If we’re interested in r = 0 (outperformance of current price) then as vol increases, the probability goes down.
(There’s actually quite a bit of intuitive stuff which drops out of this model (if we’re required to hit a given r, then increasing volatility makes it easier (up until vol = sqrt(2*r)))
So to recap, I was right in that riskier assets can have higher avg returns, but I was missing the usually bigger and opposing effect where as the assets gets riskier, the same avg returns rely more and more on lucky very big gains while doing worse more often (at least if they are sort of lognormal).
My second point I still think was correct, right? -- i.e., that if Scott believed ETH had some chance of total collapse (a mixture distribution), then this skews it to the other side and pushes the median below the mean, and gives some reason to think ETH is more likely to outperform BTC. Does this make sense?
It is possible that Scott believed that ETH is negatively-skewed (ie small chance of collapsing, large chance of small increase) but this would be inconsistent with his probability that ETH is going to 5k.
I think the vast majority of people think crypto is positively-skewed.
tl;dr “some sort of median vs mean distinction”
No, there’s two things going on which act against each other:
Riskier assets have higher returns on average
Riskier assets are more skewed (mean higher than median)
I’ve made the (I think safe) assumption that the skewness of the assets are more important than the relative differences in their expected return.
You can have a play with some toy models for this, for example, fixed Sharpe, lognormal assets you will have something which looks like:
log(X) ~ N(sharpe * vol—vol^2/2, vol)
P(return larger than r) = P( Z*vol + (sharpe*vol—vol^2/2) > r) = P(Z > (-sharpe + vol/2 + r/vol))
If we’re interested in r = 0 (outperformance of current price) then as vol increases, the probability goes down.
(There’s actually quite a bit of intuitive stuff which drops out of this model (if we’re required to hit a given r, then increasing volatility makes it easier (up until vol = sqrt(2*r)))
Niice, it makes sense! Thanks!
So to recap, I was right in that riskier assets can have higher avg returns, but I was missing the usually bigger and opposing effect where as the assets gets riskier, the same avg returns rely more and more on lucky very big gains while doing worse more often (at least if they are sort of lognormal).
My second point I still think was correct, right? -- i.e., that if Scott believed ETH had some chance of total collapse (a mixture distribution), then this skews it to the other side and pushes the median below the mean, and gives some reason to think ETH is more likely to outperform BTC. Does this make sense?
It is possible that Scott believed that ETH is negatively-skewed (ie small chance of collapsing, large chance of small increase) but this would be inconsistent with his probability that ETH is going to 5k.
I think the vast majority of people think crypto is positively-skewed.
Right! My untrained intuition still resists a bit; I should play with the numbers.