  Kelly's criterion is a betting strategy that maximizes the long term growth
rate, but which is known to be risky. Here, we find optimal betting strategies
that gives the highest capital growth rate while keeping a certain low value of
risky fluctuations. We then analyze the trade-off between the average and the
fluctuations of the growth rate, in models of horse races, first for two horses
then for an arbitrary number of horses, and for uncorrelated or correlated
races. We find an analog of a phase transition with a coexistence between two
optimal strategies, where one has risk and the other one does not. The above
trade-off is also embodied in a general bound on the average growth rate,
similar to thermodynamic uncertainty relations. We also prove mathematically
the absence of other phase transitions between Kelly's point and the risk free
strategy.
