To follow from my previous post, I can illustrate my point by looking into an earlier post by Paul Bouchey:

"Take uncertainty out of the picture: imagine an investment that oscillates between +30% and -10% return with certainty. Average return = 10%. Growth rate = 8%. Volatility reduces the growth rate, relative to an investment that gave 10% each period with certainty."

Firstly this is a very hypothetical investment, needless to say you will never find one like this! But let us put that to one side for now.

It is true that the arithmetic average return, if you assume 30% up one period, -10% down the next, is 10%. But if you are assuming that there is compounding of returns, you should not even be looking at the arithmetic return as a basis for comparison. Arithmetic return is relevant if you are extracting gains or making good losses at the end of each period, and are not assuming a reinvestment of net gains, they just sit in the bank. So with a $100 investment you bank $30 after the first period and pay back in $10 after the next, net gain is $20 over 2 periods i.e. average $10 per period.

But here we are assuming compounding so we should be using geometric returns. The geometric growth rate in this example is 8.17% per period.

Does increased volatility reduce the expected geometric growth rate?

It should not, but the return assumptions in this example are not useful for demonstrating this because they do not explicitly assume a lognormal distribution of returns. If you use arbitrary return assumptions that do not conform to a lognormal distribution, you will arrive at the wrong answer.