Report: Uniswap and other automated market makers' impermanent losses are permanent

Alexis Direr, a researcher at the University of Orleans in France, has published a paper summarizing the mathematical underpinnings of Uniswap and other automated market maker-based exchanges.

Automated Market Maker (AMM) is a term referring to a type of decentralized exchange represented by Uniswap, which has gained wide 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 most illiquid tokens, but every trade affects the price of the asset they are trading — a phenomenon known as “slippage.”

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Mathematical formulas define how prices vary with the size of a particular order. For example, the formula might say that exchanging 10 Ethereum (ETH) for Dai (DAI) yields $3,500, but trading 10 ETH yields only $3,400. This means that in the former case, the price of 1 ETH is $350, but in the latter case it is only $340. This formula is often referred to as the "bonding curve" because the various possible combinations describe a specific price curve. Uniswap is hyperbolic, although other AMMs may have more complex shapes to optimize for different scenarios.

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

Although this sounds like a good deal, liquidity providers need to bear the "impermanent" loss. When prices fluctuate wildly in one direction, liquidity providers may end up with less money than they initially invested. Compared to a 50:50 portfolio of underlying assets, this portfolio has significantly underperformed, with large price deviations.

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The researchers explain that this phenomenon is due to the presence of arbitrage traders. External market prices do not obey the bonding curve, so continuous action must be taken to keep Uniswap's price in balance with the rest of the market. However, when arbitrageurs readjust the liquidity pool to the correct value, they trade at the "suboptimal rate" defined by the bonding curve. This also allows arbitrageurs to capture value from liquidity providers.

This kind of loss is often called "impermanent" loss, and liquidity providers can even compare it to a 50:50 portfolio. Leaving aside the case of prices reaching a new equilibrium permanently, Direr asks the question:

“The fact that both strategies produce the same outcome is troubling at first glance. In the liquidity pool strategy, the liquidity pool incurs the arbitrage cost twice. In the holding strategy, the investor avoids the arbitrage cost entirely, but end up with the same ultimate wealth. How is that possible?"

The researchers' answer is that the way benchmarks are often done is misleading. Uniswap is constantly rebalancing liquidity pools as they grow or shrink, so liquidity providers can have fewer assets that are rising and relatively more assets that are falling.

Liquidity providers effectively use profit and cost averaging in both ways. They lock in some profits when the price of an asset rises and gradually buy more as 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 figure. In contrast, the value of the liquidity pool just remains at the original level.

So "impermanent loss" seems to be a misleading term. The loss is always permanent, but in the optimistic case it just reduces the gain that the same strategy could have made.

Bancor V2 and Mooniswap have implemented techniques to reduce these types of losses. Bancor V2 uses oracles to obtain true market prices and balances liquidity pools accordingly, while Mooniswap delays price updates to minimize profits for arbitrage traders.

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