[HN Gopher] Kelly Criterion - how to calculate optimal bet sizes
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Kelly Criterion - how to calculate optimal bet sizes
Author : fernandohur
Score : 132 points
Date : 2021-06-08 07:37 UTC (1 days ago)
(HTM) web link (fhur.github.io)
(TXT) w3m dump (fhur.github.io)
| YossarianFrPrez wrote:
| It's well worth implementing and testing out the Kelly Criterion.
| It's super simple to code up in a Jupyter Notebook so that you
| get to enter an amount to bet each time. When I tried it, I found
| my own psychology changing as the bets continued, even when I
| knew the coin's bias. It's a really great demonstration of the
| difference between a) intellectually knowing the optimal
| strategy, and b) what actually happens.
|
| A bet on a biased coin paradigm was actually tried in the real
| world with finance professionals, with a cap on the maximum
| payout. The results are described here:
| https://arxiv.org/pdf/1701.01427.pdf It's pretty interesting.
| (Note though that the "average returns" reported hide a lot of
| variation.)
| bugzz wrote:
| A great investing blog I follow is
| https://breakingthemarket.com/. He talks about the Kelly
| Criterion and extending it to making correlated bets
| (investments).
| piyh wrote:
| There's a loose vs lose typo in the second paragraph. It's like
| school teachers whipped they're, their, there into our heads and
| where one typo door closes another opens.
| mwcremer wrote:
| Its definately a loosing battle.
| epapsiou wrote:
| This is so timely. I made this with a buddy of mine to help us
| figure out optimal allocation for stocks in a portfolio using
| Kelly. https://engine.oracled.com/
| tel wrote:
| Be careful using this formula too naively. Predicting
| tomorrow's expected return is quite difficult, though
| predicting tomorrow's expected volatility is doable.
| nostromo wrote:
| It's telling me I should short SPY and go long GameStop and
| AMC... Sounds suicidal to me.
| allyourhorses wrote:
| This is beta hedging, it works on the assumption that when
| SPY performs positively, your risky basket will outperform
| SPY, but if there is some systemic risk-off event, your SPY
| short will at least dampen if not fully cover any losses made
| in your risky basket
|
| Good day, you lose -0.5% on SPY but gain +2% on AMC
|
| Bad day, you maybe gain 0.5-1%% on SPY and lose -2% on AMC
| behrlich wrote:
| Another word of caution. The Kelly Criterion depends on each
| event being independent. Lets say I'm told to allocate 50% to
| QQQ and 50% to SPY. Those may independently be correct, but
| since the NASDAQ and S&P are highly correlated, this wouldn't
| be the correct allocation. You've essentially allocated 100% of
| your portfolio to one probability, rather than 50% to two
| independent probabilities.
|
| This is an obvious example. But really all stocks (or at least
| sectors) are correlated just like this. So other examples
| wouldn't be so obvious.
| hatsunearu wrote:
| I have no idea how this website works, could you explain?
| deertick1 wrote:
| How does kelly criterion apply to sizing portfolio positions?
| From what I understand trying to use it in the stock market is
| a fools errand.
| Etheryte wrote:
| Some estimates end up being negative, is that intentional? E.g
| AAPL.
| jyriand wrote:
| How you will go bust in favorable bet by N N Taleb -
| https://youtu.be/91IOwS0gf3g
| titanomachy wrote:
| Am I missing something? It doesn't seem counterintuitive to me
| that repeatedly making an all-or-nothing bet with a non-zero
| chance of losing will eventually cause my expected value of
| capital to go to zero.
|
| I like the presentation style though, and the allusion to
| Shannon's theorem although I didn't quite grasp the connection.
| oarabbus_ wrote:
| Right, I don't think that's the part that's counterintuitive.
| It's the claim he makes "if someone offers you a bet 70%
| chance to win a dollar, 30% loss, it's better not to take it
| in some cases" is what is counterintuitive to people.
| tel wrote:
| I forget what he said in the video, but generally this thing
| can be a little surprising because even a pretty decent
| looking bet can blow you up with high probability if you size
| it wrong.
|
| The other side of this is Shannon's Demon: a sort of crappy
| game can be made into a profitable one by sizing it properly
| (and rebalancing).
| [deleted]
| murbard2 wrote:
| The Kelly criterion is to financial math as the Fibonacci
| sequence is to mathematics. Yes it's neat, no it's not special,
| please stop bringing it up all the time.
| ltbarcly3 wrote:
| This so much. Like anything that is 'optimal' it is optimal
| with respect to some criterion. For the Kelly Criterion it is
| to maximize the logarithm of the weighted sum of the expected
| value across all outcomes.
|
| This is probably not what you actually want in almost any
| situation.
|
| The one time it actually makes sense to bring it in is if you
| are forced to make a certain number of wagers in some game AND
| you have good-enough knowledge of the odds of the game AND your
| payout is proportional to how much money you have left at the
| end of the game, AND the wager size does not effect the outcome
| of the game in any way. This scenario never happens.
|
| Even when you meet many of the necessary prerequisites to use
| Kelly, it still doesn't make sense at all. For example,
| Blackjack tournaments. You are given a set amount of starting
| money, you play for a set number of hands (or time), and you
| know the odds perfectly. However, the payout structure isn't
| proportional to your final amount of money, so using Kelly has
| more or less no chance of winning any money in the tournament.
| They usually pay for the top N results, which means you have to
| go with a very high variance strategy AND win consistently to
| place.
|
| Poker: Nope, not even close, bet size is a direct input into
| the dynamics of the game. Predictable betting of any kind is a
| maximally bad strategy.
|
| Stock Market: utility of money isn't logarithmic, so it is not
| worth maximizing, even if you knew probabilities and were
| forced to make wagers, which you don't and aren't. If you could
| even approximate probabilities you could use that power to
| basically print money on derivatives, so even if Kelly applied,
| the prerequisites are too strong and would make far far better
| strategies available.
|
| The only valid use case for bringing in the Kelly Criteria is
| for gamblers to feel better about burning money at the tables
| by improperly applying it.
| wizzwizz4 wrote:
| > _For the Kelly Criterion it is to maximize the logarithm of
| the weighted sum of the expected value across all outcomes._
|
| It's actually to maximise the expected logarithm of the
| monetary amount, and it's a pretty good heuristic (in most
| circumstances) given that most opportunities are exponential.
| (It usually takes a given amount of effort to double your
| money.)
| concreteblock wrote:
| You obviously wouldn't use Kelly criterion during a poker
| game because the assumptions don't fit. But on a larger scale
| it can be used for 'bankroll management' - what proportion of
| your wealth should you use on a tournament entry fee. Of
| course you don't have the exact parameter p but you can use
| an estimate to make sure you are not making a grossly
| over/under-sized bet.
| arthurcolle wrote:
| Kelly must be the name of the wife in the WSB "wife's
| boyfriend" memes.
|
| More seriously, great analysis. Couldn't have said it better
| myself.
| concreteblock wrote:
| The first sentence of your post, while technically true,
| misses the point. This misunderstanding undermines many of
| your other points.
|
| The kelly criterion happens to be optimal with respect to log
| wealth but that's not the main reason why it's interesting.
| Many explanations, including the original post, make this
| mistake. Maybe because 'maximizing expected utility' is a
| more common idea.
|
| The first sentence of the wikipedia article:
|
| "In probability theory and intertemporal portfolio choice,
| the Kelly criterion (or Kelly strategy or Kelly bet), also
| known as the scientific gambling method, is a formula for bet
| sizing that leads almost surely (under the assumption of
| known expected returns) to higher wealth compared to any
| other strategy in the long run".
|
| In other words, pick a strategy. I'll pick the kelly
| strategy. There will be some point in time, after which I
| will have more money than you, and you will never overtake
| me. No logarithms involved. This is something you can easily
| check by simulation, but requires some heavier math to
| formulate precisely and prove.
|
| See also posts by spekcular and rssoconnor elsewhere in this
| thread.
| thenaturalist wrote:
| Hi there, you seem to know a good deal about the intricacies
| of where the Kelly Criterion lies within the range of
| possible options to calculate outcomes.
|
| Could you point me to any good resource you know of to learn
| more about the range itself and other options?
| lordnacho wrote:
| When I was a young guy who happened to be partner in a hedge
| fund, I used to ask candidates about this.
|
| I kinda regret it now. I'm not sure what I was trying to find
| out from people. I guess in some way it's a cultural question
| masquerading as a technical question, because it reveals
| whether you've heard of it and whether you have heard of the
| standard stuff that's said about it:
|
| - Don't do full Kelly because if you're on the wrong side (too
| big bets) that's definitely bad.
|
| - Depends on the probabilities being static.
|
| - You can find a continuous form of it. You can also find the
| implied leverage from the Sharpe ratio.
|
| I wonder if my memory is even correct on these. But the point
| remains that it's not terribly useful as a thing to evaluate
| people with. I guess the question "How do you size your
| positions?" needs to start somewhere.
| bobbylarrybobby wrote:
| The Fibonacci sequence is not special because it is just one
| element of the family of linear recurrences. Is there a larger
| family to which the Kelly criterion belongs? (Not being snide
| -- I'm genuinely asking)
| paulpauper wrote:
| the entire field of martingale pricing
| arthurcolle wrote:
| I can hear your sigh in answering this all the way from
| here :D
| murbard2 wrote:
| Yes, any utility function will give you a Kelly-like
| criterion. Kelly is log utility.
| spekcular wrote:
| Kelly is special though, it's not just log utility. In
| fact, viewing it as maximizing log utility is ahistorical;
| the original derivation was in terms of information theory.
| (The paper's title is "A new interpretation of information
| rate" [0].) Further, and most importantly, Kelly betting
| has certain favorable asymptotic properties that betting
| strategies motivated by other utility functions don't have.
| See Breiman's "Optimal gambling systems for favorable
| games" [1].
|
| This point is well known in the literature, for instance
| see [2]:
|
| > Perhaps one reason is that maximizing E log S suggests
| that the investor has a logarithmic utility for money.
| However, the criticism of the choice of utility functions
| ignores the fact that maximizing E log S is a consequence
| of the goals represented by properties P1 and P2, and has
| nothing to do with utility theory.
|
| [0]
| https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf
|
| [1] http://www-
| stat.wharton.upenn.edu/~steele/Resources/FTSResou...
|
| [2]
| https://pubsonline.informs.org/doi/abs/10.1287/moor.5.2.161
| rssoconnor wrote:
| Exactly. The Kelly strategy is the strategy that, with
| probability 1, eventually and permanently beats any other
| strategy. This isn't true for any other strategy and this
| criterion has nothing to do with utility.
| LudwigNagasena wrote:
| That's wrong. The assumptions behind the Kelly criterion
| lead to the log utility only in a particular case.
|
| What if each week you can bet $1 that in 50% of cases will
| triple your bet and in 50% of cases will give you $0?
| According to the log utility whether you should bet depends
| on your current wealth. According to the assumptions behind
| the Kelly criterion you should take the bet every week.
| jsw97 wrote:
| Paul Samuelson wrote an article in 1979 on this. It
| contains only one-syllable words, except for the last word,
| which is "syllable". Title: "Why We Should Not Make Mean
| Log of Wealth Big though Years to Act Are Long."
|
| His 1971 paper, "The 'Fallacy' of Maximizing the Geometric
| Mean in Long Sequences of Investing or Gambling" is more
| readable.
| ArtWomb wrote:
| I believe it's come into fashion again with the rise of legal
| sportsbook in the US (DraftKings, etc). Particulaly, with the
| fascination over long shot "parlays" or chain bets that can
| achieve very high payouts with very little outlays. Better odds
| than buying a lottery ticket by far, or so it is perceived.
|
| I think you have to have a few jokers short of a full deck to
| get into sports gaming and expect any outcome other than ruin.
| But the parlays are interesting. And the simplicity one could
| devise a winning strategy is enticing.
|
| Consider a typical NBA season: 120 game nights, about 8 games
| per night. Let's say you constructed a parlay strategy in which
| you pick the under to hit on every game played that night. If
| the payout is large, say $1000 for a $3 bet. That's $360 for
| the season risk. And a high probability that eventually it'll
| cash ;)
| confidantlake wrote:
| If the odds of an under happening is 50% then that is indeed
| a great bet! The odds of winning would be (0.5)^8 or 1 in
| 256. So the EV of the bet would be
|
| E(x) = (1/256 * 1000) - (255/256 * 3)
|
| E(x) = .91796875
|
| So you are expected to make nearly 92 cents on every 3
| dollars bet. Over a 30% return! Any bookie offering these
| odds would quickly go broke.
| pradn wrote:
| Bloom filters are the equivalent in computer science. Just
| beyond the basics, but ubiquitous enough to be annoying when
| it's on the front page every other week.
| aborsy wrote:
| But how much is it actually used in trading?
| fighterpilot wrote:
| It's not practical when the outcome is distributed in a very
| weird and unknown way and each bet isn't really iid.
|
| Moreover you can find the same result with a simple grid search
| over bet sizes, obviating the need to estimate any parameters
| of the outcome distribution.
|
| It would be more useful in professional gambling where outcome
| distributions are more knowable.
| evo wrote:
| It's useful as an upper limit--if you're leveraging past what
| Kelly suggests you're almost certainly overleveraged.
|
| In theory, Kelly is optimal--if you knew the exact probability
| density function of your returns, it would give you the right
| leverage to take.
|
| In practice, you're always playing with risks, some you're
| factoring into your models, some you're choosing not to because
| they're intractable, some you're not even aware of until they
| occur. The most basic premise--today's returns will be a
| function of hypotheses that I've derived from looking at past
| observations--is an approximation at best.
|
| This mismatch between model and reality can lead to expensive
| lessons learned when using the full Kelly model, so often
| traders will "half-kelly" or something like that, to
| incorporate the basic idea of risk scaling proposed by the
| Kelly model but with more safety margin.
| 6gvONxR4sf7o wrote:
| > In theory, Kelly is optimal...
|
| It's optimal for one utility function (log of future $), but
| that doesn't make it necessarily optimal for everything,
| right?
| evo wrote:
| That's a great callout as well: it's optimal for maximizing
| your expected growth in the long term, but does carry
| significant volatility. That's fine for an emotionless
| immortal robot investor, but as we're human.
|
| If we're close to an investment goal, like saving a house
| downpayment or retiring, the calculus is quite different.
| Even outside that, loss aversion is a real thing and we're
| likely happier trading some upside for being able to sleep
| at night.
| rezahussain wrote:
| I used it in live trading last year, I couldn't really make it
| work.
|
| I precalc my stoplosses + stopgains then use a simulation to
| get the win/loss probabilities on training data.
|
| What I observed is the kelly formula really prefers the tiny
| stoplosses, so when you sort your predictions by kelly score it
| will pick the ones with tiny stoplosses.
|
| What happened to me in the live test is the tiny stoplosses
| triggered, when a stopgain would have triggered later.
|
| I know someone is going to say "thats a problem with your
| stoplosses+stopgains OOS performance" and they are right, but
| OOS stoploss+stopgain calculation isn't trivial for me to
| calculate :\
| aborsy wrote:
| Where did you get the distribution of the real data from?
| rezahussain wrote:
| Yes, model predictions on volume+price data from 2019+2020
| paulpauper wrote:
| I am sure it is used but the challenge is calculating the
| necessary parameters
| SatvikBeri wrote:
| Almost no one uses it directly - in practice, Kelly is too
| aggressive - but variations of it are quite common.
| syntaxing wrote:
| Would love to hear about the variations!
| paulpauper wrote:
| bet less than the formula recommends. depends on personal
| preferences
| dcolkitt wrote:
| Kelly criterion is just the square of Sharpe ratio. Sharpe is
| used pretty extensively from a strategy selection and risk
| management perspective.
| arthurcolle wrote:
| Please share this calculation, dying to know what kind of
| inter-universal Teichmuller space theoretic math you're using
| to come up with this.
| afryer wrote:
| https://news.ycombinator.com/item?id=27453365
|
| Edward Thorpe and Ralph Vince both conclude that the Kelly
| Criterion in the continuous case is excess returns divided
| by variance, which is pretty close to the Sharpe Ratio,
| correct?
|
| Asking to understand better, not to be combative. Your
| comment made it seem like that formula is way off.
| SpicyP wrote:
| This is not correct. Kelly criterion is not the square of the
| sharpe ratio
| afryer wrote:
| https://ifta.org/public/files/journal/d_ifta_journal_11.pdf
|
| Page 27: Vince(vi) and independently Thorpv(ii) provide a
| solution that satisfies the Kelly Criterion for the
| continuous finance case, often quoted in the financial
| community to the effect that "f should equal the expected
| excess return of the strategy divided by the expected
| variance of the excess return:"
|
| f = (m-r) / s^2
|
| so it's Sharpe with variance instead of standard deviation
| in the denominator, correct?
| dcolkitt wrote:
| Yes, that's correct.
|
| An intuitive way to think about this is that Sharpe
| depends on the specific horizon you're using. E.g.
| annualized Sharpe will be sqrt(252) larger than daily
| Sharpe. It would not make sense to change the Kelly
| criterion based on a substitution of variables. In
| contrast variance, like returns, scales linearly with
| time horizon. Therefore the variance ratio is invariant
| to the time horizon.
| Solstinox wrote:
| Knowingly? Not often. You don't find long-run successful
| traders who don't knowingly/unknowingly use it or some
| variation of it.
| mywacaday wrote:
| Does anyone know of any formulas that would accommodate p
| changing on every bet?
| hervature wrote:
| Well, Kelly is infinite horizon, so any derivation is going to
| depend on the exact payouts. If it is not infinite horizon, you
| can do dynamic programming to figure it out but you will have
| to be careful with myopic reasoning (betting it all in the last
| stage). In practice, just plug your changing payoff into the
| formula. The rationale being every bet is growth optimal in the
| long run even if you only bet it once.
| dcolkitt wrote:
| If the distribution of p is ergodic, then Kelly criterion (re-
| sized at every p) still maximizes expected growth rate.
| pge wrote:
| Fortune's Formula by William Poundstone is a fun read on the
| origin of the Kelly criterion, as well as the role of gangsters
| and bookies in the funding of the early communications
| infrastructure in the US.
| ArtWomb wrote:
| Thanks! I'm looking for "history of probability" mass market
| books and this looks spot on ;)
| sigmaskipper wrote:
| Used an abbreviated version to of the kelly criterion along with
| Markowitz portfolio optimization and applied it to sports
| betting. All I can say is that past results do not indicate
| future returns
| windsignaling wrote:
| The part that all these nice theories miss is that you actually
| do not know the distribution p(win) (in the case of Kelly) or
| the expected return and covariance (in the case of Markowitz).
| contravariant wrote:
| Well 'knowing' the 'true probability' is a philosophical can
| of worms anyway.
|
| The good news is that you don't need to know it exactly, you
| just need to make a better guess than the bookies (w.r.t. the
| Kullback Leibler divergence or cross-entropy, whichever takes
| your fancy).
| zorked wrote:
| I know I would never be a writer of seminal papers because I
| would never publish a formula that includes unknowable
| parameters.
|
| Same goes for Black-Scholes which includes _future_
| volatility.
| LeegleechN wrote:
| The fact that Black-Scholes has only one unknowable
| parameter makes it quite usable, more so than more
| complicated option pricing models. You can work backwards
| from the market price to solve for the implied volatility,
| treating it as a generalized 'price' for the option after
| factoring out things that are easily adjusted for. You can
| also abuse the implied volatility (adjusting it up or down)
| to account for factors outside of the idealized model.
| tel wrote:
| That's something of a moot argument, though, since BSM
| computes prices in terms of the future volatility. Since we
| have the actual price, the volatility is actually what we
| solve for with BSM.
|
| Even if we had neither price nor volatility, we can still
| talk about the surface of possible (price, volatility)
| pairs which are compatible with the model.
| hgibbs wrote:
| But how do we get the prices? If I tell you the at-the-
| money front-month call is $1000 will you tell me the vol
| is 100pa?
| tel wrote:
| Yeah, basically. Also need the strike, spot, an estimate
| of the risk free rate (probably not today).
|
| The implied vol is a useful way to make sense of the
| actual market prices of options. We also might have some
| predictions about the market's implied vol changing going
| forward and we can reverse those errors back into
| expected price changes (and maybe trade on them).
| arthurcolle wrote:
| It's a real concern, but more realistically you can just
| compute many potential outcomes and at least get a sense of
| the structure of the surface
| zucker42 wrote:
| I wonder how professional gamblers approach bet sizing. It seems
| to me that for most applications the Kelly Criterion is not the
| right choice. The utility of money is asymmetric; gaining $25000
| is worse than not losing $25000. Relatedly, most actual gamblers
| want to ensure good returns while not going broke, so minimizing
| risk of ruin is often more important than maximizing return rate.
| Further complicating the matter is that in real life you don't
| know your actual probability of success, but you may have an
| estimation. And finally, though this is less commonly
| significant, your rate of return in a given game might depend on
| your the amount you bet.
|
| From what I've seen in the poker community, no one has really
| approached this type of bet sizing from a rigorous perspective
| beyond the relatively simple Kelly Criterion.
| mewse-hn wrote:
| I read about this on HN and then lost it for months when I wanted
| to apply it to a crappy game on a random discord server.
|
| The game lets you bet on chicken fights and tells you the
| probability your chicken will win (starts at like 62% chance to
| win), so this kelly criterion is perfect. It's a bit incredible
| how reliable it is.
| sigstoat wrote:
| read the original paper. the gambler's formulation is garbage,
| which obscures all insight.
| awaythrowact wrote:
| https://news.ycombinator.com/item?id=26834333
| fairity wrote:
| Perhaps I'm misunderstanding, but the post says that the optimal
| bet size f = 1-2p (where p is the probability of winning). But,
| this seems backwards. As p goes up, you should be betting more.
|
| Shouldn't the optimal bet size, f, be 2p-1?
| mgraczyk wrote:
| Yes, looks like a typo. The plots show the correct direction
| though (f* is an increasing function of p)
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