Statement
For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the Kelly bet is:
where:
- f* is the fraction of the current bankroll to wager;
- b is the net odds received on the wager ("b to 1"); that is, you could win $b (plus the $1 wagered) for a $1 bet
- p is the probability of winning;
- q is the probability of losing, which is 1 − p.
As an example, if a gamble has a 60% chance of winning (p = 0.60, q = 0.40), but the gambler receives 1-to-1 odds on a winning bet (b = 1), then the gambler should bet 20% of the bankroll at each opportunity (f* = 0.20), in order to maximize the long-run growth rate of the bankroll.
If the gambler has zero edge, i.e. if b = q / p, then the criterion recommends the gambler bets nothing. If the edge is negative (b < q / p) the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in standard American roulette, the bettor is offered an even money payoff (b = 1) on red, when there are 18 red numbers and 20 non-red numbers on the wheel (p = 18/38). The Kelly bet is -1/19, meaning the gambler should bet one-nineteenth of the bankroll that red will not come up. Unfortunately, the casino doesn't allow betting against red, so a Kelly gambler could not bet.
The top of the first fraction is the expected net winnings from a $1 bet, since the two outcomes are that you either win $b with probability p, or lose the $1 wagered, i.e. win $-1, with probability q. Hence:
For even-money bets (i.e. when b = 1), the first formula can be simplified to:
Since q = 1-p, this simplifies further to
A more general problem relevant for investment decisions is the following:
1. The probability of success is .
2. If you succeed, the value of your investment increases from to .
3. If you fail (for which the probability is ) the value of your investment decreases from to . (Note that the previous description above assumes that a is 1).
In this case, the Kelly criterion turns out to be the relatively simple expression
Note that this reduces to the original expression for the special case above for .
Clearly, in order to decide in favor of investing at least a small amount, you must have
which obviously is nothing more than the fact that your expected profit must exceed the expected loss for the investment to make any sense.
The general result clarifies why leveraging (taking a loan to invest) decreases the optimal fraction to be invested, as in that case . Obviously, no matter how large the probability of success, is, if is sufficiently large, the optimal fraction to invest is zero. Thus using too much margin is not a good investment strategy, no matter how good an investor you are.
Read more about this topic: Kelly Criterion
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