Effects of Kurtosis ? None !

From Stanford Wong's BJ21

Posted by MathProf on 3 Jul 1998, 8:40 a.m.

Kurtosis Study:

I decided to look at this briefly look at this issue of Kurtosis (mentioned in a thread below) and its effect on Kelly Calculation. I did a combinatorial study. What I did was to put together an hypothetical "Hot" Shoe and play an optimal game against it with DAS and Resplitting allowed. (Unfortunately, I did not also consider insurance). I had the computer track the probabilities of each possible outcome. (Probability of winning 8 units, 6 units, 4 units, 1.5 units, etc.). We then computed the required Kelly banks several different ways: using the Variance Approximation, using the SumOfSquares Approximation, and finding the exact value to maximize the Log function. The latter calculation requires trial and error, I used Newton's method which converges almost immediately.

Now the hard part of this study is dealing with splits. Figuring out the probability of winning the first split hand, loosing the second, winning double on the third, etc. However there is an easy fudge. First we pretend that the split hands are 100% correlated. That is, resolve the first split hand and simlpy double, triple, and quadruple the results for the other lists. This should overstate the Variance and the Kurtosis. Another easy fudge is to pretend the split hands are independent (0% correlated). This should understate both. Finally we can an average of the probabilities for a real fudge. Fortunately, the differences between all these are all fairly small.

Now after ranting about Kurtosis having a minor effect, I had anticipated that the exact Kelly bank would be higher than the approximation. Incredibly it was shade lower. Presumably, this is due to the positive effects of skew.

I list below the results.

Mean is     2.65% 

SD range is   [1.151,1.185]

    Estimated (Fudge) is  ...    1.168
Kelly Banks for Full List (100% correlation)

 Mean is    2.646%  SD =   1.1852

 Var Approximation is 53.0990

  SS Approximation is 53.1254

             Exact is 52.9380




Kelly Banks for Independent List

 Mean is    2.646%  SD =   1.1512

 Var Approximation is 50.0899

  SS Approximation is 50.1164

             Exact is 49.9350




Kelly Banks for  Fudged List

 Mean is    2.646%  SD =   1.1683

 Var Approximation is 51.5945

  SS Approximation is 51.6209

             Exact is 51.4367

PS: I believe I now have a better algorithm for doing the splits, but I just haven't implemented it yet. May look at it this weekend.