[dan-robertson]

1. Im not sure that’s what the Kelly criterion is but I didn’t look it up.

2. Arithmetic mean of log returns is the same as the geometric mean of returns. Indeed it’s pretty typical to work with log returns for this reason as adding is easier/better for computers than multiplying. This equivalence is easy to prove:

  gm(returns) = prod(returns)^(1/N)
  log(gm(returns)) = 1/N * log(prod(returns))
                   = 1/N * sum(log(returns))
                   = mean(log(returns))
  gm(returns) = exp(mean(log(returns)))

Where returns is a list of the multipliers to go from the values before/after the returns, eg it has 1.01 not 1%.

[danuker]

This is beautiful. I now realize both go to 0 if you lose all money (i.e. any one of "returns" as you define them is 0). Thank you!

[jasonfarnon]

you mean (as your proof shows) "is the same as the [log of] geometric mean of returns."

[dan-robertson]

Yeah, I should have been more clear. The point is that you can convert between them without needing any other information (like the original values that were averaged)