Variance is the squared difference - why not to the 3, or 4 instead? So there is this question about  why variance  is squared. 
And the answer seems to be "because we get to do groovy maths when it is squared". Ok, that's cool, I can dig.
However, I'm sitting reading some financial maths stuff, and a lot of the equations on pricing and risk are based on variance. It doesn't seem to be the best basis for, you know, pricing exotic vehicles that are worth in the millions (or billions) that a formula for variance is used "because the maths is better".  
To make a point  then, why not have the variance be from the cubed, abs cubed, or the 4th power (or even a negative power)? 
eg (apologies, I don't know Latex)
Sum 1/N * |(x - mean)^3| 
OR
Sum 1/N * (x - mean)^4 
Would using variance-to-a-different power measurably alter pricings/valuations if the equations still used variance as usual (but the variance was calculated with the different power)?
Is there a reason why we stopped  at "power of 2", and are there any implications of using a variance concocted from a different (higher or lower) power?

In principle, decisions involving large amounts of money should be made using the nonlinear utility of money.  However, that is subjective and hard to quantify.
