LVR: Quantifying the Value of Offering Liquidity to Automated Market Makers


There are two varieties of contributors in an automatic market maker (AMM): merchants, who trade one of many AMM’s tokens for one more (as a operating instance, say ETH and USDC); and liquidity suppliers (LPs), who present tokens to the AMM within the first place, typically in trade for a share of the buying and selling charges.

When does it make financial sense to take part as an LP? When does the profit exceed the price? The profit facet of this comparability is straightforward to grasp: income from shared buying and selling charges, plus in some circumstances further token rewards. This submit summarizes a brand new means to consider the price facet, which facilities round a amount that we name LVR (“loss versus rebalancing,” pronounced “lever”). We’ll say extra about LVR and its implications for LPs and AMM designers under, however first let’s assessment how AMMs behave as market costs evolve.

Arbitrage and opposed choice in AMMs

Liquidity suppliers in automated market makers endure losses from opposed choice, which is a part of the value of doing enterprise as an LP. By advantage of providing to take both facet (purchase or promote) of a commerce at a given worth, each LP in an AMM runs the chance of taking the mistaken facet of a commerce by a dealer with higher or newer details about a token’s market worth. For instance, if the value of ETH on the open market instantly will increase, a speedy arbitrageur could purchase ETH from an AMM (at a stale, cheaper price) and resell it on a centralized trade like Binance (on the new, larger market worth), pocketing a revenue. As a result of there are solely two varieties of contributors in an AMM, revenue to merchants should correspond to losses to LPs.

To motive about the price of liquidity provision, and thereby inform each LP participation choices and AMM design, let’s begin with the easier query of assessing the previous. Suppose we simply completed supplying liquidity to an ETH-USDC AMM. In hindsight, was this a good suggestion? Suppose we deposited 1 ETH and 1000 USDC into the AMM, and upon withdrawal obtained 0.5 ETH and 2000 USDC. (In most AMMs, what you get again could differ from what you set in, relying on how the market worth of the AMM’s tokens have moved within the meantime.) Suppose additional that it was a very good month for ETH, with the market worth leaping from $1000 to $4000 over the course of the month. On this case, the choice to offer liquidity would have doubled your cash from a $2000-value portfolio on deposit to a $4000-value one upon withdrawal. 

This may occasionally seem to be a terrific transfer, however that is sloppy considering. Offering liquidity to the AMM concerned holding some quantity of ETH for the month. Provided that the value of ETH quadrupled through the month, just about any technique that concerned holding some ETH would look fairly nice in hindsight.

The sharper and extra essential query is: how did the precise technique of AMM liquidity provision evaluate to all the opposite ways in which you would have “gone lengthy ETH”? Equivalently, how does the choice take care of setting apart the earnings (or loss) ensuing purely from the evolution of the value of ETH?

The only method to guess on an rising ETH worth is to — look forward to it — purchase some ETH and maintain it. And in our operating instance, that technique would have resulted in an end-of-month portfolio (nonetheless 1 ETH and USDC 1000, however now with an ETH worth of $4000) value $5000, a full $1000 greater than what was withdrawn from the AMM. This hole of $1000 is an instance of what’s also known as “impermanent loss” or “divergence loss.”

The issue with impermanent loss 

Whereas impermanent loss does no less than evaluate LP earnings to what may have been made below a reference technique, it fails to isolate the opposed choice prices confronted by AMM LPs. To see this, let’s change our operating instance in order that the value of ETH is $1000 at each the start and the tip of the month. On this case, in most AMMs, you’ll get again the identical mixture of tokens as in your preliminary deposit (holding, in impact), which implies that the impermanent loss shall be zero. That is true whether or not the ETH worth stayed fixed all month or jumped round earlier than returning to $1000.

The independence of impermanent loss on the value trajectory (aside from its preliminary and remaining values) ought to strike you as fishy. For instance, we’ve already mentioned arbitrage on AMMs, whereby merchants revenue on the expense of LPs. It will appear, then, that LP prices must be rising within the variety of alternatives for AMM arbitrage. And the frequency of such alternatives may be very totally different if the value stays the identical (no arbitrage) versus if it jumps round so much (a lot of arbitrage).

Introducing LVR

We suggest a brand new means to consider the prices suffered by the LPs of an AMM, which facilities round a amount that we name LVR (“loss versus rebalancing,” pronounced “lever”). LVR could be interpreted in a number of alternative ways (which is usually signal for a definition). The one we spotlight right here is as a substitute for impermanent loss that makes use of a extra applicable and nuanced reference strategy, rebalancing. (One other interpretation of LVR is because the loss to an LP after appropriately hedging its market publicity to the value of ETH. Nonetheless one other is because the best-case earnings that might be made by arbitrageurs.)

Rebalancing is AMM-specific, so let’s introduce it within the canonical particular case of the constant-product market maker (CPMM) that was made well-known by Uniswap (v1 and v2). The particular case of a two-token CPMM — also called an “x*y=ok” curve — maintains reserves of two tokens, say x models of ETH and y models of USDC. The spot worth — the value of an infinitesimal commerce — is outlined as y/x, which has the impact of equalizing the market worth of the 2 reserves. (On this sense, such an AMM successfully carries out a rebalancing technique.) In follow, this spot worth is outlined implicitly by permitting solely trades that go away that product x*y of the 2 token portions invariant.

LVR could be outlined on a trade-by-trade foundation, so let’s have a look at a single commerce. Contemplate a CPMM with 1 ETH and 1000 USDC, and suppose that the market worth of ETH jumps instantly from $1000 to $4000. We anticipate some arbitrageur to purchase 0.5 ETH from the CPMM at an efficient per-ETH worth of 2000 USDC, thereby protecting x*y fixed whereas shifting the spot worth to 2000/0.5=4000 USDC/ETH (and equalizing the market worth of the 2 reserves at $2000). 

Right here’s the reference rebalancing, ranging from the identical preliminary portfolio of 1 ETH and 1000 USDC: copy the CPMM’s commerce (that means promote 0.5 ETH, similar to the CPMM) however execute it on the present market worth of $4000 (e.g., on Binance). As a result of this different technique ends in a portfolio value $1000 greater than that of the CPMM ($5000 vs. $4000), we are saying that the LVR of this commerce is $1000.

Persevering with the instance, suppose the value of ETH instantly jumps again right down to $1000. The CPMM will return (post-arbitrage) proper again to its unique state of 1 ETH and 1000 USDC, in impact shopping for again the identical 0.5 ETH for a similar per-ETH worth of 2000 USDC. The rebalancing reference technique copies the commerce (shopping for 0.5 ETH) however executes it on the market worth ($1000). The market worth of the rebalancing technique’s portfolio is now $1500 greater than that of the CPMM ($3500 vs. $2000), with the second commerce contributing a further $500 to the cumulative LVR. 

This calculation is intuitively satisfying: not like impermanent loss, LVR depends upon the value trajectory (LVR is 0 if the value stays fixed however not if it jumps up after which again down) and accumulates trade-by-trade (as each commerce is perhaps on the mistaken facet, resulting in further opposed choice prices).

LVR: A basic definition 

After seeing the previous instance, the final definition of LVR shouldn’t shock you: given an arbitrary sequence of trades on an arbitrary AMM, the LVR of the sequence is the sum of the losses incurred by executing the trades through the AMM somewhat than on the open market. Every time period of this sum is of the shape a(p – q), the place a denotes the amount of ETH bought within the commerce (e.g., in our first and second trades above, 0.5 and -0.5), p denotes the market worth at the moment (above, 4000 and 1000), and q denotes the per-unit worth of the AMM commerce (above, 2000 and 2000). 

A variant of the definition is to rebalance periodically (e.g., hourly or every day) somewhat than trade-by-trade, in impact batching transactions and copying the web commerce of every batch. This variant can simplify the empirical evaluation of LVR, and is arguably extra pure within the hedging interpretation of LVR talked about above.

Reasoning concerning the previous …

LVR isolates the opposed choice prices borne by an LP. Was a call to offer liquidity a good suggestion in hindsight? To first order, this query boils down as to whether the charges collected exceeded the LVR suffered, and thus is usually straightforward to reply utilizing publicly accessible knowledge (e.g., the on-chain file of an AMM’s trades and historic worth knowledge on Binance).

… and concerning the future

To motive about future somewhat than previous LP choices, we can’t rely straight on knowledge and should undertake some mathematical mannequin of how costs would possibly evolve. (Do not forget that LVR relies upon crucially on the value trajectory.) We may use quite a lot of totally different fashions, however maybe probably the most pure place to begin is the usual Black-Scholes mannequin from finance, with the value of ETH evolving constantly in keeping with a geometrical Brownian movement (with respect to an acceptable martingale measure). 

When you’re unfamiliar with this mannequin, the important thing factor to know is that it has primarily just one essential parameter, the value volatility σ. If σ=0 the value stays fixed, whereas if σ is massive, the value jumps round wildly. When you’re considering of returns as a random stroll, σ could be loosely interpreted as the everyday size of a step.

LVR could be characterised on this mannequin exactly. As a result of LVR accumulates trade-by-trade, and since it is a continuous-time mannequin with trades taking place on a regular basis, LVR accumulates as an integral of the instantaneous LVR. Instantaneous LVR seems to scale quadratically with σ and the present market worth, and linearly with the marginal liquidity of the AMM at that worth.

This mathematical characterization could sound barely intimidating, however most of the frequent AMMs are so easy that LVR is given by an elementary closed-form components. 

For instance, with a CPMM, the instantaneous LVR, when normalized by the CPMM’s market worth, seems to be precisely σ²/8. Plugging in numbers, if a Uniswap v2 ETH-USDC pool has a every day volatility of 5%, then in keeping with our mannequin LPs lose 3.125 bps to LVR daily (for a roughly 11% loss yearly). Will payment income compensate for this loss? The reply depends upon the buying and selling charges and buying and selling quantity. For instance, if this AMM fees a set 30 bps buying and selling payment, then LPs will break even offered the every day quantity is roughly 10.4% of the AMM’s belongings. Had the every day volatility been 10%, the required quantity would have been 4 occasions as excessive. (Do not forget that LVR scales quadratically with σ.)

Implications for AMM designers

LVR is essential not just for potential liquidity suppliers but in addition for AMM designers. An AMM could be profitable provided that it has comfortable LPs, which implies that payment revenues must scale with LVR. 

One implication of our work is that, as a result of LVR depends upon volatility and payment income on buying and selling quantity, AMMs ought to think about dynamic charges that modify with quantity, volatility, or empirically noticed LVR. A second is that AMM designers ought to examine strategies for minimizing LVR (and therefore the LP incentives required), for instance by incorporating a high-quality pricing oracle to cite closer-to-market costs. Subsequent-generation AMMs are already exploring these and associated concepts, and we will’t wait to see the way it performs out.


For deeper technical evaluation and dialogue of LVR, please see our unique paper, “Automated Market Making and Loss-Versus-Rebalancing.” And right here is Tim Roughgarden giving a chat on LVR at SBC’22.


Jason Milionis is a Ph.D. pupil within the Pc Science Division at Columbia College, the place he’s suggested by Christos Papadimitriou and Tim Roughgarden. He’s broadly serious about Recreation Principle, particularly along with Machine Studying, and Decentralized Finance (DeFi).

Ciamac Moallemi is the William von Mueffling Professor of Enterprise within the Resolution, Threat, and Operations Division of the Graduate Faculty of Enterprise at Columbia College.

Tim Roughgarden is a Professor of Pc Science and a member of the Knowledge Science Institute at Columbia College, and Head of Analysis at a16z crypto.

Anthony Lee Zhang is an assistant professor of finance on the College of Chicago Sales space Faculty of Enterprise. His analysis covers matters similar to banking and monetary intermediation, family finance, cash markets, housing markets, and crypto/DeFi. 

Editor: Tim Sullivan @tim_org 


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