Investors tend to think in terms of 1) volatility 2) exposure/diversification.

1) What _really really_ really matters is the Sharpe Ratio, as in "how much returns you get per unit of volatility".

The returns themselves are meaningless if not compared to the volatility to earn them.

Also, you want to discount the risk free rate (at least), as your benchmark.

2) The market as a whole is the biggest exposure you can have, you'd want to discount it as being X% of your portfolio

The Sharpe ratio gives you a good score if you lose everything (-100%) and then you gain a 20% return for 10 years.

But once you lost everything, there is no capital to invest, so the score should be infinitely bad.

Arithmetic averages are dangerous in geometric environments.

Use the expected log-return or the geometric mean instead of the arithmetic one in Sharpe's formula.

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

Maxmizing log returns is very good in many respects. It has nice mathematical properties. It's not too far off from what people subjectively value. But it is an approximation. For instance, how upset would you be if you woke up to find your bank account was $0? Pretty upset I'm sure. Infinitely? Doubt it. Now some one gives you a single cent out of pity. Feeling a lot better? Hardly.

The subjective satisfaction we get from a certain amount of money is something that would take a lot of experimental science to figure out, and subject to change as society changes. How high up Maslows hierarchy of needs can you climb, and how long can you stay there until age brings you low?

Now where log returns really shine is if you make a very large number of similar bets. Thats where the asymptotic behavior dominates. But if you make a big once in a lifetime decision of whether to bet the farm on a new business idea, that's where you have to figure out your own values.

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

Unless you are risk adverse. Which it's probably rational to be.

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.

> 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.

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.

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%.

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!

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

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)