[FooBarBizBazz]

That's the thing. The Sharpe Ratio looks at a catastrophic situation and says it's ok. It's not appropriately scoring risk!

Let's say the risk-free rate of return is 3%.

Asset 1:

Every year, with 99% probability you get 8% return, and with 1% probability you get -100% return, i.e., you lose everything. This has an expected return of 7%, which is 4% above risk-free; the standard deviation is 0.1; and the Sharpe Ratio is 0.36. But the exponential of the mean log annual multiplier is zero; you will eventually lose everything.

Asset 2:

With 90% probability you get the risk-free rate of 3%, and with 10% probability, you get a 10,000% return (multiply balance by 101). Yes, this has a good average return of 1,000%, but it also has a giant standard deviation of 30, so its Sharpe Ratio is slightly worse, at 0.33. But, the exponential of the mean log multiplier is 1.62, which means that over time it will have a 62% annual return. Moreover, it literally never goes down; there's no risk.

Asset 3:

You just take the "risk free rate of return" at 3%.

Surely, the best choice is Asset 2. It's literally Asset 3 plus free lottery tickets. But it has the worst Sharpe Ratio of the three. And Asset 1, which has the flavor of some prudent tradeoff, is actually guaranteed to bankrupt you eventually.

[geysersam]

> But maximizing log-return was proven by Kelly to be optimal, and you don't need to further penalize volatility.

This is what I'm questioning. We do need to further penalize volatility, if that is our preference.

The criteria is optimal in the sense of greatest expected return, in the limit of infinite number of bets. But we don't make infinite numbers of bets, and the variance matters.

Any truly optimal strategy has to factor in subjective preferences.

Example: We play a game where you are ill and need to pay for medical treatment. At the beginning of the game you obtain a sum of money exactly enough to pay for the treatment. Then you are allowed to place (a finite number of) bets in some gambling, possibly increasing your payoff, or losing part of it. I'd argue that in this scenario the "optimal" strategy is not playing, no matter what criteria is used to select the size of the bets.

[FooBarBizBazz]

It would make sense to allow a risk-averse utility function in our framework (say a concave function of total dollars at the end).

I don't think the identification "volatility" = "standard deviation" = "risk" matches anyone's actual preferences. So that part doesn't make sense to me.

But I like your example with the medical treatment. That could be modeled with a step utility function. Mixing it with my example, there'd be no problem choosing my Asset 2 or 3, since both guarantee that your capital will be preserved so you can pay for your treatment. If your utility function were truly a step, you'd be indifferent between Assets 2 and 3. More realistically, you'd assign minus infinity to values beneath the threshold and some monotonic function to values above (e.g. just the number of dollars), and you'd prefer Asset 2: It guarantees your medical treatment, which is what you really care about, but it throws in a free lottery ticket, so why not take that.