Report: Impermanent loss on Uniswap and other automated market makers is permanent

Alexis Direr, a researcher at the University of Orleans in France, published a paper summarizing the mathematical foundations of Uniswap and other exchanges based on automated market makers .

Automated market maker (AMM) is a term that refers to a type of decentralized exchange represented by Uniswap that has gained widespread popularity in 2020.

In short, these exchanges do away with traditional order books and instead rely on liquidity pools governed by mathematical formulas. Traders can always trade with liquidity pools, even for the least liquid tokens, but every trade affects the price of the asset they’re trading — a phenomenon known as “slippage.”

The mathematical formula defines how the price changes with the size of a specific order. For example, the formula might say that exchanging 10 Ethereum (ETH) for Dai (DAI) will get you $3,500, but trading 10 Ethereum will only get you $3,400. This means that in the former case, the price of 1 Ethereum is $350, but in the latter case, it is only $340. This formula is often called a "bonding curve" because the various possible combinations describe a specific price curve. Uniswap is a hyperbola, although other AMMs may have more complex shapes to optimize different scenarios.

AMMs rely on liquidity providers — individuals and entities that put their funds into liquidity pools to facilitate transactions and reduce slippage. In return, liquidity providers receive transaction fees paid by users.

While this sounds like a good deal, liquidity providers are subject to "impermanent" losses. When prices move sharply in one direction, liquidity providers may end up with less money than they initially invested. Compared to a 50:50 portfolio of the underlying assets, this combination significantly underperformed, with larger price deviations.

The researchers explained that this phenomenon is caused by the presence of arbitrage traders. External market prices do not follow the joint curve, so continuous action must be taken to keep Uniswap prices in balance with other markets. However, when arbitrageurs rebalance the liquidity pool to the correct value, they trade at the "suboptimal rate" defined by the joint curve. This also allows arbitrageurs to extract value from liquidity providers.

This loss is often referred to as “impermanent” loss, and liquidity providers can even compare it to a 50:50 portfolio. Leaving aside the case where the price permanently reaches a new equilibrium, Direr raises a question:

“The fact that both strategies produce the same outcome seems troubling at first glance. In the pooling strategy, the liquidity pool incurs arbitrage costs twice. In the holding strategy, investors avoid arbitrage costs entirely but end up with the same final wealth. How is this possible?”

The researchers’ answer is that the way benchmarking is usually done is misleading. Uniswap constantly rebalances its liquidity pools as they increase or decrease, so that liquidity providers can own fewer assets that have risen and more assets that have fallen in relative terms.

Liquidity providers effectively use profit and cost averaging in both ways. When the price of an asset rises, they lock in some profits and gradually buy more when the price falls.

Similar to how this averaging works, a 50:50 portfolio that is constantly rebalancing will make a profit even if the price returns to the initial number. In contrast, the value of the liquidity pool simply remains at the original level.

Thus, "impermanent loss" appears to be a misleading term. The loss is always permanent, but in the optimistic case, it simply reduces the gains that would have been obtained with an equivalent strategy.

Bancor V2 and Mooniswap have adopted technologies to reduce such losses. Bancor V2 uses oracles to obtain real market prices and balance liquidity pools accordingly, while Mooniswap minimizes the profits of arbitrage traders by delaying price updates.