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Variance over the short-term is a useful way of comparing the risk of investments. Got a home purchase or large medical expense coming up? Better not to have too high variance on your invested savings. 20 with a well-paying job and decades until retirement? Sure, risk the high variance for higher returns.

But if you've got decades or centuries in the market, and you're confident your investments won't fold entirely (though they very well may lose a large portion of their value), how do you reason about comparing variances on potential investments?

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Typically, the higher the variance, the higher the long-term average returns.
In a graph (x-axis: variance; y-axis: long-term average return), most investments will line up nicely to form an upward sloped line. Investments that fall below the line are considered inferior and might die out; investments above the line simply don’t exist (or everyone would run for them).

As a result, every long-term investor should be in the wildest markets, but - risk aversion typically limits this. Many people can’t sleep well with high variance in their investments, so everyone just goes up to his risk limit. For some that’s global share markets, for some it’s CDs, and fir some it’s cash under the mattress.

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  • Nice! I'll comment though that something here smells fishy. Am I to understand that the only thing holding long-term investors from the mathematically best investors is emotions? May 8 at 21:48
  • That is my understanding, from 30 years of experience, and 4 semesters of economics. I was just too lazy to pull the graphs and sources, and I’m on the road. We’ll see what others say.
    – Aganju
    May 8 at 22:12
  • @TheEnvironmentalist Risk aversion is not necessarily an emotional behavior. Some situations require risk aversion. College savings, for example. It's fine to have some risk to increase expected return and improve your college options, but you don't want so much risk that there's a chance you won't be able to afford the cheapest college you want to go to. Pension funds have limits on the amount of risk they can take as well.
    – D Stanley
    Oct 6 at 2:34
  • @DStanley So the ultimate limiting factor is time? If you could were willing to wait out any storm, you could afford to take on the highest return investments? Oct 6 at 21:11
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Typically, variance of an investment is correlated with expected return in practice. However, correlation is not causation. There is the idea of a mean-preserving spread. In other words, you can increase the variance without affecting the expected return. An investment that has the same expected return as another but has higher variance is said to be "second-order stochastically dominated" by the other. Even a risk neutral investor would prefer the investment with the lower variance, because it "stochastically dominates" the other investment. However, a risk-taking individual (in the sense that they have a convex utility function over risk taking behavior) would prefer the higher variance investment.

The right way to think about comparing variance on investments is to see if investment A first order stochastically dominates another, which means that the probability distribution for a higher outcome is greater than another one at every point, this equates to one has a higher average return in practice. Then assuming that neither first order stochastically dominates another, you should look to see if one second order stochastically dominates the other.

Now, that only gets you half the story, because then you need to account for the fact that you can take advantage of the volatility. Essentially, you will have more opportunities to buy low and sell high. That's where determining your risk tolerance is important, do you have a concave or convex utility function. Only you can answer that. Good investing occurs 40% in the brain and 60% in the stomach. If you can honestly ride the roller coaster without getting sick, and the average returns are the same, you can do better by riding the roller coaster.

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  • > "the probability distribution for a higher outcome is greater than another one at every point" I don't understand this. Why does the probability of a higher outcome have to be greater at every point, and not most points? Can't the two lines cross? Oct 6 at 21:09
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    That's just the definition of first order stochastic dominance. It guarantees that the expected value is greater under one case. Whereas, if they cross, you are no longer guaranteed that one will have a higher expected return. Again, that is just the way that the concept is defined.
    – Ryan
    Oct 6 at 23:09

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