12 ms·

Kelly Criterion (2007)

164 points by kwikiel 8y ago

avvt4avaw 8y ago
This massively oversells the usefulness of the Kelly Criterion. The opening lines are > One should buy stock when it is undervalued. What I have always wondered about is how much stock one should buy. A few months ago I stumbled upon the answer which is given by the Kelly criterion. But the rest of the post analyses a mathematical game which has nothing to do with buying stocks, and is in fact only useful in theoretical situations where you know the precise distribution of outcomes.
- D_Alex 8y ago
  Heh... The most value I got out of applying the Kelly Criterion was having the Exploration Manager for an oil and gas company admitting that he does not believe the estimates that came out of his own department (applying the Kelly Criterion to the numbers he provided would have led to the company carrying the exploration project internally, rather than looking for joint venture partners to share the risks).
- phkahler 8y ago
  >> But the rest of the post analyses a mathematical game which has nothing to do with buying stocks, and is in fact only useful in theoretical situations where you know the precise distribution of outcomes. You might want to think about that. In the real world you don't know anything with precision which means you probably can't do better. One of the unrealistic aspects of it was the idea that you can play as many times as you like, but that's not possible in the real world. But we can move toward playing many times through diversification. That's a good lesson in itself.
  - np_tedious 8y ago
    I don't think diversification is much like playing multiple games at all. One expands on the axis of time, with the game constant. The other expands on the number of unique games all played simultaneously
lordnacho 8y ago
Ex hedge fund guy here. You can extend this into continuous space and use that to tell you how much leverage you should have, given some Sharpe ratio. Results may surprise you (it's a lot for even a modest Sharpe). But also most practitioners aren't going to use the full number. If you've overestimated it you are always worse off on the right side of that.
- joosters 8y ago
  It's common to use some percentage of the Kelly-calculated stakes in betting, Kelly might be the optimal sized bets but it is extremely aggressive. IIRC, even if you are calculating your % edge correctly, Kelly staking means that any point, you've got a 50% chance of losing 50% of all your existing bank at some point in the future. Not many investors/gamblers can stomach that!
  - iopq 8y ago
    I mean, it's not acceptable to use it if there are no stakes below a certain point and no way to borrow more money. Or if the stakes below a certain point have worse returns (higher house take)
- jrx 8y ago
  I've written a blog post about exactly that some time ago, I hope it's at least a bit enjoyable read: https://blog.millionintegrals.com/kelly-criterion-and-investing/ https://blog.millionintegrals.com/kelly-criterion-and-invest...
- evrydayhustling 8y ago
  This. The article concludes: > So what is the answer to the question, how much should stock should I buy? The answer is probably less than you think. But in my experience, optimal log-wealth portfolios look like the craziest guy on your desk. Kelly accepts huge drawdowns that most people would get fired (or fire themselves) for. Half Kelly or quarter Kelly looks more like most professionals' instincts about acceptable risk.
  - 8y ago
deleted 8y ago
[deleted]
OscarCunningham 8y ago
I think the Kelly criterion doesn't apply as widely as people think it does. Its derivation is based on on maximising the growth rate of your fortune. But this is equivalent to assuming that money has logarithmic utility for you. If you don't value money in a logarithmic way then you shouldn't use the Kelly criterion. Personally I feel that my utility function is sublogarithmic. If I'm just spending on myself then beyond a certain point additional money makes me absolutely no happier. Note that the usual justification of progressive taxation also assumes sublogarithmic utility. So based on this we should be more conservative than Kelly. On the other hand, if I plan to give money to charity then my utility function is almost linear. Big charites can absorb a lot of money without becoming less effective. So in this case you should be maximally aggressive, betting everything at every opportunity. Sometimes people say that because the Kelly criterion maximises growth rate it will be the best "in the long run" even if your utility function isn't logarithmic. But I've never seen any evidence of this. Does anybody know of a toy model where you can prove the Kelly criterion is optimal even if your utility is linear?
- whitepoplar 8y ago
  Not sure I follow. Unless I'm misunderstanding, wouldn't Kelly apply especially for your case? As in, Kelly is concerned with avoiding absorbing barriers and increasing bets when they're on "house money" (given a known edge, of course).
  - OscarCunningham 8y ago
    Kelly isn't about either of those things. It's a consequence of Kelly that it never hits the absorbing barrier, but lots of other strategies also have that property. What I'm saying is that among all of those strategies, Kelly isn't optimal.
    - whitepoplar 8y ago
      I misunderstood then, gotcha.
- tim333 8y ago
krackers 8y ago
I've only barely looked into the Kelly Criterion, but can someone explain the intuition behind maximizing the expected value of the log of your wealth? Trying the same derivation mentioned in the article but without the logarithm: the expected value comes out to 0.5×(1 + 1.1×f) + 0.5×(1 − f) = 1 + 0.05f which would make it seem that betting the entire fraction always maximizes your expected value. But why does this reasoning break down in the long term, and why does maximizing the log seem to make it work?
- OscarCunningham 8y ago
  The reasoning doesn't break down in the long term. If you repeatedly bet your entire fortune then you will have a very high probability of losing all of it, but you will also have a very small probability of having a huge amount of money. If your utility really is linear then this is worth it.
- n4r9 8y ago
  Maximising the expected return of a single throw is different to maximising the growth rate over many throws. This is perhaps a bit unintuitive, for the reason that the expectation doesn't tell you a great amount about a distribution. You could imagine a similar game in which one choice of bet results in an enormous return with a tiny probability, or else ruin. Whilst your expected return method would tell you to repeatedly make this choice, doing so is clearly insane.
- chrischattin 8y ago
  You're right if you only have one coin flip. But, this assumes a series of flips. The way I understand the Kelly Criterion it is it's similar to a fancy Martingale. It's statistically inevitable that X number of coin flips will go against you. So, the smaller % of your capital at risk, the less your risk of ruin. And, if you have an edge in the coin flip, the more flips you get, the better you'll perform in the long run. But, it's not that great of a proxy for real life as explained by other commenters in this thread.
- imh 8y ago
  Don't look at the expectation of a single throw. Look at the distribution of the outcome after many throws. Relate that to the product of each throw's relative return. Look at how the log expectation of the product of n random variables converges as n gets big, and you see what to optimize.
praptak 8y ago
The correct number of individually picked stocks you should buy is most probably zero: httpedmarkovich.blogspot.com/2013/12/why-i-dont-trade-stocks-and-probably.html?m=1
- iopq 8y ago
  If you plan on tax loss harvesting through direct indexing, you can buy 500 stocks directly and sell one stock to buy a highly correlated one to decrease your tax payments in taxable accounts.
soVeryTired 8y ago
The the main flaw of the kelly criterion (along with a number of other results in investment theory like Markowitz allocation) is that in practice it's extremely difficult to know the distribution of the result you're betting on. The mean and variance of a prospective investment are not observable. But more to the point, if you try to use some sort of proxy like a sample mean or standard deviation, you'll get inconsistent results over time. We're a long way from the clean, simple, i.i.d world that theorists like to play in.
- miketery 8y ago
  In such cases what are models that are more effective?
  - soVeryTired 8y ago
    I don't know of anything that's guaranteed to be effective out of the box. But as a rule, simpler is better. Any assumptions should be picked apart with a fine-tooth comb and tested for robustness. But in the end, there's not a whole lot that you can do: just cross your fingers and hope.
  - iopq 8y ago
    You could probably model something that uses sampled variance less aggressively
    - sjg007 8y ago
      Something bayesian
- downandout 8y ago
  in practice it's extremely difficult to know the distribution of the result you're betting on. This of course depends on your use case. For example, the Kelly Criterion is a staple of advantage gambling. For most casino games, probability distribution can be easily calculated with absolute accuracy. I'm sure there are other use cases for it where probability distribution is also fairly stable.
freddie_mercury 8y ago
The Kelly Criterion was the subject of an incomprehensibly bitter argument in the 1970s/1980s. Paul Samuelson, considered by many to be the greatest economist of the 20th century, believed the Kelly Criterion was wrong. And not just wrong but SO WRONG that anyone who believed it was an idiot. The kind of idiot who could only understand single syllable words. So he wrote a paper in the Journal of Finance and Banking in words of only a single syllable saying why no one should use the Kelly Criterion. http://www-stat.wharton.upenn.edu/~steele/Courses/434/434Context/Kelly%20Resources/Samuelson1979.pdf http://www-stat.wharton.upenn.edu/~steele/Courses/434/434Con...
- n4r9 8y ago
  Interesting. I don't know much about economics or the Kelly criterion, but my understanding of the argument is that maximising the growth rate is distinct from maximising actual profit arbitrarily far into the future. Would be cool if there was a simple toy example demonstrating a situation where these differ. Perhaps it even differs for the example in OP.
  - norswap 8y ago
    How is the growth rate determined? Since presumably it varies, is it averaged in some ways?
    - imh 8y ago
      If a big growth rate is a composition of a bunch of small changes with their own growth rate, then the overall change is the product of the individual ones, so geometric mean is appropriate.
    - n4r9 8y ago
      I wrote a reply, but accidentally made it top-level: It's the expected growth rate, so yes it would vary in a real-world instance. I wondered for a while why they've taken the logarithm, but I think it's just because growth models are normally defined exponentially (with the rate being a parameter inside the exponential) - it shouldn't make a difference to the result. As for varying, that Samuelson article says: > For N as large as one likes, your growth rate can well (and at times must) turn out to be less than mine - and turn out so much less that my tastes for risk will force me to shun your mode of play.
anonu 8y ago
Always a good topic to discuss. Many applications to high-frequency trading due to the probabilistic nature of outcomes. Here are 2 previous HN discussions: https://news.ycombinator.com/item?id=13143821 https://news.ycombinator.com/item?id=13143821 https://news.ycombinator.com/item?id=2504222 https://news.ycombinator.com/item?id=2504222
evrydayhustling 8y ago
One thing I find really interesting about the Kelly Criterion is that it exposes a very stealthy and fundamental "rich get richer" phenomenon. Most real-life risks have a minimum and maximum investment amounts, meaning that you can't just size the bet exactly as Kelly says. So if your wealth is low, you cannot rationally participate in many risky but positive-expected-value investments. Simply put, the poor can't take many worthwhile risks (think college!) without rising ruin (and sub-optimal growth). Conversely, the rich can come closer to maximizing EV in many risky markets at once, increasing income and growth while even decreasing variance.
haliax 8y ago
The Kelly Criterion is the subject of an absolutely incredible book by William Poundstone called "Fortune's Formula". In the course of discussing the formula, the book takes you through the birth of the MIT blackjack team, the genesis of statistical arbitrage, and mini biographies of people like Claude Shannon and Ed Thorpe. I can't recommend it highly enough.
- r00fus 8y ago
  Don't forget the organized crime connection. It was an absolutely fascinating read. My dad was a daytrader (read: armchair gambler) but this helped him curb his trading - he wasn't ready to do all the statistical analyses to keep rigorously investing.
rlander 8y ago
A similar (but more useful) position sizing strategy is the Optimal F formula, described by Ralph Vince in his book Portfolio Management Formulas. But its real value is in showing you your 'cliff of death' curve: how close you can get to bankruptcy given your position sizing. In my opinion, position sizing is more way more important (and less understood) than market timing.
bedhead 8y ago
There was a hedge fund manager named Mark Sellers who blew up his fund in 2008 by following the Kelly Formula exactly, which had told him to put 90% of his fund in a single stock, which was a small offshore oil/gas driller. It was real money too, over $200 million.
- ignoranceprior 8y ago
  Nominative determinism? Although perhaps Mark Buyers would have been even more appropriate.
pyrex41 8y ago
The reasoning behind the Kelly Criterion was explored recently in a more broad context, showing that the logarithmic utility is not required: https://aip.scitation.org/doi/10.1063/1.4940236 https://aip.scitation.org/doi/10.1063/1.4940236 Taleb has a good discussion here: https://medium.com/incerto/the-logic-of-risk-taking-107bf41029d3 https://medium.com/incerto/the-logic-of-risk-taking-107bf410...
pyrex41 8y ago
Most of the examples of Kelly criterion application are either concrete bets with discrete payoff/loss odds and values, or assumed to be normally distributed. This paper discusses how extremely skewed outcomes (eg, stock options) should affect the Kelly calculation: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2956161 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2956161
n4r9 8y ago
It's the expected growth rate, so yes it would vary in a real-world instance. I wondered for a while why they've taken the logarithm, but I think it's just because growth models are normally defined exponentially (with the rate being a parameter inside the exponential) - it shouldn't make a difference to the result. As for varying, that Samuelson article says: > For N as large as one likes, your growth rate can well (and at times must) turn out to be less than mine - and turn out so much less that my tastes for risk will force me to shun your mode of play.
- btilly 8y ago
  The real reason to take logarithms is that investment strategies repeatedly multiply our net worth by random factors. Taking the logarithm turns multiplication into addition, and we know a lot about the statistics of adding lots of random things together. (Thanks to the Weak Law of Large Numbers, the Strong Law of Large Numbers and the Central Limit Theorem.)
  - arnioxux 8y ago
    I think the log is just the utility function. You could substitute it with any non-linear utility function and it would've gave you another answer that makes sense for that utility. The main takeaway is that a linear expected utility doesn't make sense. It would've told you to bet all your wealth every game, which does result in a higher linear expected value, where you win (1+1.1)^N with probability 1/2^N at time N, but 0 otherwise. But no real human would take the bet of extreme high payoff at extremely rare chances with ruin otherwise. Also see St. Petersburg paradox for a similar "paradox" resolved with expected utility theory.
    - btilly 8y ago
      I'm sorry, but you are plain wrong. The log has nothing to do with utility. And there is no chance of really understanding the result if you're confusing yourself with that bad idea. To start, EVERY utility function that is both increasing and sublinear will agree that Kelly is the best strategy. Whether square root, log, or bounded - it doesn't matter. The details of your utility function are unimportant. What matters is that each iteration of an investment strategy multiplies your net worth by a random factor. But log turns multiplication into addition. And statistics has very strong results about sums of independent variables. The result is that with 100% odds, a player following Kelly will eventually wind up ahead of any other static strategy that you could choose. Both wind up ahead and eventually remain ahead. Which is why a wide variety of utility functions will conclude that Kelly is the optimal strategy.
dafty4 8y ago
"If all the economists in the world were placed end to end they would not reach a conclusion" -Isaac Marcoson (attrib. 1933 by O.O. McIntyre) http://www.systemicrisk.ac.uk/sites/default/files/downloads/publications/dp-52.pdf http://www.systemicrisk.ac.uk/sites/default/files/downloads/...
aidenn0 8y ago
The most interesting thing to me about the Kelly criterion is that it demonstrates that the martingale system[1] is a bad strategy even if the odds are in your favor! While it's immediately obvious that the martingale is bad if the odds are in the house's favor, it's less obvious that you are likely to go bankrupt with the martingale even if the odds are slightly in your favor (assuming the house's bankroll is much greater than yours). 1: https://en.wikipedia.org/wiki/Martingale_(betting_system) https://en.wikipedia.org/wiki/Martingale_(betting_system)
bigpicture 8y ago
Wow, this is the 2nd time in two weeks that something has appeared on my probability homework and then made the front page of HN almost immediately. Coincidence? Who else is taking Stat 110?

164 points by kwikiel 8y ago

This massively oversells the usefulness of the Kelly Criterion. The opening lines are > One should buy stock when it is undervalued. What I have always wondered about is how much stock one should buy. A few months ago I stumbled upon the answer which is given by the Kelly criterion. But the rest of the post analyses a mathematical game which has nothing to do with buying stocks, and is in fact only useful in theoretical situations where you know the precise distribution of outcomes.

D_Alex 8y ago
Heh... The most value I got out of applying the Kelly Criterion was having the Exploration Manager for an oil and gas company admitting that he does not believe the estimates that came out of his own department (applying the Kelly Criterion to the numbers he provided would have led to the company carrying the exploration project internally, rather than looking for joint venture partners to share the risks).
phkahler 8y ago
>> But the rest of the post analyses a mathematical game which has nothing to do with buying stocks, and is in fact only useful in theoretical situations where you know the precise distribution of outcomes. You might want to think about that. In the real world you don't know anything with precision which means you probably can't do better. One of the unrealistic aspects of it was the idea that you can play as many times as you like, but that's not possible in the real world. But we can move toward playing many times through diversification. That's a good lesson in itself.
- np_tedious 8y ago
  I don't think diversification is much like playing multiple games at all. One expands on the axis of time, with the game constant. The other expands on the number of unique games all played simultaneously

lordnacho 8y ago

Ex hedge fund guy here. You can extend this into continuous space and use that to tell you how much leverage you should have, given some Sharpe ratio. Results may surprise you (it's a lot for even a modest Sharpe). But also most practitioners aren't going to use the full number. If you've overestimated it you are always worse off on the right side of that.

joosters 8y ago
It's common to use some percentage of the Kelly-calculated stakes in betting, Kelly might be the optimal sized bets but it is extremely aggressive. IIRC, even if you are calculating your % edge correctly, Kelly staking means that any point, you've got a 50% chance of losing 50% of all your existing bank at some point in the future. Not many investors/gamblers can stomach that!
- iopq 8y ago
  I mean, it's not acceptable to use it if there are no stakes below a certain point and no way to borrow more money. Or if the stakes below a certain point have worse returns (higher house take)
jrx 8y ago
I've written a blog post about exactly that some time ago, I hope it's at least a bit enjoyable read: https://blog.millionintegrals.com/kelly-criterion-and-investing/ https://blog.millionintegrals.com/kelly-criterion-and-invest...
evrydayhustling 8y ago
This. The article concludes: > So what is the answer to the question, how much should stock should I buy? The answer is probably less than you think. But in my experience, optimal log-wealth portfolios look like the craziest guy on your desk. Kelly accepts huge drawdowns that most people would get fired (or fire themselves) for. Half Kelly or quarter Kelly looks more like most professionals' instincts about acceptable risk.
- 8y ago

deleted 8y ago

[deleted]

OscarCunningham 8y ago

I think the Kelly criterion doesn't apply as widely as people think it does. Its derivation is based on on maximising the growth rate of your fortune. But this is equivalent to assuming that money has logarithmic utility for you. If you don't value money in a logarithmic way then you shouldn't use the Kelly criterion. Personally I feel that my utility function is sublogarithmic. If I'm just spending on myself then beyond a certain point additional money makes me absolutely no happier. Note that the usual justification of progressive taxation also assumes sublogarithmic utility. So based on this we should be more conservative than Kelly. On the other hand, if I plan to give money to charity then my utility function is almost linear. Big charites can absorb a lot of money without becoming less effective. So in this case you should be maximally aggressive, betting everything at every opportunity. Sometimes people say that because the Kelly criterion maximises growth rate it will be the best "in the long run" even if your utility function isn't logarithmic. But I've never seen any evidence of this. Does anybody know of a toy model where you can prove the Kelly criterion is optimal even if your utility is linear?

whitepoplar 8y ago
Not sure I follow. Unless I'm misunderstanding, wouldn't Kelly apply especially for your case? As in, Kelly is concerned with avoiding absorbing barriers and increasing bets when they're on "house money" (given a known edge, of course).
- OscarCunningham 8y ago
  Kelly isn't about either of those things. It's a consequence of Kelly that it never hits the absorbing barrier, but lots of other strategies also have that property. What I'm saying is that among all of those strategies, Kelly isn't optimal.
  - whitepoplar 8y ago
    I misunderstood then, gotcha.
tim333 8y ago

krackers 8y ago

I've only barely looked into the Kelly Criterion, but can someone explain the intuition behind maximizing the expected value of the log of your wealth? Trying the same derivation mentioned in the article but without the logarithm: the expected value comes out to 0.5×(1 + 1.1×f) + 0.5×(1 − f) = 1 + 0.05f which would make it seem that betting the entire fraction always maximizes your expected value. But why does this reasoning break down in the long term, and why does maximizing the log seem to make it work?

OscarCunningham 8y ago
The reasoning doesn't break down in the long term. If you repeatedly bet your entire fortune then you will have a very high probability of losing all of it, but you will also have a very small probability of having a huge amount of money. If your utility really is linear then this is worth it.
n4r9 8y ago
Maximising the expected return of a single throw is different to maximising the growth rate over many throws. This is perhaps a bit unintuitive, for the reason that the expectation doesn't tell you a great amount about a distribution. You could imagine a similar game in which one choice of bet results in an enormous return with a tiny probability, or else ruin. Whilst your expected return method would tell you to repeatedly make this choice, doing so is clearly insane.
chrischattin 8y ago
You're right if you only have one coin flip. But, this assumes a series of flips. The way I understand the Kelly Criterion it is it's similar to a fancy Martingale. It's statistically inevitable that X number of coin flips will go against you. So, the smaller % of your capital at risk, the less your risk of ruin. And, if you have an edge in the coin flip, the more flips you get, the better you'll perform in the long run. But, it's not that great of a proxy for real life as explained by other commenters in this thread.
imh 8y ago
Don't look at the expectation of a single throw. Look at the distribution of the outcome after many throws. Relate that to the product of each throw's relative return. Look at how the log expectation of the product of n random variables converges as n gets big, and you see what to optimize.

praptak 8y ago

The correct number of individually picked stocks you should buy is most probably zero: httpedmarkovich.blogspot.com/2013/12/why-i-dont-trade-stocks-and-probably.html?m=1

iopq 8y ago
If you plan on tax loss harvesting through direct indexing, you can buy 500 stocks directly and sell one stock to buy a highly correlated one to decrease your tax payments in taxable accounts.

soVeryTired 8y ago

The the main flaw of the kelly criterion (along with a number of other results in investment theory like Markowitz allocation) is that in practice it's extremely difficult to know the distribution of the result you're betting on. The mean and variance of a prospective investment are not observable. But more to the point, if you try to use some sort of proxy like a sample mean or standard deviation, you'll get inconsistent results over time. We're a long way from the clean, simple, i.i.d world that theorists like to play in.

miketery 8y ago
In such cases what are models that are more effective?
- soVeryTired 8y ago
  I don't know of anything that's guaranteed to be effective out of the box. But as a rule, simpler is better. Any assumptions should be picked apart with a fine-tooth comb and tested for robustness. But in the end, there's not a whole lot that you can do: just cross your fingers and hope.
- iopq 8y ago
  You could probably model something that uses sampled variance less aggressively
  - sjg007 8y ago
    Something bayesian
downandout 8y ago
in practice it's extremely difficult to know the distribution of the result you're betting on. This of course depends on your use case. For example, the Kelly Criterion is a staple of advantage gambling. For most casino games, probability distribution can be easily calculated with absolute accuracy. I'm sure there are other use cases for it where probability distribution is also fairly stable.

freddie_mercury 8y ago

The Kelly Criterion was the subject of an incomprehensibly bitter argument in the 1970s/1980s. Paul Samuelson, considered by many to be the greatest economist of the 20th century, believed the Kelly Criterion was wrong. And not just wrong but SO WRONG that anyone who believed it was an idiot. The kind of idiot who could only understand single syllable words. So he wrote a paper in the Journal of Finance and Banking in words of only a single syllable saying why no one should use the Kelly Criterion. http://www-stat.wharton.upenn.edu/~steele/Courses/434/434Context/Kelly%20Resources/Samuelson1979.pdf http://www-stat.wharton.upenn.edu/~steele/Courses/434/434Con...

n4r9 8y ago
Interesting. I don't know much about economics or the Kelly criterion, but my understanding of the argument is that maximising the growth rate is distinct from maximising actual profit arbitrarily far into the future. Would be cool if there was a simple toy example demonstrating a situation where these differ. Perhaps it even differs for the example in OP.
- norswap 8y ago
  How is the growth rate determined? Since presumably it varies, is it averaged in some ways?
  - imh 8y ago
    If a big growth rate is a composition of a bunch of small changes with their own growth rate, then the overall change is the product of the individual ones, so geometric mean is appropriate.
  - n4r9 8y ago
    I wrote a reply, but accidentally made it top-level: It's the expected growth rate, so yes it would vary in a real-world instance. I wondered for a while why they've taken the logarithm, but I think it's just because growth models are normally defined exponentially (with the rate being a parameter inside the exponential) - it shouldn't make a difference to the result. As for varying, that Samuelson article says: > For N as large as one likes, your growth rate can well (and at times must) turn out to be less than mine - and turn out so much less that my tastes for risk will force me to shun your mode of play.

anonu 8y ago

Always a good topic to discuss. Many applications to high-frequency trading due to the probabilistic nature of outcomes. Here are 2 previous HN discussions: https://news.ycombinator.com/item?id=13143821 https://news.ycombinator.com/item?id=13143821 https://news.ycombinator.com/item?id=2504222 https://news.ycombinator.com/item?id=2504222

evrydayhustling 8y ago

One thing I find really interesting about the Kelly Criterion is that it exposes a very stealthy and fundamental "rich get richer" phenomenon. Most real-life risks have a minimum and maximum investment amounts, meaning that you can't just size the bet exactly as Kelly says. So if your wealth is low, you cannot rationally participate in many risky but positive-expected-value investments. Simply put, the poor can't take many worthwhile risks (think college!) without rising ruin (and sub-optimal growth). Conversely, the rich can come closer to maximizing EV in many risky markets at once, increasing income and growth while even decreasing variance.

haliax 8y ago

The Kelly Criterion is the subject of an absolutely incredible book by William Poundstone called "Fortune's Formula". In the course of discussing the formula, the book takes you through the birth of the MIT blackjack team, the genesis of statistical arbitrage, and mini biographies of people like Claude Shannon and Ed Thorpe. I can't recommend it highly enough.

r00fus 8y ago
Don't forget the organized crime connection. It was an absolutely fascinating read. My dad was a daytrader (read: armchair gambler) but this helped him curb his trading - he wasn't ready to do all the statistical analyses to keep rigorously investing.

rlander 8y ago

A similar (but more useful) position sizing strategy is the Optimal F formula, described by Ralph Vince in his book Portfolio Management Formulas. But its real value is in showing you your 'cliff of death' curve: how close you can get to bankruptcy given your position sizing. In my opinion, position sizing is more way more important (and less understood) than market timing.

bedhead 8y ago

There was a hedge fund manager named Mark Sellers who blew up his fund in 2008 by following the Kelly Formula exactly, which had told him to put 90% of his fund in a single stock, which was a small offshore oil/gas driller. It was real money too, over $200 million.

ignoranceprior 8y ago
Nominative determinism? Although perhaps Mark Buyers would have been even more appropriate.

pyrex41 8y ago

The reasoning behind the Kelly Criterion was explored recently in a more broad context, showing that the logarithmic utility is not required: https://aip.scitation.org/doi/10.1063/1.4940236 https://aip.scitation.org/doi/10.1063/1.4940236 Taleb has a good discussion here: https://medium.com/incerto/the-logic-of-risk-taking-107bf41029d3 https://medium.com/incerto/the-logic-of-risk-taking-107bf410...

pyrex41 8y ago

Most of the examples of Kelly criterion application are either concrete bets with discrete payoff/loss odds and values, or assumed to be normally distributed. This paper discusses how extremely skewed outcomes (eg, stock options) should affect the Kelly calculation: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2956161 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2956161

n4r9 8y ago

It's the expected growth rate, so yes it would vary in a real-world instance. I wondered for a while why they've taken the logarithm, but I think it's just because growth models are normally defined exponentially (with the rate being a parameter inside the exponential) - it shouldn't make a difference to the result. As for varying, that Samuelson article says: > For N as large as one likes, your growth rate can well (and at times must) turn out to be less than mine - and turn out so much less that my tastes for risk will force me to shun your mode of play.

btilly 8y ago
The real reason to take logarithms is that investment strategies repeatedly multiply our net worth by random factors. Taking the logarithm turns multiplication into addition, and we know a lot about the statistics of adding lots of random things together. (Thanks to the Weak Law of Large Numbers, the Strong Law of Large Numbers and the Central Limit Theorem.)
- arnioxux 8y ago
  I think the log is just the utility function. You could substitute it with any non-linear utility function and it would've gave you another answer that makes sense for that utility. The main takeaway is that a linear expected utility doesn't make sense. It would've told you to bet all your wealth every game, which does result in a higher linear expected value, where you win (1+1.1)^N with probability 1/2^N at time N, but 0 otherwise. But no real human would take the bet of extreme high payoff at extremely rare chances with ruin otherwise. Also see St. Petersburg paradox for a similar "paradox" resolved with expected utility theory.
  - btilly 8y ago
    I'm sorry, but you are plain wrong. The log has nothing to do with utility. And there is no chance of really understanding the result if you're confusing yourself with that bad idea. To start, EVERY utility function that is both increasing and sublinear will agree that Kelly is the best strategy. Whether square root, log, or bounded - it doesn't matter. The details of your utility function are unimportant. What matters is that each iteration of an investment strategy multiplies your net worth by a random factor. But log turns multiplication into addition. And statistics has very strong results about sums of independent variables. The result is that with 100% odds, a player following Kelly will eventually wind up ahead of any other static strategy that you could choose. Both wind up ahead and eventually remain ahead. Which is why a wide variety of utility functions will conclude that Kelly is the optimal strategy.

dafty4 8y ago

"If all the economists in the world were placed end to end they would not reach a conclusion" -Isaac Marcoson (attrib. 1933 by O.O. McIntyre) http://www.systemicrisk.ac.uk/sites/default/files/downloads/publications/dp-52.pdf http://www.systemicrisk.ac.uk/sites/default/files/downloads/...

aidenn0 8y ago

The most interesting thing to me about the Kelly criterion is that it demonstrates that the martingale system[1] is a bad strategy even if the odds are in your favor! While it's immediately obvious that the martingale is bad if the odds are in the house's favor, it's less obvious that you are likely to go bankrupt with the martingale even if the odds are slightly in your favor (assuming the house's bankroll is much greater than yours). 1: https://en.wikipedia.org/wiki/Martingale_(betting_system) https://en.wikipedia.org/wiki/Martingale_(betting_system)

bigpicture 8y ago

Wow, this is the 2nd time in two weeks that something has appeared on my probability homework and then made the front page of HN almost immediately. Coincidence? Who else is taking Stat 110?