Impermanent loss and loss-vs-rebalancing I: some statistical properties

TL;DR

This paper analyzes the statistical properties of impermanent loss and loss-vs-rebalancing in automated market makers, revealing they have identical expected values but different distribution characteristics under Brownian motion.

Contribution

It demonstrates that IL and LVR are more similar than previously thought, providing a detailed statistical comparison based on properties of random walks and CFMM mechanics.

Findings

IL and LVR have identical expectation values for Brownian motion.

IL and LVR have vastly different distribution functions.

The analysis uses properties of a random walk and statistical integrals.

Abstract

There are two predominant metrics to assess the performance of automated market makers and their profitability for liquidity providers: 'impermanent loss' (IL) and 'loss-versus-rebalance' (LVR). In this short paper we shed light on the statistical aspects of both concepts and show that they are more similar than conventionally appreciated. Our analysis uses the properties of a random walk and some analytical properties of the statistical integral combined with the mechanics of a constant function market maker (CFMM). We consider non-toxic or rather unspecific trading in this paper. Our main finding can be summarized in one sentence: For Brownian motion with a given volatility, IL and LVR have identical expectation values but vastly differing distribution functions.