The Kelly formula is widely used in gambling. It is a variable staking approach where stakes depend on the profitability of a bet. The more profitable a bet, the higher the stake. Kelly has demonstrated that it is the staking approach that maximizes the geometric growth rate of a gambler’s portfolio.
Kelly = (P * odds – 1)/(odds – 1)
Assuming you start with a bank of 100, Kelly states that to maximize your returns you should bet a proportion of your bank equal to (P * odds – 1)/(odds – 1), where odds are decimal and P is the winning probability of a runner. The bank value should be updated at each bet.
The numerator is equal to the expected return of the bet. So, the more profitable the bet, the higher the fraction invested. The denominator is a function of the odds. Risky bets tend to come at higher odds so, intuitively, the Kelly criterion establishes a trade-off between expected return and risk.