Impermanent Loss in AMMs

• Impermanent loss (IL) is the relative underperformance of an AMM liquidity position compared to simply holding assets, driven by reserve rebalancing.
• It arises from the geometric-arithmetic gap in payoffs as the constant-product mechanism adjusts asset balances during price fluctuations.
• Mitigation strategies such as fee adjustments, hedging with options, and focused liquidity designs help manage IL risks.

Impermanent loss (IL) is the benchmark-relative underperformance of an automated market maker (AMM) liquidity position versus simply holding the initially deposited assets outside the pool. In the constant-product setting, IL arises because arbitrage and user trading continuously rebalance reserves along the AMM curve: the liquidity provider (LP) sells part of the appreciating asset and accumulates more of the depreciating asset. Across the literature, the core economic object is stable, but the sign convention, normalization, and benchmark can differ: some papers write IL as a non-positive payoff difference, some as a nonnegative loss magnitude, and some distinguish between a terminal hold-value normalization and an initial-capital normalization (Aigner et al., 2021, Tiruviluamala et al., 2022).

1. Definition, benchmark, and normalization

A general definition for a fixed AMM surface compares the final value of the pool inventory with the value of passively holding the initial inventory. In one formalization,

$$IL=\frac{p^f\cdot x^f-p^f\cdot x^i}{p^f\cdot x^i} =\frac{p^f\cdot x^f}{p^f\cdot x^i}-1,$$

where $x^i$ and $x^f$ are the initial and final stable states and $p^f$ is the final external price vector. This makes IL explicitly a relative underperformance versus HODL, evaluated at final prices (Tiruviluamala et al., 2022).

In the two-asset constant-product literature, the most common scalar formula is

$$IL(r)=\frac{2\sqrt{r}}{1+r}-1,\qquad r=\frac{p_m^T}{p_m^1},$$

which uses the final value of the hold portfolio in the denominator. A different normalization, argued to be more intuitive from an investor P&L perspective, measures underperformance relative to initial capital and yields

$\sqrt{R}-\frac{R+1}{2},\qquad R=\frac{P_1}{P_0}.$

The distinction matters because the first expression is symmetric in $R$ and $1/R$, whereas the second is not; the latter keeps the denominator fixed at initial capital (Aigner et al., 2021, Hafner et al., 2024).

The sign convention is not universal. Some papers define IL as a shortfall $\text{HODL} - \text{pool}$, while others define

$IL = V_{\text{pool}}-V_{\text{hold}},$

so IL is typically non-positive and becomes zero only at the entry price (Bergault et al., 8 Sep 2025, Gonzalez et al., 27 Mar 2025). This convention difference is substantive when comparing reported magnitudes, but not when comparing economic meaning.

A further distinction concerns realization. In one interpretation, IL is more accurately an unrealized loss: if price leaves the entry level and later returns, the reserve composition implied by the initial price is restored and IL vanishes. It becomes permanent when liquidity is withdrawn after the adverse move, thereby crystallizing the underperformance relative to holding (Aigner et al., 2021).

2. Constant-product mechanics and the source of convexity

For Uniswap v2-style constant-product AMMs, the reserve law is

$x^i$0

with pool price imposed by the reserve ratio,

$x^i$1

Solving gives

$x^i$2

Once external price changes and arbitrage realigns the AMM, the LP’s inventory is mechanically altered according to these reserve equations (Aigner et al., 2021).

This reserve transformation is the core mechanism of IL. The LP “is buying the asset that is dropping in value and selling the asset that is rising in value,” so the LP underperforms a buy-and-hold portfolio in both directions away from the initial price. In valuation terms, the pool value in a 50/50 constant-product pool is

$x^i$3

whereas the HODL benchmark is linear in terminal price,

$x^i$4

The gap between the geometric-mean payoff and the arithmetic-mean payoff is exactly the IL curve (Gonzalez et al., 27 Mar 2025).

The resulting exposure is nonlinear. Several papers characterize the LP as short volatility, short convexity, short gamma, or synthetically short variance while earning theta-like fee income (Aigner et al., 2021, Bardoscia et al., 2023). In one option-theoretic formulation, impermanent loss is the Gamma component of the associated self-financing strategy, and the LP’s value function has strictly negative second derivative in price (Loesch et al., 2021, Bardoscia et al., 2023).

The same idea extends beyond the two-asset 50/50 case. For an $x^i$5-asset constant-product pool, IL can be written as a geometric-mean to arithmetic-mean ratio,

$x^i$6

with $x^i$7 the relative exchange-rate moves. By the arithmetic mean-geometric mean inequality, this is always non-positive. More generally, the literature describes IL as a curvature effect: the final AMM value function lies below its tangent-plane approximation at the initial state (Tiruviluamala et al., 2022).

3. General CFMMs, weighted pools, and concentrated liquidity

A broad formulation replaces the constant-product law with a general CFMM

$x^i$8

with internal marginal prices

$x^i$9

In that setting, IL is still the difference between the value of remaining in the pool and the value of holding the same assets outside the pool, but the mapping from external prices to final reserves depends on the geometry of $x^f$0, the number of assets, the weights, and whether liquidity is concentrated (Tangri et al., 2023).

One general result is that all AMMs in the framework are price-level independent: IL depends on exchange rates rather than absolute price levels. A stronger property, exchange rate level independence, holds for geometric mean market makers (G3Ms), which is why G3Ms are described as “the simplest class of AMMs from an impermanent loss viewpoint.” Curve-like invariants generally do not share this property, so IL there requires more parameters than a single relative price move (Tiruviluamala et al., 2022).

Uniswap v3 changes the problem by restricting liquidity to a finite interval

$x^f$1

The shifted reserve system is described by

$x^f$2

with

$x^f$3

At the lower bound the LP is entirely in one asset; at the upper bound entirely in the other (Aigner et al., 2021). This makes the v3 payoff piecewise: inside the range the position has CFMM curvature, below the lower bound it becomes linear in the risky asset, and above the upper bound it becomes flat in that asset’s upside direction (Khakhar et al., 2022, Loesch et al., 2021).

The economic consequence is leverage. Narrower ranges increase fee efficiency but steepen the local IL profile. For a $x^f$4 move, one reported comparison gives about $x^f$5 and $x^f$6 for v2, versus $x^f$7 and $x^f$8 for a much narrower v3 range of $x^f$9 (Aigner et al., 2021). This is why concentrated liquidity is repeatedly described as more capital efficient but also more exposed to IL (Loesch et al., 2021).

4. Fees, arbitrage, dynamics, and the relation to LVR

In static no-fee analysis, any permanent deviation of the relative price away from its initial level implies IL under the standard constant-product benchmark, with equality only at the initial price ratio (Hafner et al., 2024). This is the familiar textbook result. Dynamic analyses qualify it by emphasizing that realized LP outcomes combine two forces: rebalancing losses and fee gains (Hafner et al., 2024).

A central simulation result is that once fee accrual is modeled explicitly, price changes do not necessarily lead to net losses. In a calibrated Uniswap v2 WETH/USDC setting with a 0.3% protocol fee, LPs reportedly outperform or at least do not underperform holding as long as market prices do not drop by more than $p^f$0 or increase by more than $p^f$1 within a year (Hafner et al., 2024). In that framework, arbitrage is not purely extractive: more competitive, lower-friction arbitrage increases fee-paying flow without increasing the terminal rebalancing loss implied by the final price ratio, so an arbitrage-friendly environment benefits LPs (Hafner et al., 2024).

Dynamic control formulations go further by embedding IL in an optimal stopping problem. In one such model, IL is

$p^f$2

and the LP chooses an exit time $p^f$3 to maximize

$p^f$4

The stopping rule is state dependent: the LP tends to remain when the external price and the AMM price are close, and exits when the mispricing becomes too large or too little time remains to earn offsetting fees (Bergault et al., 8 Sep 2025).

Another important line of work compares IL with loss-versus-rebalancing (LVR). In one Brownian framework, IL and LVR have identical expectation values but very different distribution functions: IL is endpoint-based, whereas LVR is path-dependent and accumulates over every rebalancing step (Alexander et al., 2024). A later treatment refines this into three regimes: at very short times IL and LVR are identical; at intermediate times they have distinct distributions but the same expectation; at long times both distribution functions and averages diverge (Alexander et al., 6 Feb 2025). This suggests that IL is the natural withdrawal-time benchmark, while LVR is a sharper measure of dynamic adverse-selection loss.

5. Hedging, replication, and design responses

Because the LP payoff is concave, several papers show that it admits static option replication. For a constant-product pool,

$p^f$5

and the pool value can be represented by the Carr–Madan/Breeden–Litzenberger static spanning formula as a combination of a bond, linear exposure, and a continuum of short calls and puts across strikes. This makes the “short volatility” interpretation exact rather than heuristic (Gonzalez et al., 27 Mar 2025).

The same logic yields direct hedges for IL. One practical proposal overlays the LP position with a long strangle—long calls above spot and long puts below spot—so that option gains offset the two-sided loss of the LP relative to HODL over a chosen terminal-price interval (Gonzalez et al., 27 Mar 2025). For concentrated liquidity, static replication formulas exist with strike support restricted to the liquidity provision interval: right-side positions can be replicated with calls only, and left-side positions with puts only (Deng et al., 2022).

A related critique is that IL is not always the right hedge target. One paper argues that the relevant object for risk management is Liquidity Position PNL, defined as the change in the mark-to-market value of the LP position itself. In the uniform-liquidity case this price-driven return is

$p^f$6

whereas the usual IL metric compares that same position to a different strategy, namely HODL (Khakhar et al., 2022).

Design responses extend beyond exogenous hedging. In G3Ms with proportional transaction fees, one result shows that IL can be super-hedged pathwise by a model-free rebalancing strategy, while LVR vanishes in continuous time because the fee-induced no-arbitrage band makes the AMM exchange rate a finite-variation process (Fukasawa et al., 2023). Fee design itself is another mitigation route: block-adaptive, deal-adaptive, and oracle-based asymmetric fees reportedly outperform fixed-fee baselines in reducing IL while maintaining uninformed trading activity (Lebedeva et al., 3 Jun 2025). At the most structural end, path-independent fee rules can be chosen so that IL is exactly zero for a given initial CPMM state, although no universal fee function can eliminate IL for all initial states simultaneously (Voronin et al., 30 Apr 2026).

6. Empirical findings, strategy dependence, and limitations

Empirical work generally finds that fee income and IL must be evaluated jointly, and that outcomes are highly heterogeneous across pools, fee tiers, ranges, and holding periods. In one Uniswap v3 study covering 17 pools and 43% of TVL, total fees earned since inception until the cut-off date were $p^f$7m; in aggregate, those LPs would have been better off by $p^f$8m had they simply HODLd (Loesch et al., 2021).

A separate historical study of 700 days of Uniswap v3 data reports average realized IL per position of $p^f$9, with $$IL(r)=\frac{2\sqrt{r}}{1+r}-1,\qquad r=\frac{p_m^T}{p_m^1},$$0 of positions generating negative returns. Average total returns were $$IL(r)=\frac{2\sqrt{r}}{1+r}-1,\qquad r=\frac{p_m^T}{p_m^1},$$1 for stable-stable pools, $$IL(r)=\frac{2\sqrt{r}}{1+r}-1,\qquad r=\frac{p_m^T}{p_m^1},$$2 for stable-risky pools, and $$IL(r)=\frac{2\sqrt{r}}{1+r}-1,\qquad r=\frac{p_m^T}{p_m^1},$$3 for risky-risky pools. Narrow ranges had higher daily IL than wide ranges, and only range sizes wider than $$IL(r)=\frac{2\sqrt{r}}{1+r}-1,\qquad r=\frac{p_m^T}{p_m^1},$$4 produced slightly positive returns; durations longer than 360 days were the only clearly profitable bucket (Drossos et al., 14 Jan 2025).

Cross-sectional evidence for G3Ms is similarly mixed. One study concludes that the median liquidity pool had a net nil ROI once IL is taken into account, although cross-sectional dispersion was high and pool net ROI ranking was significantly autocorrelated for several weeks (Boueri, 2021). Another line of work reframes the problem in terms of profitability zones: no-arbitrage, impermanent gain (IG), and impermanent loss (IL). Under that view, a fee can be chosen to target a fixed probability of remaining in the IG zone within a block, so fee selection becomes a risk-control parameter rather than only a revenue parameter (Melnikov et al., 30 Apr 2026).

The literature also repeatedly states its own limits. Many analyses abstract from dynamic fee accrual paths, gas costs, active hedging, stochastic volume, smart-contract risk, and token failure scenarios (Aigner et al., 2021). Concentrated-liquidity backtests often exclude gas fees and use only closed positions, so realized IL estimates can omit both implementation costs and open-position losses (Drossos et al., 14 Jan 2025). This suggests that IL is best understood as one component of LP risk: a benchmark-relative rebalancing loss produced by the AMM invariant, partially offset by fee income, and modified by arbitrage intensity, range selection, and exit timing.