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probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, the Kelly criterion (or Kelly strategy or Kelly bet), is a formula that determines the optimal theoretical size for a bet. It is valid when the expected returns are known. The Kelly bet size is found by maximizing the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. J. L. Kelly Jr, a researcher at
Bell Labs Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984), then AT&T Bell Laboratories (1984–1996) and Bell Labs Innovations (1996–2007), is an American industrial research and scientific development company owned by mult ...
, described the criterion in 1956. Because the Kelly Criterion leads to higher wealth than any other strategy in the long run (i.e., the theoretical maximum return as the number of bets goes to infinity), it is a scientific gambling method. The practical use of the formula has been demonstrated for gambling and the same idea was used to explain
diversification Diversification may refer to: Biology and agriculture * Genetic divergence, emergence of subpopulations that have accumulated independent genetic changes * Agricultural diversification involves the re-allocation of some of a farm's resources to n ...
in investment management., page 184f. In the 2000s, Kelly-style analysis became a part of mainstream investment theory and the claim has been made that well-known successful investors including
Warren Buffett Warren Edward Buffett ( ; born August 30, 1930) is an American business magnate, investor, and philanthropist. He is currently the chairman and CEO of Berkshire Hathaway. He is one of the most successful investors in the world and has a net ...
and Bill Gross use Kelly methods. William Poundstone wrote an extensive popular account of the history of Kelly betting. Also see
Intertemporal portfolio choice Intertemporal portfolio choice is the process of allocating one's investable wealth to various assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines ...
.


Optimal betting example

In a study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250. But the behavior of the test subjects was far from optimal: Using the Kelly criterion and based on the odds in the experiment (ignoring the cap of $250 and the finite duration of the test), the right approach would be to bet 20% of one's bankroll on each toss of the coin, which works out to a 2.034% average gain each round. This is a
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
, not the arithmetic rate of 4% ( r=(1+ 0.2 \cdot 1.0)^ \cdot (1-0.2 \cdot 1.0)^). The theoretical expected wealth after 300 rounds works out to $10,505 (= 25 \cdot (1.02034) ^ ) if it were not capped. In this particular game, because of the cap, a strategy of betting only 12% of the pot on each toss would have even better results (a 95% probability of reaching the cap and an average payout of $242.03).


Gambling formula

Where losing the bet involves losing the entire wager, the Kelly bet is: : f^* = p-\frac = p - \frac where: * f^ is the fraction of the current bankroll to wager. * p is the probability of a win. * q is the probability of a loss ( q = 1 - p). * b is the proportion of the bet gained with a win. E.g. If betting $10 on a 2-to-1 odds bet, (upon win you are returned $30, winning you $20), then b = \$20/\$10 = 2.0. As an example, if a gamble has a 60% chance of winning (p = 0.6, q = 0.4), and the gambler receives 1-to-1 odds on a winning bet (b=1), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity (f^ = 0.6-\frac = 0.2). If the gambler has zero edge, i.e. if b = q / p, then the criterion recommends for the gambler to bet 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
American roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...
, 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 their bankroll that red will ''not'' come up. There is no explicit ''anti-red'' bet offered with comparable odds in roulette, so the best a Kelly gambler can do is bet nothing.


Investment formula

A more general form of the Kelly formula allows for partial losses, which is relevant for investments: : f^ = \frac-\frac where: * f^ is the fraction of the assets to apply to the security. * p is the probability that the investment increases in value. * q is the probability that the investment decreases in value ( q = 1 - p). * a is the fraction that is lost in a negative outcome. If the security price falls 10%, then a = 0.1 * b is the fraction that is gained in a positive outcome. If the security price rises 10%, then b = 0.1. Note that the Kelly Criterion is valid only for ''known'' outcome probabilities, which is not the case with investments.
Risk averse In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more ...
investors should not invest the full Kelly fraction. This formula can result in Kelly fractions higher than 1. In this case, it is theoretically advantageous to use leverage to purchase additional securities on
margin Margin may refer to: Physical or graphical edges * Margin (typography), the white space that surrounds the content of a page *Continental margin, the zone of the ocean floor that separates the thin oceanic crust from thick continental crust *Leaf ...
.


Proof

Heuristic proofs of the Kelly criterion are straightforward. The Kelly criterion maximizes the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of the logarithm of wealth (the expectation value of a function is given by the sum, over all possible outcomes, of the probability of each particular outcome multiplied by the value of the function in the event of that outcome). We start with 1 unit of wealth and bet a fraction f of that wealth on an outcome that occurs with probability p and offers odds of b. The probability of winning is p, and in that case the resulting wealth is equal to 1+fb. The probability of losing is q=1-p, and in that case the resulting wealth is equal to 1-fa. Therefore, the expected geometric growth rate r is: : r=(1+fb)^p\cdot(1-fa)^ We want to find the maximum ''r'' of this curve (as a function of ''f''), which involves finding the derivative of the equation. This is more easily accomplished by taking the logarithm of each side first. The resulting equation is: : E = \log(r) = p \log(1+fb)+q\log(1-fa) with E denoting logarithmic wealth growth. To find the value of f for which the growth rate is maximized, denoted as f^, we differentiate the above expression and set this equal to zero. This gives: : \left.\frac\_=\frac+\frac=0 Rearranging this equation to solve for the value of f^ gives the Kelly criterion: : f^ = \frac-\frac Notice that this expression reduces to the simple gambling formula when a=1=100\%, when a loss results in full loss of the wager.


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 In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is different (Bernoulli wanted to resolve the
St. Petersburg paradox The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the theoretical lottery game approaches infinity but nevertheless seems to be worth only a very small amount to t ...
). An English-language translation of the Bernoulli article was not published until 1954, but the work was well known among mathematicians and economists.


Application to the stock market

In mathematical finance, if security weights maximize the expected geometric growth rate (which is equivalent to maximizing log wealth), then a portfolio is ''growth optimal.'' Computations of growth optimal portfolios can suffer tremendous garbage in, garbage out problems. For example, the cases below take as given the expected return and covariance structure of assets, but these parameters are at best estimates or models that have significant uncertainty. If portfolio weights are largely a function of estimation errors, then ''Ex-post'' performance of a growth-optimal portfolio may differ fantastically from the ''ex-ante'' prediction. Parameter uncertainty and estimation errors are a large topic in portfolio theory. An approach to counteract the unknown risk is to invest less than the Kelly criterion (e.g., half).


Criticism

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 In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
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.


Advanced mathematics

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 with b = 1 (a 50:50 "even money" bet) to show the general idea and provide some insights. When b = 1, a Kelly bettor bets 2p - 1 times their initial wealth W, as shown above. If they win, they have 2pW after one bet. If they lose, they have 2(1 - p)W. Suppose they make N bets like this, and win K times out of this series of N bets. The resulting wealth will be: : 2^Np^K(1-p)^W \! . Note that the ordering of the wins and losses does not affect the resulting wealth. Suppose another bettor bets a different amount, (2p - 1 + \Delta)W for some value of \Delta (where \Delta may be positive or negative). They will have (2p + \Delta)W after a win and (1-p)-\Delta after a loss. After the same series of wins and losses as the Kelly bettor, they will have: : (2p+\Delta)^K (1-p)-\DeltaW Take the derivative of this with respect to \Delta and get: : K(2p+\Delta)^ (1-p)-\DeltaW-(N-K)(2p+\Delta)^K (1-p)-\DeltaW The function is maximized when this derivative is equal to zero, which occurs at: : K (1-p)-\Delta(N-K)(2p+\Delta) which implies that : \Delta=2\left(\frac-p\right) but the proportion of winning bets will eventually converge to: : \lim_\frac=p according to the
weak law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
. So in the long run, final wealth is maximized by setting \Delta to zero, which means following the Kelly strategy. 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: : \left(2\frac-1\right)W 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 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 f of your capital, if your strategy succeeds, your capital at the end of the trial increases by the factor 1-f + f(1+b) = 1+fb, and, likewise, if the strategy fails, you end up having your capital decreased by the factor 1-fa. Thus at the end of N trials (with pN successes and qN failures), the starting capital of $1 yields : C_N=(1+fb)^(1-fa)^. Maximizing \log(C_N)/N, and consequently C_N, with respect to f leads to the desired result : f^=p/a-q/b .
Edward O. Thorp Edward Oakley Thorp (born August 14, 1932) is an American mathematics professor, author, hedge fund manager, and blackjack researcher. He pioneered the modern applications of probability theory, including the harnessing of very small correlatio ...
provided a more detailed discussion of this formula for the general case. There, it can be seen that the substitution of p for the ratio of the number of "successes" to the number of trials implies that the number of trials must be very large, since p is defined as the limit of this ratio as the number of trials goes to infinity. In brief, betting f^ each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and p and b are the same for each trial. In practice, this is a matter of playing the same game over and over, where the probability of winning and the payoff odds are always the same. In the heuristic proof above, pN successes and qN failures are highly likely only for very large N.


Multiple outcomes

Kelly's criterion may be generalizedSmoczynski, Peter; Tomkins, Dave (2010) "An explicit solution to the problem of optimizing the allocations of a bettor’s wealth when wagering on horse races", Mathematical Scientist", 35 (1), 10-17 on gambling on many mutually exclusive outcomes, such as in horse races. Suppose there are several mutually exclusive outcomes. The probability that the k-th horse wins the race is p_k, the total amount of bets placed on k-th horse is B_k, and :\beta_k=\frac=\frac , where Q_k are the pay-off odds. D=1-tt, is the dividend rate where tt is the track take or tax, \frac is the revenue rate after deduction of the track take when k-th horse wins. The fraction of the bettor's funds to bet on k-th horse is f_k. Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set S^o of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions f^o_k of bettor's wealth to be bet on the outcomes included in the optimal set S^o. The algorithm for the optimal set of outcomes consists of four steps: # Calculate the expected revenue rate for all possible (or only for several of the most promising) outcomes: er_i = \frac = p_i(Q_i + 1) # Reorder the outcomes so that the new sequence er_k is non-increasing. Thus er_1 will be the best bet. # Set S = \varnothing (the empty set), k = 1, R(S)=1. Thus the best bet er_k = er_1 will be considered first. # Repeat: #:If er_k=\fracp_k > R(S) then insert k-th outcome into the set: S = S \cup \, recalculate R(S) according to the formula: R(S) = \frac and then set k = k+1 , Otherwise, set S^o=S and stop the repetition. If the optimal set S^o is empty then do not bet at all. If the set S^o of optimal outcomes is not empty, then the optimal fraction f^o_k to bet on k-th outcome may be calculated from this formula: : f_i = p_i - \beta_i \frac. One may prove that :R(S^o)=1-\sum_ where the right hand-side is the reserve rate. Therefore, the requirement er_k=\fracp_k > R(S) may be interpreted as follows: k-th outcome is included in the set S^o of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction f^o_k may be interpreted as the excess of the expected revenue rate of k-th horse over the reserve rate divided by the revenue after deduction of the track take when k-th horse wins or as the excess of the probability of k-th horse winning over the reserve rate divided by revenue after deduction of the track take when k-th horse wins. The binary growth exponent is :G^o=\sum_ + \left(1-\sum_\right)\log_2(R(S^o)) , and the doubling time is : T_d=\frac. This method of selection of optimal bets may be applied also when probabilities p_k are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that : \sum_i < 1 , and : \sum_i < 1.


Stock investments

The second-order Taylor polynomial can be used as a good approximation of the main criterion. Primarily, it is useful for stock investment, where the fraction devoted to investment is based on simple characteristics that can be easily estimated from existing historical data –
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
and variance. This approximation leads to results that are robust and offer similar results as the original criterion. For single assets(stock, index fund, etc.), and a risk-free rate, it is easy to obtain the optimal fraction to invest through geometric Brownian motion. The value of a lognormally distributed asset S at time t (S_t) is : S_t = S_0\exp\left( \left(\mu - \frac \right)t + \sigma W_t\right), from the solution of the geometric Brownian motion where W_t is a Wiener process, and \mu (percentage drift) and \sigma (the percentage volatility) are constants. Taking expectations of the logarithm: :\mathbb \log(S_t)=\log(S_0)+\left(\mu- \frac \right)t. Then the expected log return R_s is :R_s = \left(\mu -\frac\,\right) s. For a portfolio made of an asset S and a bond paying risk-free rate r, with fraction f invested in S and (1-f) in the bond, the expected one-period return is given by : \mathbb = \mathbb + (1 - f) r however people seem to deal with the expected log return G(f) for one-period instead in the context of Kelly: :G(f) = f\mu - \frac + ((1-f)\ r). Solving \max (G(f)) we obtain :f^* = \frac . f^* is the fraction that maximizes the expected logarithmic return, and so, is the Kelly fraction. Thorp arrived at the same result but through a different derivation. Remember that \mu is different from the asset log return R_s. Confusing this is a common mistake made by websites and articles talking about the Kelly Criterion. For multiple assets, consider a market with n correlated stocks S_k with stochastic returns r_k, k= 1, \dots, n, and a riskless bond with return r. An investor puts a fraction u_k of their capital in S_k and the rest is invested in the bond. Without loss of generality, assume that investor's starting capital is equal to 1. According to the Kelly criterion one should maximize :\mathbb\left \ln\left((1 + r) + \sum\limits_^n u_k(r_k -r) \right) \right Expanding this with a Taylor series around \vec = (0, \ldots , 0) we obtain :\mathbb \left \ln(1+r) + \sum\limits_^ \frac - \frac\sum\limits_^\sum\limits_^ u_k u_j \frac \right Thus we reduce the optimization problem to quadratic programming and the unconstrained solution is : \vec = (1+r) ( \widehat )^ ( \widehat - r ) where \widehat and \widehat are the vector of means and the matrix of second mixed noncentral moments of the excess returns. There is also a numerical algorithm for the fractional Kelly strategies and for the optimal solution under no leverage and no short selling constraints.


See also

*
Risk of ruin Risk of ruin is a concept in gambling, insurance, and finance relating to the likelihood of losing all one's investment capital or extinguishing one's bankroll below the minimum for further play. For instance, if someone bets all their money on a s ...
*
Gambling and information theory Statistical inference might be thought of as gambling theory applied to the world around us. The myriad applications for logarithmic information measures tell us precisely how to take the best guess in the face of partial information. In that sens ...
* Proebsting's paradox * Merton's portfolio problem


References


External links

{{Authority control Optimal decisions Gambling mathematics Information theory Wagering Articles containing proofs 1956 introductions Portfolio theories