The math here isn’t the point, although you might like it if that’s your type of thing. Just stay with me.
I gave Claude this old post Fun Analogy Between Combinations and a Correlation Matrix. It’s just some math candy pointing out that number of pairwise correlations from n numbers is (n² – n)/2 which is what ₙC₂ reduces to.
You can see the math visually if you arrange, say, A to J in a 10×10 matrix.
There are 100 boxes, but the 10 “on the diagonal” are the same, and of the remaining 90 boxes, only 45 are unique (since the correlation of A to B is the same as B to A).
So if n = 100, the number of combinations is:
(100 – 10 diagonals) / 2 = 45
Later in the post, I also point out how the sum of the numbers 1 to n would also be represented by (n² + n)/2.
So if n is 9, the sum of 1 through 9 is 45.
You can visualize that as n/2 pairs (round down) that each sum to n, with the (n+1)/2 number left over:
1, 9 = 10
2,8 = 10
3,7 = 10
4,6 = 10
5
Reminds you of the correlation matrix. Numbers pair off, but in this case, there’s a middle number remaining that needs to be added rather than the diagonal that needs to be lopped off (the correlation example thus gets the “minus” sign in n² – n).
Anyway, I told Claude that the rebalance factor for a levered ETF is x²-x where x is the leverage factor (ie 2 if it’s a double long or -2 if it’s a double short).
I asked if there was a conceptual throughline between these algebras taking the form of x²±x and it said things:
Then I said things:
Umm, the honest version is these things embody “don’t ask for permission, ask for forgiveness”. And this makes them more successful products. Except, they need surveillance.
Wouldn’t it be ironic if the products people, who embody this ethos were in charge of surveillance?
Scratch that.
Ironic is not the right word.
And no matter what happens, nobody is asking for forgiveness.
It didn’t say, “I’m sorry. I made things up because you’re in a position of power, I didn’t wanna look stupid, I want you to love me, my cat is sick, I’m not even supposed to be here today…”
It said:
Fair. I dressed it up without really knowing…
FAIR?!
I’m not sure where we are collectively on truth and decency these days, but the default posture of the fastest growing product on Earth is:
Lie, and if caught, noncommittally concede “I can see how you might think I lied” without admitting guilt.
Sleep tight.