Report: Impermanent loss in Uniswap and other WMAs is always permanent

Is it time to look for another name for this phenomenon?

Alexis Direr, a researcher at the University of Orleans in France, has published an article summarising the mathematical foundations of Uniswap and other exchanges based on Automated Market Makers.

Automated Market Maker is the term for a kind of decentralized exchange that reached significant popularity in 2020, led by Uniswap.

In short, these exchanges eliminate traditional order books and instead are based on liquidity groups governed by a mathematical formula. Traders can always trade the group for even the most illiquid tokens, but each order will affect the price of the asset they are trading, a phenomenon called ‘slippage’.

The mathematical formula defines how the price changes in response to the size of a particular order. For example, the formula may say that trading 10 Ether (ETH) for Dai (DAI) yields USD 3500, but trading 100 ETH yields only USD 3,400. The formula is often called the “linkage curve”, as the various possible combinations describe a particular price curve. In the case of Uniswap, the curve is a hyperbola, although other MMAs may have more complex ways of optimizing for different scenarios.

MMAs depend on liquidity providers: individuals and entities that commit their capital to liquidity groups to facilitate transactions and reduce slippage. In return, LPs get commercial rates paid by users.

While this may seem like good business, liquidity providers must deal with “impermanent” losses. LPs can end up with less money than they initially invested when the price fluctuates significantly in one direction. Compared to a 50:50 portfolio of the assets in question, the portfolio has a significantly lower return with large price deviations.

The researcher explains that this phenomenon is caused by the presence of arbitrage traders. External market prices do not follow the linkage curve, so constant action is needed to keep the Uniswap price in balance with the rest of the market. But when arbitrageurs rebalance the group to the correct value, they do so at a “sub-optimal exchange rate” defined by the tie curve. This action extracts value from the liquidity providers in favour of the arbitrageurs.

The loss is generally referred to as “impermanent” because if the price were to return to its initial value, the liquidity providers would be completely flat compared with the 50:50 benchmark portfolio. Discounting the case where the price moves permanently to a new equilibrium, Direr raises the question:

“The fact that the two strategies give the same result seems disturbing at first. In the pooling strategy, the 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 researcher’s answer is that the way benchmarking is commonly done is misleading. Uniswap constantly rebalances the group as it goes up or down, so that liquidity providers have fewer units of the asset that went up in price and more units of the asset that went down in relative terms.

LPs effectively perform a two-way profit and cost averaging technique. They secure part of the profits as the price of an asset rises and progressively buy more as it falls again.

Similar to how such an averaging technique would work, a 50:50 portfolio which is constantly rebalancing will generate profits, even though the price returns to the initial number. In comparison the value of the liquidity fund simply remains where it was.

Therefore “temporary loss” appears to be a misleading term. The loss is always permanent, but in the optimistic scenario it simply cuts into the gains that an equivalent strategy would have made.

Bancor V2 and Mooniswap have adopted techniques to mitigate this type of loss. The former uses oracles to read the actual market prices and balance the group accordingly, while the latter introduces a gradual time delay to minimise arbitrage traders’ profits.