# Optimality of Kelly criterion in non-normal environment

It is a not so well known fact that the Kelly criterion is only optimal in a nice and well-behaved Merton-world. It is far from optimal when things are getting non-(log)normal (i.e. more realistic!).

My question
My question is simply: Why is this so? What are the reasons and is there some intuition for this fact.

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Is this a fact? Since it's not so well known, is there a paper you can reference, or is this from personal observation? I don't mean to be cynical, I'm genuinely curious... of course any proof that this a "fact" seems likely also to carry the explanation you're looking for, so perhaps that is the question. – Greg Jun 30 '11 at 19:27
I attended a presentation yesterday where this was stated - see for an abstract here: frankfurt-school.de/content/en/news/newsfolder/2011/06/… – vonjd Jun 30 '11 at 19:32
According to the talk slides, they argued that Fractional Kelly strategies were only optimal in a Merton world. – Joshua Ulrich Jun 30 '11 at 20:13
@Joshua, Even though those slides use the term "Kelly Strategies", keep in mind that this is NOT what most people refer to when they discuss "Kelly" methods/strategies. The terminology in that paper would be less confusing if it used something like "Log Utility Strategies". "Kelly Criterion" as it has been known for decades is based on two-outcome events. Here's a typical example.... en.wikipedia.org/wiki/Kelly_criterion – bill_080 Jun 30 '11 at 20:36
@bill, In finance, the terms used are "growth optimal portfolio" or "geometric mean maximization", but mathematically they're identical to the standard Kelly Criterion - maximize the expected value of the log of future wealth. – joshayers Jul 1 '11 at 12:31
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## 2 Answers

The Kelly Criterion was derived for two-outcome events (binomial). Assuming it "works" for anything else (including "normal" events) is asking for trouble.

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That's not an issue, it generalizes easily to a continuous distribution. The danger lies instead in not knowing the precise distribution of your returns. There's an excellent summary of the good and bad properties of the Kelly Criterion here: edwardothorp.com/sitebuildercontent/sitebuilderfiles/… – mpeac Jul 1 '11 at 16:03

It helps to think about what the Kelly criterion is attempting to achieve. The purpose of the Kelly criterion is to find a betting strategy that maximizes the geometric growth rate. In a portfolio management context where the investment universe contains a risk-free asset, it would be equivalent to (ignoring constraints) $$w\equiv argmax\left\{ median\left(\mu_{p}\right)\right\}$$ where $\mu_{p}$ is the arithmetic return of the portfolio over the horizon and $w$ is a vector of weights.

When one adds the assumption that all security prices are log-normally distributed, people will often say that the above is equivalent to $$w\equiv argmax\left\{ w'm-\frac{1}{2}w'Vw\right\}$$ where $m$ is the mean log return and $V$ is the log covariance matrix. This is typically considered to be equivalent to the mean-variance optimization with a risk aversion coefficient equivalent to 1. While it is true in the univariate case that by accounting for the formula for the median of the log normal distribution and how to convert a normal distribution and log distribution a version of this formula would be created, there are aggregation issues when switching from the multivariate normal to multivariate log normal. Also, when extending this to a non-normal case, it is inconvenient even if you assume away the aggregation issue.

Another option is to instead optimize the arithmetic mean-variance problem (which is the second formula, but with the arithmetic mean and covariance matrices replacing the log versions) to construct the efficient frontier. Then, using the discrete points on the frontier, one can calculate the median return given each efficient portfolio (this is easy to do once the portfolio has been transformed into a univariate mean and variance, the problem above is the aggregataion of the whole portfolio). The investor would then select the optimal portfolio as the one with the largest median return. It would also be trivial to add constraints to the mean-variance problem in case the investor does not wish to have too much leverage (equivalent to a fractional Kelly-like solution).

This can easily be extended to the non-normal case. The investor would calculate the efficient frontier as above, but when calculating the median return could use a simulation-based set of returns at the horizon. All that is required is to calculate the median of each portfolio using these returns and then find the portfolio with the largest median. A simple approach to non-normal Kelly.

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