Glossary term
Kelly fraction
Definition
f* = p − (1 − p) / b. The growth-optimal fraction under assumed odds. Fragile to bad p and b. Never an order size for a person.
How it works
The Kelly criterion gives the fraction of capital that maximizes long-run compound growth for a repeated bet. With win probability p and net odds b (the amount won per unit risked), f* = p − (1 − p) / b. The edge disappears when p = 1 / (1 + b); below that, the formula returns zero or a negative number.
Its assumptions are the whole story: it takes p and b as known, assumes the bet repeats under identical conditions, and optimizes the long-run limit with no regard for how deep intermediate falls get. Practitioners often study half or quarter Kelly because over-betting costs far more than under-betting saves.
Worked example
Assume p = 0.55 and b = 1.5. Then f* = 0.55 − 0.45 / 1.5 = 0.55 − 0.30 = 0.25, or 25%. Half Kelly is 12.5%. With exactly correct inputs, growth per bet at 25% is 0.55 × ln(1.375) + 0.45 × ln(0.75) ≈ 4.57%, against about 3.44% at half Kelly.
Now suppose the true win rate was 47% while the bet was still sized at 25%. Growth per bet is 0.47 × ln(1.375) + 0.53 × ln(0.75) ≈ −0.28%, so the book shrinks, while 12.5% still compounds at about +1.00%. A mis-estimate of eight points, easy to make from a small sample, flips the result.
Hypothetical, for education only. Not a recommendation and not a prediction.
Common mistake
The mistake is quoting f* to a decimal place and treating it as an order size. The precision comes from the arithmetic, never from the evidence. Small changes in p move the output a long way, and an output of 25.0% usually hides an honest range. In this course the formula stops at a fraction: turning it into a share count needs a person's balance, and that step is advice.