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

ArXiv ID: 2410.00854 “View on arXiv”

Authors: Unknown

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.

Keywords: impermanent loss, loss-versus-rebalance, automated market makers, constant function market maker, Cryptocurrency

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats
  • Why: The paper is heavily theoretical with advanced mathematical derivations involving stochastic calculus (Brownian motion, differential equations) and statistical analysis of distribution functions. It lacks empirical rigor, showing no real data, backtests, or implementation details beyond simulated random walks for conceptual validation.
  flowchart TD
    Start["Research Goal<br>Clarify statistical relationship between<br>Impermanent Loss (IL) and Loss-vs-Rebalance (LVR)"] --> Inputs["Key Inputs & Assumptions<br>- Constant Function Market Maker (CFMM)<br>- Non-toxic (unspecific) trading<br>- Random walk / Brownian motion<br>- Asset price volatility (σ)"]
    Inputs --> Analysis["Methodology<br>Statistical analysis & calculus<br>1. Define IL and LVR metrics<br>2. Apply Brownian motion properties<br>3. Compute analytical expectations"]
    Analysis --> Computation["Computational Process<br>- Derive closed-form expectation values<br>- Analyze distribution functions<br>- Compare statistical properties"]
    Computation --> Outcome["Key Findings<br>1. IL and LVR have identical expectations<br>2. Distribution functions differ vastly<br>3. Statistical similarity > conventional appreciation"]
    Outcome --> End["Conclusion<br>IL and LVR are fundamentally linked by<br>volatility, though their risk distributions differ"]