Kelly Criterion with Time

How do we compare the performance of trading strategies each using the Kelly Investment System?

Please consider the paper by Q. J. Zhu:

http://homepages.wmich.edu/~ledyaev/eir.pdf

In the last paragraph of Section 2, Zhu proposes using the annualized return for the systems.

The above is the only paper I know of which addresses the question.

I will appreciate any references or comments.

Thanks,
Jim Murphy
 
Hi Jim,

A couple of problems here. Equation 2.1 here is correct for determining an optimal fraction, but on the top of page 3, he tries to express it in terms of the sum of natural logs -- the two are NOT equivalent except in what I call the "special case" which is frequently seen in gambling and rarely in trading (for example, a short position is never a candidate for the special case. Similarly, any forex transaction can never be a candidate for the special case, nor can any spread transaction. By extension, those which are members of the special case, actually are themselves members of the non-special case by virtue of the fact that every transaction is, in effect, a spread transaction). Thus, for the second form of this to yield and optimal "fraction" is virtually impossible, and it is the second form of this that is the expression of the Kelly Criterion. So once again there is the false presumption that the Kelly Criterion solution results in the expected geometric growth optimal fraction of ones capital to bet asymptotically. It simply is not true.

That aside, his conclusion is interesting and very interestingly developed. I would say though, that it, as with everything looking at optimal growth fractions, misses the mark in that it assumes one's criterion is to maximize asymptotic expected geometric growth. Knowing where the peak of the curve is, as well as the nature of the curve itself -- it's important geometrical nuances, and the acceptance of the inescapable fact that we are somewhere on this curve, and likely moving about it from holding period to holding period, allows us the luxury of now seeking out algorithmic paths along this curve to satisfy other criteria. -Ralph Vince
 
Ralph,

Thanks for your response. I would also like to express my appreciation to you for your work in generalizing the Kelly Criterion concept.

Quote from rvince99:

Equation 2.1 here is correct for determining an optimal fraction, but on the top of page 4, he tries to express it in terms of the sum of natural logs -- the two are NOT equivalent except in what I call the "special case"... .
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Please consider the paper by Ralph Vince, page 21, at:

http://ifta.org/public/files/journal/d_ifta_journal_11.pdf

The rhs of Vince Equation (1) is identical in form to the rhs of Zhu equation top of page 4. This form will be referred to as the sum of logs form.

The rhs of Vince Equation (1a) is identical in form to the rhs of Zhu Equation (2.1). This form will be referred to as the product form.

I paraphrase the statement by Vince regarding his Equation (1) and his Equation (1a):

"Rather than taking the sum of the logs of the returns, we can take the product of those returns. Thus, the value for f that maximizes (1) will also maximize (1a)."

Please notice that:

ln(product form) = sum of logs form

and:

exp(sum of logs form) = product form

Because of the strictly increasing and continuity properties of the ln and exp functions, a maximum is attained at a value of f for one of the forms if and only if a maximum is attained at that same value of f for the other form.

In light of this, it can be said that the two forms are equivalent for the purpose of finding values of f which maximize either form. I do not understand what the phrase, “the two are NOT equivalent”, was intended to convey. Any enlightenment would be appreciated.

Thanks,
Jim Murphy
 
Jim,

You're absolutely right! Where you say:

<< The rhs of Vince Equation (1a) is identical in form to the rhs of Zhu Equation (2.1). This form will be referred to as the product form. >>

I had mistakenly took Zhu's equation (2.1) to be my equation (2) from my paper, when in fact it is (1a) from my paper.

I apologize for the confusion, and appreciate your pointing this out. I'm at the age where I shouldn't do anything after about 3 p.m. that requires any thinking! -Ralph Vince
 
I don't have time to go over the details but I see a serious problem IMO with equation (3.1) in Zhu's paper.

Specifically, the units of s come out [currency]^-1

The numerator is the sum of probability weighted gain/loss but the denominator is gain x loss and that leaves the above strange unit. In that case s cannot be the size of each trade as a percentage of available capital as he claims.

On the contrary, Kelly percent is dimensionles, as a true ratio should be.

Am I missing something?

Edit: I just figured it out that c and t are percentages. In that case the number is dimensionless.
 
Bill,

Remember though that the answer to both of Zhu's equations here, the Kelly Criterion itself, is NOT a percentage (it just often masquerades as one, very meritricioiusly) but rather is a leverage factor. It might equal what is the optimal percentage in the "special case" (i.e. is is a long trade only, and the most one can lose is the price of the underlying), but is it NOT a percentage (and this becomes really a problem when the answer returned by the Kelly Criterion solution is < 1, but > the actual optimal fraction, a think an example is provided in the paper I wrote a few years ago, cited a few posts earlier) Ralph Vince
 
Here's a writeup I did some time ago: http://mayer.pro/documents, have a look at the section "Gambles that are offered with lower frequency" of the online document "The Kelly criterion and fixed fraction betting".
The way I looked at it back then was to assume a 'betting frequency', i.e. per time interval bets are offered with a certain probability, the result is that the Kelly gain (the growth rate at optimal betting fraction) has to be multiplied by the betting frequency.
In the short run the betting frequency concept doesn't apply here, but in the long run it's close enough and it provides some insight:
The important intuition to take from this viewpoint is that the Kelly gain is reduced proportionally to the 'idleness'. (And the optimal betting fraction doesn't change.)
 
Quote from PHOENIX TRADING:

IMHO the kelly criterion is a formula guaranteed to blow you up.
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First of all, Kelly had a betting scenario in mind when he came up with his fixed fraction betting idea. In betting you can lose no more than how much you bet.

In the betting scenario, if you use the Kelly criterion correctly you cannot blow up. This is how it works: You define an initial amount of money which is your capital. According to the Kelly criterion you bet a fraction of that capital. You cannot lose all your money unless you bet 100%, which implies a bet with certain outcome. (I.e. you can only blow up if you think some event is certain when in fact it isn't.)

In real world trading scenarios the Kelly criterion doesn't strictly apply. Usually the loss distribution is not bounded nor is the price continous (i.e. it jumps). But you can get close by being conservative (because of the jumps) and defining stop losses (thereby limiting the losses).

I agree that naive application of the Kelly "formula" leads to overly aggressive trading.
 
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