Kelly stake, the fraction, and where the formula turns against you
Your data
The first field is the one nobody can check, and it is the one the answer hangs on. It is your estimate of how often this wins, not a fact about the world, and the page treats it as the guess it is.
Nothing here recommends betting, and nothing here says an edge exists. The page sizes a bet that somebody has already decided to make, and its most useful output is the point at which the formula turns against them.
Results
| Edge, as the formula measures it | — |
| Full formula stake, as a share of the money, in percent | — |
| The same in money | — |
| What your chosen fraction stakes instead | — |
| Growth per bet at the full stake, in percent | — |
| Growth per bet at your fraction, in percent | — |
| Share of the growth your fraction keeps, in percent | — |
| Stake at which growth reaches zero, in percent of the money | — |
| How many times the full formula that is | — |
The reason to stake less than the formula says is not timidity, it is that the curve is flat at the top and a cliff on the far side. With the numbers this page starts from, half the stake keeps about three quarters of the growth and three quarters of the stake keeps about ninety four per cent of it. You give up very little and you halve what a bad run does to you.
The same bet at different fractions
| Times the full formula | Money staked | Growth per bet, in percent | Share of the growth kept, in percent |
|---|
Now read the bottom of that table, because it is the part the formula is famous for and the part people skip. Past a stake of a little under twice the full figure the growth per bet is negative, and negative growth with a real edge still ends at nothing. Betting more than the formula says does not buy speed, it buys a slower version of losing.
The page works that crossing out exactly rather than repeating the rule of thumb, because it moves with the odds. It is not always at twice: with these numbers it lands at one point nine five, and a different price puts it somewhere else.
There is a third thing, and it is the real argument for using a fraction. The stake is far more sensitive to your estimate of winning than to anything else on the page. At even money the formula reduces to twice your probability minus one, so an estimate of sixty asks for twenty per cent of the money and an estimate of fifty five asks for ten. Five points of error, half the stake.
Nobody knows their own probability to five points. That is not a criticism of anyone, it is what estimating is; and it means the full formula is being fed a number with more uncertainty in it than the difference between a sensible stake and a reckless one. Staking a fraction is what you do when you know the input is soft.
What does staking half the formula cost me?
About a quarter of the growth, for half the exposure. With the numbers this page starts from the full stake grows at 0.02014 per bet and half of it grows at 0.01504, which is 74.7 per cent of the same thing.
Three quarters of the stake keeps 93.6 per cent. The curve is flat near the top, so the first cuts cost almost nothing while removing a great deal of the swing.
| Times the full formula | Share of the growth kept, in percent |
|---|---|
| 0,25 | 43,5 |
| 0,50 | 74,7 |
| 0,75 | 93,6 |
| 1,00 | 100,0 |
| 1,50 | 73,2 |
| 2,00 | -12,2 |
Can I still lose money with a real edge?
Yes, by staking too much of it. Past a little under twice the full figure the growth per bet is negative even though the edge never went away, and negative growth compounds to nothing just as reliably as positive growth compounds upward.
With these numbers that crossing sits at 1.95 times the full stake. It is not always at twice, which is why the page works it out from your odds instead of repeating the rule of thumb.
Why use a fraction rather than the full formula?
Because the stake is far more sensitive to your estimate of winning than to anything else. At even money the formula reduces to twice your probability minus one, so sixty per cent asks for twenty per cent of the money and fifty five asks for ten.
Five points of error halves the stake, and nobody estimates their own chances to five points. A fraction is what you use when you know the input is soft, which it always is.
What if the odds do not beat my estimate?
Then there is no edge and the formula asks for a stake of nothing, which the page says plainly rather than printing a negative number.
A negative result would only mean the other side of the bet looks good, and sizing that is a different question from the one asked here.
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