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# The Kelly Criterion, from Ralph Vince

November 7, 2010 |

As far as I have ever been able to ascertain, Larry Williams was the first to attempt to apply the Kelly Criterion to outright position trading, and the first to openly discuss it. His pursuit in this regard not only was my initial immersion to the ideas, but he funded those attempts. Whatever I've uncovered along the way is a product of that — Larry's unquenchable curiosity, fearlessness regarding risk, and willingness to fund pursuits others would never touch.

A couple of points further in the post worth mentioning here because I think the other interested members deserve to have light shed on some misconceptions, some of which are a little dangerous to ascribe to, but are widely held.

"One "plays" forever, or practically forever."

But no one does and no one can, and it is this very notion of there being a finite "horizon," that changes not only the calculation of a growth optimal fraction, but every other metric related to it, giving rise to an entirely new discipline in and of itself.

"If one is somewhat risk averse, one can establish a half Kelly criterion, essentially betting half one's full Kelly bet. This results in a lower probability of one's bankroll halving."

But why "half?" Why this arbitrary number? (Or any other arbitrary dilution for that matter?) Remember, we're dealing with a function that has an optimal point, implying a curve, and it is the nature of this curve that is important to us. Being at different points on the curve has vastly different implications to us. Further, the various and important watershed points almost all are a function of that "horizon" mentioned earlier, i.e. the points migrate about this curve as a function of that horizon. Advocates of a "Half Kelly," or other arbitrary point along this chronomorphic curve (with respect to the horizon and events transpired) are seemingly unaware of the implications of their arbitrarily-chosen points.

"The criterion is to maximize the expected value of the logarithm of one's bankroll."

Yes, that is the Kelly Criterion which, in trading, does NOT result in the growth optimal fraction but a far more aggressive (and dangerous, without growth-commensurate benefit) number. No one seems to understand this.. The number returned in determining the value that satisfies the Kelly Criterion can be converted into a growth optimal number (which I call the optimal fraction. or optimal f) but in and of itself, the value that satisfies the Kelly Criterion is NOT the growth optimal fraction in trading. Incidentally, the so-called Kelly Formulas (put forth by Thorp I believe, and market applications attempted by Larry Williams in the mid-1980s) do NOT satisfy the Kelly Criterion in trading applications, but DO in gambling ones (that is, in trading applications they will not yield the same results as the value which satisfies the Kelly Criterion. The Kelly Formulas do, for dual-outcome situations, return the growth optimal fraction). For more on this I can only refer those interested to the most recent Journal of the International Federation of Technical Analysts 11 (available at admin at ifta dot org) or the 2-day course on Risk-Opportunity Analysis I am having in Tampa Nov 13 & 14 see http://ralphvince.com)

"The biggest issue of application is that one makes many assumptions about statistical distributions, correlations, returns, etc. that are all wrong."

I agree. In a strange, ironic twist to my modest participation to this story, it was (again, but some decades later) Larry Williams (rather recent) insistence of a way to apply what I know of growth maximization in a robust way. As a result of the pollenization of these ideas by Larry, I can state unequivocally that there are clear, simple, mathematical solutions to these impediments — in short, if someone wishes to apply a growth optimal approach to their future trading, these impediments ARE readily surmountable. But be certain your criterion is growth optimality, and be sure you really want to get into the cage and fight the gorilla. Most just want to sit and watch Dancing with the Stars.

## Nick White comments:

Dancing with the stars….brilliant and well said.

We're all fortunate beneficiaries of Mr. Vince's investigations into the intricacies of these issues.

## Phil McDonnell writes:

Kelly originally wrote his paper based on race track examples with binary outcome. You won or lost with assumed probabilities and you knew the wager size and payoff. So strictly speaking his formula only applies to wagers with two outcomes. Even a blackjack hand has at least five possible outcomes (win, lose, blackjack, double down, split) and not just two so strictly speaking Kelly's formula does not apply. Some people have erroneously tried to modify the binary Kelly formula by using average win size and average loss size to compute. All such formulas are dead wrong. The reason is that, in general, the average log does not equal the log of the average.

As Larry Williams pointed out most people do not feel comfortable using the optimum log approach even if the math is done correctly. I believe there is a simple reason for this. Most people do not have a simple logarithmic utility function. Rather they seek to maximize ln( ln w), where w is wealth. This is an iterated log function and results in a much more conservative ride. I talk about this distinction toward the end of my book. Ralph Vince also has written extensively on this subject using his term optimal f.

There is another issue with simply maximizing returns and that is it may not really take into account risk in a proper manner. It is true that the log function weights the largest loss the most in a non-linear manner and reduces the weights of gains so that the largest gains are weighted sub-linearly. But that may still not be enough to satisfy one's real risk aversion. That is part of my argument for the iterated log form but it may be that an explicit metric such as standard deviation is still needed.

## Larry Williams writes:

Optimal or Maximum Wealth (possible gain) only comes with Maximum Risk; therein lies the problem. Not loosing…risk…is more important than gain in the art of speculation business.

## Chris Cooper writes:

More important, as far as my practical experience goes, is that one's estimate of the edge is always subject to uncertainty. The reasons have been discussed on this list before, but certainly include changing regimes, limited history for the models, curve fitting, flexionic machinations, scaling nonlinearity, etc. I relied on the Kelly formula extensively in the mid-'70s when gambling, and uncertainty in your edge was no less important then. The problem arises because overestimating your edge is so destructive to your terminal wealth.

It might be interesting academically to consider an approach, such as Bayesian, where your estimate of the edge is not stationary, but in fact must decrease when you hit a losing streak.

## James Arveson writes:

I am a newbie on this site, but I can assure y'all that any finite amounts of outcomes can easily be handled by maximizing the expected value of the logarithm of one's fortune. I have also executed these theoretical outcomes for many years in AC and LV in BJ, and yes, in Bethlehem, PA in Texas Hold 'Em. See Mathematics of Poker for a better exposition of these issues than I could ever present.

Remember that each bet is a single bet, and one can bet forever. Leo Breiman has actually proved that (in the most general cases) that this approach DOMINATES all other strategies.

Now, IMHO, this approach is irrelevant to the market. NO ONE can get all the statistical assumptions correct-statistical distribution, EV, correlation, return, s.d., etc.

Have fun until we get to the next level. Same goes for Markowitz. Check out www.styleadvisor.com. I have no piece of their puzzle but wish I did (I might be able to get a write-off ski trip to Lake Tahoe where they are located).

Actualizing all of this crap may be the next Nobel Prize in Econ, but it will probably not help schlepers make money in the markets.

## Ralph Vince replies:

James,

Pursuing awards is for schlepers like Krugman or other academic dweebs –it's an award voted upon by dweebs for dweebs, and its pursuit bridles and constrains the mind (as *any* political pursuit will. Usually, the truth lies with things that - people off). To-wit, the lack of challenge to the notion that Kelly presents on p925 in the conclusion of his now-famous paper wherein he asserts that geometric growth is maximized by the gambler betting a fraction such that "at every bet he maximizes the the expected value of the logarithm of his capital."

This is accepted by the gambling community, and, by extension (falsely, mistakenly) accepted by the trading community. HOWEVER, a critical analysis of this notion reveals that it does NOT result in the growth optimal fraction, but rather in a multiplier of one's account to risk (the two are different indeed, the latter being less than or equal to the former, resulting often in over-wagering). In fact, the multiplier on one's stake equals the optimal fraction to risk only in certain, specific instances which manifest in gambling, but are rare still in trading (e.g. only on long positions, etc.). I would gladly go into this in depth put I cannot publicly do so as the paper on this has been publish in a current issue of a journal, and I have agreed to refer those interested to the article instead. The upshot is, that the Kelly Criterion, as specified above, is not what Kelly and others thought it was except in the special case I just mentioned — it is NOT the growth-optimal fraction, but something different, equal to the growth-optimal fraction only in the special case — a case that manifests in gambling with ubiquity, and oddly, in trading very rarely.

Again, the gambling community has accepted it for reasons mentioned –because it does give you the same answer for the optimal fraction to bet as the formulations for the optimal fraction in the gambling situations. But just because it gives you he same answer as the optimal fraction in special situations does not mean it is the formulation for the optimal fraction –it isn't.

Secondly, even the "optimal fraction"it is never optimal. Suppose you are playing a game with a 50% probability and odds of 2 to 1. Your optimal fraction is .25 (if you were to play forever). However, after the first pay, the phone rings, it;s your wife, and she informs you of an emergency and you have to bolt the game (with your winnings from the one play, make you a popular guy). If you knoew beforehand you were going to only play for 1 play, you should have bet 100% of your stake to maximize your gain. If the call came after two plays in this game, you should have bet .5.

Tomorrow, you come back to this game — and you bet .25, reconciling yourself that yesterday you bet .25. (so….the game possesses memory?)

Wait, it gets worse in trading, where we see that each individual bet is, in fact, NOT a bet. Let's say you trade only XYZ stock, and you put on 300 shares. Let's say you have a stop below your buy price but it;s a different level for each of 100 shares, so you have three stops below the proce for 100 shares each at different levels. Now, let's say onyl the closest stop, for 100 shares, gets hit, resulting in a loss on 100 of the three hundred shares. Weeks later, you sell out 100 shares at a profit, and, a few weeks after that, another 100 shares at a price higher than that.

But these are NOT three separate trades. This is ONE trade, one wager for the purposes of growth-optimal calculations. And the reason is because you are ONLY trading XYZ — there has been NO recalculation of positions to put on until the entire thing has been closed out. IF, on the other hand, you were having other trades throughout the course of your aggregate position in XYZ, then you WOULD consider each of these a separate trade.

Trading is not the same as gambling. There are similarities, but don't make the assumption that because you risk something and gain something that it is the same. There are things which are proxies for truth, that asymptotically appear to be truth, but they are only proxies (such as the Kelly Criterion) as well as the widely-held (in the gambling community, and hence the trading one as well) but incorrect notion that a wager should be assessed based on it's asymptotic mathematical expectation. This too is a mere proxy and an incorrect one that can, in extreme cases, lead one to accept bad wagers and reject favorable ones.

Again, critical thinking has been absent and trumped by the acceptance of industry catechism.

Finally, you speak of SD's and EV (mean-variance is dead incidentally, as dead as dead can be everywhere BUT academia) correlations, and Chris mentions the (valid) problems of assessing the edge in the future and the problem of non-stiationarity.

The solution to growth optimality in the markets, lies in NOT accepting the Kelly Criterion, but instead accepting what IS the growth optimal fraction– because that then reveals (in the simplest of ways!) how to address the problem of non-stationarity in the future and it doesn't require any of these parameters, or even a computer, it's really THAT simple if you want to attempt growth optimality in the future.

## Phil McDonnell comments:

Ralph raises a lot of interesting philosophical questions. On some points I disagree, so let me elaborate. For the purposes of this piece I will assume one is entirely risk averse and seeks only to maximize expected wealth on a compounded growth basis.

First he raises the point that there is no guarantee that a game or investment opportunity will continue. Certainly a true statement. However it is also true that there will be a succession of such opportunities available in one's lifetime. Thus some rational basis for choosing bet size each time should include consideration of expected logs of the outcomes.

Philosophically I disagree with Ralph's analysis of bet it all on the last bet. His math is correct, in that it will maximize the expected dollar outcome. But there will always be other bets, so one's lifetime objective should still be to maximize the expected log not simply the arithmetic expected value. I believe Kahneman and Tversky made the same error in their Nobel winning papers.

I have an alternate take on Ralph's argument that it is hard to define a trade because you can put on 300 shares and exit 3 times at different prices, unknowable in advance. Rather than look on each trade as the basic metric one should look on the portfolio as the metric and a basic unit of time as the portfolio re balancing decision point. For example if you invested .25 of your wealth in a trade that doubled you know have .50 of your 1.25 wealth in the trade. That is too much if you want to maintain the .25 ratio so you need to sell .1875 to get back to your optimal ratio. But the simplest way to look at it is to look at the investment portfolio in each time period, be it a day, week or whatever.

One of the reasons the mean covariance model is in disfavor is that it seems to fail when everything hits the fan. In fact the model is incomplete in the sense that EV and COV are stochastic variables and vary over time. (I am implicitly including VAR here.) You need to explicitly include the correlations somehow in order to take into account how an entire portfolio will vary together. Using the formulas for optimal bet size on a trade level will always lead to serious over trading if there are multiple trades put on at the same time except in the case of a negative correlation between the trades. So it is misleading to calculate an optimal trade size for one system or one trade without consideration of any others that might be on at the same time. At best it is a dangerous upper bound for any single trade size. But it will almost always be an estimate too high. Optimization of expected log of wealth can only be done at the portfolio level.

## Ralph Vince responds:

Philip,

I am not raising ANY "philosophical questions." Just because people may have to think about them doesn't make them philosophical questions as opposed to facts:

**1.** The value that satisfies the Kelly Criterion is NOT the (growth) optimal fraction of ones stake to risk (although, in special circumstances which we find ubiquitously in gambling and not in trading, it is an equivalent value to the value that IS the optimal fraction). And the pervasive mistake by those attempting growth maximization in the marketplace of using the Kelly Criterion result puts then OVER exposed, to their unwitting peril. They are NOT growth optimal. In fact, the value that satisfies the Kelly Criterion NEVER returns the growth optimal fraction. This was a mistake on the part of Kelly and Shannon. The very fact that it is still accepted by others is testimony to the absence of critical thinking in this matter.

**2.** Further, what IS the growth optimal fraction is a function of the horizon of the game — and all games have a horizon, including the game of evolution on earth. Further, all metrics, including the analysis of drawdowns (including VAR where a horizon of 1 is implicit), even the analysis of whether a wager should be accepted or not, are a function of horizon. Disregarding the horizon leads us to incorrect conclusions at every turn in risk-opportunity analysis. In fact, it is the necessary introduction of "horizon" that gives rise to this entire burgeoning discipline.

**3.** Once we accept points 1 and 2 above, the obvious solution to solving for the non-stationarity of the distribution of outcomes we are dealing with becomes obvious. Growth-maximization, unlike attempts at it in the past, now CAN be performed with informed assessments of what the best growth optimal fraction value to use in the future will be.

# Comments

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Wonderful thread. Thanks for sharing.

Dr. Phil writes that one should “seek to maximize ln( ln w), where w is wealth.”

Hmm, isn’t there something un-kosher about that? If we measure wealth in units of billions of dollars, some of us end up with wealth less than 1. So we get a negative number when we take the logarithm the first time. How then do you take the log a second time, when you’d be taking the log of a negative number?

I believe Ed Thorp was using the Kelly Formula for betsizing in his hedge funds significantly earlier than Larry Williams. Check out www.FortunesFormula.com for the story.