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| 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 [[John Larry Kelly, Jr|J. L. Kelly, Jr]] in 1956.<ref name="original Kelly article">{{cite doi|10.1002/j.1538-7305.1956.tb03809.x}}</ref> The practical use of the formula has been demonstrated.<ref name="Thorp talk">{{Citation |last=Thorp |first=E. O. |date=January 1961 |title=Fortune's Formula: The Game of Blackjack |work=American Mathematical Society }}</ref><ref name="Beat the Dealer">{{Citation |last=Thorp |first=E. O. |year=1962 |title=Beat the dealer: a winning strategy for the game of twenty-one. A scientific analysis of the world-wide game known variously as blackjack, twenty-one, vingt-et-un, pontoon or Van John |publisher=Blaisdell Pub. Co }}</ref><ref name="Beat the Market">{{Citation |last=Thorp |first=Edward O. |last2=Kassouf |first2=Sheen T. |year=1967 |title=Beat the Market: A Scientific Stock Market System |publisher=Random House |isbn=0-394-42439-5 | url = http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/beatthemarket.pdf}}{{page needed|date=July 2012}}</ref>
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| 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.<ref name="Poundstone book article">{{citation |last=Poundstone |first=William|title=Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street |year=2005 |publisher=Hill and Wang |location=New York |isbn=0-8090-4637-7 }}</ref> The conventional alternative is [[expected utility]] theory which says bets should be sized to [[Optimization (mathematics)|maximize]] the [[Expectation-maximization algorithm|expected]] utility of the outcome (to an individual with [[logarithm]]ic utility, the Kelly bet maximizes expected utility, so there is no conflict). 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.<ref name="Wilmott I">{{citation |last=Thorp |first=E. O. |title=The Kelly Criterion: Part I |date=May 2008 |journal=Wilmott Magazine }}</ref>
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| In recent years, Kelly has become a part of mainstream investment theory<ref name="Handbook of Asset and Liability Management">{{citation |last=Zenios |first=S. A. |last2=Ziemba |first2=W. T. |title=Handbook of Asset and Liability Management |year=2006 |publisher=North Holland |location= |isbn=978-0-444-50875-1 }}</ref> and the claim has been made that well-known successful investors including [[Warren Buffett]]<ref name="The Dhandho Investor">{{citation |last=Pabrai |first=Mohnish |title=The Dhandho Investor: The Low-Risk Value Method to High Returns |year=2007 |publisher=Wiley |location= |isbn=978-0-470-04389-9}}</ref> and [[William H. Gross|Bill Gross]]<ref name="Wilmott II">{{citation |last=Thorp |first=E. O. |title=The Kelly Criterion: Part II |date=September 2008 |journal=Wilmott Magazine }}</ref> use Kelly methods. [[William Poundstone]] wrote an extensive popular account of the history of Kelly betting.<ref name="Poundstone book article" />
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| == Statement ==
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| 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:
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| : <math> f^{*} = \frac{bp - q}{b} = \frac{p(b + 1) - 1}{b}, \! </math>
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| where:
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| * ''f''* is the fraction of the current bankroll to wager;
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| * ''b'' is the net odds received on the wager ("''b'' to 1"); that is, you could win $b (and get a return of your $1 wagered) for a $1 bet
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| * ''p'' is the probability of winning;
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| * ''q'' is the probability of losing, which is 1 − ''p''.
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| 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.
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| 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.
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| 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:
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| : <math> f^{*} = \frac{\text{expected net winnings}}{\text{net winnings if you win}} \! </math>
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| For even-money bets (i.e. when ''b'' = 1), the first formula can be simplified to:
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| : <math> f^{*} = p - q . \! </math>
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| Since q = 1-p, this simplifies further to | |
| : <math> f^{*} = 2p - 1 . \! </math>
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| A more general problem relevant for investment decisions is the following:
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| 1. The probability of success is <math>p</math>.
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| 2. If you succeed, the value of your investment increases from <math>1</math> to <math>1+b</math>.
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| 3. If you fail (for which the probability is <math>q=1-p</math>) the value of your investment decreases from <math>1</math> to <math>1-a</math>. (Note that the previous description above assumes that a is 1).
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| In this case, the Kelly criterion turns out to be the relatively simple expression
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| : <math> f^{*} = p/a - q/b . \! </math>
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| Note that this reduces to the original expression for the special case above (<math>f^{*}=p-q</math>) for <math>b=a=1</math>.
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| Clearly, in order to decide in favor of investing at least a small amount <math>(f^{*}>0)</math>, you must have
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| : <math> p b > q a . \! </math>
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| which obviously is nothing more than the fact that your expected profit must exceed the expected loss for the investment to make any sense.
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| The general result clarifies why leveraging (taking a loan to invest) decreases the optimal fraction to be invested, as in that case <math>a>1</math>. Obviously, no matter how large the probability of success, <math>p</math>, is, if <math>a</math> 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.
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| == Proof ==
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| Heuristic proofs of the Kelly criterion are straightforward.<ref>{{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 14.7 (Example 2.) | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=757}}</ref>
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| For a symbolic verification with [[Python (programming language)|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'':
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| <source lang="python">
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| >>> from sympy import *
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| >>> x,b,p = symbols('x b p')
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| >>> y = p*log(1+b*x) + (1-p)*log(1-x)
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| >>> solve(diff(y,x), x)
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| [-(1 - p - b*p)/b]
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| </source>
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| For a rigorous and general proof, see [[John Larry Kelly, Jr|Kelly's]] original paper<ref name="original Kelly article" /> or some of the other references listed below. Some corrections have been published.<ref>{{cite jstor|1402118}}</ref>
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| 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.<ref name="original Kelly article" />
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| When ''b'' = 1, the Kelly bettor bets 2''p'' - 1 times initial wealth, ''W'', as shown above. If he wins, he has 2''pW''. 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:
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| : <math> 2^Np^K(1-p)^{N-K}W \! .</math>
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| Suppose another bettor bets a different amount, (2''p'' - 1 + <math>\Delta</math>)W for some positive or negative <math>\Delta</math>. He will have (2p + <math>\Delta</math>)''W'' after a win and [2(1 - ''p'')- <math>\Delta</math>]''W'' after a loss. After the same wins and losses as the Kelly bettor, he will have:
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| : <math> (2p+\Delta)^K[2(1-p)-\Delta]^{N-K}W \! </math>
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| Take the derivative of this with respect to <math>\Delta</math> and get:
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| : <math> K(2p+\Delta)^{K-1}[2(1-p)-\Delta]^{N-K}W-(N-K)(2p+\Delta)^K[2(1-p)-\Delta]^{N-K-1}W\! </math>
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| The turning point of the original function occurs when this derivative equals zero, which occurs at:
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| : <math> K[2(1-p)-\Delta]=(N-K)(2p+\Delta) \! </math>
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| which implies:
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| : <math> \Delta=2(\frac{K}{N}-p) \! </math>
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| but:
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| : <math> \lim_{N \to +\infty}\frac{K}{N}=p \! </math>
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| so in the long run, final wealth is maximized by setting <math>\Delta</math> to zero, which means following the Kelly strategy.
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| 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 K<sup>th</sup> win knowing that the rest of the bets will lose), one will end up with the most money if one bets:
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| : <math> \left(2\frac{K}{N}-1\right)W \! </math>
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| 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 {{nowrap|''K'' > ''pN''}} for a stretch; someone who bets less than Kelly can do better if {{nowrap|''K ''< ''pN''}} for a stretch, but in the long run, Kelly always wins.
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| The heuristic proof for the general case proceeds as follows.{{Citation needed|date=April 2012}}
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| In a single trial, if you invest the fraction <math>f</math> of your capital, if your strategy succeeds, your capital at the end of the trial increases by the factor <math>1-f + f(1+b) = 1+fb</math>, and, likewise, if the strategy fails, you end up having your capital decreased by the factor <math>1-fa</math>. Thus at the end of <math>N</math> trials (with <math>pN</math> successes and <math>qN</math> failures ), the starting capital of $1 yields
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| : <math>C_N=(1+fb)^{pN}(1-fa)^{qN}.</math>
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| Maximizing <math>\log(C_N)/N</math>, and consequently <math>C_N</math>, with respect to <math>f</math> leads to the desired result
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| : <math>f^{*}=p/a-q/b .</math>
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| For a more detailed discussion of this formula for the general case, see http://www.bjmath.com/bjmath/thorp/ch2.pdf. There, it can be seen that the substitution of <math>p</math> for the ratio of the number of "successes" to the number of trials implies that the number of trials must be very large, since <math>p</math> is defined as the limit of this ratio as the number of trials goes to infinity. In brief, betting <math>f^{*}</math> each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and <math>p</math> and <math>b</math> 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, <math>pN</math> successes and <math>qN</math> failures are highly likely only for very large <math>N</math>.
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| == Reasons to bet less than Kelly ==
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| A natural assumption is that taking more risk increases the probability of both very good and very bad outcomes. One of the most important ideas in Kelly is that betting more than the Kelly amount ''decreases'' the probability of very good results, while still increasing the probability of very bad results. Since in reality we seldom know the precise probabilities and payoffs, and since overbetting is worse than underbetting, it makes sense to err on the side of caution and bet less than the Kelly amount.
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| Kelly assumes sequential bets that are [[statistical independence|independent]] (later work generalizes to bets that have sufficient independence). That may be a good model for some gambling games, but generally does not apply in investing and other forms of risk-taking.
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| The Kelly property appears "in the long run" (that is, it is an [[asymptotic analysis|asymptotic]] property). To a person, it matters whether the property emerges over a small number or a large number of bets. It makes sense to consider not just the long run, but where losing a bet might leave one in the short and medium term as well. A related point is that Kelly assumes the only important thing is long-term wealth. Most people also care about the path to get there. Kelly betting leads to highly volatile short-term outcomes which many people find unpleasant, even if they believe they will do well in the end.
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| The criterion assumes you know the true value of ''p'', the probability of the winning. The formula tells you to bet a positive amount if ''p'' is greater than ''1/(b+1)''. In many situations you cannot be sure ''p'' is the true probability. For example if you are told there are just 100 tickets ($1 each) to a raffle, and the prize for winning is $110, then Kelly will tell you to bet a positive fraction of your bank. However, if the information of "100 tickets" was a lie or mis-estimate, and if the true number of tickets was 120, then any bet needs to be avoided. Your optimal investement strategy will need to consider the statistical distribution for your ''estimate'' for ''p''.
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| == Bernoulli ==
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| In a 1738 article, [[Daniel Bernoulli]] suggested that when one has a choice of bets or investments that one should choose that with the highest [[geometric mean]] of outcomes. This is mathematically equivalent to the Kelly criterion {{Citation needed|date=June 2013}}, although the motivation is entirely different (Bernoulli wanted to resolve the [[St. Petersburg paradox]]). The Bernoulli article was not translated into [[English (language)|English]] until 1956,<ref name="Bernoulli translation">{{cite jstor|1909829}}</ref> but the work was well-known among mathematicians and economists.
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| == Many horses ==
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| Kelly's criterion may be generalized
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| <ref name="many horses">Smoczynski, 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</ref> 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 <math>p_k</math>, the total amount of bets placed on k-th horse is <math>B_k</math>, and
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| :<math>\beta_k=\frac{B_k}{\sum_i B_i}=\frac{1}{1+Q_k} ,</math> | |
| where <math>Q_k</math> are the pay-off odds. <math>D=1-tt</math>, is the dividend rate where <math>tt</math> is the track take or tax, <math>\frac{D}{\beta_k}</math> 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 <math>f_k</math>. Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set <math>S^o</math> of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions <math>f^o_k</math> of bettor's wealth to be bet on the outcomes included in the optimal set <math>S^o</math>.
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| The algorithm for the optimal set of outcomes consists of four steps.<ref name="many horses"/>
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| Step 1 Calculate the expected revenue rate for all possible (or only for several of the most promising) outcomes:
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| <math>er_k=\frac{D}{\beta_k}p_k=D(1+Q_k)p_k.</math> | |
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| Step 2 Reorder the outcomes so that the new sequence <math>er_k</math> is non-increasing. Thus <math>er_1</math> will be the best bet.
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| Step 3 Set <math> S = \varnothing </math> (the empty set), <math>k = 1</math>, <math>R(S)=1</math>. Thus the best bet <math>er_k = er_1</math> will be considered first.
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| Step 4 Repeat:
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| If <math>er_k=\frac{D}{\beta_k}p_k > R(S)</math> then insert k-th outcome into the set: <math>S = S \cup \{k\}</math>, recalculate <math>R(S)</math> according to the formula:
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| <math>R(S)=\frac{1-\sum_{i \in S}{p_i}}{1-\sum_{i \in S } \frac{\beta_i}{D}}</math> and then set <math>k = k+1 </math>,
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| Else set <math>S^o=S</math> and then stop the repetition.
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| If the optimal set <math>S^o</math> is empty then do not bet at all. If the set <math>S^o</math> of optimal outcomes is not empty then the optimal fraction <math>f^o_k</math> to bet on k-th outcome may be calculated from this formula: <math>f^o_k=\frac{er_k - R(S^o)}{\frac{D}{\beta_k}}=p_k-\frac{R(S^o)}{\frac{D}{\beta_k}}</math>.
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| One may prove<ref name="many horses"/> that
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| :<math>R(S^o)=1-\sum_{i \in S^o}{f^o_i}</math>
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| where the right hand-side is the reserve rate{{clarify|reason=not mentioned before|date=June 2012}}. Therefore the requirement <math>er_k=\frac{D}{\beta_k}p_k > R(S)</math> may be interpreted<ref name="many horses"/> as follows: k-th outcome is included in the set <math>S^o</math> of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction <math>f^o_k</math> 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
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|
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| :<math>G^o=\sum_{i \in S}{p_i\log_2{(er_i)}}+(1-\sum_{i \in S}{p_i})\log_2{(R(S^o))} ,</math>
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| and the doubling time is
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| :<math>T_d=\frac{1}{G^o}.</math>
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| This method of selection of optimal bets may be applied also when probabilities <math>p_k</math> are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that <math>\sum_i{p_i} < 1</math> and <math>\sum_i{\beta_i} < 1</math>.
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| == Application to the stock market ==
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| Consider a market with <math>n</math> correlated stocks <math>S_k</math> with stochastic returns <math>r_k</math>, <math>k= 1,...,n</math> and a riskless bond with
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| return <math>r</math>. An investor puts a fraction <math>u_k</math> of his capital in <math>S_k</math> and the rest is invested in bond. Without loss of generality, assume that investor's starting capital is equal to 1.
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| According to Kelly criterion one should maximize
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| <math>\mathbb{E}\left[ \ln\left((1 + r) + \sum\limits_{k=1}^n u_k(r_k -r) \right) \right]</math>
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| <br>
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| Expanding it to the Taylor series around <math>\vec{u_0} = (0, \ldots ,0)</math> we obtain
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| <br>
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| <math>\mathbb{E} \left[ \ln(1+r) + \sum\limits_{k=1}^{n} \frac{u_k(r_k - r)}{1+r} -
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| \frac{1}{2}\sum\limits_{k=1}^{n}\sum\limits_{j=1}^{n} u_k u_j \frac{(r_k
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| -r)(r_j - r)}{(1+r)^2} \right]</math>
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| <br>
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| Thus we reduce the optimization problem to [[quadratic programming]] and the unconstrained solution
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| is
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| <math>
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| \vec{u^{\star}} = (1+r) ( \widehat{\Sigma} )^{-1} ( \widehat{\vec{r}} - r )
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| </math>
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| <br>
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| where <math>\widehat{\vec{r}}</math> and <math>\widehat{\Sigma}</math> are the vector of means and the matrix of second mixed noncentral moments of the excess returns.<ref>[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2259133 Nekrasov, Vasily(2013) "Kelly Criterion for Multivariate Portfolios: A Model-Free Approach"]</ref>
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| There are also numerical algorithms for the fractional Kelly strategies and for the optimal solution under no leverage and no short selling constraints.
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| == See also ==
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| * [[Gambling and information theory]]
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| * [[Proebsting's paradox]]
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| == References ==
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| {{Reflist}}
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| ==External links==
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| * [http://rldinvestments.com/Kelly%20Calculator/js/Money%20Management.html Online Kelly Calculator]
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| [[Category:Decision theory]]
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| [[Category:Games (probability)]]
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| [[Category:Information theory]]
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| [[Category:Wagering]]
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| [[Category:Articles containing proofs]]
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| [[Category:1956 introductions]]
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| [[bg:Критерий на Кели]]
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| [[da:Kelly kriteriet]]
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| [[ru:Критерий Келли]]
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