays the stated odds. There are exactly 200 bets, each sized as a fixed share of the bankroll at that moment, with no minimum or maximum stake, no rounding to whole units, no commissions, fees or taxes, and no borrowing. One hundred futures give an estimate with the sampling margin shown, not a forecast.

Restrictions and non-computing states

This calculator enforces its numeric limits before it accepts a result. The odds are capped at 20 to 1 so that every simulated bankroll stays a finite number, and the share of the Kelly bet is capped at 100%: betting more than Kelly is shown on the growth curve, not simulated. A retained seed remains part of every simulation.

Errors and warnings

A request rejected by the page's input limits is not a computed result. A transport or service failure cannot establish a fresh workbook answer. Review the workbook's References and Disclaimer, and check any plan with a qualified professional before using a result for a real decision.

Sources and further reading

The chance of winning, the odds, the bankroll and the limits come from the inputs, not market data. The workbook's References sheet identifies the calculation sources. The links below explain the ideas behind them; they do not endorse this calculator.

• J. L. Kelly Jr., “A New Interpretation of Information Rate”, Bell System Technical Journal 35(4), 1956, introduces the bet that maximises the long-run growth rate of a bankroll.
• E. O. Thorp, “The Kelly criterion in blackjack, sports betting and the stock market”, in Handbook of Asset and Liability Management, volume 1, 2006, derives the Kelly fraction p − q ÷ b for a bet paying b to 1 and discusses betting a fraction of it.
• L. C. MacLean, E. O. Thorp and W. T. Ziemba (eds.), The Kelly Capital Growth Investment Criterion: Theory and Practice, World Scientific, 2011, collects the work on the trade-off between growth and falls from a peak at fractions of Kelly.
• S. K. Park and K. W. Miller, “Random number generators: good ones are hard to find”, Communications of the ACM 31(10), 1988, and the Park–Miller–Stockmeyer 1993 correspondence describe the generator family. The workbook uses multiplier 48,271 and modulus 2³¹ − 1.
• Microsoft Support's PERCENTILE.INC and COUNTIF documentation describe the percentile and counting functions used by the workbook.

This is an educational estimate, not financial, investment, betting, tax or legal advice. Check any plan with a qualified professional.