Disclaimer: Am a software engineer by training, diving into finance. Am still a noob xD

Hey team, I have been reading one of Spitznagel's whitepapers:


Here, Spitznagel explains how adding an asset that has 0% overall return, when combined with another asset, can produce better results than just holding the other asset itself.

i.e: holding 100% SPY performs worse then 97% SPY 3% insurance, even though insurance has a neutral return by itself.

I can understand this intuitively: the insurance offsets large drawdowns in capital, which helps the compound growth.

But I don't know how I could think about this mathematically. I would guess it's even possible for the asset to have a negative return, and this would still work.

How could I model this? What are the variables at play, and how do they come together to determine the overall capital growth? Any beginning pieces of intuition I can start with, or textbooks to look into?


This is a subset of https://en.wikipedia.org/wiki/Parrondo%27s_paradox

This paper goes into a more in-depth explanation: https://projecteuclid.org/download/pdf_1/euclid.ss/1009212247


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