For anyone actively managing investment portfolios, a deep understanding of the Kelley criterion is very important. For example, it is common practice to use "Half Kelly" to size positions, but most sources only provide a hand-wavey intuitive explanation. Thorp's paper[2] quantifies the benefits for any fraction of the "full Kelly" bet and its implications. In addition to Poundstone's book [1] I strongly recommend Ed Thorp's highly readable paper[2].
Understanding Kelly criterion is almost useless in practical investment management. I’m a professional trader and former quant and I don’t know a single actual pro who uses anything like Kelly to size bets.
I’m not saying understanding the methodology isn’t commonplace and useful, I’m saying this isn’t how portfolios are structured in the real world. Securities are not like a deck of cards.
This seems to be discussed at greater length among retail traders who have no way of even knowing their odds than any professional.
I don't have the source at hand but by looking at what data we have from successful investors, many of them have returns that statistically seem like what you'd expect from E log X strategies.
In fact, it's not even a point of debate. If you target growth, you are using the Kelly criterion whether you know it or not. It's just the name for the thing you do when you optimise for growth.
Kelly can work if you can properly model your uncertainty over the probability of outcomes and take this into account. You can either do some sort of Bayesian averaging over your posterior belief of the risk, or you can use the pessimistic side of the confidence interval of the actual risk probability.
The key understanding of the Kelly Criterion is that you need to scale your investment size with risk; riskier investments require smaller investments. How you estimate risk and how that informs your investments is rather fluid, but understanding it is the cornerstone of professional investing.
If you don't understand that, then you are going to go eventually go bust.
For any kind of trading activity, the most important skill to have is risk management. You can get everything else completely wrong, but if you have your risk management down you're still in the game and can learn from your mistakes. If you don't you're liable to be ruined and be out of the game until you can build back a bankroll some other way.
Ed Thorp AND Claude Shannon! One of the best nontechnical finance books ever written.
In practice though, positioning doesn’t work like that in modern times because a lot of your entries and exits happen around liquidity events. However, it is very pertinent for biotech stocks and special situations where you are dealing with discrete outcomes.
The problem with anything that isn't 1/n is the large estimator variance of the mean of asset returns. There's such little signal there that Markowitz et al invariably fit to mostly noise, which reduces diversification, increases transaction costs, among other problems.
A similar phenomenon occurs in ensemble methods in statistics. It's often better to equal weight many estimates than try to fit weights to them, since that fitting process introduces lots of variance.
I'm not sure what you mean by using 1/n, but the Kelly criterion optimised on past returns for common portfolios of thickly traded assets does suggest something very close to 1/n very often.
I've always attributed this to market efficiency (if it suggested anything else, that's what investors would do until the mispricing went away) but maybe there's a deeper reason it happens.
This random person's thesis describes 1/N in a way I think is understandable:
> In circa 400 A.D. Jewish Rabbi Issac Bar Aha recommended always to invest a third into land, a third into merchandise and to keep a third at hand. This method later became well-known under the name “1/n asset allocation strategy”, “equal asset allocation strategy” or “naïve strategy” and is further defined by DeMiguel et al.(2009) as ”the one in which a segment 1/n of wealth is allocated to each of N assets available for investment at each rebalancing data.” The strategy requires investing an equal part of the capital in the different present assets. Nowadays this rule is often labelled as naïve and too simple, by McClatchy and VandenHul (2005) for example.
Gerd Gigerenzer has a number of books, the one I recently read was, "Risk Savvy" and he goes into some detail about the topic. All I'd do here is write a terrible book review, so if you're curious, I definitely recommend taking a look at the book. I'm not sure I totally agree with his arguments (I had a hard time understanding how he would suggest accounting for human bias), but they're definitely interesting.
I am confused so hope you will clarify. I thought the article argues that Markowitz mean variance has problems and 1/n is a reasonable estimator. You seem to be arguing for the opposite? Or perhaps you mean 1/n vs. Kelly but that article does not talk about Kelly.
Sorry, I didn't mean to argue any point really, just expose folks to Gerd Gigerenzer's work, as it seems relevant to this topic. He makes the arguments much more strongly than I ever could.
Any confusion or inconsistency I'm presenting is my fault, and I apologize!
Many here are correct that the Kelly criterion is relatively useless compared to standard portfolio management techniques for a basket of assets.
However... I will say that it's incredible useful when deciding on more high risk bets based on binary outcomes which is not something portfolio managers would dream of doing for their clients. Consider a long dated call spread on the SPY that goes out to 12/2023.
Say you think the SPY will be over $600. Today, for $140 of risk, you stand to make $1,860 if you're right if you buy a $570 call and sell a $590.
This is exactly what Kelly was made for.
The proper strategy, IMO, is to find a comfortable allocation for trades of this sort as a portion of an overall portfolio (Say 1-2%), then of that percentage use Kelly to allocate capital to different bets of this nature to lower the variance.
So sure, Kelly isn't useful for portfolio management writ large, but for managing a portfolio of binary trades, it's a useful metric.
> In one 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.
> 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.
To be fair, as a participant in psychology experiments I go in aware that it's plausible, even likely that I am being misled about what's really going on. That's even necessary in some experiments. Maybe I'm not technically lied to but if deliberately engineering a false impression is the goal, psychologists are the people to do it in a controlled experiment. The experimenters aren't (ethically) allowed to cause you harm, and they'll probably tell you exactly what was really going on afterwards at least if you ask, but during the experiment everything is potentially suspect. Maybe the task you're focused on was just a distraction and they really care whether you notice the clocks in the room are running too fast so that "five minutes" to do the task is really only 250 seconds - but equally maybe the apparent "time pressure" to complete the task is the distraction and they really care whether you lie about completing it properly given an opportunity to cheat.
So if the experimenter in a psych experiment tells me the coin is biased 60% heads, I don't consider that the same way I would if the friend I play board games with says it.
As a result chances are my first few dozen bets are confirming this unusual claim about the world. Biased coins are hard to make, is this coin really biased? Maybe I try fifty bets in rapid succession, $1 on heads each time. Apparently that's expected to take about five minutes of my half an hour, and before that's done I won't feel comfortable even assuming it's really 60% heads.
And at the end of those five minutes on average I turn $25 into $35 and feel comfortable it's really 60% heads or that I can't tell what's wrong.
Now, why gamble on tails? Well like I said, Psychologists mislead you intentionally during experimentation. Maybe the experimenter tells you it's 60% likely to be Heads. If the gamer told me that, I believe it's 40% likely to be Tails because that's logical, but when an experimenter tells me that, I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check.
I kinda feel sorry for psychology and related social science fields. They have an immense hurdle to clear when designing experiments. Both protocol and statistical analysis.
50 or 100 years ago, a study participant might have gone in oblivious to the possibility of subterfuge. Totally unaware that the "taste test" they're participating in for the "marketing majors" was really a study on how political party affiliation affects choices between lemon cake and chocolate chip cookies. Or whatever.
But I have a feeling that college students are much more aware of how these things go today. The experiment is tainted from the get-go by all the participants looking for the "real" data being collected.
I know for damn sure that if I'm recruited for an experiment where I'm taking some sort of test, when a "fellow student" suggests we cheat, that this is an honesty test. Or maybe if the clock runs out before I'm done, I'm being watched for how I handle stress. Wait, is it kind of cold in here? Ah, they must be gauging performance as a function of comfort.
And of course, study participants are way too often 18-24 year olds who happen to go to college. Such a tiny slice of the general population.
So I could see myself placing bets on the "40%" outcome. I wonder if the coordinators straight up told the participants, "Look, we're really testing your betting decisions. This coin really has a 60/40 bias. This isn't a ruse. Please treat this info as true; we're not doing deception testing here" if that would eliminate the kind of second-guessing we're talking about. (I guess we need to study that:) But if that became a norm, then it would further highlight the deceptive tests when that statement is missing.
"I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check."
Only if you were clueless, or perhaps if the experimenter said "if you bet on heads it has a 60% chance of winning". Being unstated what would happen if you bet on tails, you might forget that the coin has know knowledge of how you bet, thus making it impossible for there to be any different outcome than a 60% chance of loss by betting on tails.
Even worse, the experimenters didn't actually provide real coins. They just sent around links to a website that they said was simulating a biased coin. Participants presumably had no actual way to know whether the flips were actually 60% biased towards heads, whether the results were truly independent from one flip to the next, or even whether their bet might impact the outcome.
Sometimes an experiment to see if you can go five minutes without eating the marshmallow is just an experiment to see if you can go five minutes without eating the marshmallow, and not a trick to see what happens if they give you three marshmallows after eating the first one.
Here's a related yet totally different take: your comment demonstrates flawlessly the reason why sufficiently intelligent people must be weeded out of these experiments (or at least the results). And that in turn helps explain why we end up with people who bet tails.
(Note that the thrill of gambling is another explanation; I'm not claiming "those people are less intelligent, it's the only explanation" but rather "a bias against a certain kind of intelligence could lead to an increase in the observed outcome".)
Did they know that it was biased towards heads? With only a 60-40 split I probably wouldn't notice it unless I was actually keeping track, which could take a while. A 6-4 split on 10 tosses doesn't tell you anything. If you told me it was a fair coin and I thought the experiment was about something else, it might take a very long time before it occurred to me to test the hypothesis that the coin wasn't fair.
If they knew it was biased... I'm sure there's an optimal strategy, but a simple strategy would be "bet half of what you have on heads every time". Any idea how much worse that is than the optimal strategy?
"Prior to starting the game, participants read a detailed description of the game, which included a clear statement, in bold, indicating that the simulated coin had a 60% chance of coming up heads and a 40% chance of coming up tails."
> I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not an add thing to bet on tails is it?
I realize you're probably joking, but since this argument is intuitively appealing to many people, I will answer as if it was serious: if you have a weighted coin that is 60% likely to land on heads, that means it's 60% likely to land on heads on any given toss. On the first toss. On the second toss. Any given toss. Even after you have tossed it 6 times and seen 6 heads in a row, the coin is still 60% likely to land on heads. The coin has no "memory". Previous results have no effect on future results.
That's the gambler's fallacy in action. So long as each event is independent, the prior ones have no impact on the likelihood of future events. If you've flipped the coin 60 times and they've all been heads, there's no reason to expect the next 40 will be tails. They still have better odds of being heads.
Your friend walks up while you're playing. They haven't seen the game, so think heads is coming up.
Your other friend has been playing longer, before you even started. They saw 13 tails and then your 6 heads. The next throw should be heads to even it out for them.
Why is your history more of an influence than theirs?
Yes it is irrational. That's a common statistical misconception, the key thing here is that every flip has a 60% chance of being heads.
The result of each flip is completely independent of what came before it. In your example the 7th flip is just as likely to be heads as the first flip, or any of the other 5 flips that landed on heads.
While this is irrational in this experiment, but it is likely that the biological systems in which humans evolved, tend to not have truly independent events - hence our intuition.
The probability of a coin flip being heads or tails is completely independent from the previous flips. If the coin lands 6 heads in a row, the next coin flip still has a 60% chance of being heads, hence it is always unwise to bet on tails in this experiment. This is an example of the Gambler's fallacy [1].
Each toss is independent of prior (and subsequent) tosses, so no matter what, a given tosshas 60% chance of landing heads. Rationally, one should bet heads on any given toss.
Those are independent variables. The fact you've had X heads has no bearing on the future flips. It is irrational to bet on tails statistically speaking, though psychologically that line of reasoning is common.
you're not betting on the number of heads/tails per 10 trials though, each trial is independent with a 60% of heads. In a striaght-up prediction you should always choose heads, it the how much to wager that is the question.
For anyone interested in practising your Kelly estimation, I made a game inspired by Bernoulli's original paper on the subject for a lunch and learn at my job: https://static.loop54.com/ship-investor.html
Before my parental leave is over, I hope to make two more sequels, one with futures and one with options. Maybe also a fixed-income version, but I'd have to learn more about that myself first.
This has been posted 5 other times on HN with no real discussion [1].
I'll add my 2 cents: I used to use the principles of kelly betting back when I designed systematic HFT strategies. It gives you a good framework to think about how much to bet based on the batting average of a particular pattern you recognize in the market...
You may be interested to know that Kelly's work was instrumental in a company called Axcom in the 60s. Elwyn Berlekamp, previously an assistant to Kelly at Bell Labs, implemented Kelly et al's work in early financial trading at Axcom, which was later turned into the Medallion Fund at Renaissance Technologies. Wikipedia [1] has some info on this, but I also highly recommend "The Man Who Solved The Market" (Zuckerman, 2019) for more history.
You may be interested to know that Ed Thorps - Princeton Newport Partners/ the Santa Fe school work lives on at an even better performing fund called TGS Management based in Irvine.
> I used to use the principles of kelly betting back when I designed systematic HFT strategies.
possibly a dumb question, but how did this work exactly? the kelly criterion assumes you know the amount by which the coin is weighted, how would you know the equivalent for the stock market in the very near term?
Kelly goes beyond binary outcomes. The underlying principle is the same, though: you maximise expected logarithmic wealth.
To do that you need the joint distribution of outcomes (what are the possible future scenarios and how likely are they?) Estimating this well is the trick to successful application of the Kelly criterion.
the binary outcome formulation you see everywhere is just "real" kelly boiled down. the real thing, which is contained fully in the first paragraph ("The Kelly bet size is found by maximizing the expected value of the logarithm of wealth"), has no such restrictions.
A simple description of the Kelly criterion is that if you want to grow wealth over a long period time, at each decision point take the one that maximizes your average expected log wealth.
I'm trying to use it in real life, though sometimes the decisions are quite scary, as it's hard to estimate the probability of outcomes. Also my wealth is much more volatile than most people can stomach, but I look at it like a game.
It doesn't seem obvious that this is a good strategy for personal wealth management because besides maximizing expected wealth, there's another very important criterion: minimizing probability of going broke. I only get to play one game, after all. Obviously you can't go entirely broke if you always bet a fraction of your portfolio, but are there results of how these strategies compare in, say, the probability of dipping below 10%, or 1%, of the starting value?
I can't tell you about the 1% version, but when it dipped to 15%, it was a strange feeling that I made a bad decision with the thinking that I'm making a great decision (or more trying not to think about it and trust the decision that I made earlier). It's a mental game at that point that you have to wait through. At least with investing it's just about waiting through those periods, being a CEO of a company and making decisions in that state would have been much harder.
Betting with 'full' Kelly-calculated stakes is highly volatile. If I'm remembering correctly, if you get your probabilities/edge exactly right, you will still have a 50/50 chance of losing half of your bank at some point in the future (i.e. after some number of future bets) It's very common to bet just some fraction of the Kelly stakes in order to smooth out the roller coaster ride.
Sure, I've gone through losing more than 80% of my wealth multiple times by being 100% in BTC, so I got used to that already. At the same time it stresses my friends out a lot. I'm expecting to lose more than 50% of my wealth, but at this point it doesn't really change my life style.
E log X strategies are known for Being very volatile.
However, there are two things that take the scariness out of estimating probabilities for me:
- You're often maximising something that looks like a quadratic function. This means you're aiming at a plateau more than a peak: if you make small errors in either direction it doesn't affect growth that much.
- You always have the safe option of underestimating. The E log X strategy forms an "efficient frontier" (to borrow terminology from MPT) of linear combinations from the risk-free rate to the full Kelly bet (and even past it into leveraged Kelly strategies.) You can always mix in more of the risk-free rate and get lower growth but at higher safety.
These two properties makes the Kelly criterion very forgiving to estimation. (In contrast to MPT style mean--variance estimations, and other less principled strategies.)
I find both mean variance and Kelly to be very poor in practice due to the dependence on the expected return term. Like, if I knew that, I wouldn't be wasting my time with all this math! (half joking)
One simple example is buying 2X S&P index ETF instead of 1x. There was a great article about the Kelly optimal S&P allocation, and with all the fees included it's about 2x. Of course there's increased execution risk for the ETF itself, which needs to be estimated.
Another thing where I may look stupid from outside is that I started to take some loan against my BTC and use that to finance my lifestyle, as currently (under $100k BTC price) my estimate of the Kelly optimal BTC allocation is more than 1. This is of course a personal estimate, I don't suggest other people to do the same thing, and again there's a lot of execution risk, so I do this only with a part of my portfolio.
For people who earn a wage and don't just make money by investing, the Kelly Criterion can't be applied in its basic form, since it means your capital gain has both constant and linear components, instead of just being linear as the formula assumes, which complicates matters a lot.
Plus for low probability high reward bets you have the additional complication that you probably can't make them often enough to get a decent chance of hitting the jackpot.
The bookie odds go into the formula as b- the net fractional odds received on the wager.
We have our own models for our confidence, and the Kelly criterion decides our wager size (though we don't use a full Kelly bet).
Yes, the sportsbook minimizes their own risk by setting a spread or odds with respect to how patrons are wagering. This actually makes it easier to make money if your model is much better than the average bettor. There will be games where public opinion and the majority of bets are on the wrong side of a matchup, and the bookie adjusts the odds accordingly, so the correct bet's payout is bigger than it should be.
In high school I tried to do more what you are asking- use one bookie's odds (which I deemed the most accurate) as the "true probability", and another as the payout. This was not successful, but theoretically could be if the two bookies' clientele were consistently better or worse than each other, therefore influencing their odds consistently.
I used this to win AI Rock Paper Scissors competition in undergrad. I just played random symbols, but used Kelly criterion to compute my bid. This worked well because the game wouldn’t allow your bankroll to go to 0 — the floor was 1.
Can you explain why the Kelly criterion wouldn't have you bet 0 every time? The chance of winning a round of rock-paper-scissors when throwing a random symbol seems to be 50% (if ties cause re-dos), so wouldn't that work out to 50 * 2 - 100 = 0?
It’s a good question. You’re right, if I followed it strictly it wouldn’t work. I suppose I rationalized offsetting it because I couldn’t actually go bankrupt. If I was better at math there’s probably some other criterion that takes into account how hard it would be to get back to where you are, given that you couldn’t go below 1.
That's strange, given that the Kelly criterion maximizes the expectation of the log of wealth--that is, it's maximizing over multiplicative percent gains in a scenario where you can go bankrupt.
I don’t get why it’s strange? What I learned from that competition was that bid sizing was way more important than the symbol selection strategy. Trying to beat the other students at iocaine powder wasn’t really a winning proposition.
I first heard about it from him. He summarizes it as follows:
> Naval: The Kelly criterion is a popularized mathematical formulation of a simple concept. The simple concept is: Don’t risk everything. Stay out of jail. Don’t bet everything on one big gamble. Be careful how much you bet each time, so you don’t lose the whole kitty.
And when you're done with that, the Kelly Capital Growth Investment Criterion is one of the better books I've read. But it's a much more advanced read.
And when you are done with that, I highly suggest reading Edward Thorp's autobiography "Man for all markets" where he employs the Kelly Criterion in adventure after adventure. He not only developed the first card counting system for blackjack but he also created the first wearable computer to beat roulette (with Claude Shannon).
If anyone wants to see Kelly in action, I made an app where you define an edge and a wager (% of pot or absolute amount) and see how you fare compared to the optimal bet strategy.
Fortune’s Forumla by William Poundstone is an excellent book. Edward Thorpe has most of his papers published too, they’re all good for a read [though I cannot be held responsible if you’re going to beat the dealer at Blackjack, I don’t need Kelly to know how that will turn out :)]
I made a streamlit app about Kelly last year, showing how to bet when you have an "edge" over a toy market of coin flippers: https://kelly-streamlit.herokuapp.com/
Other references I found interesting:
- Cover and Thomas's "Elements of Information Theory" shows some interesting connections between Kelly betting and optimal message encoding.
- Ed Thorp, the inventor of card counting, has a nice compendium of papers on this in "The Kelly Capital Growth Investment Criterion".
Kelley betting could probably be applied with some success to momentum trading strategies. Momentum trading is more deterministic than purely speculative strategies since it is based on observed/historical behavior.
I remember the first time I read about this. I put in the numbers for the lottery and a negative number came out. Of course! Your expected winnings are negative and you shouldn't play the lottery.
Well, most of the time, anyway. If you do find a lottery game where the odds are in your favor, something resembling the Kelly criterion is a reasonable starting point for a bankroll-management strategy.
It’s worth mentioning that Kelly was an associate of Claude Shannon (the father of information theory) at Bell Labs. Kelly’s criterion is in fact based on Shannon’s theory.
It seems they developed the approach together. Shannon, his wife and Ed Thorp later went to Las Vegas gambling using this method, and apparently made some money.
If they made some money gambling in Vegas it was clearly not thanks to Kelly’s criterion, because Kelly’s criterion clearly (and correctly) states that the optimal bet in Vegas is zero dollar.
They played, among other things, blackjack. The rules of blackjack are such that the edge varies around zero. It's mostly a tiny bit negative, but sometimes creeps over zero and that's when you apply bigger than minimum bets, following the Kelly criterion.
Note that the martingale, a common betting strategy, does exactly the opposite of the Kelly criterion. If you have a small edge and bet with the martingale against a very wealthy house, you have a fairly large chance of going bankrupt!
Kelly criterion (or Kelly strategy or Kelly bet) is a formula for bet sizing that leads almost surely to higher wealth. It was described by J. L. Kelly Jr, a researcher at Bell Labs, in 1956. The Kelly Criterion is to bet a predetermined fraction of assets, and it can seem counterintuitive. In one study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. 18 of the participants bet everything on one toss, while two-thirds gambled on tails. The average payout was just $91.36.
Kelly bet: Bet one-nineteenth of the bankroll that red will not come up. If gamble has 60% chance of winning, gambler should bet 20% of bankroll at each opportunity. In American roulette, the bettor is offered an even money payoff (even money payoff, with a 95% probability of reaching the cap) If edge is negative, then the formula gives a negative result, indicating that the gambler shouldn't bet. The top of the first fraction is the expected net winnings from a $1 bet, since the two outcomes are that you either win with probability or lose the $1 wagered, i.e. win $ + $1, with probability $+.1. For even-money bets, the formula can be simplified to the first simplified formula: b=p-q-q/p, b=1, and b=0.20.
Probability of success is: If you succeed, the value of your investment increases. If you fail (for which the probability is): In that case, the price of the investment decreases. Kelly criterion maximizes the expected value of the logarithm of wealth. Using too much margin is not a good investment strategy when the cost of capital is high, even when the opportunity appears promising. The general result clarifies why leveraging (taking out a loan that requires paying interest in order to raise investment capital) decreases the optimal fraction to be invested, as in that case the odds of winning are less than 0.1. The expected profit must exceed the expected loss for the investment to make any sense. The Kelly criterion can be used to determine the optimal amount of money to invest.
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, that the rest of the bets will lose), one will end up with the most money if one bets. In the long run, final wealth is maximized by setting the derivative to zero, which means following the Kelly strategy. The function is maximised when this derivative is equal to zero. For a rigorous and general proof, see Kelly's original paper[1] or some of the other references listed below. Some corrections have been published[12] .
In the long run, Kelly always wins. This is true whether N is small or large. 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. Betting each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and the odds of winning are the same for each trial. 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) Exponential mechanism (differential privacy)
Please stop doing this (with any account). HN threads are supposed to be conversations. Copying a mass of content from someplace else is not conversation.
For anyone actively managing investment portfolios, a deep understanding of the Kelley criterion is very important. For example, it is common practice to use "Half Kelly" to size positions, but most sources only provide a hand-wavey intuitive explanation. Thorp's paper[2] quantifies the benefits for any fraction of the "full Kelly" bet and its implications. In addition to Poundstone's book [1] I strongly recommend Ed Thorp's highly readable paper[2].
[1] Poundstone, William (2005), Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street, [2]https://wayback.archive-it.org/all/20090320125959/http://www...
Understanding Kelly criterion is almost useless in practical investment management. I’m a professional trader and former quant and I don’t know a single actual pro who uses anything like Kelly to size bets. I’m not saying understanding the methodology isn’t commonplace and useful, I’m saying this isn’t how portfolios are structured in the real world. Securities are not like a deck of cards.
This seems to be discussed at greater length among retail traders who have no way of even knowing their odds than any professional.
I don't have the source at hand but by looking at what data we have from successful investors, many of them have returns that statistically seem like what you'd expect from E log X strategies.
In fact, it's not even a point of debate. If you target growth, you are using the Kelly criterion whether you know it or not. It's just the name for the thing you do when you optimise for growth.
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Kelly can work if you can properly model your uncertainty over the probability of outcomes and take this into account. You can either do some sort of Bayesian averaging over your posterior belief of the risk, or you can use the pessimistic side of the confidence interval of the actual risk probability.
The key understanding of the Kelly Criterion is that you need to scale your investment size with risk; riskier investments require smaller investments. How you estimate risk and how that informs your investments is rather fluid, but understanding it is the cornerstone of professional investing.
If you don't understand that, then you are going to go eventually go bust.
https://blog.alphatheory.com/2013/01/kelly-criterion-in-prac...
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Second this. Nobody actually uses it. It's a bit too theoretical.
For any kind of trading activity, the most important skill to have is risk management. You can get everything else completely wrong, but if you have your risk management down you're still in the game and can learn from your mistakes. If you don't you're liable to be ruined and be out of the game until you can build back a bankroll some other way.
Ed Thorp AND Claude Shannon! One of the best nontechnical finance books ever written.
In practice though, positioning doesn’t work like that in modern times because a lot of your entries and exits happen around liquidity events. However, it is very pertinent for biotech stocks and special situations where you are dealing with discrete outcomes.
Kelly himself ended up using 1/n for his own personal portfolio management.
Gerd Gigerenzer has a lot to say about how harmful this model has been to finance and the world, because it creates "false certainty".
https://news.ycombinator.com/item?id=26325425 has further discussion.
The problem with anything that isn't 1/n is the large estimator variance of the mean of asset returns. There's such little signal there that Markowitz et al invariably fit to mostly noise, which reduces diversification, increases transaction costs, among other problems.
A similar phenomenon occurs in ensemble methods in statistics. It's often better to equal weight many estimates than try to fit weights to them, since that fitting process introduces lots of variance.
I'm not sure what you mean by using 1/n, but the Kelly criterion optimised on past returns for common portfolios of thickly traded assets does suggest something very close to 1/n very often.
I've always attributed this to market efficiency (if it suggested anything else, that's what investors would do until the mispricing went away) but maybe there's a deeper reason it happens.
This random person's thesis describes 1/N in a way I think is understandable:
> In circa 400 A.D. Jewish Rabbi Issac Bar Aha recommended always to invest a third into land, a third into merchandise and to keep a third at hand. This method later became well-known under the name “1/n asset allocation strategy”, “equal asset allocation strategy” or “naïve strategy” and is further defined by DeMiguel et al.(2009) as ”the one in which a segment 1/n of wealth is allocated to each of N assets available for investment at each rebalancing data.” The strategy requires investing an equal part of the capital in the different present assets. Nowadays this rule is often labelled as naïve and too simple, by McClatchy and VandenHul (2005) for example.
http://arno.uvt.nl/show.cgi?fid=129399
Gerd Gigerenzer has a number of books, the one I recently read was, "Risk Savvy" and he goes into some detail about the topic. All I'd do here is write a terrible book review, so if you're curious, I definitely recommend taking a look at the book. I'm not sure I totally agree with his arguments (I had a hard time understanding how he would suggest accounting for human bias), but they're definitely interesting.
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I am confused so hope you will clarify. I thought the article argues that Markowitz mean variance has problems and 1/n is a reasonable estimator. You seem to be arguing for the opposite? Or perhaps you mean 1/n vs. Kelly but that article does not talk about Kelly.
Sorry, I didn't mean to argue any point really, just expose folks to Gerd Gigerenzer's work, as it seems relevant to this topic. He makes the arguments much more strongly than I ever could.
Any confusion or inconsistency I'm presenting is my fault, and I apologize!
Many here are correct that the Kelly criterion is relatively useless compared to standard portfolio management techniques for a basket of assets.
However... I will say that it's incredible useful when deciding on more high risk bets based on binary outcomes which is not something portfolio managers would dream of doing for their clients. Consider a long dated call spread on the SPY that goes out to 12/2023.
Say you think the SPY will be over $600. Today, for $140 of risk, you stand to make $1,860 if you're right if you buy a $570 call and sell a $590.
This is exactly what Kelly was made for.
The proper strategy, IMO, is to find a comfortable allocation for trades of this sort as a portion of an overall portfolio (Say 1-2%), then of that percentage use Kelly to allocate capital to different bets of this nature to lower the variance.
So sure, Kelly isn't useful for portfolio management writ large, but for managing a portfolio of binary trades, it's a useful metric.
Some interesting psychology here:
> In one 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.
> 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.
To be fair, as a participant in psychology experiments I go in aware that it's plausible, even likely that I am being misled about what's really going on. That's even necessary in some experiments. Maybe I'm not technically lied to but if deliberately engineering a false impression is the goal, psychologists are the people to do it in a controlled experiment. The experimenters aren't (ethically) allowed to cause you harm, and they'll probably tell you exactly what was really going on afterwards at least if you ask, but during the experiment everything is potentially suspect. Maybe the task you're focused on was just a distraction and they really care whether you notice the clocks in the room are running too fast so that "five minutes" to do the task is really only 250 seconds - but equally maybe the apparent "time pressure" to complete the task is the distraction and they really care whether you lie about completing it properly given an opportunity to cheat.
So if the experimenter in a psych experiment tells me the coin is biased 60% heads, I don't consider that the same way I would if the friend I play board games with says it.
As a result chances are my first few dozen bets are confirming this unusual claim about the world. Biased coins are hard to make, is this coin really biased? Maybe I try fifty bets in rapid succession, $1 on heads each time. Apparently that's expected to take about five minutes of my half an hour, and before that's done I won't feel comfortable even assuming it's really 60% heads.
And at the end of those five minutes on average I turn $25 into $35 and feel comfortable it's really 60% heads or that I can't tell what's wrong.
Now, why gamble on tails? Well like I said, Psychologists mislead you intentionally during experimentation. Maybe the experimenter tells you it's 60% likely to be Heads. If the gamer told me that, I believe it's 40% likely to be Tails because that's logical, but when an experimenter tells me that, I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check.
Spot on.
I kinda feel sorry for psychology and related social science fields. They have an immense hurdle to clear when designing experiments. Both protocol and statistical analysis.
50 or 100 years ago, a study participant might have gone in oblivious to the possibility of subterfuge. Totally unaware that the "taste test" they're participating in for the "marketing majors" was really a study on how political party affiliation affects choices between lemon cake and chocolate chip cookies. Or whatever.
But I have a feeling that college students are much more aware of how these things go today. The experiment is tainted from the get-go by all the participants looking for the "real" data being collected.
I know for damn sure that if I'm recruited for an experiment where I'm taking some sort of test, when a "fellow student" suggests we cheat, that this is an honesty test. Or maybe if the clock runs out before I'm done, I'm being watched for how I handle stress. Wait, is it kind of cold in here? Ah, they must be gauging performance as a function of comfort.
And of course, study participants are way too often 18-24 year olds who happen to go to college. Such a tiny slice of the general population.
So I could see myself placing bets on the "40%" outcome. I wonder if the coordinators straight up told the participants, "Look, we're really testing your betting decisions. This coin really has a 60/40 bias. This isn't a ruse. Please treat this info as true; we're not doing deception testing here" if that would eliminate the kind of second-guessing we're talking about. (I guess we need to study that:) But if that became a norm, then it would further highlight the deceptive tests when that statement is missing.
I feel sorry for social science experimenters.
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Maybe once you've started to perceive the meta-patterns between psych experiments, you've taken too many tests to be a good subject.
"I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check."
Only if you were clueless, or perhaps if the experimenter said "if you bet on heads it has a 60% chance of winning". Being unstated what would happen if you bet on tails, you might forget that the coin has know knowledge of how you bet, thus making it impossible for there to be any different outcome than a 60% chance of loss by betting on tails.
Even worse, the experimenters didn't actually provide real coins. They just sent around links to a website that they said was simulating a biased coin. Participants presumably had no actual way to know whether the flips were actually 60% biased towards heads, whether the results were truly independent from one flip to the next, or even whether their bet might impact the outcome.
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Biased coins are *impossible" to make if the coin is flipped not spun.
I doubt any story about a biased coins in the real world.
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Sometimes an experiment to see if you can go five minutes without eating the marshmallow is just an experiment to see if you can go five minutes without eating the marshmallow, and not a trick to see what happens if they give you three marshmallows after eating the first one.
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Here's a related yet totally different take: your comment demonstrates flawlessly the reason why sufficiently intelligent people must be weeded out of these experiments (or at least the results). And that in turn helps explain why we end up with people who bet tails.
(Note that the thrill of gambling is another explanation; I'm not claiming "those people are less intelligent, it's the only explanation" but rather "a bias against a certain kind of intelligence could lead to an increase in the observed outcome".)
I made a little playground for this, you can fiddle with the numbers. https://parsebox.io/dthree/lnumtuenmskr
Did they know that it was biased towards heads? With only a 60-40 split I probably wouldn't notice it unless I was actually keeping track, which could take a while. A 6-4 split on 10 tosses doesn't tell you anything. If you told me it was a fair coin and I thought the experiment was about something else, it might take a very long time before it occurred to me to test the hypothesis that the coin wasn't fair.
If they knew it was biased... I'm sure there's an optimal strategy, but a simple strategy would be "bet half of what you have on heads every time". Any idea how much worse that is than the optimal strategy?
You can plot
g = 0.6 log (1 + 2f) + 0.4 log (1 - f)
And locate f=0.5 and compare to the maximum g.
Edit: I wanted to check my intuition so I did: https://www.wolframalpha.com/input/?i=plot++0.6+log+%281+%2B...
Looks like 0.5 is a slight overbet, but still very, very good.
> Did they know that it was biased towards heads?
"Prior to starting the game, participants read a detailed description of the game, which included a clear statement, in bold, indicating that the simulated coin had a 60% chance of coming up heads and a 40% chance of coming up tails."
If only there was a link to the study so we could see how it was setup.
The paper is pretty awesome and accessibly-written: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963
The PDF is free-to-read.
> two-thirds gambled on tails at some stage in the experiment
I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not irrational to bet on tails is it?
> I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not an add thing to bet on tails is it?
I realize you're probably joking, but since this argument is intuitively appealing to many people, I will answer as if it was serious: if you have a weighted coin that is 60% likely to land on heads, that means it's 60% likely to land on heads on any given toss. On the first toss. On the second toss. Any given toss. Even after you have tossed it 6 times and seen 6 heads in a row, the coin is still 60% likely to land on heads. The coin has no "memory". Previous results have no effect on future results.
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No, the next toss still has a 60% chance of being heads. The coin doesn't remember how it landed last time.
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That's the gambler's fallacy in action. So long as each event is independent, the prior ones have no impact on the likelihood of future events. If you've flipped the coin 60 times and they've all been heads, there's no reason to expect the next 40 will be tails. They still have better odds of being heads.
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Your friend walks up while you're playing. They haven't seen the game, so think heads is coming up.
Your other friend has been playing longer, before you even started. They saw 13 tails and then your 6 heads. The next throw should be heads to even it out for them.
Why is your history more of an influence than theirs?
This wiki page can explain why better than me: https://en.wikipedia.org/wiki/Gambler%27s_fallacy
Yes it is irrational. That's a common statistical misconception, the key thing here is that every flip has a 60% chance of being heads.
The result of each flip is completely independent of what came before it. In your example the 7th flip is just as likely to be heads as the first flip, or any of the other 5 flips that landed on heads.
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While this is irrational in this experiment, but it is likely that the biological systems in which humans evolved, tend to not have truly independent events - hence our intuition.
The probability of a coin flip being heads or tails is completely independent from the previous flips. If the coin lands 6 heads in a row, the next coin flip still has a 60% chance of being heads, hence it is always unwise to bet on tails in this experiment. This is an example of the Gambler's fallacy [1].
[1] https://en.wikipedia.org/wiki/Gambler%27s_fallacy
Each toss is independent of prior (and subsequent) tosses, so no matter what, a given tosshas 60% chance of landing heads. Rationally, one should bet heads on any given toss.
But most people would agree with the irrational bet. This tendency is known as the Gambler’s fallacy (https://en.wikipedia.org/wiki/Gambler's_fallacy).
No, the coin doesn't have a memory, so the chance of tails is still 40% making it still optimal to choose heads.
Those are independent variables. The fact you've had X heads has no bearing on the future flips. It is irrational to bet on tails statistically speaking, though psychologically that line of reasoning is common.
> If you've just had 6 heads in a row the next 4 "should" be tails
That's not how this works. Each toss is independent, so you should never pay attention to previous results if you know the true odds.
you're not betting on the number of heads/tails per 10 trials though, each trial is independent with a 60% of heads. In a striaght-up prediction you should always choose heads, it the how much to wager that is the question.
You've just discovered the Gambler's Fallacy.
For anyone interested in practising your Kelly estimation, I made a game inspired by Bernoulli's original paper on the subject for a lunch and learn at my job: https://static.loop54.com/ship-investor.html
There's also a sequel for the case of continuous outcomes: https://static.loop54.com/ship-investor-2.html
Before my parental leave is over, I hope to make two more sequels, one with futures and one with options. Maybe also a fixed-income version, but I'd have to learn more about that myself first.
Hey, this is really cool! What's the optimal strategy? Would love to learn more
I found good success with going for the smallest investment in Bering and the second smallest investment in the other two straits.
The optimal strategy would be to estimate which investment maximises your log returns :) but I don't have time for that.
Edit: First investment always fails. The next 5 always succeed. Predetermined randomness?
Not intentional, but I expect there to be some bugs -- didn't want to spend too much time on it.
This has been posted 5 other times on HN with no real discussion [1].
I'll add my 2 cents: I used to use the principles of kelly betting back when I designed systematic HFT strategies. It gives you a good framework to think about how much to bet based on the batting average of a particular pattern you recognize in the market...
[1] https://hn.algolia.com/?q=https%3A%2F%2Fen.wikipedia.org%2Fw...
You may be interested to know that Kelly's work was instrumental in a company called Axcom in the 60s. Elwyn Berlekamp, previously an assistant to Kelly at Bell Labs, implemented Kelly et al's work in early financial trading at Axcom, which was later turned into the Medallion Fund at Renaissance Technologies. Wikipedia [1] has some info on this, but I also highly recommend "The Man Who Solved The Market" (Zuckerman, 2019) for more history.
[1] https://en.wikipedia.org/wiki/John_Larry_Kelly_Jr.
You may be interested to know that Ed Thorps - Princeton Newport Partners/ the Santa Fe school work lives on at an even better performing fund called TGS Management based in Irvine.
> I used to use the principles of kelly betting back when I designed systematic HFT strategies.
possibly a dumb question, but how did this work exactly? the kelly criterion assumes you know the amount by which the coin is weighted, how would you know the equivalent for the stock market in the very near term?
You make a conservative guess. The Kelly criterion is somewhat forgiving about guessing it wrong.
Your question is not dumb: you figured out exactly what's hard about this stuff.
There's a bit of a discussion here: https://news.ycombinator.com/item?id=18484631
How did you apply Kelly to a HFT strategy? Usually those strats don't have a binary outcome so standard Kelly wouldn't fit.
Kelly goes beyond binary outcomes. The underlying principle is the same, though: you maximise expected logarithmic wealth.
To do that you need the joint distribution of outcomes (what are the possible future scenarios and how likely are they?) Estimating this well is the trick to successful application of the Kelly criterion.
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For continuous payoffs, Kelly sizing reduces to the square of Sharpe ratio.
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the binary outcome formulation you see everywhere is just "real" kelly boiled down. the real thing, which is contained fully in the first paragraph ("The Kelly bet size is found by maximizing the expected value of the logarithm of wealth"), has no such restrictions.
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Not sure if it's how they did it, but there's this: https://en.wikipedia.org/wiki/Kelly_criterion#Multiple_outco...
Hi I work at a small hft firm and would love to discuss this more in detail, please contact me if you have the time.
Thank you
A simple description of the Kelly criterion is that if you want to grow wealth over a long period time, at each decision point take the one that maximizes your average expected log wealth.
I'm trying to use it in real life, though sometimes the decisions are quite scary, as it's hard to estimate the probability of outcomes. Also my wealth is much more volatile than most people can stomach, but I look at it like a game.
It doesn't seem obvious that this is a good strategy for personal wealth management because besides maximizing expected wealth, there's another very important criterion: minimizing probability of going broke. I only get to play one game, after all. Obviously you can't go entirely broke if you always bet a fraction of your portfolio, but are there results of how these strategies compare in, say, the probability of dipping below 10%, or 1%, of the starting value?
I can't tell you about the 1% version, but when it dipped to 15%, it was a strange feeling that I made a bad decision with the thinking that I'm making a great decision (or more trying not to think about it and trust the decision that I made earlier). It's a mental game at that point that you have to wait through. At least with investing it's just about waiting through those periods, being a CEO of a company and making decisions in that state would have been much harder.
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Betting with 'full' Kelly-calculated stakes is highly volatile. If I'm remembering correctly, if you get your probabilities/edge exactly right, you will still have a 50/50 chance of losing half of your bank at some point in the future (i.e. after some number of future bets) It's very common to bet just some fraction of the Kelly stakes in order to smooth out the roller coaster ride.
Sure, I've gone through losing more than 80% of my wealth multiple times by being 100% in BTC, so I got used to that already. At the same time it stresses my friends out a lot. I'm expecting to lose more than 50% of my wealth, but at this point it doesn't really change my life style.
E log X strategies are known for Being very volatile.
However, there are two things that take the scariness out of estimating probabilities for me:
- You're often maximising something that looks like a quadratic function. This means you're aiming at a plateau more than a peak: if you make small errors in either direction it doesn't affect growth that much.
- You always have the safe option of underestimating. The E log X strategy forms an "efficient frontier" (to borrow terminology from MPT) of linear combinations from the risk-free rate to the full Kelly bet (and even past it into leveraged Kelly strategies.) You can always mix in more of the risk-free rate and get lower growth but at higher safety.
These two properties makes the Kelly criterion very forgiving to estimation. (In contrast to MPT style mean--variance estimations, and other less principled strategies.)
I find both mean variance and Kelly to be very poor in practice due to the dependence on the expected return term. Like, if I knew that, I wouldn't be wasting my time with all this math! (half joking)
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Do you have examples of how that would be used in real life decisionmaking?
One simple example is buying 2X S&P index ETF instead of 1x. There was a great article about the Kelly optimal S&P allocation, and with all the fees included it's about 2x. Of course there's increased execution risk for the ETF itself, which needs to be estimated.
Another thing where I may look stupid from outside is that I started to take some loan against my BTC and use that to finance my lifestyle, as currently (under $100k BTC price) my estimate of the Kelly optimal BTC allocation is more than 1. This is of course a personal estimate, I don't suggest other people to do the same thing, and again there's a lot of execution risk, so I do this only with a part of my portfolio.
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For people who earn a wage and don't just make money by investing, the Kelly Criterion can't be applied in its basic form, since it means your capital gain has both constant and linear components, instead of just being linear as the formula assumes, which complicates matters a lot.
Plus for low probability high reward bets you have the additional complication that you probably can't make them often enough to get a decent chance of hitting the jackpot.
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I have used the Kelly Criterion successfully in automated sports gambling. It's relevant anywhere you are doing confidence-based arbitrage.
Did you determine the probability of winning based merely on the sport-book odds are do something more sophisticated?
The sport-book odds, as I understand, are merely trying to divide the bets on each side evenly (i.e., they don't necessarily represent a probability).
The bookie odds go into the formula as b- the net fractional odds received on the wager.
We have our own models for our confidence, and the Kelly criterion decides our wager size (though we don't use a full Kelly bet).
Yes, the sportsbook minimizes their own risk by setting a spread or odds with respect to how patrons are wagering. This actually makes it easier to make money if your model is much better than the average bettor. There will be games where public opinion and the majority of bets are on the wrong side of a matchup, and the bookie adjusts the odds accordingly, so the correct bet's payout is bigger than it should be.
In high school I tried to do more what you are asking- use one bookie's odds (which I deemed the most accurate) as the "true probability", and another as the payout. This was not successful, but theoretically could be if the two bookies' clientele were consistently better or worse than each other, therefore influencing their odds consistently.
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I used this to win AI Rock Paper Scissors competition in undergrad. I just played random symbols, but used Kelly criterion to compute my bid. This worked well because the game wouldn’t allow your bankroll to go to 0 — the floor was 1.
Can you explain why the Kelly criterion wouldn't have you bet 0 every time? The chance of winning a round of rock-paper-scissors when throwing a random symbol seems to be 50% (if ties cause re-dos), so wouldn't that work out to 50 * 2 - 100 = 0?
It’s a good question. You’re right, if I followed it strictly it wouldn’t work. I suppose I rationalized offsetting it because I couldn’t actually go bankrupt. If I was better at math there’s probably some other criterion that takes into account how hard it would be to get back to where you are, given that you couldn’t go below 1.
That's strange, given that the Kelly criterion maximizes the expectation of the log of wealth--that is, it's maximizing over multiplicative percent gains in a scenario where you can go bankrupt.
I don’t get why it’s strange? What I learned from that competition was that bid sizing was way more important than the symbol selection strategy. Trying to beat the other students at iocaine powder wasn’t really a winning proposition.
I'd like to share a video I prepared on Kelly betting: https://youtu.be/6xhjbgREGDA
Naval Ravikant has a small post about this here: https://nav.al/kelly-criterion
I first heard about it from him. He summarizes it as follows:
> Naval: The Kelly criterion is a popularized mathematical formulation of a simple concept. The simple concept is: Don’t risk everything. Stay out of jail. Don’t bet everything on one big gamble. Be careful how much you bet each time, so you don’t lose the whole kitty.
Lol. He must have never met anyone who has bet full Kelly.
Seriously. Full Kelly betting involves the use of significant leverage. The correct Kelly bet on the S&P index would be long 2.5x your total wealth.
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If you enjoyed this, I highly recommend reading Fortune's Formula.
And when you're done with that, the Kelly Capital Growth Investment Criterion is one of the better books I've read. But it's a much more advanced read.
And when you are done with that, I highly suggest reading Edward Thorp's autobiography "Man for all markets" where he employs the Kelly Criterion in adventure after adventure. He not only developed the first card counting system for blackjack but he also created the first wearable computer to beat roulette (with Claude Shannon).
Hi, I wrote an essay about Kelly Criterion a while back based on a review of paper by Edward Thorpe. Cheers.
https://medium.com/from-the-diaries-of-john-henry/an-optimal...
If anyone wants to see Kelly in action, I made an app where you define an edge and a wager (% of pot or absolute amount) and see how you fare compared to the optimal bet strategy.
https://kelly-criterion.netlify.app/
https://github.com/breeko/kelly-criterion
Fortune’s Forumla by William Poundstone is an excellent book. Edward Thorpe has most of his papers published too, they’re all good for a read [though I cannot be held responsible if you’re going to beat the dealer at Blackjack, I don’t need Kelly to know how that will turn out :)]
I made a streamlit app about Kelly last year, showing how to bet when you have an "edge" over a toy market of coin flippers: https://kelly-streamlit.herokuapp.com/
Other references I found interesting:
Kelley betting could probably be applied with some success to momentum trading strategies. Momentum trading is more deterministic than purely speculative strategies since it is based on observed/historical behavior.
Momentum itself is as speculative as it gets.
I would say pure speculation is not based on tangible data.
Pure speculation: I think consumer space travel will be popular in the future, let me buy some SpaceX shares.
Momentum: SpaceX seems to be trading higher in pre-market, let me buy some SpaceX shares at market open.
Edit: I know SpaceX is not public, this is just an example.
Hey, I made a game recently for people to gain an intuition for Kelly: https://beatkelly.celebi.me/ Let me know what you think :)
It sounds great, but I like instant results, can you write out after each round how much Kelly put up and has?
I remember the first time I read about this. I put in the numbers for the lottery and a negative number came out. Of course! Your expected winnings are negative and you shouldn't play the lottery.
Well, most of the time, anyway. If you do find a lottery game where the odds are in your favor, something resembling the Kelly criterion is a reasonable starting point for a bankroll-management strategy.
A Nobel winning economist was not impressed by the Kelly criterion. http://www-stat.wharton.upenn.edu/~steele/Courses/434F2005/C...
It’s worth mentioning that Kelly was an associate of Claude Shannon (the father of information theory) at Bell Labs. Kelly’s criterion is in fact based on Shannon’s theory.
It seems they developed the approach together. Shannon, his wife and Ed Thorp later went to Las Vegas gambling using this method, and apparently made some money.
If they made some money gambling in Vegas it was clearly not thanks to Kelly’s criterion, because Kelly’s criterion clearly (and correctly) states that the optimal bet in Vegas is zero dollar.
They played, among other things, blackjack. The rules of blackjack are such that the edge varies around zero. It's mostly a tiny bit negative, but sometimes creeps over zero and that's when you apply bigger than minimum bets, following the Kelly criterion.
And after Vegas Ed Thorp started a hedge fund and made even more money ;)
Edward Thorp used the Kelly Criterion for success in blackjack strategies and later the stock market. He has articles on his statistical methods.
http://www.edwardothorp.com/articles/
Note that the martingale, a common betting strategy, does exactly the opposite of the Kelly criterion. If you have a small edge and bet with the martingale against a very wealthy house, you have a fairly large chance of going bankrupt!
Also worth looking at is this previous discussion on HN,
https://news.ycombinator.com/item?id=13143821
Sadly this only works in games with a positive expected outcome, so it's not actually useful in a casino unless you're a card counter.
Nah, you just use the technique to conclude that the optimal percentage of your bankroll to bet to maximize expected logarithmic wealth is 0% ;)
Ok, fair point. :)
Its really fun to learn something new and realize how incredibly naive you've been your whole life.
brb updating the stardew valley wiki https://stardewvalleywiki.com/Stardew_Valley_Fair
Edit: whoops they already had used this strategy
Kelly criterion (or Kelly strategy or Kelly bet) is a formula for bet sizing that leads almost surely to higher wealth. It was described by J. L. Kelly Jr, a researcher at Bell Labs, in 1956. The Kelly Criterion is to bet a predetermined fraction of assets, and it can seem counterintuitive. In one study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. 18 of the participants bet everything on one toss, while two-thirds gambled on tails. The average payout was just $91.36.
Kelly bet: Bet one-nineteenth of the bankroll that red will not come up. If gamble has 60% chance of winning, gambler should bet 20% of bankroll at each opportunity. In American roulette, the bettor is offered an even money payoff (even money payoff, with a 95% probability of reaching the cap) If edge is negative, then the formula gives a negative result, indicating that the gambler shouldn't bet. The top of the first fraction is the expected net winnings from a $1 bet, since the two outcomes are that you either win with probability or lose the $1 wagered, i.e. win $ + $1, with probability $+.1. For even-money bets, the formula can be simplified to the first simplified formula: b=p-q-q/p, b=1, and b=0.20.
Probability of success is: If you succeed, the value of your investment increases. If you fail (for which the probability is): In that case, the price of the investment decreases. Kelly criterion maximizes the expected value of the logarithm of wealth. Using too much margin is not a good investment strategy when the cost of capital is high, even when the opportunity appears promising. The general result clarifies why leveraging (taking out a loan that requires paying interest in order to raise investment capital) decreases the optimal fraction to be invested, as in that case the odds of winning are less than 0.1. The expected profit must exceed the expected loss for the investment to make any sense. The Kelly criterion can be used to determine the optimal amount of money to invest.
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, that the rest of the bets will lose), one will end up with the most money if one bets. In the long run, final wealth is maximized by setting the derivative to zero, which means following the Kelly strategy. The function is maximised when this derivative is equal to zero. For a rigorous and general proof, see Kelly's original paper[1] or some of the other references listed below. Some corrections have been published[12] .
In the long run, Kelly always wins. This is true whether N is small or large. 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. Betting each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and the odds of winning are the same for each trial. 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) Exponential mechanism (differential privacy)
Please stop doing this (with any account). HN threads are supposed to be conversations. Copying a mass of content from someplace else is not conversation.
(This appears to be a copy-paste of the Wikipedia article.)