Kelly and Logarithmic Utility

You talk a lot about utility functions and investor objectives in asset allocation books (see Meucci's Risk and Asset Allocation).  Log utility is defined by the function: U(w)=ln^(-w), where w = wealth.  A good mathematical betting scheme that came to finance as a byproduct of information theory.  John Kelly's formula has been used by Bill Gross at PIMCO and Ed Thorp at Princeton/Newport Partners, to name a few.  In its simplest form: Pct of Capital to Allocate=P - (1-P)/W
    where P = Pr(Win)
              W= E[Win]/E[Loss]

Assumptions: 1) E[Win] and E[Loss] are known (unrealistic)
                     2)Pr(win) is known (also unrealistic)

Still, according to the law of large numbers, if you have lots and lots of trades or investments to sample from, you can use statistics to come up with these values.  Further, if you can enhance anything Bayesian forecasting (or updating), you can further enhance your knowledge for E[Win],E[Loss], and Pr(Win).

Here is a Matlab function with three input parameters to calculate an equity curve:

function [equity] = Kelly(p,PF,initialEquity)
%% Kelly criteria for logartihmic utiliity

optimalF=(p-((1-p)/PF));
equity=zeros(1,1000);
equity(1)=initialEquity;
  
for i=2:1000
    equity(i) = equity(i-1)+ (randn*optimalF);
end

And here is a script that generates 16 equity curves for different E[Win]/E[Loss] ratios and Probabilities for winning.
initialEquity=100000;
p=.35:.10:.65;
PF=.5:.5:2.0;

equityCurves=zeros(length(p)*length(PF),1000);

i=0;
for b=1:4
    for a=1:4
        i=i+1;
        equityCurves(i,:)=Kelly(p(a),PF(b),initialEquity)
        subplot(4,4,i);plot(equityCurves(i,:))
    end
end
   

Lastly, a graph of the plots: