In this article, we will observe the theoretical profits arbitrageurs collect at the expense of Liquidity Providers (LP) on Uniswap V2, one of the biggest Constant Function Automated Market Maker (CFAMM) deployed on a blockchain in terms of Total Value Locked and Volume. We will also observe which asset classes theoretically accumulate the most LVR, and possible implications of these findings.

Uniswap V2 relies on arbitrageurs to keep the exchange rate between tokens in a pair up to date, or "fresh". They provide this service for a profit and, as rational agents, will always find ways to maximise their profits. As a result, arbitrageurs take advantage of stale AMM prices and the Constant Function formula of CFAMMs to maximise profits.

This article uses a model proposed in [2], where arbitrageurs purchase tokens (provided for by LPs) at the stale AMM price and sell them for profit on Centralised Exchanges (CEXs). This is at the cost of LPs who, not counting the fees they earn for their services as LPs, would lose relative to performing the same trade on a CEX instead of LP'ing.

This adverse selection cost LPs incur was given the name Loss-Versus-Rebalancing (LVR, pronounced as 'lever') by the research in [2]. LVR is the loss LPs incur by providing liquidity in a CFAMM vs. performing the same set of trades on a Central Limit Order Book exchange (CLOB).

For example, if a trader buys WETH for USDC in a pool where a LP provides liquidity for, this LP might have incurred fewer losses if they had sold their USDC for WETH on a CLOB exchange instead. The existence of LVR poses a threat to LPs, who may prefer to pull liquidity if they are not earning enough fees to offset this loss. For a more in depth example of LVR refer to this post from CoW DAO.

But how prevalent is LVR on Uniswap V2? What are some of the determinants of LVR, and how does LVR compare across various blockchains where Uniswap V2 is deployed?

In this article, we aim to provide additional insight to help answer these questions using data analytics.

All data comes from Dune Analytics and contains data accumulated from the past three months of trading on Uniswap V2 across observed chains. Trades were partitioned by two classes, those involving stablecoins being sold for volatile assets, and those involving WETH being sold for volatile assets. The cumulative LVR, from a model based on research in [2], is found for each pair for each class on a selected list of observed blockchains. Charts and figures were compiled to help answer the questions posed in this article. For additional background on calculations and assumptions made for this study, as well as the rationale for selecting the pairs observed for this study, see the Calculations and Assumptions section at the end of this article.

The first section examines LVR on Ethereum. The second section examines LVR on all blockchains observed for this study. The third section analyses the profitability of LP'ing across the observed chains. The fourth section attempts to analyse the behaviour of arbitrageurs on Uniswap V2 deployed on Ethereum. The last section concludes this study and provides possible improvements for further studies on this subject.

The Bibliography follows this conclusion, as well as a section detailing the calculations and assumptions used for this study and a section logging the edits made to this article since publishing.

Uniswap V2 was initially deployed on Ethereum, and it continues to be the biggest market in which Uniswap V2 operates, according to data from DeFiLlama. But how does LVR affect the profitability of LPs on this chain?

Examining the top forty pairs on the Ethereum deployment of Uniswap V2 that include a stablecoin (e.g. USDT, USDC, DAI, etc), we can see that most of the LVR incurred by five pairs: DAI-MKR pair, DAI-WETH, sUSD-WETH, USDC-WETH and Okinami-WETH (Okinami being a community token), with LVR ranging from $12k - $18k. The total LVR observed for the top forty pairs with stablecoins was ~$81k, so it's fair to say that these pairs took the lion's share of LVR.

See the "Assumptions" section below for how Uniswap V2 pairs were ranked in this study.

This study also examines token pairs where WETH was sold for other classes of tokens. These include what I refer to as "community tokens" (aka meme tokens), as well as utility tokens. Examples of utility tokens include governance tokens and currency, wrapped or derivative tokens such as stETH and WBTC. This pairing of tokens was also examined and found to accumulate fewer LVR than pairs with stablecoins.

The WBTC-WETH pair dominates the pie for LVR... but has a total accumulated LVR for the past three months of ~$708. PAXG-WETH accumulated ~$4 in this timeframe, coming in second for cumulative LVR (fyi, PAXG represents tokenised Gold). Total accumulated LVR for the top 40 WETH pairings is ~$712.

So far, it seems like stable pairings dominate the accumulation of LVR (this could be due to the difference in volatility between the two asset classes, we study this possibility later), but is this consistent with other chains as well? And how does LVR compare amongst other chains where Uniswap V2 is deployed?

Note:

Due to low TVL of Uniswap V2 pairs (with < $1.1k in TVL) on other chains, some chains had to be omitted from this study. These include Optimism, Avalanche, BSC and Polygon.

Due to how large the Uniswap V2 market is on Ethereum compared to other chains (not just in terms of TVL, but also number of pairs where TVL is > $1.1k being quite low), I couldn't make a fair comparison between these chains. And so, just like I did with Uniswap V2 on Ethereum in the last section, I will compare LVR between asset classes for each chain.

For pairs with stablecoins, Ethereum still accumulated the most LVR at ~$81k, Arbitrum One accumulated ~64 cents, and Base accumulated just a tenth of a cent.

Ethereum had the same 40 pools which were eligible for this study from last section, Arbitrum had only two pools eligible, but was able to amass ~$770 in fees, with most of it coming from the OPUL-USDT pair (fyi, OPUL is the utility token for Opulous, a tokenised music and fundraising platform for musicians).

For pairs where WETH was sold for community or utility tokens, Ethereum accumulated, as mentioned previously, ~$712 LVR. Base and Arbitrum One were the only other viable chains to study. LVR was a fraction of a cent for this class of pairs for both.

So, when comparing the two asset classes, we again see that the stablecoin-volatile pairs accumulated more LVR than the WETH-volatile pairings. One reason for this could be that the volatility between the assets in stablecoin-volatile pairs is greater than the volatility between assets in WETH-volatile pairs, since volatility is a determinant of LVR (see the Calculation section for more).

The covariance between the assets in each asset class was also studied, and the average covariances for pairs on each chain was found and compiled into the following table:

On Ethereum, the pairs which incurred the most LVR were DAI-MKR, DAI-WETH and sUSD-WETH, the covariance for these pairs were in the top 5 in absolute value covariance for stablecoin-volatile pairings. So, we may consider covariance to be one determinant of LVR; pairs with low covariance may be associated with high LVR.

But what about the profitability of LPs on Uniswap V2 for these classes of tokens across the observed chains? Next, we examine the profitability of LPs on Uniswap V2 for these token classes across the observed chains. This is provided that LPs engage in a hedging strategy for liquidity provisioning on an AMM, called the Δ hedged LP strategy. We will explain this strategy a bit further, and use it, and its terms, to measure the profitability of pairs on Uniswap V2.

Given the differences in accumulated LVR observed between stablecoin pairing and WETH pairings, we are particularly interested in the profitability of tokens paired with stablecoins, as these pairs tend to accumulate more LVR.

The Δ hedged Profit and Loss (Δ hedged PnL) is defined by the research in [2]. as the difference between LP fees accumulated and LVR, provided that LPs dynamically copy trades made on their liquidity to a CEX. If the difference is negative, LPs lose money from LVR, otherwise LPs earn a profit from fees.

To give an example, let a LP hold a Uniswap V2 position with 1000 USDC and 0.47 WETH (let's assume this proportion also represents the current exchange rate for ETH/USD). An arbitrageur buys 0.0051 WETH for ~10 USDC after the price of ETH rises, due to the dynamics of a CFAMM (aka slippage).

At the same time, the LP owner sells the same 0.0051 WETH they sold to the arbitrageur, but at the market price of a CEX rather than the CFAMM they LP for. The result is that the slippage losses incurred from being an LP are mitigated by this hedging strategy. As the price of ETH rises, the LPs and arbitrageurs continue to behave this way. When the price of ETH drops, both agents perform the opposite trade for each asset.

This strategy mitigates losses from LVR and maximises profit from fees earned from LP'ing. If you're still a bit confused about this process, these articles from CoW DAO and a16z go more in-depth into a similar example.

Now back to the study. We will look at the profitability of LP'ing using this Δ hedged strategy. For pairs with stablecoins on Ethereum, the Δ hedged PnL was $6,618,540, with cumulative fees equating to $6,700,214 and LVR, as mentioned previously, being ~$81k. For pairs with WETH, Δ hedged PnL was $7,456,774, with this category of pairs accumulating just ~$700 in LVR, and amassing $7,457,476 in fees.

On Base, we can observe the Δ hedged PnL for pairs with stablecoins was just over $50, while on Arbitrum One it was just over $736.

For pairs with WETH, Base accumulated ~$750k Δ hedged PnL after amassing the same amount in fees, Arbitrum One amassed ~$361 Δ hedged PnL , same amount in fees. Both Arbitrum One and Base pairs incurred less than a cent in LVR for this class of pairs as explained previously.

These findings suggest that LVR, also known as arbitrage profits, is indeed more prevalent in pairs with stablecoins. This would make sense logically, as volatility (measured using standard deviation) tends to be higher in pairs where stablecoins are paired with community or utility tokens, leading to greater LVR. Indeed, we observed that stablecoin-volatile pairs have a covariance close to zero, while it was quite high on average for WETH-volatile pairs.

Below are tables summarising these findings for both stablecoin pairings and WETH pairings. The LVR-to-Fee ratio is simply the ratio of LVR to fees and could be used as a metric for assessing profitability of pairs. The inclusion of this ratio in this study is not meant to be used to compare blockchains, but instead to compare asset classes.

Given these findings, the prevalence of theoretical arbitrageur profits (LVR) in stablecoin-volatile pairs, how do arbitrageurs behave in these markets for tokens trading on Uniswap V2? Can we observe a correlation between number of, and volume traded for, arbitrageurs and LVR?

For pairs with stablecoins on Ethereum, speculators (aka noise traders) dominate this market for the past 3 months. There were ~267k speculators trading in this market vs. ~92k arbitrageurs.

Speculators also dominate arbitrageurs for volume traded. Speculators were found to trade at a volume ~$1.4B, meanwhile arbitrageurs traded ~$440M in volume.

For information on how arbitrageurs were classified, see the Assumptions section.

Surprisingly, tokens paired with WETH were observed to have more activity from arbitrageurs.

Note: Only Ethereum was observed for this study since it accumulated the most LVR and fees by such a large magnitude, and had the most liquid pools when assessing its TVL and Volume, I thought it sufficient for this study.

Both the proportion of arbitrageurs and the volume traded by them were greater for WETH pairings. This is when `arbitrageurs` are defined as the entity taking (or buying; aka "takers") the volatile asset for either a stablecoins or WETH, which means that the `arbitrageur` could've bought these volatile tokens in the trace of a transaction which was executed by an entity which may not be an arbitrageur.

However, even when `arbitrageurs` are defined as the entity which initiated the transaction, the outcomes are the same: there is a greater proportion of arbitrageurs, and arbitrageur trading volume, for WETH pairings vs. stablecoin pairings.

This was surprising to observe, because stablecoin pairings were found to accumulate more LVR theoretically.

This raises a few questions which were left unanswered:

• Are arbitrageurs well informed about the level of volatility for stablecoin-volatile pairs vs WETH-volatile pairings?

• Are these arbitrageurs solely trading between DEXs, mostly DEXs, or are they mostly trading between CEXs and DEXs?

  • Is the inter-DEX arbitrage happening across chains as well (cross-chain swaps)?

These questions were not answered in this study, but could be explored in further research on the topic of LVR or LP profitability in general.

This study reveals that there is a significant concentration of LVR, but also Δ hedged PnL, on Ethereum, this is mainly due to its dominance in TVL and volume in Uniswap V2 deployments. The magnitude of this dominance makes it unfair to compare with other chains Uniswap V2 is deployed on.

This could be due to competition from other DEXs deployed on chains other than Ethereum. It could also be due to the popularity of Ethereum paired with the complexities and costs (both monetary and temporal) of bridging tokens for end users. These hypotheses were not studied for this article but could be explored in further research. Comparing LVR between all the major DEXs on the biggest chains by TVL and Volume could also yield interesting insights.

The findings from this study shows that it is still profitable to LP on Uniswap V2, even on Ethereum, despite having higher transactions costs and slower settlement of transactions than other chains observed for this study. This is provided that LPs engage in a Δ hedged strategy. However, if LVR is mitigated, or removed entirely, it would theoretically further incentivise LP'ing. Therefore, there is still a need for improvement of existing AMM designs.

It would also appear that stablecoin pairings accumulate the most LVR compared to pairs with WETH. Specifically, pairs where a USD stablecoin is paired with a volatile asset, such as the DAI-MKR, DAI-WETH and sUSD-WETH pairs on Ethereum, all of which accumulated the most LVR on the Uniswap V2 instance deployed on Ethereum.

When comparing volatilities (daily standard deviation), stablecoin-volatile pairings were observed to have higher volatility on average than WETH-volatile pairings. However, the covariances of stablecoin-volatile pairings were nearly zero, while they were significantly greater for WETH-volatile pairings. From this, we can classify the type of pairings which generate the most LVR: pairings where a stable asset (DAI, USDC, etc.) is paired with a highly volatile asset (WETH, WBTC, etc.), and these asset pairings tend to have significantly low covariance because moments of high volatility in one asset does not coincide with high volatility in its paired asset.

Another interesting avenue for further research would be to compare LVR and Loss-versus-Holding (or LVH) in these markets. This research could explore whether similar trends emerge, and how long LPs typically experience LVH losses?

The research in [7] also explored LVR in the presence of trading fees, which this study does not. It would thus also be interesting to explore the effects of LVR when arbitrageurs are faced with trading fees.

So, what are the implications of this study?:

• Arbitrageurs could use this research to target stablecoin-volatile pairings more than they are now.

• LPs, both current and potential, could opt to LP for WETH-volatile pairings, particularly on Ethereum.

Implications being that TVL would gradually concentrate in WETH-volatile pairs, and Uniswap V2 would specialise in this asset class. This is all provided that LPs and arbitrageurs only have the choice of those two asset classes.

AMM designers could also use this and further research on LVR and the profitability of liquidity provisioning to help design mechanisms for automated market making, and resource allocation in general. This could lead to a minimising of LVR on their platforms, maximising profit for LPs and still being able to incentivise arbitrageurs to keep AMM prices fresh.

I hope you found this article insightful. You can find the charts and figures used for this study on this dashboard: https://dune.com/takeabreath/univ2-lvr.

I encourage readers to checkout this dashboard and delve deeper into these findings if it interests you as much as it did for me. Further research on LVR, its determinants, and it's impacts on Uniswap could help us understand these new paradigms for resource allocation further and help in its future iterations to attract more users.

Whether they be LPs, speculators or arbitrageurs.

[1] Wikipedia Contributors, “Constant function market maker,” Wikipedia, Sep. 06, 2024. Available: https://en.wikipedia.org/wiki/Constant_function_market_maker.

[2] J. Milionis, C. Moallemi, T. Roughgarden, and A. Zhang, “Automated Market Making and Loss-Versus-Rebalancing *,” May 2024. Available: https://arxiv.org/pdf/2208.06046.

[3] “What is Loss-Versus-Rebalancing (LVR)? - CoW DAO,” Cow.fi, 2023. Available: https://cow.fi/learn/what-is-loss-versus-rebalancing-lvr.

[4] “Dashboards,” dune.com. Available: https://dune.com/

[5] “Uniswap V2 - DeFiLlama,” DeFiLlama. Available: https://defillama.com/protocol/uniswap-v2#information

[6] “LVR: Quantifying the Cost of Providing Liquidity - a16z crypto,” a16z crypto, Sep. 19, 2022. Available: https://a16zcrypto.com/posts/article/lvr-quantifying-the-cost-of-providing-liquidity-to-automated-market-makers/.

[7] J. Milionis, C. Moallemi, and T. Roughgarden, “Automated Market Making and Arbitrage Profits in the Presence of Fees,” May 2023. Available: https://arxiv.org/pdf/2305.14604.

[8] A. Hayes, “Adverse Selection Definition,” Investopedia, Apr. 29, 2024. Available: https://www.investopedia.com/terms/a/adverseselection.asp.

• LVR = $\frac{\sigma ^2}{8}$

• $\sigma$ = STDDEV_POP($\frac{\text{swapped in token volume}}{\text{swapped out token volume}} \times \text{hourly USD price of swapped in token}$)

  • this is calculated in a daily timeframe.

  • STDDEV_POP is the TrinoSQL/DuneSQL's aggregate function for calculating the standard deviation of a population, where the "population" is "all swaps occurring within a day".

• Δ hedged PnL = Daily Fees accumulated - LVR

• Daily Fees accumulated = $\frac{\text{swapped in token volume}}{0.997} \times 0.003$

  • the 0.3% fee taken from each swap. This is calculated in a daily timeframe.

• Arbitrageurs are rational, profit maximising, and can trade under any legal jurisdiction

• Arbitrageurs do not pay trading fees.

• Arbitrageurs constantly monitor for arbitrage opportunities and take whatever profits available at any time

• LPs are passive: they do not mint or burn their position after the initial mint/deposit

• Pairs w/ stables are examined where the token being sold in swaps is always a stablecoin

• Pairs w/ WETH are examined where the token being sold in swaps is always WETH

• The top 40 pools were chosen, ranked by TVL

  • Most pools on chains other than Ethereum had < $1.1k TVL, these pools caused a lot of issues during analysis since these pools tend to have high volatility which skewed results

  • Many chains were therefore omitted by applying a > $1.1k requirement, most notably BNB and Avalanche

  • Volume was not used as a metric, but was attempted. Issues arose when trying to use volume as a metric; many pairs which sieved through using the top 25 percentile for cumulative volume in the past 30 days were likely manipulated by pump and dump schemes, skewing results.

• Stable coins were chosen based on Dune spell, `labels.stablecoins`, maintained by Dune wizard hildobby.

  • Canonical USDC deployed on optimism, base and others were omitted from this spell, but were added manually for this study

• A list of arbitrageur addresses were given from a Dune spell, `labels.arbitrage_traders`, maintained by Dune wizards alexth and hosuke.

  • "arbitrageurs" were identified as entities who take liquidity from LPs by swapping in WETH or stablecoins (takers) as well as entities who initiated the transactions that initiate one or more swaps

    • in the `dex.trades` spell on Dune Analytics, we thus used both `taker` and `tx_from` to identify these classes of entities, respectively

    • this distinction is explained in the article

• Assumptions/limitations added

  • arbitrageurs do not pay fees for trades

  • arbitrageurs obey myopically: they grab whatever LVR they can get at any time

    • arbitrageurs thus monitor the market constantly

• "LVR on Ethereum" section

  • Added DefiLlama link showing Uniswap V2 usage metrics

  • "with LVR ranging from $12k - $18." changed to "with LVR ranging from $12k - $18k."

  • Cumulative LVR for ethereum WETH-volatile pairs was updated, as well as figures for this section

• "LVR on Significant Chains" section

  • Updated charts and figures for this section, based on previous changes

  • Added covariance study

• "Profitability of LP'ing" section

  • Updated figures for this section, based on previous changes

  • Expanded on the Δ hedged LP strategy, provided examples

• "Arbitrageur Prevalence" section

  • Added another class of arbitrageurs trading in the model, those who initiated the posted transaction

• Conclusion

  • Provided additional context for conclusions based on previous changes, added Note

  • Added the paper [7] iterating on the research from [2], to include arbitrage fees into [2]'s model