Impermanent loss

Reading depth

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BeginnerCore purpose and mental modelPractitionerAdds mechanisms and economicsEngineer or auditorAdds state and verification

Dependency check

Advanced detail assumes prerequisite knowledge

Before using practitioner mechanisms or engineer and auditor checks, review any unfamiliar dependency. These paths come from canonical depends_on relationships; Web3 Academy does not track completed lessons.

Review 11 dependencies

• AssetRequired through Asset versus claim

  1. Impermanent loss
  2. Arbitrage
  3. Decentralized exchange
  4. Asset versus claim
  5. Asset

• ClaimRequired through Asset versus claim

  1. Impermanent loss
  2. Arbitrage
  3. Decentralized exchange
  4. Asset versus claim
  5. Claim

• TokenRequired through Asset versus claim

  1. Impermanent loss
  2. Arbitrage
  3. Decentralized exchange
  4. Asset versus claim
  5. Token

• Asset versus claimRequired through Decentralized exchange

  1. Impermanent loss
  2. Arbitrage
  3. Decentralized exchange
  4. Asset versus claim

• Decentralized exchangeRequired through Arbitrage

  1. Impermanent loss
  2. Arbitrage
  3. Decentralized exchange

• Liquidity poolRequired through Automated market maker

  1. Impermanent loss
  2. Arbitrage
  3. Price impact
  4. Automated market maker
  5. Liquidity pool

• Liquidity providerDirect prerequisite

  1. Impermanent loss
  2. Liquidity provider

• SwapRequired through Price impact

  1. Impermanent loss
  2. Arbitrage
  3. Price impact
  4. Swap

• Automated market makerRequired through Price impact

  1. Impermanent loss
  2. Arbitrage
  3. Price impact
  4. Automated market maker

• Price impactRequired through Arbitrage

  1. Impermanent loss
  2. Arbitrage
  3. Price impact

• ArbitrageDirect prerequisite

  1. Impermanent loss
  2. Arbitrage

Impermanent loss is the value difference between an AMM liquidity position and a stated hold benchmark after relative asset prices change.

Category: TradingOpposed arrows category cue

System record

Start with the economic purpose, participants, resources, and entitlements before studying implementation detail.

Why it exists

The metric isolates how rule-based inventory rebalancing changes a liquidity provider's outcome compared with keeping the benchmark asset quantities outside the pool.

Traditional-finance analogy

Rebalancing shortfall against a buy-and-hold benchmark is the closest comparison recorded for this concept.

Where the analogy stops

• Impermanent loss is benchmark-relative rather than a complete profit-and-loss measure, and fees or incentives may offset or exceed it.
• The familiar closed-form expression assumes a full-range, two-asset, constant-product position with no fees; concentrated and other AMMs require different models.

Main actors

• ActorLiquidity provider
• ActorTrader
• ActorArbitrageur
• ActorPool and pricing mechanism

Assets and claims

Assets — controlled or transformed resources

Assets are resources the mechanism moves, holds, values, or transforms.

• AssetLiquidity-position reserve assets
• AssetBenchmark hold assets
• AssetAccrued fees or incentives

Claims — entitlements and corresponding dependencies

Claims are rights to value, repayment, redemption, control, or another party's performance; each depends on an obligation or system that must honor it.

• ClaimPool share or range-specific liquidity position

Impermanent loss compares an AMM liquidity position with a specific hold benchmark after the assets’ relative price changes.

Why it exists

An AMM automatically changes its inventory as traders move along the pricing rule. The LP therefore ends with different asset quantities than someone who held the starting quantities outside the pool. Impermanent loss isolates that benchmark difference; it is not total profit and loss.

Traditional-finance analogy

Rebalancing shortfall against buy-and-hold is a useful analogy. The measure is conditional on a benchmark, interval, valuation price, and AMM model. “Impermanent” does not mean unrealized, guaranteed to reverse, or harmless after withdrawal.

Full-range constant-product example

For an initially balanced, two-asset, full-range constant-product position with no fees, let:

r = ending external price / initial external price
impermanentLossFraction = 2 × sqrt(r) / (1 + r) - 1

r and the result are unitless. At r = 1, the difference is zero. For any other positive r, this expression is negative relative to holding the original quantities. The symmetry between r and 1/r depends on this selected model.

Do not apply the formula unchanged to concentrated ranges, unequal weights, multi-asset pools, stable-swap curves, dynamic fees, rebalancing strategies, or positions entered off the reference ratio.

Position balance-sheet and benchmark ledger

Table: Position balance-sheet and benchmark ledger

Quantity	LP position	Hold benchmark
Starting assets	Assets deposited under pool rules	Same declared token quantities held outside
Ending quantities	Determined by pool state and LP share or range	Starting quantities unchanged
Ending value	Current quantities × one reference-price set	Held quantities × the same prices
Fees and incentives	Add separately unless the comparison explicitly includes them	Include only comparable external income
Costs	Gas, rebalancing, hedging, financing	Comparable custody or transaction costs if selected

The metric is invalid when the two columns use different times, price sources, units, or fee treatments.

Mechanism and flows

1. An LP contributes the starting asset mix and receives a position claim.
2. Relative external prices change.
3. Traders and arbitrageurs alter pool reserves toward executable market prices.
4. The LP claim now represents a different inventory mix.
5. Value both the LP position and original hold quantities at the same prices.
6. Report their difference separately from fees, incentives, and costs.

Capital flow is the inventory traded into and out of the pool. Claim flow is the LP position. Return flow includes trader-paid fees and explicit incentives. Risk flow reaches the LP through inventory rebalancing, adverse selection, token loss, ranges, contract failure, and strategy costs.

Who pays and who loses

Impermanent loss is a benchmark difference, not an invoice from one named payer. Counterparties and arbitrageurs trade against the changing inventory; their economics, combined with external price movement, produce the LP’s reserve mix. The LP bears the difference unless fees or other income offset it. A positive total LP return can coexist with negative impermanent loss, and vice versa.

Assumptions and failure modes

The comparison assumes identical starting quantities, valuation time, price source, numeraire, fee treatment, and cost treatment for the LP position and the hold benchmark. The closed-form equation additionally assumes a balanced, full-range, two-asset constant-product pool with positive prices and no fees. Using mismatched inputs, a different AMM curve, concentrated liquidity, or an unreconstructed position makes the result inapplicable. Separately, token, oracle, pool-contract, or strategy failure can reduce the LP’s actual value; those losses are not impermanent loss and should be reported on their own.

Engineer or auditor lens

Require the benchmark quantities, entry state, exit state, valuation currency, price source, time interval, AMM model, fee treatment, ranges, and rounding. Test r = 1, reciprocal price changes, very small and large positive ratios, zero/invalid prices, and comparisons against independently reconstructed reserves. Keep impermanent loss, LVR, fees, incentives, and total P&L separate.

Uniswap v2’s returns documentation derives this formula for its simplified model. The research on loss-versus-rebalancing studies a related but distinct adverse-selection cost. Neither source justifies applying the formula to every AMM or claiming fees will offset the loss. The living Uniswap documentation was reviewed 2026-08-10; the research paper is a dated analytical artifact, not a current protocol-behavior source.

Common misunderstandings

• “Impermanent means it is not real.” Withdrawal can realize the position outcome, and reversal is not guaranteed.
• “Impermanent loss equals total LP loss.” It excludes or separately treats fees, incentives, gas, financing, and token-level gains or losses.
• “One formula works for every pool.” Curve, weights, ranges, entry state, fees, and management rules change the model.

Return to the liquidity-provider page to place this benchmark inside total LP economics.

Machine-readable model

Key equations

Canonical expressions come from the structured concept record. KaTeX renders the notation, while the plain-text expression and variable table keep its meaning and units inspectable without JavaScript. Read the narrative above for the model's domain, assumptions, and rounding rules.

1. Equation 1

   $$\mathrm{impermanentLossFraction} = \frac{2 \times \sqrt{\mathrm{priceRatio}}}{\left(1 + \mathrm{priceRatio}\right)} - 1$$

   Plain-text equation: impermanentLossFraction = 2 * sqrt(priceRatio) / (1 + priceRatio) - 1

   Variables and units for equation 1

   Symbol	Meaning	Unit
   impermanentLossFraction	Impermanent Loss Fraction	dimensionless fraction (1 = 100%)
   priceRatio	Price Ratio	dimensionless ratio

Assurance contract

Security properties

These structured statements define desired behavior. Their stable IDs can bind tests, invariants, specifications, audit findings, or proof results without turning descriptive review advice into an assurance claim.

Desired · not evaluated: No test, audit, or proof result is implied until scoped evidence is linked to this property.

1. SP-01Desired · not evaluated

   Calculations state the benchmark quantities, valuation currency, interval, price source, fee treatment, and AMM model

2. SP-02Desired · not evaluated

   The selected formula is used only inside its stated domain and assumptions

3. SP-03Desired · not evaluated

   Position value, hold value, fees, incentives, gas, and realized withdrawals remain separately inspectable

4. SP-04Desired · not evaluated

   User interfaces do not describe a benchmark-relative metric as a guaranteed temporary loss

Prerequisites

Direct dependencies recorded by this concept's canonicaldepends_on relationships.

• Liquidity providerPath 22

  A liquidity provider commits assets to a market-making system and receives a position whose value changes with inventory, fees, prices, and accounting rules.

• Automated market makerPath 24

  An automated market maker is a trading mechanism that accepts or rejects exchanges according to a programmed rule over market state and liquidity.

• ArbitragePath 27

  Arbitrage trades related markets when their executable prices differ enough to cover fees, gas, latency, financing, and execution risk.

Learn next

Concepts that directly depend on this one, ranked by the recommended learning path. Check each page for additional prerequisites.

No concept directly builds on this page in the current graph; use the guided learning paths to choose another branch.

Knowledge check

Quiz

Answer in your own words, then open the model answer.

What problem does Impermanent loss exist to address?

Model answer

The metric isolates how rule-based inventory rebalancing changes a liquidity provider's outcome compared with keeping the benchmark asset quantities outside the pool.

Mark as understood