In probability theory and intertemporal portfolio choice, the Kelly criterion, Kelly strategy, Kelly formula, or Kelly bet, is a formula used to determine the optimal size of a series of bets. In most gambling scenarios, and some investing scenarios under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run (that is, over a span of time in which the observed fraction of bets that are successful equals the probability that any given bet will be successful). It was described by J. L. Kelly, Jr, a researcher at Bell Labs, in 1956. The practical use of the formula has been demonstrated.
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The Kelly Criterion is to bet a predetermined fraction of assets and can be counterintuitive. In one study, each participant was given $25 and asked to bet on a coin that would land heads 60% of the time. The prizes were capped at $250. "Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment. Neither approach is in the least bit optimal." Using the Kelly criterion and based on the odds in the experiment, the right approach would be to bet 20% of the pot on each throw (see first example in Statement below). If losing, the size of the bet gets cut; if winning, the stake increases.
Although the Kelly strategy's promise of doing better than any other strategy in the long run seems compelling, some economists have argued strenuously against it, mainly because an individual's specific investing constraints may override the desire for optimal growth rate. The conventional alternative is expected utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations.
In recent years, Kelly has become a part of mainstream investment theory and the claim has been made that well-known successful investors including Warren Buffett and Bill Gross use Kelly methods. William Poundstone wrote an extensive popular account of the history of Kelly betting.
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:
As an example, if a gamble has a 60% chance of winning (p = 0.60, q = 0.40), and the gambler receives 1-to-1 odds on a winning bet (b = 1), then the gambler should bet 20% of his 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 his bankroll that red will not come up. Unfortunately, the casino doesn't allow betting against something coming up, so a Kelly gambler cannot place a 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
3. If you fail (for which the probability is
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 (
Clearly, in order to decide in favor of investing at least a small amount
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
Proof
Heuristic proofs of the Kelly criterion are straightforward. For a symbolic verification with Python and SymPy one would set the derivative y'(x) of the expected value of the logarithmic bankroll y(x) to 0 and solve for x:
For a rigorous and general proof, see Kelly's original paper or some of the other references listed below. Some corrections have been published.
We give the following non-rigorous argument for the case b = 1 (a 50:50 "even money" bet) to show the general idea and provide some insights.
When b = 1, the Kelly bettor bets 2p - 1 times initial wealth, W, as shown above. If he wins, he has 2pW. If he loses, he has 2(1 - p)W. Suppose he makes N bets like this, and wins K of them. The order of the wins and losses doesn't matter, he will have:
Suppose another bettor bets a different amount, (2p - 1 +
Take the derivative of this with respect to
The turning point of the original function occurs when this derivative equals zero, which occurs at:
which implies:
but:
so in the long run, final wealth is maximized by setting
This illustrates that Kelly has both a deterministic and a stochastic component. If one knows K and N and wishes to pick a constant fraction of wealth to bet each time (otherwise one could cheat and, for example, bet zero after the Kth win knowing that the rest of the bets will lose), one will end up with the most money if one bets:
each time. This is true whether N is small or large. The "long run" part of Kelly is necessary because K is not known in advance, just that as N gets large, K will approach pN. Someone who bets more than Kelly can do better if K > pN for a stretch; someone who bets less than Kelly can do better if K < pN for a stretch, but in the long run, Kelly always wins.
The heuristic proof for the general case proceeds as follows.
In a single trial, if you invest the fraction
Maximizing
For a more detailed discussion of this formula for the general case, see. There, it can be seen that the substitution of
Bernoulli
In a 1738 article, Daniel Bernoulli suggested that, when one has a choice of bets or investments, one should choose that with the highest geometric mean of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is entirely different (Bernoulli wanted to resolve the St. Petersburg paradox).
The Bernoulli article was not translated into English until 1954, but the work was well-known among mathematicians and economists.
Multiple horses
Kelly's criterion may be generalized on gambling on many mutually exclusive outcomes, like in horse races. Suppose there are several mutually exclusive outcomes. The probability that the k-th horse wins the race is
where
Step 1 Calculate the expected revenue rate for all possible (or only for several of the most promising) outcomes:
Step 2 Reorder the outcomes so that the new sequence
Step 3 Set
Step 4 Repeat:
If
Else set
If the optimal set
One may prove that
where the right hand-side is the reserve rate. Therefore the requirement
and the doubling time is
This method of selection of optimal bets may be applied also when probabilities
Single Asset
Considering a single asset (Stock, index etc.) and a risk-free rate, it is easy to obtain the optimal fraction to invest through the Geometric Brownian Motion. Taking the value of asset
Then the expected log return
For a portfolio made of an asset
Solving
Being
Thorp arrived at the same result but through a different deduction / derivation.
Remember that
Many Assets
Consider a market with
Expanding it to the Taylor series around
Thus we reduce the optimization problem to quadratic programming and the unconstrained solution is
where