Impermanent loss vs. loss-versus-rebalancing (LVR)

1. TL;DR

Impermanent loss (IL) compares an LP position with holding the same tokens and, for $x\cdot y = k$, equals $2\sqrt{r}/(1+r) - 1$ where $r$ is the exit-to-entry price ratio. Loss-versus-rebalancing (LVR) instead compares the pool with a trader who holds the same inventory but trades at market prices; Milionis, Moallemi, Roughgarden and Zhang show LVR is exactly what arbitrageurs extract, accrues at rate $\sigma^2 V/8$ for a constant-product pool, and is the only part of LP returns that is not market exposure. Fees and discrete blocks shrink what arbitrageurs capture, which is why faster chains and dynamic fees are the main mitigations.

2. Explain it simply

Analogy

An airport currency booth updates the rate on its board only when a customer walks up, while the real exchange rate moves every second. Whenever the rate jumps, the first person to notice buys from the booth at the stale rate and sells elsewhere; the booth loses a little every time. Comparing the booth’s drawer at year end with a drawer that never traded is impermanent loss. Comparing it with a booth that made the same trades but always at the live rate is loss-versus-rebalancing—the cost of being late, paid to whoever noticed first.

ELI5

When you put two coins into a swap box, the box sells whichever coin is getting more valuable and buys whichever is getting cheaper, always a little late. So compared with keeping the coins in your pocket, you end up with less whenever prices move a lot in either direction. People call that “impermanent loss” because it shrinks if prices come back. A sharper way to measure the damage is to ask how much the box lost by trading late, compared with a friend who bought and sold the same amounts but at the real market price. That number only grows, never shrinks, and the trading fees the box collects are what pays for it.

Step-by-step walkthrough

Scenario: an ETH/USDC pool with 1,000 ETH and 2,000,000 USDC ($k = 2\times10^{9}$); an LP owns 1% of it. ETH rises from 2,000 to 3,000 on external markets.

1. Before: LP share = 10 ETH + 20,000 USDC = $40,000 at 2,000.
2. Arbitrage: traders buy ETH from the pool until its spot equals 3,000: reserves become $\sqrt{k/3000} = 816.5$ ETH and $\sqrt{k\cdot 3000} = 2{,}449{,}490$ USDC.
3. After: LP share = 8.165 ETH + 24,494.9 USDC = $48,990 at 3,000. Holding would be $10 \times 3{,}000 + 20{,}000 = 50{,}000$. IL =$-1{,}010 = -2.02%$.
4. LVR view: the pool sold 1.835 ETH (the LP’s share) on the way up at prices between 2,000 and 3,000, always slightly below market. A rebalancing trader who sold the same 1.835 ETH at market prices would hold $50,000 − $48,990 more: the $1,010 gap is the cumulative LVR on this path, captured by arbitrageurs.
5. Fees: if the LP’s share of fees over the period exceeds $1,010, the LP beats the rebalancing benchmark; hedged P&L = fees − LVR.

Common misconceptions

• Myth: impermanent loss is “impermanent” because it goes away. Reality: it goes to zero only if the price returns exactly to the entry price; the arbitrage losses along the way are permanent and are what LVR measures.
• Myth: IL is a loss of money. Reality: IL is an opportunity cost versus holding; an LP can have positive absolute returns and large IL at the same time (Milionis et al. 2022: 99.991% of Uniswap v2 ETH-USDC P&L variance is just ETH price exposure).
• Myth: LVR is a new name for IL. Reality: IL depends on an arbitrary start date and does not add up over sub-periods; LVR is additive, non-decreasing and equals arbitrageur profits (Milionis et al. 2022, §1).
• Myth: high fees always protect LPs. Reality: in an all-arbitrage model Robinson, White, Noyes and Tassy find the fee should be “as low as possible without being zero”; fees matter through which flow they attract, not just their rate.

If you only remember one thing

Hedge out the market exposure and an LP’s return is fees minus LVR, where LVR for $x\cdot y = k$ accrues at $\sigma^2/8$ of pool value per unit time.

3. How it works

Impermanent loss for x·y=k

Let the LP enter at price $P_0$ with reserves $x_0 = \sqrt{k/P_0}$, $y_0 = \sqrt{k P_0}$, and let the price move to $P_1 = r P_0$. After arbitrage the pool holds $x_1 = \sqrt{k/P_1}$, $y_1 = \sqrt{k P_1}$, so its value in Y is $V_{\text{pool}} = 2\sqrt{k P_1} = 2\sqrt{k P_0}\sqrt{r}$. Holding the original tokens would be worth $V_{\text{hold}} = x_0 P_1 + y_0 = \sqrt{k P_0}\,(r + 1)$. Hence

$\text{IL}(r) = \frac{V_{\text{pool}}}{V_{\text{hold}}} - 1 = \frac{2\sqrt{r}}{1 + r} - 1 \le 0.$

$r = P_1/P_0$	0.5	1.25	1.5	2	3	4	5
IL	−5.72%	−0.62%	−2.02%	−5.72%	−13.40%	−20.00%	−25.46%

IL is the concavity of the pool value $2\sqrt{kP}$ (see CFMM math). Milionis et al. list its defects as a metric: “It does not aggregate cleanly over time: it can be positive on two short time intervals, but zero on the two intervals combined. Its value, and even its sign, can be manipulated through the analyst’s arbitrary choice of start point.”

The LVR decomposition

Model the risky asset’s price as geometric Brownian motion $dP_t = \sigma_t P_t\, dW_t$ (the paper allows stochastic volatility), a CFMM with invariant $f(x, y) = L$ whose risky holdings at market price $P$ are $x^*(P)$, and arbitrageurs who pay no fees so the pool price always equals the market price. Define the rebalancing strategy as the self-financing strategy that holds exactly $x^*(P_t)$ of the risky asset at all times but trades at market prices; its value is $R_t$. Then (Milionis et al. 2022, eq. 6–8):

$$V_t - V_0 = (R_t - V_0) - \text{LVR}_t, \qquad \text{LVR}_t = \int_0^t \ell(\sigma_s, P_s), ds, \qquad \ell(\sigma, P) = \frac{\sigma^2 P^2}{2},\big|x^{*\prime}(P)\big|.$$

Here $|x^{*\prime}(P)|$ is the pool’s marginal liquidity: how much of the risky asset it sells per unit price rise. Three interpretations coincide: LVR is (1) arbitrageurs’ cumulative profit, (2) the pool’s shortfall versus rebalancing, and (3) the option value the LP gives up by committing to a demand curve. Because the rebalancing strategy is pure market exposure (a martingale under the risk-neutral measure), hedging it away leaves fees minus LVR.

The pool's price is stale between blocks; each time an arbitrageur re-aligns it, the gap between the stale price and the market price is paid out of LP reserves. Summed over time, those gaps are LVR.

Constant product: LVR = σ²V/8

For $x\cdot y = k$ with $L = \sqrt{k}$: $x^*(P) = L/\sqrt{P}$, so $x^{*\prime}(P) = -L/(2P^{3/2})$ and the pool value is $V(P) = 2L\sqrt{P}$. Substituting,

$\ell = \frac{\sigma^2 P^2}{2}\cdot\frac{L}{2P^{3/2}} = \frac{\sigma^2 L\sqrt{P}}{4} = \frac{\sigma^2}{8}\,V(P).$

With daily volatility $\sigma = 5\%$ this is $0.0025/8 = 3.125$ basis points of pool value per day (the ethresear.ch LVR-minimisation post uses the same figure), about 11.4% per year. Angeris et al. (2019, Appendix C) found the same $e^{-\sigma^2 T/8}$ factor in expectation; the LVR result is path-by-path and holds for any CFMM (concentrated positions have larger $|x^{*\prime}|$ in range, hence larger LVR per dollar).

Fees and discrete blocks

Arbitrageurs do pay fees and can only act when a block is produced. Milionis, Moallemi and Roughgarden (2023) add a fee $\gamma$ and Poisson block arrivals at rate $\lambda$ (mean block time $\Delta t = 1/\lambda$). The log-mispricing between pool and market then lives inside a no-trade band $[-\gamma, +\gamma]$ except at block times, when an arbitrageur jumps it back to the band edge. Its stationary distribution gives the fraction of blocks with a profitable arbitrage:

$P_{\text{trade}} = \frac{1}{1 + \sqrt{2\lambda}\,\gamma/\sigma},$

and in the fast-block limit (Theorem 3–4)

$$\text{ARB} \approx \text{LVR}\times P_{\text{trade}}, \qquad \text{FEE} \approx \text{LVR}\times(1 - P_{\text{trade}}), \qquad \text{ARB} + \text{FEE} \approx \text{LVR}.$$

So LVR is split between arbitrageurs and fee-paying arbitrage flow according to $P_{\text{trade}}$. With $\gamma > 0$, arbitrage profits scale as $\sqrt{\Delta t}\,\sigma^3/\gamma$: halving the block time cuts arbitrageur profits by about 29%, and “faster blockchains will result in reduced LP losses.” Adding a fixed gas cost $\delta$ widens the band to $\gamma + \delta$; in the fast-block limit arbitrageurs’ profits are competed away to validators, so “LPs still lose a constant amount of money, but this is taken by validators rather than arbitrageurs.” Elsts (2024) argues that burned base fees break the clean split, so $\sqrt{\Delta t}$ over-predicts the benefit on mainnet; see the LP toxicity and JIT page.

4. Worked numeric example

Pool: 1,000 ETH / 2,000,000 USDC, $k = 2\times10^{9}$, value $V = 4{,}000{,}000$ USDC at 2,000.

IL for a move to 3,000 ($r = 1.5$): pool value $2\sqrt{2\times10^{9}\times 3000} = 4{,}898{,}979$; hold value $1000 \times 3000 + 2{,}000{,}000 = 5{,}000{,}000$; IL $= -101{,}021 = -2.02\%$, matching $2\sqrt{1.5}/2.5 - 1$.

LVR per day with $\sigma = 5\%$ daily: $\ell = 0.0025/8 \times 4{,}000{,}000 = 1{,}250$ USDC per day, regardless of direction. Break-even fee volume at 30 bp: $1{,}250 / 0.003 \approx 416{,}700$ USDC of fee-paying volume per day, about 10% of pool value.

Split between arbitrageurs and fees (model estimate using $P_{\text{trade}}$; $\sigma$ converted to per-second units, $0.05/\sqrt{86{,}400}$):

Chain / fee	$\sqrt{2\lambda}\,\gamma/\sigma$	$P_{\text{trade}}$	Arbitrageur share of $1,250/day	Paid back as fees
Ethereum, 12 s, 30 bp	7.2	12%	≈ $150	≈ $1,100
Ethereum, 12 s, 5 bp	1.2	45%	≈ $570	≈ $680
Solana, 0.4 s, 30 bp	39	2.5%	≈ $31	≈ $1,220
Solana, 0.4 s, 5 bp	6.6	13%	≈ $165	≈ $1,085

These are model outputs at the asymptotic approximation, not measurements; they ignore gas, jumps and non-Poisson blocks. The qualitative lesson: on a 12-second chain a 5 bp pool hands almost half its LVR to arbitrageurs; on a sub-second chain most of it comes back as fees.

5. Where it’s used

Ethereum

• Uniswap v2 ETH-USDC — the empirical testbed of the LVR paper: daily P&L swings above $10M are common but 99.991% of variance is beta; subtracting a 1-minute-to-1-hour rebalancing strategy cuts variance by four orders of magnitude (as of the paper’s 2026-05 version). arXiv
• Uniswap v4 hooks — “LVR-minimization in Uniswap V4” (ethresear.ch, 2023-06) sketches a Diamond-style hook that discounts each block’s first swap; simulations on a $300M pool at 5% daily volatility show a 1.058 relative return over 180 days at full retention. See v4 hooks and am-AMM.
• am-AMM — Adams, Moallemi, Reynolds and Robinson (2024) auction the right to set fees and capture arbitrage on a constant-product pool, returning rent to LPs.
• CoW AMM (FM-AMM) — batch execution intended to eliminate LVR; an ethresear.ch analysis (2024-07) finds the benefit depends on volatility, transaction costs and pool size. See batch auctions.

Solana

• Meteora DLMM — charges a variable fee that rises with a volatility accumulator based on bin crossings, a direct on-chain response to arbitrage-heavy flow; base fee is tied to bin step (docs, accessed 2026-08). docs
• Lifinity — an oracle-based AMM (launched 2022-01) that re-centres its curve on an oracle price instead of waiting for arbitrage, explicitly to avoid selling stale quotes (Helius, 2025-08).
• Proprietary AMMs (SolFi, HumidiFi, ZeroFi, Tessera V) — refresh quotes several times per second via 143-CU oracle updates, a cheap “cancel priority” against toxic takers; more than 60% of SOL/USDC volume (as of 2025-07). Helius

6. Risks, attacks, and incidents

• Underestimating LVR in concentrated positions. Uniswap’s v3 example position (1,000–2,250 DAI/ETH) uses 5.44× less capital for the same fees, and suffers about 5.5× the IL per dollar for the same move (−5.6% vs −1.0% from 1,500 to 2,000; see concentrated liquidity).
• Toxic flow and JIT liquidity. Because LVR is paid to whoever arbitrages first, Ethereum block builders internalise it: “the first, guaranteed cost that a DEX must pay each block is that of arbitraging the DEX’s stale reserves” (ethresear.ch, 2023-06).
• Measurement disputes. The ethresear.ch FM-AMM analysis (2024-07) records a dispute over markout as a metric for claimed LVR reduction; unhedged loss claims are unreliable (Milionis et al. 2022 estimate ~11,664× more data needed).
• No exploits. IL and LVR are structural costs, not attack vectors; no hack in the sources is attributed to them.

7. Open problems

• What to measure. Milionis et al. note that Augustin et al. (2024) measure IL from a fixed start while Aquilina et al. (2024) and Fang (2024) sum it over short intervals, with “no theoretical guidance on which measurement choice is correct”; LVR is the interval-sum limit.
• Do faster blocks help LPs net of gas? The fees paper says yes in the model; Elsts (2024) argues burned base fees and fixed costs break the zero-sum assumption and that “other factors are more important, including the transaction fees.”
• Who should capture LVR? McAMM (2022), LVR-minimisation hooks (2023), am-AMM (2024) and FM-AMM (2024) redirect it to LPs or auction it; the ethresear.ch thread asks whether a pool retaining $\beta$ of LVR can deploy more than $1-\beta$ of its liquidity.
• Fee level. Alchemy (2020) says minimise the fee; the fees paper says low fees raise arbitrage profits; both omit price-sensitive retail flow, which am-AMM tries to endogenise.

8. Ethereum vs Solana

Aspect	Ethereum	Solana
Mean block time in the LVR model	~12 s	~0.4 s slots
Arbitrageur share of LVR at 30 bp fee (model)	≈ 12%	≈ 2.5%
Arbitrageur share at 5 bp fee (model)	≈ 45%	≈ 13%
Who arbitrages first	searcher–builder pipeline with public mempool	leader-ordered flow, Jito bundles, oracle-updated market makers
Dominant mitigation in practice	hooks, batch auctions, am-AMM research	short slots plus proprietary oracle-quoted AMMs

The LVR rate $\sigma^2 V/8$ is chain-independent; its division is not. On Ethereum, 12-second blocks and a public mempool let a large share of LVR leave as arbitrage profit split among searchers, builders and burned base fees, which is why Ethereum research focuses on clawing it back with hooks or auctions. On Solana the same volatility produces many small mispricings that fee-paying arbitrage mostly returns to LPs, and market makers who refresh quotes for 143 compute units avoid quoting stale prices at all; passive LPs there are outcompeted rather than arbitraged.

9. Reference doc

The reference

Automated Market Making and Loss-Versus-Rebalancing — Jason Milionis, Ciamac C. Moallemi, Tim Roughgarden, Anthony Lee Zhang; initial version 2022-07-31, current version 2026-05-03 (arXiv v6). https://arxiv.org/abs/2208.06046

Summary of the reference

§1 Introduction frames AMMs as both trading venues and investment vehicles and asks how LP returns decompose. The answer mirrors asset pricing: a “beta-like” component—the rebalancing strategy that holds the AMM’s risky inventory but trades at CEX prices—and an “alpha-like” residual equal to fees from noise traders minus adverse-selection losses to arbitrageurs. The introduction previews the empirical result (99.991% of Uniswap v2 ETH-USDC P&L variance is beta; subtracting a 1-minute-to-1-hour rebalancing strategy cuts variance by four orders of magnitude) and criticises impermanent loss for not aggregating over time and depending on an arbitrary start date. §1.1 surveys microstructure models (Aoyagi, Capponi–Jia, Lehar–Parlour, Hasbrouck et al.) and empirical papers that measure “impermanent loss” inconsistently. §2 Model sets up a risky asset with GBM price (possibly stochastic volatility), an infinitely deep CEX, fee-paying noise traders, and fee-free arbitrageurs who keep the CFMM on its level curve $f(x,y) = L$ at the market price; the pool’s value function $V(P)$ and holdings $x^*(P)$ come from the reserve curve. §3 Rebalancing and LVR defines the rebalancing strategy $R_t$ and proves Lemma 2: $V_t - V_0 = (R_t - V_0) - \text{LVR}_t$ with $\text{LVR}_t = \int_0^t \frac{\sigma_s^2 P_s^2}{2}|x^{*\prime}(P_s)|\,ds$, non-negative, non-decreasing and predictable, and equal to arbitrageurs’ cumulative profit (Figure 2 gives the slippage picture). Theorem 1 then states the beta/alpha decomposition and Corollary 1 shows the rebalancing strategy is the quadratic-variation-minimising projection of LP returns onto price moves, so LVR is the projection residual. For constant product pools this yields $\text{LVR} = \sigma^2 V/8$. §4 Empirics implements the decomposition on Uniswap v2 ETH-USDC using only holdings, mints, burns and a price series, at rebalancing frequencies from one minute to months. §5 shows why one-period models cannot see market risk and why summing IL over short intervals converges to LVR. §6 connects LP positions to options (Clark 2020, Fukasawa et al. 2022, Deng et al. 2023), delta hedging, and a liquidity-weighted variance swap. The appendix shows results hold under both physical and risk-neutral measures.

Key quotes

• “over 99.991% of LP return variance is driven by beta exposure to market risk.” (Abstract)
• “Impermanent loss has a number of undesirable features as a loss metric. It does not aggregate cleanly over time: it can be positive on two short time intervals, but zero on the two intervals combined.” (§1)
• “the CFMM systematically loses money relative to the rebalancing strategy due to price slippage: every trade made by the CFMM is made at slightly worse prices than the rebalancing strategy.” (§3, after Lemma 2)
• “LVR is large when volatility σt is high, so prices move more; and when |x∗′(Pt)| is large, so the CFMM trades more of the risky asset when prices move” (§3)
• “Our approach fixes these issues simply by frequently updating the holdings benchmark” (§1)

How to read the original

Background: Itô calculus at the level of $d(\sqrt{P})$, the idea of a self-financing strategy, and the CFMM value function $V(P)$ from Angeris and Chitra. Read the abstract, §1, Figure 2 and Lemma 2 first; that is the whole idea. Skip §1.1, the measure-theoretic footnotes and §6 on a first pass. The hardest paragraph is the proof sketch of Lemma 2: apply Itô’s lemma to $V(P_t)$, note the first-order term is exactly the rebalancing strategy’s gain $x^*(P_t)\,dP_t$, and observe that the second-order term $\frac{1}{2}V''(P)\sigma^2P^2\,dt$ is negative because $V$ is concave—that negative drift, with $V'' = x^{*\prime}$, is LVR.

What changed since

• 2023-02 / 2025-07: Milionis, Moallemi and Roughgarden add fees and Poisson blocks; ARB ≈ LVR × $P_{\text{trade}}$, arbitrage profits ∝ $\sqrt{\Delta t}$, gas leaks losses to validators.
• 2023-06: LVR-minimisation hook designs for Uniswap v4 (ethresear.ch).
• 2024: am-AMM (Adams, Moallemi, Reynolds, Robinson) and FM-AMM/CoW AMM go from paper to deployment; Fritsch and Canidio (2024) empirically confirm the block-time relationship; Elsts (2024-05) adds base-fee friction.
• 2025: Nezlobin and Tassy extend to general block-time distributions; Hasbrouck et al. model equilibrium concentrated liquidity; the paper’s v6 (2026-05) reframes results around beta/alpha decomposition.
• On Solana, the practical response has been proactive quoting (Lifinity 2022, proprietary AMMs 2024–2025) rather than LVR capture mechanisms.

Secondary references

1. Milionis, Moallemi, Roughgarden, Automated Market Making and Arbitrage Profits in the Presence of Fees (2023/2025) — read if you want the fee and block-time corrections.
2. Robinson, White, Noyes, Tassy, Uniswap’s Financial Alchemy (2020-12) — read if you want the growth-rate view and why nonzero fees can beat holding.
3. Elsts, CEX/DEX arbitrage, transaction fees, block times, and LP profits (ethresear.ch, 2024-05) — read if you want the mainnet-specific critique with gas.
4. The-CTra1n, LVR-minimization in Uniswap V4 (ethresear.ch, 2023-06) — read if you want a concrete hook design and simulation numbers.
5. Angeris et al., An analysis of Uniswap markets (2019), Appendix C — read if you want the earlier $e^{-\sigma^2T/8}$ result.

10. Sources

• Automated Market Making and Loss-Versus-Rebalancing — Milionis, Moallemi, Roughgarden, Zhang — 2022-08-11 (v6 2026-05-03) — https://arxiv.org/abs/2208.06046
• Automated Market Making and Arbitrage Profits in the Presence of Fees — Milionis, Moallemi, Roughgarden — 2023-05-24 (v2 2025-07-23) — https://arxiv.org/abs/2305.14604
• Uniswap’s Financial Alchemy — Robinson, White, Noyes, Tassy — 2020-12-01 — https://www.paradigm.xyz/writing/uniswaps-alchemy
• An analysis of Uniswap markets — Angeris, Kao, Chiang, Noyes, Chitra — 2019-11-08 — https://arxiv.org/abs/1911.03380
• LVR-minimization in Uniswap V4 — The-CTra1n — 2023-06-16 — https://ethresear.ch/t/lvr-minimization-in-uniswap-v4/15900
• CEX/DEX arbitrage, transaction fees, block times, and LP profits — atiselsts — 2024-05-02 — https://ethresear.ch/t/cex-dex-arbitrage-transaction-fees-block-times-and-lp-profits/19444
• Notes on the LVR of FM-AMM — kosunghun317 — 2024-07-26 — https://ethresear.ch/t/notes-on-the-lvr-of-fm-amm/20151
• MEV capturing AMM (McAMM) — josojo — 2022-08-10 — https://ethresear.ch/t/mev-capturing-amm-mcamm/13336
• Tech to Make Impermanent Loss Impermanent Again — Planck — 2025-04-03 — https://ethresear.ch/t/tech-to-make-impermanent-loss-impermanent-again/22082
• am-AMM: An Auction-Managed Automated Market Maker — Adams, Moallemi, Reynolds, Robinson — 2024-03-05 — https://arxiv.org/abs/2403.03367
• Introducing Uniswap v3 — Uniswap Labs — 2021-03-23 — https://blog.uniswap.org/uniswap-v3
• Solana’s Proprietary AMM Revolution — Helius — 2025-08 — https://www.helius.dev/blog/solanas-proprietary-amm-revolution
• What is Raydium — Raydium docs — accessed 2026-08-29 — https://docs.raydium.io/introduction/what-is-raydium
• DLMM formulas — Meteora docs — accessed 2026-08-29 — https://docs.meteora.ag/core-products/dlmm/formulas.md