a16z: A Detailed Explanation of the New Method for Measuring AMM LP Costs - LVR

Original authors: Jason Milionis, PhD in Computer Science at Columbia University; Cimac Moallemi, Professor at Columbia Business School; Tim Roughgarden, Head of Research at a16z crypto, Professor of Computer Science at Columbia University and member of the Data Science Institute; Anthony Lee Zhang, Assistant Professor of Finance at the University of Chicago Booth School of Business

Source: LVR: Quantifying the Cost of Providing Liquidity to Automated Market Makers

Original translation: Babywhale, Foresight News

Automated Market Makers (AMMs) have two types of participants: one is traders who exchange one token for another (e.g., ETH and USDC); the other is liquidity providers (LPs) who provide token liquidity to the AMM in exchange for a share of the trading fees.

When does it make economic sense to participate as an LP? When do the returns exceed the costs? LPs earn from trading fees and, in some cases, additional token rewards. This article summarizes a new method of calculation that we call LVR (Loss Versus Rebalancing). We will detail LVR and its implications for LPs and AMM design below, but first, let’s review how AMMs perform during market price evolution.

Arbitrage and "Adverse Selection" in AMMs

LPs in automated market makers may suffer losses due to adverse selection, which is one of the main costs of becoming an LP. By providing liquidity for trading at a given price (buying or selling), each LP in an AMM risks becoming the counterparty to traders who have better or more immediate information about token prices.

For example, if the price of ETH on the open market suddenly rises, fast arbitrageurs may buy ETH from the AMM (at the lower old price) and then resell it on centralized exchanges like Binance (at the new higher market price) to make a profit. Since there are only two types of participants in the AMM, the profits of traders correspond to the losses of LPs.

To reason about the costs for LPs and provide insights for LP participation decisions and AMM design, we start with a simple question about past evaluations. Suppose we have just finished providing liquidity to an ETH-USDC AMM. Assume we deposited 1 ETH and 1000 USDC into the AMM and received 0.5 ETH and 2000 USDC upon withdrawal (in most AMMs, what you receive may differ from what you deposited, depending on the market price fluctuations of the AMM tokens during that time).

Further assume that the price of ETH rose during the month, jumping from $1000 to $4000. In this case, the decision to provide liquidity would double your investment from a portfolio worth $2000 at the time of deposit to a portfolio worth $4000 at the time of withdrawal.

Providing liquidity to an AMM involves holding a certain amount of ETH during the month. Given that the price of ETH quadrupled during this month, in hindsight, almost any strategy involving holding some ETH looks quite good.

But the more important question is: how does the specific strategy of AMM LPs compare to all other ways of "going long on ETH"? Similarly, how should we view the decision to provide liquidity when stripping away the profits (or losses) generated purely by changes in ETH prices?

The simplest way to bet on the rise of ETH prices is to buy some ETH and hold it. In the example above, the holding strategy would result in a month-end portfolio (still 1 ETH and 1000 USDC, but now with ETH priced at $4000) worth $5000, which is $1000 more than the amount withdrawn from the AMM. This $1000 gap is an example of what is commonly referred to as "impermanent loss."

Example of Impermanent Loss

Impermanent loss compares the profits of LPs to the profits that could have been earned under a reference strategy, but it fails to isolate the adverse selection costs faced by AMM LPs. To see this, let’s change our example so that the price of ETH is $1000 at both the beginning and the end of the month. In this case, in most AMMs, you would receive the same token combination as your initial deposit, meaning the impermanent loss would be zero. Whether the price of ETH remained unchanged throughout the month or fluctuated up and down before returning to $1000, the result would be the same.

The independence of impermanent loss from the price trajectory (except for its initial and final values) should raise some suspicion. For example, we have discussed arbitrage in AMMs, where traders profit at the expense of LPs. Thus, LP costs should seem to increase with the number of arbitrage opportunities in AMMs, and the frequency of opportunities with stable prices (no arbitrage) should be very different from those with significant price increases (lots of arbitrage).

What is LVR

We propose a new way to think about the costs borne by AMM LPs, with the core metric we call LVR (Loss Versus Rebalancing). LVR can be interpreted in several different ways. Here, we emphasize it as an alternative to impermanent loss, with a more nuanced calculation. (Another interpretation of LVR is the losses LPs incur after appropriately hedging their exposure to ETH prices, and another interpretation is the maximum profit that arbitrageurs can earn.)

Rebalancing is unique to AMMs, so let’s introduce it in the context of the well-known constant product market maker (CPMM) of Uniswap (v1 and v2). A dual-token CPMM, also known as the "xy=k" curve, maintains reserves of two tokens, such as x units of ETH and y units of USDC. The spot price is defined as y/x, which has the effect of equalizing the market value of the two reserves (in this sense, such an AMM effectively executes a rebalancing strategy). In practice, this spot price is defined by transactions that only allow the product of the two token quantities, xy, to remain constant.

LVR can be defined on a per-transaction basis, so let’s look at a single transaction. Consider a CPMM with 1 ETH and 1000 USDC, assuming the market price of ETH suddenly rises from $1000 to $4000. We expect some arbitrageurs to buy 0.5 ETH from the CPMM at an effective price of 2000 USDC, thereby keeping x*y constant while moving the spot price to 2000/0.5=4000 USDC/ETH (and making the value of both reserves $2000).

At this point, referencing rebalancing, starting from the same initial portfolio of 1 ETH and 1000 USDC: replicate the CPMM transaction (meaning selling 0.5 ETH, just like the CPMM), but execute it at the current market price of $4000 (e.g., on Binance). Because this alternative strategy results in a portfolio value that is $1000 higher than the CPMM (5000 vs. 4000), we say that the LVR for this transaction is $1000.

Continuing this example, suppose the price of ETH suddenly falls back to $1000. The CPMM will immediately return to its original state of 1 ETH and 1000 USDC (after arbitrage), effectively repurchasing 0.5 ETH for the same 2000 USDC. The rebalancing strategy replicates the transaction (buying 0.5 ETH), but executes it at the market price ($1000). The portfolio value of the rebalancing strategy is now $1500 more than the CPMM (3500 vs. 2000), contributing an additional $500 to the LVR for the second transaction.

This calculation is intuitively reasonable; unlike impermanent loss, LVR depends on the price trajectory (if the price remains unchanged, LVR is 0, but if the price rises and then falls, it is not) and accumulates per transaction (since each transaction may be on the wrong side, leading to additional adverse selection costs).

General Definition of LVR

From the previous examples, we define LVR as follows: given any sequence of transactions on any AMM, LVR is the total loss incurred by executing transactions through the AMM rather than on the open market. Each term in this total is of the form a(p - q), where a represents the amount of ETH sold in the transaction (e.g., 0.5 and -0.5 in our first and second transactions above), p represents the market price at that time (4000 and 1000 above), and q represents the unit price of the AMM transaction (2000 and 2000 above).

This definition can also vary to periodic (e.g., hourly or daily) rebalancing rather than per-transaction. This variation can simplify the empirical analysis of LVR and also allow for a more natural interpretation of hedging against the aforementioned LVR.

Thinking About Past and Future Strategies

LVR isolates the adverse selection costs borne by LPs. In hindsight, was the decision to provide liquidity a good idea? First, this question boils down to whether the fees collected exceed the LVR, so it is generally easy to answer using public data (e.g., on-chain records of AMM transactions or historical price data on Binance).

To reason about future LP decisions rather than past ones, we cannot rely directly on data; we must adopt some mathematical models of how prices might evolve (LVR critically depends on the price trajectory). We can use various models, but perhaps the most natural choice is the Black-Scholes model, where the price of ETH evolves according to a geometric Brownian motion.

If you are not familiar with this model, the key point to know is that it fundamentally has only one important parameter, the price volatility σ. If σ=0, the price remains unchanged, while if σ is large, it indicates significant price fluctuations.

LVR can be precisely represented in this model. Because LVR accumulates per transaction and because this is a continuous-time model where transactions are always occurring, LVR can be expressed as the integral of the instantaneous LVR. The instantaneous LVR has an exponential relationship with σ and the current market price and a linear relationship with the marginal liquidity of the AMM at that price.

This mathematical expression may sound a bit daunting, but many common AMMs are so simple that LVR is given by a basic formula.

For example, for CPMM, the instantaneous LVR, when normalized by the market value through the CPMM, results in σ²/8. If the daily volatility of the Uniswap v2 ETH-USDC pool is 5%, then according to our model, LPs incur an LVR loss of 3.125 basis points per day (note: one basis point is 0.01%) (resulting in an annual loss of about 11%). Can trading fee income compensate for this loss? The answer depends on trading fees and trading volume. For example, if the AMM charges a fixed trading fee of 30 basis points, LPs will break even if the daily trading volume is about 10.4% of the AMM's assets. If the daily volatility is 10%, the required trading volume will be four times that.

Implications for AMM Design

LVR is important not only for potential liquidity providers but also for AMM designers. AMMs can only succeed if they allow LPs to earn sufficient returns, meaning that fee income needs to scale alongside LVR.

One insight from our research is that since LVR depends on the volatility of trading volume and fee income, AMMs should consider dynamic fees that adjust with trading volume, volatility, or observed LVR. Another is that AMM designers should explore ways to minimize LVR (and thus the required LP incentives), such as by combining high-quality price oracle feeds to quote prices closer to the market. Next-generation AMMs are already exploring these ideas and related concepts, and we can’t wait to see how they play out.