Works quite well in many ways, but has serious limitations when used for trading. If the dice bias were less, at 53%, the Kelly criterion recommends staking 6%. The Kelly Criterion is a formula used to determine the optimal size of a bet when the expected returns are known. g. As I mentioned earlier, the formula is a mainstay of the gambling and investing worlds to help manage risk in asset management. The main goal of the Kelly Criterion strategy is to maximise your profits and minimise the downside (risk associated with your betting). This user should likely bet about $1,000 * 0. The picture above has 2 simulations of betting at 20%, 50%, and 75% and 4 at the Kelly Criterion amount. e the value available) and the size of your bankroll, so as to minimise risk and. To avoid complicated. Well, say hello to Kelly’s Criterion! 14. In the third event on the Belmont program, you find an angle play that has a 15 percent strike rate and 8-1 average winning odds. In this post, I’ll apply it to a EURUSD breakout strategy and explain some of its potential shortcomings when applied to forex trading. If the player has no edge, or the house holds an edge, a bet cannot be made with this system. 067 or 6. The Kelly criterion is a money management system that’s used by many professional traders and hedge funds. zurück, der sie 1956 veröffentlichte. Conclusion. When the odds are 1:1 (a=1 and b=1), the optimal betting size becomes f∗ = p − q f ∗ = p − q. The Kelly Criterion formula allows bettors to maximize profits from a particular bankroll, based on the value the bet holds. g. Chapter 2 presents some of the earlier studies of applications of the Kelly criterion and also clarifies more in depth what the Kelly criterion is and how it works. The Kelly Criterion has many critics; primarily because we cannot exactly measure the probability of winning (in horse racing, at least) and is only works where we have a large number of identical events (no two horse races are identical. The question arises because it is widely believed that the Kelly criterion is best for betting online, while, in fact, its use is appropriate only if our betting system has been actually proved profitable. (2009) show the Kelly criterion to de ne an investment strategy that ensures the survival of agents following this strategy and yields global evolutionary stability. We need to maximize E (G) = (1 + (O-1) * X) p * (1 - X) 1-p - 1 with respect to X, subject to X lying on the unit interval [0,1]. Source: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market by Edward Thorp. So which is the most OPTIMAL bet to make , that grows your bankroll the. How can the Kelly Criterion be adjusted for making Angel Investment Decisions? 1. s = b * (o / f. 60 – 0. For example, if you have a 60% chance of winning and a 40% chance of losing, and your payoff is 2 times your loss, your edge is 0. A reasonable criterion would be to maximise the compound gain at the end of the sequence. 124 = 0. So let's say I have a portfolio X. My 1962 book Beat the Dealer explained the detailed theory and practice. The Kelly criterion: exploiting favorable bets and the stock market In many ways this is my favorite topic to talk about. This is what the Kelly Criterion calculates - the value of f* where the expected value of ln(G), and consequently G, are at their maximum. The kelly formula helps investors to determine the optimal amount to put into a single trade. Step - 2: Calculate the probability of each outcome. 00, with a winning probability of 0. 2. It is also helpful for gamblers as it determined the maximum amount. Therefore you would stake 10% of your. Theoretically, the Kelly criterion states that the optimal strategy is to allocate a fraction of available capital to each borrower. Kelly studied at the University of Texas at Austin and earned a PhD in Physics in 1953. Follow these steps to determine your optimal bet stake using the Kelly Criterion:1. , “fair” odds), a gambler can use the knowledge given him by the received symbols to cause his money to grow. It doesn’t take asymmetric utility into account. 00. Understanding Kelly criterion is almost useless in practical investment management. 2 in his paper The Kelly Criterion in Blackjack Sports Betting, and the Stock Market (which everybody reading article this. Kelly is made for known outcomes such as coin flips, table games, etc. 1. The kelly formula helps investors to determine the optimal amount to put into a single trade. The Kelly Criterion is a money management formula that calculates the amount you should bet when there is a difference between the “true” odds and the given odds. This value should be a number between 0 and 100. The Kelly Criterion: Comparison with Expected Values. the Kelly Criterion to the stock market. The calculator will tell you whether or not you should be betting and how much to stake. For vertical spreads with well-defined profit-loss profiles, I imagine you would use the probability of the spread expiring at max profit as your “probability of success” in the Kelly Criterion equation. 38 percent Kelly bet, or $53 of your current bankroll ($1,220 x 4. The Kelly Criterion is a strategy that can be used in several. Kelly criterion tells you exactly what you want to know -- what percentage of my bankroll should I be betting every turn in order to maximize my gains while minimizing my losses. 1, 2. Library "FunctionKellyCriterion" Kelly criterion methods. Full Kelly has an interesting property: there is an X% chance of your bankroll dropping to X% of what you started with 5. This announced the discovery of favorable card counting systems for blackjack. a. 50. W = Dollars won per dollar wagered (i. This is Thorp’s equation 8. e. 62. The formula was derived by J. 2 in his paper The Kelly Criterion in Blackjack Sports Betting, and the Stock Market (which everybody reading article this. 20-1)*100 = 0. It takes into account your win probability and the odds being offered by the sportsbook, and provides a recommendation for the optimal bet size based on your bankroll size. Wettrechner und Quotenrechner - Überprüfen Sie Ihre potenziellen Gewinne auf den Akkumulatoren mit unserem Wettrechner, Quotenrechner, Akkumulator berechnen, Lucky 15, Each Way, Doubles, Trebles und mehr. So if I have two simultaneous bets. If you invest in an early-stage startup, let's make the following assumptions: - The probability that this startup is successful is 2%. 098 0. E. pyplot as plt import decimal as d # Probablility of winning. W T statistics Half Kelly Full Kelly Double Kelly T riple Kelly Mean ( W T ) 1. i. Let’s calculate K for our scenario:ity”). In such a case, the Kelly criterion suggests that if one were to go over 20% repeatedly on a low. For example, if you have a ROI of 5% it would look something like this: Full Kelly % of bankroll = . Kelly Criterion (kCrit) = ((odds - 1) * (1 - winP)) / (odds - 1) 'odds' is the Decimal form of the American odds after conversion. Kelly percentage number = 0. The left-hand side of the equation, f*, is the percentage of our total wealth that we should put at risk. Here, the formula is F = W – [ (1-W) / R] W = Probability of success. This gives us a Sharpe Ratio of S = 0. The standard deviation about 20%. Tim Byrnes, Tristan Barnett. Suppose player A wins b units for every unit wager. The equity balance. Thorp used the Kelly Portfolio to produce 20%. Thorp. Rechner und Wettwerkzeuge. We briefly introduce the Kelly criterion. Full Kelly allocation is achieved when portfolio volatility is equal to the Sharpe ratio of the portfolio. For example, if you think the fair odds of an outcome are 2. I intuitively think about probability in terms of r, not b. Kelly Criterion’s Relation to League Type. Developed by John Kelly, who worked at Bell labs, the Kelly Formula was created to help calculate the optimal fraction of capital to allocate on a favorable bet. Since the variance in heads up will always be 1 than the formula is easily applicable to HU SNGs. He derived a trade sizing scheme (the Kelly criterion) which showed the optimal fraction of the bankroll to be allocated to each opportunity. The Kelly Criterion and the Stock Market / LOUIS M. Using the Kelly Criterion formula, we can calculate the optimal fraction of your bankroll to bet as: f* = ( (2. Given return of a portfolio or a single asset modeled as a continuous, but not necessarily gaussian, probability distribution, what's the Kelly criterion equation? I've heard that it's simply the the ratio of the sharpe ratio to the standard deviation. The Kelly Criterion is a method of management that helps you calculate how much money you might risk on a trade, given the level of volatility in the market. 0% of similar occasions, you would expect to gain $99. The Kelly criterion or Kelly strategy is a formula used to determine position sizing to maximize profits while minimizing losses. obj. So, technically, using the Kelly Criterion perfectly will lead one to never exhausting his/her entire bankroll because, for example, even if your bankroll shrinks to $20, the Kelly Criterion math (based on a 1% player advantage) dictates your currently resized bet should now be about $2. It just happens to coincide with log-utility. This is what is now known as the Kelly Criterion. 45)/ (2. b = the decimal odds – 1. 10. The Kelly Criterion gives an optimal result for betting based on the probability of winning a bet and how much you receive for winning. It is a function of the net fractional odds received b > 0 and the probability of a win p ∈ (0, 1). We propose an evolutionary framework for optimal portfolio growth theory in which investors subject to environmental pressures allocate their wealth between two assets. As discussed, that’s pretty much the case with 50/50 leagues, but not at all the case in heads-up leagues. k. How to develop a good trading strategy to achieve long-term stable profit is the dream of many people [14]. Below is an explanation of how to apply the generalised Kelly Criterion to betting: Step - 1: List all possible outcomes for the entire set of bets. This formula is derived by maximizing the expected value of the log-growth rate of a bettor’s bankroll, assuming certain conditions are met (more on this later). 1 Introduction Kelly (1956) considered the question of how a gambler with an edge should act in order to maximize his bankroll growth. 05. We also test Kelly’s criterion by running simulations. The Kelly Criterion is a formula to determine the proper size of a bet with known odds and a definite payout. 20*0. By Paul Butler – January 27, 2019. Computing and following an exact decision tree increases earnings by $6. The Kelly Criterion formula allows bettors to maximize profits from a particular bankroll, based on the value the bet holds. The Kelly Criterion is a mathematical formula that helps gamblers determine optimal bet sizes and maximize profits. Suppose that when you win, your reward is $200. It is a simple formula that calculates the proportion of your balance to wager on a particular gamble. the Kelly criterion is optimal only under certain conditions, which are identified. Currently i risk 2% of capital. 3 b = 0. If for example your portfolio has 20 winning trades out of 40 total trades your W is 20 / 40 = 0. To be honest I have been successfully using the simple/naïve Kelly Criterion for years and haven’t bothered looking anywhere else. If you bet less than 10%, you’re not taking full advantage of. 40)-0. If you can find an exploitable, repeatable edge, Kelly's system tells the maximum you should bet based upon. Thorp is famous for his blackjack paperback, Beat the Dealer, where he explores Kelly for gambling. However, it is important to recognize that the Kelly Criterion assumes accurate probabilities and stable edge over time. Kelly criterion is meant for each game to be played one at a time. Um den Kelly Formel Rechner zu benutzen brauchen Sie nur die angebotene Quote und die Wahrscheinlichkeit in das Formular einzugeben. 60 – 0. The formula is: Kelly fraction = edge / odds. From Wikipedia, the free encyclopedia. The Kelly Criterion requires there are no other rules other than the Kelly Criterion. The Investopedia definition is P- [ (1-P)/winLossRatio], where P is the probability of a. It means 20% of your bankroll is the optimal amount to wager on this event. To calculate the “R,” divide the average. The specifics involve too much algebra for me to elucidate here. 01 e 64 1. But to truly harness its power, one must understand the…Hi, do folks here use the Kelly Criterion? Just wondering what approaches you use for translating a Kelly number into an actual allocation. However, in order to reach that conclusion, we must wager with a flat betting plan for a large number of bets. In particular, it maximizes the expected growth rate and the median of the terminal wealth. 5653e-8. The conclusion is: Kelly Criterion is the numerical expression of the attitude of the gaming company to a certain result of the game. Since the plot of g(f) is simply that of the logarithm of G(f), both of these plots will peak at the same value of f. The Kelly Criterion is a formula which accepts known probabilities and payoffs as inputs and outputs the proportion of total wealth to bet in order to achieve the maximum growth rate. $egingroup$ @elemolotiv I read a lot more after that post. 2. However, finding that amount to invest requires immense confidence in your ability to research and come up with precise and accurate probabilities and accompanying magnitudes. This equation is not analytically solvable, but may be approximated as a quadratic programming problem as described in a paper by Vasily Nekrasov. 50. 3 – [ (1 – 0. N 935 A Simple Example Of Non-unique Factorization in Integral Domains / According to the classical Kelly criterion, a bettor should bet a fraction of their bankroll equal to edge / odds, assuming edge > 0. 55)-0. The classic gambler thought to maximize expected value of wealth, which meant she The Kelly formula is: Kelly % = P - [ (1 - P) : R] As an example, let's say one has won 40 out of 100 trades and the total win is €6,000 while the total losses are €2,000. 02 - (1-0. e. 100 betting units ($5000) is nowhere near enough to avoid ruin!‘The Kelly Criterion is to bet a predetermined fraction of assets and can be counterintuitive. This implies a reward/risk ratio of 0. This value should be a number between 0 and 100. The strategy is to maximize long run wealth of the investor by maximizing the period by period expected utility of wealth with a logarithmic utility function. Add the odds quoted by the. Risking 2% does not mean buying with 2% of capital. with constant bets. I don't know how efficient markets actually are, but they are probably efficient enough that most investors can't apply the Kelly criterion in a meaningful way!The Kelly Criterion tells you what fraction f ∗ of your bankroll to wager. Hence, the problem is one of. Kelly put forward the Kelly formula while conducting research on the probabilistic errors in communications. Nicholas joins Chris for a conversation on correctly sizing your investments using the Kelly Criterion,. The Kelly Criterion calculation is used to calculate the optimum stake or position size for a given event, and is made up of four factors. The simplified Kelly Criterion has some potential flaws and can give some head-scratching results which are explained in the article above. 00 – 1) f* =0. Predicting a borrower’s probability of default is a key requirement of any credit allocation system but turning it into labeled classes leads to problems in performance measurement. This leads us to the optimal betting amount called the kelley criterion: f∗ = pb − qa ab f ∗ = p b − q a a b. Suppose we have initial capital X 0 and we want to determine the optimal betting fraction f to invest each year in S&P 500 stocks. e. We briefly introduce the Kelly criterion. Traders often search for better position sizing methods to amplify their strategy. The Kelly criterion is a theoretical formula for obtaining the best return when repeatedly investing money. In the vast landscape of sports betting, the Kelly Criterion emerges as a beacon for bettors aiming to optimize their bankroll management. To increase it to 5. The Kelly Criterion is a money-management formula that calculates the optimal amount you should bet when there’s a difference between the true odds and the given odds. rr: float, reward to risk. The Kelly Coin-Flipping Game: Exact Solutions. That is a probability of winning of 40%. Your odds of winning any one flip are 50/50. However, in practical situations, it is impossible to play a game for an unlimited time. Consider the first three bets:K% = W- [ (1-W)/R] Based on your past trading performance, the Kelly Criterion tells you the position sizes you should be taking on your next trade. In particular, it maximizes the expected growth rate and the median of the terminal wealth. Today the best prop trading firms use this formula to maximize the possible. 凱利公式、凱利方程、凱利判據、凱利策略（英語： Kelly criterion 、 Kelly strategy 、 Kelly bet ），是一種根据赌博赢或输的概率，计算出每次下注的资金占所有赌本的最佳比例的公式 ，由約翰·拉里·凱利於1956年在《 貝爾系統技術期刊 （ 英语 ： Bell System Technical. I'm trying to figure out the best way to scale my bets and I've come across the standard Kelly Criterion posts and while it seems logically to be the best there is another bet scaling practice that appears to outperform the Kelly. Kelly, Jr in 1956. It was proposed by John Kelly in the 50’s who at that point was working for AT&T’s Bell Laboratories. Unterstützt EW, Dead Heats, Regel 4. 66. According to the Kelly criterion, the probability of getting a return of -5 or less in this example is only 5. Kelly did undergraduate and graduate work at. Wettrechner und Quotenrechner - Überprüfen Sie Ihre potenziellen Gewinne auf den Akkumulatoren mit unserem Wettrechner, Quotenrechner, Akkumulator berechnen, Lucky 15, Each Way, Doubles, Trebles und mehr. Betting > 1. 'winP' in the expected winning probability of this particular bet. The Kelly Criteria is an interesting thing to play with. If only one of them is in your favor, it must more than offset the other, i. 36: Understanding the Kelly Criterion. The next line is a trick to restrict the optimizer to values that sum to 1 (100%). We've actually covered the binary Kelly Criterion and the single-asset Kelly Criterion before - so check those articles out for some more background. For example, if you invest 100 and get 10, the gain is 10%. It’s free and easy to use. The Kelly Criterion [1], [2] was initially introduced in 1956 to find the optimal betting amount in games with fixed known odds, and was later extended to the field of financial investments by E. By supplying an arbitrary probability distribution modeling the future price movement of a set of stocks, the Kelly fraction for investing each stock can be calculated by inverting a matrix involving only first and second moments. From the graph, betting with the Kelly Criterion clearly has an advantage over constant. You need to input the probability (according to your own assessment) that your selection will win. Suppose player A wins b units for eve1Y unit wager. Kelly criterion is a mathematical formula that is widely used by investors and gamblers to calculate how much money they should be dedicated to each investment by using a fixed percent of their assets. Most sources provide coverage only…The optimal allocation given in Proposition 1 coincides with the Kelly criterion (Kelly 1956; Thorp 1971) in probability theory and the portfolio choice literature. By the Kelly criterion, the bidding fraction will result in an expected return of í µí°´40µí°´40. Kelly, Jr. 0. 51. Kelly Criterion gambling is especially popular in sports betting and can potentially help you become a successful bettor. This paper examines how the Kelly criterion, a strategy for maximizing the expected log-growth of capital through informed betting, can be applied to non. In fact, the point is precisely not to do that, because maximizing expected wealth (somewhat paradoxically) leads to almost sure ruin. Kelly criterion is a money management principle that beats any other approach in many respects. 4397%. The goal was to figure out the best ways to manage signal-noise issues in long-distance telephone. ket, must begin with a bet. Orange: 1/2 Kelly to 1 Kelly is the Aggressive risk area. Add this topic to your repo. From the recent events in the financial market correction, I thought it would be a fun time to talk about risk management. 10-K Diver. Simultaneous kelly is done by taking the product of 1-kelly for all simultaneous wagers and then multiplying each kelly amount by that product. The Kelly Criterion is a bet-sizing technique which balances both risk and reward for the advantage gambler. Many adages exist: don't bet more than 2% on one trade, don't enter trades that don't have x% upside, never bet on the same stock twice in quick succession, etc. - If it's successful, you earn 100 times your money, so your odds are 100/1=100. Kelly came up with a betting system which optimizes bankroll growth based upon known odds and a definite payout. e. Finally, MintDice operates at a 1x Kelly Criterion schedule. is the author of the Kelly criterion formula from 1956. The purpose of the Kelly criterion in investing is to see how much money you should put into a single trade. The Kelly Criterion: A mathematical formula relating to the long-term growth of capital developed by John Larry Kelly Jr. )It's also the form which most directly suggests how to derive the Kelly criterion, and therefore the situations in which it will/won't apply. Equivalently, we can take the logarithm to transform the product into a sum. There can be other interpretations however. This is due to the requirement that the gambler commits to a specific outcome of the team game. My confusion is how to apply the Kelly formula once the system goes live and I am making trades based on the system signals. The Kelly criterion: How to size bets. These two factors are then input into the Kelly Criterion equation. Share. The initial relative wealth plays a critical role in determining the deviation of optimal behavior from the Kelly criterion regardless of whether the investor is myopic across a single time period or maximizing wealth over an infinite horizon. Mit einer Wette ist in diesem Zusammenhang das Riskieren eines Geldbetrages (Einsatz) gemeint, der im Gewinnfall mit einem festgelegten. 5 if you want to wager 50% of the stake recommended by the Kelly. As mentioned earlier, the Kelly Criterion is a variable after the probability and odds variables, and when this variable is expressed in the form of the index. L. 17. The Kelly Criterion maximizes your profit while eliminating your risk of ruin. With all the prior odds, the equation will add up to this: ( (4x 0. The Kelly Criterion can be used to determine the maximum size of a bet. Hence, the Kelly criterion can be very risky in the short term. 0 Kelly on even 1 bet reduces the growth rate of your capital while increasing risk of ruin. Kelly in 1956. 36. The Kelly Criterion is a powerful tool that can help sports bettors maximize their profits and minimize their risks. The Kelly criterion is a money management principle that beats any other approach in many respects. The growth rate is reduced less significantly in comparison. how much to bet; b is the net odds received on the. On 40. 100:1 odds 0. The odds are in your favor, but read the following carefully: According to the Kelly criterion your optimal bet is about 5. 05 as the equation will look like this: ((0. R = Win/Loss Ratio. The Kelly criterion, which was first introduced by John R. The expanses of the American West take center stage in this intimately observed triptych from Kelly Reichardt. I want to use actual trade data to calculate the Kelly %. 2. The Kelly wager is positive iff the expected value b p − (1 − p) is positive. arithmetic expectation). The first bettor (blue line) bets exactly $5. A 50% chance of a 50% drawdown is a lot to stomach. Further, suppose that on each trial the win probability is p > 0 and pb - q > 0 so the game is advantageous to player A. You get the only answer one can reasonably expect: if. B=2-1=1. 25%. So if you’re placing your funds on a lineup in 50/50 leagues. The Kelly criterion is a money management system that’s used by many professional traders and hedge funds. Kelly Criterion Calculator is a tool for finding the optimal investment size to maximize profits on repeated investments. Betting a Half Kelly Stake, for example, reduces bank volatility by 50%, but growth by only 25%. It is often described as optimizing the logarithm of wealth, and will do better than any other strategy in the long run. W = Historical winning percentage of a trading system. In reality, probabilities may fluctuate, and the actual edge may vary due. we choose our bet size so that our expected rate of capital growth is maximized. He gives lectures to various institutions including The World Bank, Carnegie Mellon, and billion-dollar hedge funds. 개요 [편집] 미국 의 수학자 켈리 (J. Mochkovitch. there must be a positive expected return. Strategy): def __init__. Let gt = Xt / Xt − 1 be the gain obtained after the t -th bet. The Kelly Criterion is a mathematical formula that helps investors and gamblers calculate what percentage of their money they should allocate to each investment or bet. But that was not my point. The Kelly Criterion is a relatively simple math equation to determine the percentage of your bankroll you should bet on any given circumstance, assuming you have an advantage. The Kelly Criterion is a mathematical formula that helps gamblers determine optimal bet sizes and maximize profits. By using the Kelly Criterion, and betting 20% of their. It can also be a. To calculate the “W,” divide the number of trades that brought in a positive amount by your total number of trades. As Poundstone describes, the fraction of your bankroll you should wager on any given bet in a series of bets. With Kelly Criterion we can find the optimal solution for determining the amount of investment. In other words, the fractional Kelly bet which achieves the target rate of return is said to be Kelly-optimal for that target rate. I. The Kelly Criterion is a method of betting for blackjack players who have a mathematical edge in a wager. 2 Holding Period Returns The work of Ralph Vince extends the Kelly criterion from the scenario involving afinding optimal weight using Kelly criterion. WITH KELLY. O. The Kelly Criterion can help determine the appropriate bet size to maximize long-term profitability while minimizing the risk of running out of funds. The great thing about the formula is that it’s flexible enough to work where information or skills can give you an advantage by estimating the outcome probabilities. He derived a trade sizing. 25, pick another point as invest will never be greater than 𝑥𝑥= 0. First of all, The Kelly Criterion is a function of the payoff odds and the win probability: The net odds received can be calculated by examining the betting line offered, but his confidence in. The Real Kelly) discussed in this @Pinnacle article The Real Kelly. If ever the target is to achieve a specific rate of return which is less than maximal, then the optimal bet size is said to be fractional Kelly. It is widely applied to sports betting and casino gambling. The continuous Kelly criterion states that for every i i th strategy with Sharpe ratio Si S i and standard deviation of returns σi σ i, you should be leveraged fi = mi/σ2i = Si/σi f i = m i / σ i 2 = S i / σ i. The point of the Kelly criterion has never been to maximize expected wealth. while reducing volatility. The below formula is for multiple games at the same time. 45)/1=0. The Kelly criterion calculator is a powerful tool for managing your money when gambling or investing. Three are three outcomes to the team game: Team1 wins. On average, the end capital after 40 games will be 10. Its mathematical foundation provides individuals with a systematic approach to determining appropriate bet sizes, allowing for the balancing of risk and reward. Using the Kelly Criterion formula, we can calculate the optimal fraction of your bankroll to bet as: f* = ( (2. John Larry Kelly, Jr. Revisiting the Kelly Criterion Part 2: Fractional Kelly Another common complaint about the Kelly Criterion is how to manage multiple edges on concurrent bets. It takes into account your win probability and the odds being offered by the sportsbook, and provides a recommendation for the optimal bet size based on your bankroll size. I like to break up the Kelly chart into four sections: Yellow: from Zero to 1/2 Kelly is the Conservative risk area. So with this edge the formula says to risk over a third of. So, I've built a simulation based model for betting NBA Games and I have a roughly 54% win rate across 5 seasons of backtesting. 33?1. 3. The Kelly Criterion is from the field of. If the sum of all position sizes is 1, weight. The Kelly formula is : Kelly % = W – (1-W)/R where: Kelly % = percentage of capital to be put into a single trade. While it is tempting to try to run up a big bankroll simply by knowing about the sports and teams that you are wagering on, that strategy fails to take into account important factors such as bankroll management. 00 – 1) f* =0. Our approach is to adhere most closely to Kelly’s original ap-proachof maximizing the assets overthe long-term, given a probabilistic distribution at each investment. Results. ity”).