Alexis Direr, a researcher at the University of Orleans in France, has published a paper summarizing the mathematical foundations of Uniswap and other exchanges based on Automated Market Makers.
Automated Market Maker is the term for a class of decentralized exchanges that gained significant popularity in 2020 under the leadership of Uniswap.
In short, these exchanges eliminate traditional order books and instead rely on pools of liquidity that are governed by a mathematical formula. Traders are always able to make a transaction with the pool for the illiquid tokens as well, but every order affects the price of the asset they are trading in – a phenomenon known as slippage.
The mathematical formula defines how the price changes depending on the size of a particular order. For example, the formula can say that exchanging 10 ethers (ETH) for Dai (DAI) is $ 3,500, but exchanging 100 ETH is only $ 3,400. This means that the price for 1 ETH is $ 350 in the former case but only $ 340 in the latter. The formula is often referred to as the “binding curve” because the various possible combinations describe a particular price curve. In the case of Uniswap, the curve is a hyperbola, although other AMMs can have more complex shapes to optimize for different scenarios.
AMMs rely on liquidity providers – people and companies who put their capital into pools of liquidity to make trading easier and reduce slippage. In return, LPs receive trading fees that are paid by users.
While this sounds like a sweet deal, liquidity providers have to deal with “inconsistent” losses. LPs may have less money invested than they originally invested if the price fluctuates significantly in one direction. Compared to an equivalent 50:50 portfolio of the assets in question, the pool underperforms significantly in the event of large price deviations.
Source: University of Orleans
The researcher explains that this phenomenon is due to the presence of arbitrage traders. The prices outside the market do not adhere to the binding curve. Hence, constant action is needed to keep Uniswap’s price in balance with the rest of the market. However, when arbitrageurs bring the pool back up to par, they do so at a “sub-optimal exchange rate” defined by the pegging curve. This action deprives the liquidity providers of value in favor of the arbitrageurs.
The loss is generally referred to as “volatile” as in the event that the price returns to its original value, the liquidity providers are fully compared to the 50:50 reference portfolio. Direr asks the question in which the price is permanently moving into a new equilibrium and asks the question:
“The fact that both strategies lead to the same result seems unsettling at first. With the pooling strategy, the pool incurs arbitrage costs twice […] With the holding strategy, investors avoid arbitrage costs altogether, but get the same final net worth. How is it possible?”
The answer from the researcher is that the way benchmarking is commonly done is misleading. Uniswap constantly rebalances the pool as it moves up or down, so liquidity providers have fewer units of the asset that have increased in price and more units of the asset that have decreased in relative terms.
LPs are effectively performing a profit and cost averaging technique on both types of travel. You secure part of the profit when the price of an asset goes up and gradually buy more when it goes down again.
Source: University of Orleans
Similar to how such an averaging technique would work, a 50:50 portfolio that is constantly being rebalanced will make a profit even though the price returns to the original number. In comparison, the value of the liquidity pool simply stays where it was.
Hence, “inconsistent loss” seems to be a misleading term. The loss is always permanent, but in the optimistic scenario only the profits are reduced that would have been achieved by an equivalent strategy.
Bancor V2 and Mooniswap have introduced techniques to mitigate this type of loss. The former uses oracles to read actual market prices and balance the pool accordingly, while the latter introduces a gradual time lag to minimize arbitrage traders’ profits.