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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 |
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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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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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