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53-541: [REDACTED] Look up Kelly in Wiktionary, the free dictionary. Kelly may refer to: Art and entertainment [ edit ] Kelly (Kelly Price album) , 2011 Kelly (Andrea Faustini album) Kelly (musical) , by Mark Charlap, 1965 "Kelly" (song) , by Kelly Rowland, 2018 Kelly (film) , Canada, 1981 Kelly (Australian TV series) Kelly (talk show) , Northern Ireland The Kelly Family ,

106-401: A {\displaystyle 1-fa} . Therefore, the expected geometric growth rate r {\displaystyle r} is: 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; because the logarithm is monotonic , it does not change

159-696: A constant fraction of wealth to bet each time (otherwise one could cheat and, for example, bet zero after the K 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 {\displaystyle N} is small or large. The "long run" part of Kelly is necessary because K is not known in advance, just that as N {\displaystyle N} gets large, K {\displaystyle K} will approach p N {\displaystyle pN} . Someone who bets more than Kelly can do better if K > p N {\displaystyle K>pN} for

212-624: 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. Rough estimates are still useful. If we take excess return 4% and volatility 16%, then yearly Sharpe ratio and Kelly ratio are calculated to be 25% and 150%. Daily Sharpe ratio and Kelly ratio are 1.7% and 150%. Sharpe ratio implies daily win probability of p=(50% + 1.7%/4), where we assumed that probability bandwidth

265-619: A locality Kelly Basin , Tasmania Hundred of Kelly , a cadastral unit in South Australia Azerbaijan [ edit ] Kollu, Dashkasan Trinidad and Tobago [ edit ] Kelly Village , Caroni County, Tunapuna-Piarco United Kingdom [ edit ] Kelly, Devon Kelly Bray , Cornwall United States [ edit ] Kelly, Kansas Kelly, Kentucky Kelly, Louisiana Kelly, North Carolina Kelly, Texas Kelly, West Virginia Kelly, North Dakota Kelly, Wisconsin ,

318-550: A locality Kelly Basin , Tasmania Hundred of Kelly , a cadastral unit in South Australia Azerbaijan [ edit ] Kollu, Dashkasan Trinidad and Tobago [ edit ] Kelly Village , Caroni County, Tunapuna-Piarco United Kingdom [ edit ] Kelly, Devon Kelly Bray , Cornwall United States [ edit ] Kelly, Kansas Kelly, Kentucky Kelly, Louisiana Kelly, North Carolina Kelly, Texas Kelly, West Virginia Kelly, North Dakota Kelly, Wisconsin ,

371-716: A music group Kelly Kelly (TV series) , US, 1998 "Kelly", a 2019 single by Peakboy Kelly West/ Zelena , a character on Once Upon a Time Kelly (The Walking Dead) , a character Kelly (musician) , a character portrayed by Liam Kyle Sullivan People [ edit ] Kelly (given name) Kelly (surname) Clan Kelly , a Scottish clan Kelly (murder victim) Kelly (footballer, born 1975) , Clesly Evandro Guimarães, Brazilian Kelly (footballer, born 1985) , Kelly Cristina Pereira da Silva, Brazilian Kelly (footballer, born 1987) , Kelly Rodrigues Santana Costa, Brazilian Places [ edit ] Australia [ edit ] Kelly, South Australia ,

424-716: A music group Kelly Kelly (TV series) , US, 1998 "Kelly", a 2019 single by Peakboy Kelly West/ Zelena , a character on Once Upon a Time Kelly (The Walking Dead) , a character Kelly (musician) , a character portrayed by Liam Kyle Sullivan People [ edit ] Kelly (given name) Kelly (surname) Clan Kelly , a Scottish clan Kelly (murder victim) Kelly (footballer, born 1975) , Clesly Evandro Guimarães, Brazilian Kelly (footballer, born 1985) , Kelly Cristina Pereira da Silva, Brazilian Kelly (footballer, born 1987) , Kelly Rodrigues Santana Costa, Brazilian Places [ edit ] Australia [ edit ] Kelly, South Australia ,

477-567: A prudent approach suggest a further multiplication of Kelly ratio by 50% (i.e. half-Kelly). A detailed paper by Edward O. Thorp and a co-author estimates Kelly fraction to be 117% for the American stock market SP500 index. Significant downside tail-risk for equity markets is another reason to reduce Kelly fraction from naive estimate (for instance, to reduce to half-Kelly). A rigorous and general proof can be found in Kelly's original paper or in some of

530-400: A stretch; someone who bets less than Kelly can do better if K < p N {\displaystyle 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 one invests the fraction f {\displaystyle f} of their capital, if the strategy succeeds, the capital at

583-626: A surname KELI (disambiguation) Kelley (disambiguation) Kelli (disambiguation) Kellie (disambiguation) Kellyville (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Kelly . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Kelly&oldid=1250839315 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description

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636-626: A surname KELI (disambiguation) Kelley (disambiguation) Kelli (disambiguation) Kellie (disambiguation) Kellyville (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Kelly . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Kelly&oldid=1250839315 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description

689-444: A symmetric probability density function was assumed here. 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

742-401: A town Kelly, Juneau County, Wisconsin , an unincorporated community Kelly, Wyoming Kelly Field , formerly Kelly Air Force Base, San Antonio, Texas Kelly Ridge, California , a CDP Kelly Township, Union County, Pennsylvania Other [ edit ] Kelly criterion for sizing a bet Kelly drive , part of a drilling rig kelly hose , a flexible hose that connects

795-401: A town Kelly, Juneau County, Wisconsin , an unincorporated community Kelly, Wyoming Kelly Field , formerly Kelly Air Force Base, San Antonio, Texas Kelly Ridge, California , a CDP Kelly Township, Union County, Pennsylvania Other [ edit ] Kelly criterion for sizing a bet Kelly drive , part of a drilling rig kelly hose , a flexible hose that connects

848-401: A win and [ 2 ( 1 − p ) − Δ ] W {\displaystyle [2(1-p)-\Delta ]W} after a loss. After the same series of wins and losses as the Kelly bettor, they will have: Take the derivative of this with respect to Δ {\displaystyle \Delta } and get: The function is maximized when this derivative

901-466: Is f k {\displaystyle f_{k}} . Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set S o {\displaystyle S^{o}} of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions f k o {\displaystyle f_{k}^{o}} of bettor's wealth to be bet on

954-582: Is 4 σ = 4 % {\displaystyle 4\sigma =4\%} . Now we can apply discrete Kelly formula for f ∗ {\displaystyle f^{*}} above with p = 50.425 % , a = b = 1 % {\displaystyle p=50.425\%,a=b=1\%} , and we get another rough estimate for Kelly fraction f ∗ = 85 % {\displaystyle f^{*}=85\%} . Both of these estimates of Kelly fraction appear quite reasonable, yet

1007-433: Is p {\displaystyle p} , and in that case the resulting wealth is equal to 1 + f b {\displaystyle 1+fb} . The probability of losing is q = 1 − p {\displaystyle q=1-p} and the odds of a negative outcome is a {\displaystyle a} . In that case the resulting wealth is equal to 1 − f

1060-419: Is and the doubling time is This method of selection of optimal bets may be applied also when probabilities p k {\displaystyle 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 The second-order Taylor polynomial can be used as a good approximation of the main criterion. Primarily, it

1113-490: Is almost never the case with investments. In addition, risk-averse strategies invest less than the full Kelly fraction. The general form can be rewritten as follows where: It is clear that, at least, one of the factors W L P {\displaystyle WLP} or W L R {\displaystyle WLR} needs to be larger than 1 for having an edge (so f ∗ > 0 {\displaystyle f^{*}>0} ). It

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1166-417: Is also the standard replacement of statistical power in anytime-valid statistical tests and confidence intervals, based on e-values and e-processes . In a system where the return on an investment or a bet is binary, so an interested party either wins or loses a fixed percentage of their bet, the expected growth rate coefficient yields a very specific solution for an optimal betting percentage. Where losing

1219-604: Is different from Wikidata All article disambiguation pages All disambiguation pages Kelly [REDACTED] Look up Kelly in Wiktionary, the free dictionary. Kelly may refer to: Art and entertainment [ edit ] Kelly (Kelly Price album) , 2011 Kelly (Andrea Faustini album) Kelly (musical) , by Mark Charlap, 1965 "Kelly" (song) , by Kelly Rowland, 2018 Kelly (film) , Canada, 1981 Kelly (Australian TV series) Kelly (talk show) , Northern Ireland The Kelly Family ,

1272-541: Is different from Wikidata All article disambiguation pages All disambiguation pages Kelly criterion The practical use of the formula has been demonstrated for gambling , and the same idea was used to explain diversification in investment management . 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 and Bill Gross use Kelly methods. Also see intertemporal portfolio choice . It

1325-460: Is equal to zero, which occurs at: which implies that but the proportion of winning bets will eventually converge to: according to the weak law of large numbers . So in the long run, final wealth is maximized by setting Δ {\displaystyle \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

1378-451: Is even possible that the win-loss probability ratio is unfavorable W L P < 1 {\displaystyle WLP<1} , but one has an edge as long as W L P ∗ W L R > 1 {\displaystyle WLP*WLR>1} . The Kelly formula can easily result in a fraction higher than 1, such as with losing size l ≪ 1 {\displaystyle l\ll 1} (see

1431-418: Is included in the set S o {\displaystyle 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 k o {\displaystyle f_{k}^{o}} may be interpreted as the excess of the expected revenue rate of k {\displaystyle k} -th horse over

1484-426: Is the dividend rate where t t {\displaystyle tt} is the track take or tax, D β k {\displaystyle {\frac {D}{\beta _{k}}}} is the revenue rate after deduction of the track take when k {\displaystyle k} -th horse wins. The fraction of the bettor's funds to bet on k {\displaystyle k} -th horse

1537-413: Is the fraction of available capital invested that maximizes the expected geometric growth rate, μ {\displaystyle \mu } is the expected growth rate coefficient, σ 2 {\displaystyle \sigma ^{2}} is the variance of the growth rate coefficient and r {\displaystyle r} is the risk-free rate of return. Note that

1590-478: The k {\displaystyle k} -th horse wins the race is p k {\displaystyle p_{k}} , the total amount of bets placed on k {\displaystyle k} -th horse is B k {\displaystyle B_{k}} , and where Q k {\displaystyle Q_{k}} are the pay-off odds. D = 1 − t t {\displaystyle D=1-tt} ,

1643-402: The above expression with factors of W L R {\displaystyle WLR} and W L P {\displaystyle WLP} ). This happens somewhat counterintuitively, because the Kelly fraction formula compensates for a small losing size with a larger bet. However, in most real situations, there is high uncertainty about all parameters entering the Kelly formula. In

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1696-407: The bankroll, the gambler should bet 20% of the bankroll at each opportunity ( f ∗ = 0.6 − 0.4 1 = 0.2 {\textstyle f^{*}=0.6-{\frac {0.4}{1}}=0.2} ). If the gambler has zero edge (i.e., if b = q / p {\displaystyle b=q/p} ), then the criterion recommends the gambler bet nothing. If

1749-413: The bet involves losing the entire wager, the Kelly bet is: where: As an example, if a gamble has a 60% chance of winning ( p = 0.6 {\displaystyle p=0.6} , q = 0.4 {\displaystyle q=0.4} ), and the gambler receives 1-to-1 odds on a winning bet ( b = 1 {\displaystyle b=1} ), then to maximize the long-run growth rate of

1802-399: 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). Heuristic proofs of the Kelly criterion are straightforward. The Kelly criterion maximizes the expected value of the logarithm of wealth (the expectation value of a function is given by the sum, over all possible outcomes, of

1855-415: The case of a Kelly fraction higher than 1, it is theoretically advantageous to use leverage to purchase additional securities on margin . 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

1908-426: The edge is negative ( b < q / p {\displaystyle b<q/p} ), the formula gives a negative result, indicating that the gambler should take the other side of the bet. A more general form of the Kelly formula allows for partial losses, which is relevant for investments: where: Note that the Kelly criterion is perfectly valid only for fully known outcome probabilities, which

1961-540: The end of the trial increases by the factor 1 − f + f ( 1 + b ) = 1 + f b {\displaystyle 1-f+f(1+b)=1+fb} , and, likewise, if the strategy fails, the capital is decreased by the factor 1 − f a {\displaystyle 1-fa} . Thus at the end of N {\displaystyle N} trials (with p N {\displaystyle pN} successes and q N {\displaystyle qN} failures),

2014-454: The expected geometric growth rate (which is equivalent to maximizing log wealth), then a portfolio is growth optimal. The Kelly Criterion shows that for a given volatile security this is satisfied when f ∗ = μ − r σ 2 {\displaystyle f^{*}={\frac {\mu -r}{\sigma ^{2}}}} where f ∗ {\displaystyle f^{*}}

2067-484: The general case. There, it can be seen that the substitution of p {\displaystyle 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 {\displaystyle p} is defined as the limit of this ratio as the number of trials goes to infinity. In brief, betting f ∗ {\displaystyle f^{*}} each time will likely maximize

2120-460: The locations of function extrema. The resulting equation is: with E {\displaystyle E} denoting logarithmic wealth growth. To find the value of f {\displaystyle f} for which the growth rate is maximized, denoted as f ∗ {\displaystyle f^{*}} , we differentiate the above expression and set this equal to zero. This gives: Rearranging this equation to solve for

2173-584: The optimal fraction f k o {\displaystyle f_{k}^{o}} to bet on k {\displaystyle k} -th outcome may be calculated from this formula: One may prove that where the right hand-side is the reserve rate . Therefore, the requirement e r k = D β k p k > R ( S ) {\displaystyle er_{k}={\frac {D}{\beta _{k}}}p_{k}>R(S)} may be interpreted as follows: k {\displaystyle k} -th outcome

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2226-992: The other references listed below. Some corrections have been published. We give the following non-rigorous argument for the case with b = 1 {\displaystyle b=1} (a 50:50 "even money" bet) to show the general idea and provide some insights. When b = 1 {\displaystyle b=1} , a Kelly bettor bets 2 p − 1 {\displaystyle 2p-1} times their initial wealth W {\displaystyle W} , as shown above. If they win, they have 2 p W {\displaystyle 2pW} after one bet. If they lose, they have 2 ( 1 − p ) W {\displaystyle 2(1-p)W} . Suppose they make N {\displaystyle N} bets like this, and win K {\displaystyle K} times out of this series of N {\displaystyle N} bets. The resulting wealth will be: The ordering of

2279-405: The outcomes included in the optimal set S o {\displaystyle S^{o}} . The algorithm for the optimal set of outcomes consists of four steps: If the optimal set S o {\displaystyle S^{o}} is empty then do not bet at all. If the set S o {\displaystyle S^{o}} of optimal outcomes is not empty, then

2332-406: 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 {\displaystyle f} of that wealth on an outcome that occurs with probability p {\displaystyle p} and offers odds of b {\displaystyle b} . The probability of winning

2385-408: The reserve rate divided by the revenue after deduction of the track take when k {\displaystyle k} -th horse wins or as the excess of the probability of k {\displaystyle k} -th horse winning over the reserve rate divided by revenue after deduction of the track take when k {\displaystyle k} -th horse wins. The binary growth exponent

2438-436: The standpipe to the kelly drive Kelly (gas storage) , a seamless transportable compressed gas container, often manifolded in groups See also [ edit ] [REDACTED] Search for "kelly" on Misplaced Pages. All pages with titles beginning with Kelly All pages with titles containing Kelly Earl of Kellie , title of Scottish peers Kellee Keeley (disambiguation) Keely ,

2491-436: The standpipe to the kelly drive Kelly (gas storage) , a seamless transportable compressed gas container, often manifolded in groups See also [ edit ] [REDACTED] Search for "kelly" on Misplaced Pages. All pages with titles beginning with Kelly All pages with titles containing Kelly Earl of Kellie , title of Scottish peers Kellee Keeley (disambiguation) Keely ,

2544-409: The starting capital of $ 1 yields Maximizing log ⁡ ( C N ) / N {\displaystyle \log(C_{N})/N} , and consequently C N {\displaystyle C_{N}} , with respect to f {\displaystyle f} leads to the desired result Edward O. Thorp provided a more detailed discussion of this formula for

2597-495: 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 , not the arithmetic rate of 4% (r = 0.2 x (0.6 - 0.4) = 0.04). The theoretical expected wealth after 300 rounds works out to $ 10,505 ( = 25 ⋅ ( 1.02034 ) 300 {\displaystyle =25\cdot (1.02034)^{300}} ) if it were not capped. In this particular game, because of

2650-530: The value of f ∗ {\displaystyle f^{*}} gives the Kelly criterion: Notice that this expression reduces to the simple gambling formula when a = 1 = 100 % {\displaystyle a=1=100\%} , when a loss results in full loss of the wager. If the return rates on an investment or a bet are continuous in nature the optimal growth rate coefficient must take all possible events into account. In mathematical finance, if security weights maximize

2703-786: The wealth growth rate only in the case where the number of trials is very large, and p {\displaystyle p} and b {\displaystyle 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, p N {\displaystyle pN} successes and q N {\displaystyle qN} failures are highly likely only for very large N {\displaystyle N} . Kelly's criterion may be generalized on gambling on many mutually exclusive outcomes, such as in horse races. Suppose there are several mutually exclusive outcomes. The probability that

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2756-513: The wins and losses does not affect the resulting wealth. Suppose another bettor bets a different amount, ( 2 p − 1 + Δ ) W {\displaystyle (2p-1+\Delta )W} for some value of Δ {\displaystyle \Delta } (where Δ {\displaystyle \Delta } may be positive or negative). They will have ( 2 p + Δ ) W {\displaystyle (2p+\Delta )W} after

2809-400: Was far from optimal: 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. Using the Kelly criterion and based on the odds in the experiment (ignoring the cap of $ 250 and the finite duration of

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