The fallacy of "continuous Kelly"

[kut2k2]

There is a Kelly formula floating around which is just as bad as if not worse than the Bad Kelly formula. This "continuous Kelly" formula (hereafter abbreviated as CK) has been proposed by a famous mathematician. But mathematicians understand the limitations and dangers of using certain approximations and laymen don't.

The CK formula is

CK == mu/sigma^2 ,
where
mu is the average trade return (%),
sigma is the standard deviation of trade returns.

So let's look at a practical example (www.elitetrader.com/vb/showthread.php?p=930436#post930436)

An example of a ‘system’ with 3 possible outcomes :

There is a 89% chance that you win 13% of your trade ;
there is a 10% chance that you lose 5% of your trade ;
there is a 1% chance that you lose 100% of your trade.

The Kelly equation for this system is

0 == (.89)(.13)/(1+.13k) + (.10)(-.05)/(1-.05k) + (.01)(-1)/(1-k).

The Kelly fraction is the smallest positive solution to the Kelly equation :

k == 0.89833846

If we apply the CK formula, we get ...

sum1 == 0.89(.13) + 0.10(-.05) + 0.01(-1) == +0.1007
sum2 == 0.89(.13)(.13) + 0.10(.0025) + 0.01 == +0.025291

CK == mu/(sigma^2) == sum1/(sum2 - sum1^2) == 6.64664094 !! :eek: :eek:

Now I know where the nonsense about "Kelly leverage" comes from.
http://epchan.blogspot.com/2006/10/how-much-leverage-should-you-use.html

This CK stuff is clearly dangerous and should be avoided.

 

[QuantWizard]

What do you suggest instead? Martingaling?

 

[stock777]

kelly is used as a crutch by the inept, as if it was some holy grail that will turn their incompetence into prowess.

been like that since kelly came out.

 

[QuantWizard]

I think you might take it too literally - in essence what it means is that you increase order size in case of profit and scale back in times of losses, so as to maximize long-term geometric returns and minimize the risk of ruin - this is all trivial and intuitive and cannot really be rejected.

As a principle I have applied it successfully in real trading and simulations indicate the same. Risk increase, but so does returns, meaning that risk-adjusted returns stay the same. However, if you have a crappy system from the beginning, you'll lose faster, maybe this is the reason for some people's distaste for it...

 

[kut2k2]

What do you suggest instead? Martingaling?

Of course not.

I know five separate Kelly formulae, four of which are publicly disclosed. I started this thread to point out the worst of them (CK), which for some reason a lot of people believe is "true" Kelly. As I've demonstrated, it isn't even close to the real thing.

 

[kut2k2]

I think you might take it too literally - in essence what it means is that you increase order size in case of profit and scale back in times of losses, so as to maximize long-term geometric returns and minimize the risk of ruin - this is all trivial and intuitive and cannot really be rejected.

Idiots will find cause to reject anything, no matter how reasonable and beneficial it is.

As a principle I have applied it successfully in real trading and simulations indicate the same. Risk increase, but so does returns, meaning that risk-adjusted returns stay the same. However, if you have a crappy system from the beginning, you'll lose faster, maybe this is the reason for some people's distaste for it...

Also, crappy formulae like CK that lead to ridiculous trading fractions.

 

[kut2k2]

kelly is used as a crutch by the inept, as if it was some holy grail that will turn their incompetence into prowess.

That's not a knock on Kelly, that's just a knock on the incompetent. Just because the clueless expect miracles from something that has merit, that doesn't detract from the merit that is actually present.

been like that since kelly came out.

Wrong again. The math behind the Kelly criterion was around centuries before John L Kelly published it. Daniel Bernoulli published it in 1738. There is ample evidence that Keynes used Kelly-style sizing in the 1940s. The clueless didn't expect miracles from Kelly until Thorp published an article 5 years after JL Kelly's article called "Fortune's Formula", a title that virtually begs for misinterpretation. Add to that the popular book by Poundstone of the same title, then the bad idea that Kelly not merely exploits an edge but actually creates one is pretty much a given.

Things didn't have to be this way but given human nature, no shock that it is.

 

[SplawnDarts]

I never get why anyone bothers with all these "formulas" - Kelly has always been "maximize the expected logarithm of your resulting bankroll". Whatever math you have to do in any given circumstance to make that happen is the right formula.

Then of course de-rate the hell out of it. Kelly sized bets are insanely volatile.

 

[kut2k2]

I never get why anyone bothers with all these "formulas" - Kelly has always been "maximize the expected logarithm of your resulting bankroll". Whatever math you have to do in any given circumstance to make that happen is the right formula.

You'll have to ask the creators of the inadequate formulae why they did what they did. I explained the "logic" behind the Bad Kelly formula in the "Bad Kelly" thread. I do not know (nor do I care about) the "logic" behind the CK formula.

The appeal of formulae is no mystery. Mathematical formulae are direct calculations of desired quantities. The existence of the Quadratic Formula means we don't have to resolve the Quadratic Equation every time we encounter a new example of it.

But in the case of geometric mean maximization, the formulae are necessarily approximations for all but the simplest (i.e., binary-outcome) examples. And of course trading is anything but binary-outcome.

The surprising thing is that none of these highly educated quants has developed a better Kelly formula than those available. Too busy trying to predict future price movements I guess.

 

[SplawnDarts]

The surprising thing is that none of these highly educated quants has developed a better Kelly formula than those available. Too busy trying to predict future price movements I guess.

There's certainly far more money in predicting future price (or at least price direction).