I previously wrote a note on calculating the optimal amount to leverage your portfolio to maximize compound growth using your trading strategy's historical returns.

I tried to extend it to use the actual data rather than a histogram approximation. So I just ran it with all the historical returns (not binned) assigning uniform probability to each historical return observation. It gave nearly the same results as the histogram approximation, but it was more accurate in cases where the binning had been done heavy-handedly. To figure out why this was the case, I looked back at the expected log growth formula, sum(p_i*ln(1+f*r_i)). From that it became obvious, as perhaps it should have been from the start, that just using point masses was fine. If you have multiple events with the same return you can add the probabilities if you like by expanding the sum and refactoring terms.

So now in my Kelly fraction computing code the histogram approximation (empirical density optimization) is optional.

I also improved the optimization routine so that it no longer uses brute force. The details are not important, but the basic idea is that now it does hill-climbing, and uses information about the steepness of the optimization surface at each point to jump faster.

Thirdly, I added a bit to simulate the effect of adding a stop-loss. It aggregates the probabilities of all the returns which were below the stop at the stop percentage itself. That's how it works. Easily abusable, but possibly an interesting way of bending the empirical distribution. It assumes your stop is based on a percent rather than dollar loss amount, etc.

Here is the code. First copy the kelly, empirical_distribution, and expectation functions into the command line. Then run the example code in the comment at the top. The example illustrates a nice advantage of this method of optimizing leverage rather than the analytic way which assumes Gaussian returns. You'll need the widely used numpy, matplotlib, and scipy libraries, which I believe give you access to pylab as well. If not you need that too. Notice that you can suppress the extra output by changing the disp=1 and full_output=1 arguments to 0 in scipy.optimize.fmin_cg.

The one issue it has is that sometimes the optimal leverage will be at a point that would make one of the (1+f*r_i) quantities in the sum above less than 0, which then causes the log(1+f*r_i) to be undefined, so the whole sum (i.e. objective function) is undefined, and it breaks the optimizer (it will just hang forever as far as I can tell). For example in the case of a Gaussian with mean 0.05, sd 0.10, the optimal leverage will be .05/(.1^2)=5, but of course the return values that have been observed could be, say -25% (i.e the support of the Gaussian is over (-inf,inf)), so 5x leverage pushes it under -100% and causes the undefined error. I'm not sure how to correctly handle this, or what the right answer really is.

The next step is to add a parameter that allows you to include transaction costs and releveraging frequency in the optimization since higher transaction costs result in a lower optimal leverage. This is because if you use leverage then any change in your holding's prices will require you to releverage. For example: you have 3.0 leverage and $100 in your own cash ie a $300 portfolio. Then the price goes up 100%. You now have a $600 portfolio, but $400 is your cash now. So your leverage is only 1.5. Assuming the Kelly fraction is 3.0, you need to buy more shares, which incurs a cost. Etc. The difficulty in computing effects like these on the optimal leverage are what causes people to fall back on heuristics like half-Kelly (well, that and black swan events).

I'd like to hear anything you know about this topic if you have anything to share.