Automated Market Makers (AMMs) with a conservative function, such as Uniswap, Balancer, Curve and others, are an integral part of decentralised finance. This article examines the effect of the exchange fees on the divergence losses of the automated market-making systems in public blockchain networks. The study consists of several parts: theoretical background, detailed description of the exchange mechanics, the derivation of explicit formulas, the results of modelling using the hyperparameters of pools from the Ethereum network and the analysis of the proposed approach using historical data. For the first time, the obtained closed formulas (Uniswap, Balancer) and modelling results (Uniswap, Balancer, Curve) indicate the presence of the impermanent gain for liquidity providers in the case of non-zero fees when the trading volume does not exceed a certain amount. The results indicate that the proposed methodology significantly affects the definition of ‘impermanent loss of a liquidity provider’ widely used in the blockchain community since there can always be a profitable range of values. As a practical part of the study, statistics on the share of trades with the effect of impermanent gain in Ethereum pools are provided, and the approach for managing the fee rate is considered during this observation. Explicit relationships for mostly used AMMs with non-zero trading fees are derived. The article may be useful for both practitioners and researchers in the field of decentralised finance seeking a deeper understanding of the dynamics of automated market-making in an ever-changing DeFi environment.
Vlasov et al. (Mon,) studied this question.
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