Interesting!
Kelly Criterion & Risk Of Ruin As Risk Management Tool
- Thread starter ironchef
- Start date
For those of us who lack self confidence.if you think that you need to know all about position sizing..optimal leverage..black scholes..kelly criterion..fibonacci retracements..stochastics..and so on..and on..and on..and on..and on..then..you might want to think about how it is your brain is telling you that you need all this information in order to carry out the simple acts of buying and selling
Okay, that's something entirely different from the Kelly criterion then.
That is an understatement of the year
This is why I say to not use the Kelly Criterion in trading, even if one wants to be expected growth-optimal.
The Kelly Criterion == Optimal f (the actual expected growth-optimal fraction whereas the Kelly Criterion Solution is a leverage factor on your account, as you seem to clearly see) only when the amount you can lose is the cost of the investment (i.e. long trades, no stop). In Capital Markets, we have many more dynamics (shorts, spreads and straddles [fx being a straddle in effect], volatility [which is "extreme-attracted], fixed income that is par-reverting, save for default risk, and a totally different creature if short, and on and on and on. Even the game of Blackjack, btw, for most casino rules, you do not know your worst-case loss on the hand before the deal(!) and hence "Kelly" cannot be directly applied there, contrary to conventional lore).
Since when you get away from this special case, you have Kelly > Optimal f, most people who calculate and implement Kelly then go implement it as though it was the same as Optimal f, are way beyond the peak and thus taking on extra risk for less potential gain. Use the formula for Optimal f - I don't care if people still want to call it "Kelly," I'm not out to put my name on anything - I've never named anything I have ever come up with by my name. I have just seen over-and-over where people really get messed-up when they are trying to be expected growth-optimal by using Kelly's formula.
Since in Kelly's 1956 paper, the examples used were instances of this special case, they (Kelly, Shannon, Graham) thought they were looking at a fraction, called it a fraction in the paper, when in fact it was a leverage factor, not bound on the right at 1, hence not a fraction. (It is actually bound to the right at some value, I forget what it is a function of, but it;s a different value >1 for each instance). Except it wasn't a fraction, and thus they only postulated such a fraction existed!
The first instance of a value for the (asymptotic) expected growth-optimal fraction, to my knowledge, arises with Thorp's "Kelly Formulas," closed-form equations for calculating binomially-distributed (ie. 2 possible outcome) propositions. My 1990 book from the work I was doing in the 1980s starting with Larry Williams Robbins Trading Championship victory in 1987 presents the (asymptotic) expected growth-optimal fraction for any umber of possible outcomes.
Since then I have presented it for multiple-simultaneous propositions (i.e. a portfolio) for the asymptotic as well as non-asymptotic case, that is, the actual optimal fractions for real world, capital-market implementations. (The paper referrred to earlier in this thread I believe, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2577782). Yes, they end up being computationally intense, and hence my focus in recent years on heuristics to determine this, which, as has also been edified in recent posts on this thread, are far from perfect!
This is not to say that the original formula presented in Kelly's 1956 paper isn't usable uniquely on it;s own. One of the interesting properties of the formula presented therein is that the curve about the peak is symmetrical, which is useful in may deeper applications of this material both in the markets and the natural world.
Lastly, I'm not trying to throw cold water on the "Kelly crowd." As I said, people can refer to things however they want, I'm just trying to dispel misconceptions I have seen come about over-and-over in people's application of the material, and I've pretty much devoted most of my life to studying it. I know it quite in-depth but then I've had a long time to sneak up on it!
And the over-arching issue, with anything in trading, "in the foxhole," is for simple things that work. There is an enormous disconnect between the academic endeavors around capital markets, and making money in them.
I long ago realized that the quantity decision was something I could control, was even more important in the long run than being right or wrong on any given trade, and the more I dug into it and understood, the less-suited the truth of it appeared to be to the foxhole.
So that was the real challenge, making it applicable, workable.
I long ago realized that the quantity decision was something I could control, was even more important in the long run than being right or wrong on any given trade, and the more I dug into it and understood, the less-suited the truth of it appeared to be to the foxhole.
So that was the real challenge, making it applicable, workable.
Okay, and what would you use as an alternative then?
Let's make it a case study. Here is a trading system S with the following trade return distribution:
40% probability of a 1% gain
20% probability of a 2% gain
10% probability of a 3% gain
10% probability of a 1% loss
10% probability of a 2% loss
10% probability of a 3% loss
Let's assume that this distribution is over a large number of trades and over a long period of time, and that we believe that the future distribution will be the same (or nearly the same). Let's say that our goal is to maximize the risk-adjusted rate of growth.
For this distribution,
Kelly (i.e. the risk-neutral leverage) is 13.22
Half-Kelly (i.e. the common ad-hoc risk-sensitive leverage) is 6.61
My own risk-adjusted leverage (I call it a "prudent leverage") would be 1.7
What leverage would you use?
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Then we aren't talking about being at the peak of the curve, at the expected growth-optimal point, but for "risk-adjusted rate of growth" we have two other, significant points on the curve.
Both of these points (as with the peak of the curve itself) a function of the length of time (number of trades or holding periods) we will engage this proposition.
And for either of these two points Im going to mention, this needs to be sufficiently long do as to allow these points to resolve (i.e. 1 period or trade is not enough to maximize for the risk-adjusted rate of growth in other than an ad-hoc way, like 1/2 Kelly. or something like that which lacks geometrical significance on the curve of growth with respect to risk (f) over a specified, finite number of plays or periods.
The first point would be the inflection point < peak of the curve. starting at f=0, and as f increases, the curve increases and is concave up - gain is increasing at a faster rather than the risk we are proposing. At some point (again, less than the peak) it goes to concave down. his point of marginal increase in gain increasing with respect to risk, flipping over, this inflection point, is one such point.
If the inflection point exists, then there is a point also where the slope of the line tangent to the curve (the slope of which represents the ratio of gain to risk(f)) is greatest. This second point is > than the inflection point and < the peak.
Both are significant points to maximize the risk-adjusted rate of growth, which is generally far more important to traders than the peak itself (what others often refer to as Kelly) which represents maximizing expected growth without any concern for drawdown or risk.
The actual determination of these two points can be found here:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2364092
and notes on a talk given of the subject, here:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2624329
That's the derivation, and it's quite involved. Taking that "into the foxhole," as mentioned, a different exercise. You'll forgive me if I don't do the calculation here and give you my results (I long ago stopped doing that for various reasons) but a simpler, heuristic way to achieve it can be found in the book I wrote 5 or 6 years ago, the risk-opportunity book. I'm not trying to hustle the book here, it doesn't sell a whole lot, and the proceeds from it don;t go to me, or to a "formal" charity. But it does contain how to approximate those two points mentioned above without the heavy computational burden that the papers mentioned in this post (which give you the exact answers, mathematically).
Ok, I never do this anymore, but I am in this case for the sake of clarity. Using the parameters of hte six scenario "game" you provided in this thread a few posts earlier:
Note that at a horizon of 15 periods or plays (the minimum number given teh parameters of your game, for an inflection point to appear) the inflection point is at about 1.5%. At twenty games, then quit, it is at 7.1%.....and on. Ultimately, as the number of plays or periods until you quit approaches infinity, the inflection point approaches the peak.
But this is the more conservative of the two risk-adjusted return maximizing points. (Incidentally, at 1,000 plays, the inflection point has migrated to 35.25%. closing in on the peak at 39.67%)
Note that at a horizon of 15 periods or plays (the minimum number given teh parameters of your game, for an inflection point to appear) the inflection point is at about 1.5%. At twenty games, then quit, it is at 7.1%.....and on. Ultimately, as the number of plays or periods until you quit approaches infinity, the inflection point approaches the peak.
But this is the more conservative of the two risk-adjusted return maximizing points. (Incidentally, at 1,000 plays, the inflection point has migrated to 35.25%. closing in on the peak at 39.67%)
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Thanks, Ralph. I'll take a look at your papers.