Kelly Criterion
en.wikipedia.org
en.wikipedia.org
[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...
This seems to be discussed at greater length among retail traders who have no way of even knowing their odds than any professional.
The post you replied to is right. The fundamental principle of Kelly is that you know your edge, in the markets that is mostly untrue. Funds will volatility-weight their portfolio but this isn't the same as Kelly in practice. Most fund managers will also weight their portfolio towards their "best" position but that is not necessarily based on return. Indeed, picking high return assets is only half the battle.
I also bet a lot, so I am familiar with Kelly. It is totally unusable in finance, no-one uses it in finance, and retail investors have an obsession with it.
In particular, if you Kelly-weight a value portfolio (which the firm linked to in your post is) then you are setting cash on fire. And if you Kelly-weight a long/short portfolio (again, the firm linked to appears to be doing this) then you are setting cash on fire. It is important to understand how a tool works at a practical level.
There needs to be a term for "This page you're reading is bogus horseshit theory, do not try to apply it practically".
When I place a bet, I can estimate my edge because the outcome is binary. When the outcome is continuous, it is far more tricky. It is like saying a kid who learns to ride his trike is ready for MotoGP...they are just totally different.
And yes Soros put on big bets, but what you are missing with Soros is the fact that his hit rate was still 30%. Most of the stuff he did didn't work out, macro is largely bets on skewness not returns. Buffett had a higher hit rate but trying to suggest someone optimise a strategy based on what literally the best investor of all time did is...not smart. Even if you were better than Buffett, you might not be lucky.
The reason why Kelly doesn't work with value investing in particular is because your returns are largely random, you know that your portfolio has an edge but you don't usually know which position is going to revalue.
The reason why Kelly doesn't work with long-short in particular is because you aren't only betting on return but correlation. Anyone who runs Kelly will eventually get a correlation spike and blow up (this is also roughly true of macro, again why Soros isn't a good example, he largely bet on skewness).
I was a "pro" so I am also aware of what most pros do. Again, investors don't only look at return, they have to look at correlation, volatility (note that if you are betting on sports, you don't have to worry about things like correlation).
I agree that “the inputs to the Kelly formula are imprecise and therefore we should not mindlessly implement its recommendations.”
I agree that retail investors should not model their 401k allocations like Soros and Buffett.
Having run a factor neutral long short book I’m extremely familiar with the role of correlation and volatility in portfolio management and position sizing. As others have noted, there are extensions of Kelly (and related portfolio construction formulas) that account for correlations.
I disagree that risk-reward (broadly defined) shouldn’t be the primary bet sizing metric. I think many investors ignore risk reward calculations in their sizing and they would be better off if they paid attention to it. Many of the smartest investors I know have their entire sizing strategy based on risk reward.
To suggest that active investors should ignore risk reward / odds / whatever you want to call it, is wrong, in my opinion.
[2] https://blog.alphatheory.com/2013/01/kelly-criterion-in-prac...
[0] https://alphatheory.zendesk.com/hc/en-us/articles/3600356960... has an explanation of the “Alpha Theory” which I couldn’t quickly find on the alpha theory site.
It seems the choosing the optimal strategy for allocating a portfolio to maximise growth is often called “Kelly style”, “Kelly strategies”, “Kelly methods”, and also “Kelly criterion” by some people (which is why I was confused).
The details of an optimal strategy are completely different depending upon your assumptions (how reallocation is performed as new information is received, accounting for error in predicted outcomes, blah blah blah) so there cannot be a single definition for the Kelly Criterion for a portfolio, instead there are a variety of strategies (each with different assumptions and constraints).
For example the “many assets” model you refer to looks like it models a single market correlation (alpha), and not the multiple correlations within a real market.
Disclaimer: I am not an investment professional, but a small amount of software experience with hedge fund NAV calculations.
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.
1. Investment returns are multiplicative and should be looked at as a geometric series. To optimize the portfolio, optimize for geometric mean not arithmetic mean.
2. To optimize the geometric mean of some specific games, apply some specific mathematical rules that Kelly derived.
Then 2nd part is not applicable to general market investing. The 1st part is.
If you don't understand that, then you are going to go eventually go bust.
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.
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.
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.
> 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.
I will look up your references though!
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.
Any confusion or inconsistency I'm presenting is my fault, and I apologize!
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.
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 sure you can probably tell I know next to nothing about either maths or probability, so feel free to explain why I'm wrong.
Over the long run, you expect 40% tails, but if you run the experiment an infinite amount of time there will be sequences of all-heads or all-tails.
Because the events are independent, the previous flips don't change anything about what happens next.
You can still have streaks of hundreds, thousands, millions of either heads or tails in a row.
So the situation described in that paper is that you are given the true odds of the coin, 60% heads. In this case it's just as I described - knowing previous results doesn't tell you anything useful.
> Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads.
This is true. In fact there is only about 25% chance of getting exactly 6 of the 10 to be heads (but nearly 70% chance of >= 6 heads). You can work this out with something called the binomial distribution. Chance of getting 10 heads in a row is .6%
A more interesting aspect is when you don't know the odds (or don't trust what you've been told). In this case it's definitely important what the history is. So given your 10 flips, we can ask questions like "how likely is it that this coin is fair (50/50) given the 10 flips I just saw".
It turns out the best estimation of the true probability is, pretty intuitively, (h+t)/h; this will jump aroudn for small N . In practice you are more often looking at something like P(0.55 < p < 0.65 | samples) , i.e. the probability that the true value lies between 0.55 and 0.65 heads, given the 10 flips I've seen).
Obviously in these cases, the more samples you have seen the tighter the estimate get. You can also ask questions like how many flips do I need to see to be confident at a certain the coin is really 0.6 heads.
Law of Large Numbers says that, over an arbitrarily large random sampling size, you will eventually end up with a sample that perfectly fits the probability distribution.
But the probability of each individual sample is random. This means that, if each sample is randomly-selected and independent, your history of N samples does not affect your N+1th sample.
The regression to mean curve is only predictable in the big picture, each bump is 50/50 (or 60/40 in this case).
However, until such probability is established, if I see heads in a row - my intuition would tell me that the physics is skewed towards heads. I don't think that it would be unreasonable to think that in such circumstances until one gets a larger sample of throws.
In fact, in the real world getting an unlikely string of heads (or tails, or sixes, or whatever) outside of a casino setting probably means that the coin/dice/whatever are unfairly loaded and you should adjust your expectation for the next coin toss even further towards heads.
Now, suppose you keep throwing, but somebody has opened a window, so now there's an occasionally gust of wind, which moves the paper in unexpected ways while the paper is in the air. Now you no longer hit 100% of your throws. Sometimes the paper lands in the trashcan, sometimes you miss. Regardless, the paper is still only affected by physical forces: your hand, gravity, wind.
Now, suppose you've been really unlucky the past few throws: you have missed 5 throws in a row because of the darn wind. Does it make you more likely to win the next throw, because you are "due" a win? Of course not, because the wind doesn't know or care about your paper throwing hobby. The wind does what it does, regardless of how many of your throws landed in the trashcan. If anything, missing 5 throws in a row makes it _less_ likely to land the next shot, because it may indicate conditions unfavorable to throwing (strong wind, loss of confidence, etc.)
Now, the coin flipping experiment with the weighted coin obeys the same physical laws as the paper tossing experiment. It's just a physical object that's affected by forces from your hand, gravity, air, etc. If you throw 6 heads in a row, there's no magic that somehow alters the coin's path in the air on the 7th toss to make it come down tails. The universe doesn't care about our little games.
How could the past flips of the coin possibly influence the flips you get in the future? The coin hasn't changed, the surrounding area hasn't changed, why would the coin suddenly have a different chance of turning up heads on your next flip? There's no probability god that mucks with random chance to make sure 'runs' are balanced overall. Every coin flip is independent, which means all the coin flips are also independent of the past coin flips.
If you've "had 60%", that means you've had an unlikely run of heads. Let's say the last 6 flips were 5 heads and a tail, a slightly unlikely outcome (3 in 16, I think). What physical force is acting on the coin to make it less likely to be heads, in the future? Why wouldn't it still have a 60% chance of coming up heads on the next flip?
The former implies that previous flips have an effect on future flips. Or that, if you land on heads 6 times in a row, then the probability of it landing on tails goes up. How would a coin that's weighted to increase the odds of it landing on heads, somehow start landing on tails more frequently?
If you flip a normal coin and it lands on heads 10 times, you still have a 50% chance of getting heads the 11th time. The odds of it landing on heads 10 times in a row in the first place is vanishingly small (0.5^10 or 0.097%). But if it Does, the 11th flip still has a 50% chance. The first 10 flips don't affect the 11th. Physically, how Would the first 10 flips affect the 11th?
This is all assuming that the coin flips aren't somehow magically linked or casually dependent on each other. The math changes if the previous coin flip could somehow affect the next one. But in a situation where every single roll of the dice is purely independent, then by definition (Because they are Independent ) a previous roll doesn't have an impact on future rolls
I think people who have a better-than-average understanding of statistics forget how bad their intuition is. I suspect it leads to a lot of incorrect assumptions about what a "rational" behavior for someone working from only their statistical intuition would be.
The intuition that you're going for is that if the true rate is 60% heads and you've seen more than that then to hit 60% odds you _must_ have some extra tails _eventually_. Interestingly, that isn't actually required to make the odds work out to 60% eventually. I'll try for an intuitive explanation:
Say you've gotten 10 heads in a row but that the coin really only has a 60% chance of coming up heads.
- After 1000 extra flips you'll have 610 heads and 400 tails total on average for a 60.4% chance of heads so far.
- After 10k extra flips you'll have 6010 heads and 4000 tails for a 60.04% chance of heads so far.
- After 1M extra flips you'll have 600010 heads and 400k tails for a 60.0004% chance of heads so far.
Notice how the average percentage of heads is getting closer and closer to 60% even though the extra flips don't have _any_ bias toward tails. A temporary bias toward tails would _also_ suffice, and in much less time (some games like WoW use this for their loot tables I think), but it isn't necessary, and in the example of independent coin flips it does not happen.
Nope.
> I’m sure you can probably tell I know next to nothing about either maths or probability, so feel free to explain why I’m wrong.
Lots of people have explained in terms of independence, which is correct. Another way of looking at it (definitely not more correct, but maybe more compatible with the “a series should eventually match the quoted probability” thinking) is in terms of infinity:
If you are expecting 60% of results to be heads, you expect that to hold over an infinite series of flips.
If you see any finite number of heads in a row, the probability for each of the remaining flips in the infinite series to get the total to 60% is…still 60%.
No finite series of results can change the probabilities necessary to get the infinite series to turn out as expected.
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.
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.
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).
It doesn't change the point of my original comment, regardless of the improbability of 60 heads in a row, you aren't "due" 40 tails in a row because the events are independent. That's all I was getting at before you took us on a weird tangent.
The original point of your comment is correct, at least from a probability standpoint. You don't get "owed" tails. I guess my hint was that there are sometimes other factors at play that mean the theory goes out the window. Like if someone shuffles a deck in front of you and it ends up new deck order, it's more likely they're a magician than lucky.
OTOH, the probability of some other explanation besides a fair coin isn’t consistent among all other possible sequences, so what the actual result does to your estimate of the likelihood of a fair coin depends on the actual sequence, and your basis for believing the coin was fair going in.
Things are only slightly different with, say, a coin you’ve been told has a 60% bias.
EDIT: For instance, if there is a 1:1,000,000 chance that you would be given an underestimate of bias and a 1:1,000,000,000 chance of the outcome you actually receive being true if the coin had only the bias you were informed of, its a lot more likely that you were lied to than that you just got an unusually consistent set of results.
> I’ve seen people roll a 20 on a d20 10 times in a row, and then not a single 20 the rest of the session on the same die.
People rolling dice aren’t, even when they try to be, perfect randomizers, and with a maximally favorable result and an action which demonstrably repeats it, there’s a strong incentive to repeat the action as accurately as possible rather than even trying to be a perfect randomizer.
I remember one time when I rolled really low numbers on a D20, and then there was this really important roll, where I had to get a 20. I confidently said "No problem, I rolled a few really low numbers in a row, so this is definitely going to be a 20, it's pure statistics". Also throwing some calculation in there: "I rolled a 2 and a 1, so in 3 rolls I should get a total of 30 on average, so that means I actually still need 27 to reach the average. That results in more than 100% chance of rolling a 20 right now". And then I actually rolled a 20, was able to keep my cool and a straight face "see, it's just theory". Pure gold! LOL :D
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.
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?
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 doubt any story about a biased coins in the real world.
But if you flip a coin and it turns about N times, you can't make the sum (over all k) of the probability of N+2k turns substantially more likely than thr sum of probability of N+2k+1 turns.
So what if I dial the magnetic field back just a bit? So that only when the coin is oriented flat as it lands will it maintain that orientation in spite of the opposing magnetic forces. But if the coin's orientation is near vertical, then the forces are directed to nudge it "headwards" instead of "tailwards".
Your math applies to weighting the coin. It makes sense in that context. I'm talking about a system of magnetic coin and matched magnetic landing pad.
(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 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.
It gets worse. Typically 18-24 year olds who happen to go to the same college as the researcher is working at. So, for example, if this is a large state school then it is a population selected for having SAT scores in a range. Namely above the cutoff to get into the school, but below the cutoff for more desirable schools.
Now suppose that you're doing ability testing. You should expect that any pair of unrelated abilities that help you on SATs will be inversely correlated, because being good at the one thing but landing in that range means you have to be worse at something else. And sometimes that will be the other thing you're looking at.
Several years ago I remember running into a bunch of popular science articles that I found dubious. I tracked down the paper and decided that their analysis suffered from exactly that flaw.
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.
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?
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.
"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."
The PDF is free-to-read.
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.
The optimal strategy would be to estimate which investment maximises your log returns :) but I don't have time for that.
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...
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.
Suppose you can though. For simplicity, suppose you can expose yourself to 0.4U(-1, 1.1), 40U(-1, 1.1), or any other fractional amount F U(-1, 1.1) you might like. Kelly is a technique for choosing F (maybe you had some other idea in mind like that you have to buy into a bet on U(0, 2.1) -- if so, that's nearly equivalent other than putting bounds on F -- the idea of maximizing expected logarithm will carry through to other bet structures).
Going through the motions, suppose you're starting with a bankroll B then you want to choose some ratio F=rB maximizing the expected logarithm of the bet. The distribution of your outcome is another uniform distribution U(B-rB, B+1.1rB), and you want to choose r maximizing the expected logarithm of that distribution. The details of that are probably beyond the scope of a HN comment, but you wind up with r approximately equal to 0.13624.
If you'd like you could plot the result of many instances of 100 such sequential bets with r varying. You'll find that those with r around 0.13624 will usually be much larger than for other choices of r.
The example on Wikipedia supposes we are investing in a geometric Brownian motion and a risk free asset.
optimize the value of the bet size over the expected value of the log of bankroll + betsize*outcome. you can do that for any probability distribution of outcomes.
if you can't write that in 5 minutes, then i already did half your homework for you.
> each bet has a roughly normal distribution outcome
hahaha.
for that trivial case, there's going to be a closed form solution. your nearest copy of mathematica can derive it for you.
not that having a closed form solution is relevant to anything. the answer is still the answer.
Thank you
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?
Your question is not dumb: you figured out exactly what's hard about this stuff.
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.
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.
As a business example startups are starting to apply for loans against their future subscription earnings to reinvest in their companies. Debt against your salary is the personal version of the same strategy.
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.
Optimal over my time period was 2.99x, but the expense ratio was not accounted for.
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.)
Accidentally thinking that "E log X" and "log E X" are the same thing is a common mistake, but Jensen's inequality tells us it can be a costly one.
Of course, your general point still stands: if we only knew the joint distribution of outcomes the battle would be over already.
I know for me I had the moment where things had gone down to roughly 15% and I questioned my decision making. Learning to wait through those periods is super important. Years ago I made the repeated mistakes of not waiting through those periods and missed out on log gains in favor of linear gains.
Agree that it’s easier as an investor and not a CEO to manage that experience day to day.
One thing I can tell you is that banks hate people executing the Kelly strategy, as they expect wealth of people to be predictable so that they can issue loans against it to other people.
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).
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.
As for your last points: bookies often book bets with each other in order to even out the odds. Otherwise you would be able to arbitrage bookies against each other. (Which would also result in their odds evening out, of course, but then the bookies wouldn't get the proceeds so they prefer to do it themselves.)
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.
if you include margin in your bankroll, well, that's on your head.
so 10% expected return and 10% expected volatility, optimal Kelly is 10x leverage.
https://medium.com/from-the-diaries-of-john-henry/an-optimal...
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".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.
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.
Edit: whoops they already had used this strategy