# Impermanent loss: comparing pool inventory with simply holding Impermanent loss is the shortfall of an AMM liquidity position relative to holding its original assets, under a specified price comparison. In a fee-free, full-range, equal-value constant-product pool it is 2√r ÷ (1 + r) − 1, where r is the relative price change. Fees and other costs must be assessed separately. Evidence: [Understanding returns](https://developers.uniswap.org/docs/protocols/v2/concepts/understanding-returns); [Uniswap v2 pair source](https://raw.githubusercontent.com/Uniswap/v2-core/v1.0.1/contracts/UniswapV2Pair.sol) Canonical: https://degreesofsatoshi.com/encyclopedia/impermanent-loss/ Published: 2026-10-02 Substantively modified: 2026-10-02 Independently verified by an automated reviewer: 2026-10-02T15:08:18.373Z AI-assisted research and drafting with a separate automated source-verification pass; no external expert or named human review is implied. ## Key facts - **Benchmark:** The comparison is with holding the original quantities. ([Understanding returns](https://developers.uniswap.org/docs/protocols/v2/concepts/understanding-returns)) - **Model:** The simple formula assumes full-range equal-value constant-product liquidity. ([Understanding returns](https://developers.uniswap.org/docs/protocols/v2/concepts/understanding-returns)) - **Fees:** Swap fees can offset some shortfall but do not guarantee a profit. ([Understanding returns](https://developers.uniswap.org/docs/protocols/v2/concepts/understanding-returns); [Uniswap v2 pair source](https://raw.githubusercontent.com/Uniswap/v2-core/v1.0.1/contracts/UniswapV2Pair.sol)) ## Derive a concrete fourfold-price example Start with 1 A and 100 B when A costs 100 B. Ignore fees and assume arbitrage moves the pool to the external price. If A rises to 400 B, the product remains 100 and the new balances are 0.5 A and 200 B: their ratio is 400 B per A. The pool position is now worth 400 B. Holding the original assets would instead be worth 500 B: 1 A at 400 B plus 100 B. The pool underperforms holding by 100 ÷ 500 = 20%, even though its value increased from the initial 200 B. Evidence: [Uniswap v2 pair source](https://raw.githubusercontent.com/Uniswap/v2-core/v1.0.1/contracts/UniswapV2Pair.sol); [Understanding returns](https://developers.uniswap.org/docs/protocols/v2/concepts/understanding-returns) ## Relative loss is different from cash loss The comparison needs both a benchmark and a valuation asset. A position can rise in value while falling behind holding, as above. It can also decline in value while collecting fees. Calling all fees profit ignores the inventory that produced them. The word impermanent describes the possibility that the relative price returns. It does not promise that it will return or that waiting repairs a position. A withdrawal realizes the current inventory, and later market movements occur outside that pool position. Evidence: [Understanding returns](https://developers.uniswap.org/docs/protocols/v2/concepts/understanding-returns) ## Do not apply one formula to every position The expression assumes no fees, no deposits or withdrawals, continuous repricing to the outside market and a full-range 50:50 value setup. Real results also depend on gas, token behavior, fee income and the exact execution path. A concentrated range can become entirely one asset and needs a range-specific calculation. A leveraged position adds borrowing and liquidation exposure. State these boundaries whenever quoting a loss percentage so that a compact answer does not silently change the model. Evidence: [Introducing Uniswap v3](https://blog.uniswap.org/uniswap-v3); [Uniswap v2 pair source](https://raw.githubusercontent.com/Uniswap/v2-core/v1.0.1/contracts/UniswapV2Pair.sol) ## Questions ### Is impermanent loss the same as a hacked pool? No. It describes an inventory-versus-holding comparison under normal market making. Contract exploitation is a different source of loss and is not captured by the formula. Evidence: [Understanding returns](https://developers.uniswap.org/docs/protocols/v2/concepts/understanding-returns); [Uniswap v2 pair source](https://raw.githubusercontent.com/Uniswap/v2-core/v1.0.1/contracts/UniswapV2Pair.sol) ## Claims and scope ### impermanent-loss-quick-answer Impermanent loss is the shortfall of an AMM liquidity position relative to holding its original assets, under a specified price comparison. In a fee-free, full-range, equal-value constant-product pool it is 2√r ÷ (1 + r) − 1, where r is the relative price change. Fees and other costs must be assessed separately. Scope: {"collection":"defi","dataAsOf":null,"blockHeight":null} ### impermanent-loss-fact-benchmark Benchmark: The comparison is with holding the original quantities. Scope: {"collection":"defi","dataAsOf":null,"blockHeight":null} ### impermanent-loss-fact-model Model: The simple formula assumes full-range equal-value constant-product liquidity. Scope: {"collection":"defi","dataAsOf":null,"blockHeight":null} ### impermanent-loss-fact-fees Fees: Swap fees can offset some shortfall but do not guarantee a profit. Scope: {"collection":"defi","dataAsOf":null,"blockHeight":null} ## Sources - [Understanding returns](https://developers.uniswap.org/docs/protocols/v2/concepts/understanding-returns) — Uniswap. The no-fee full-range constant-product comparison with holding both tokens. Locator: Risks; Why is my liquidity worth less than I put in?. Retrieved: 2026-10-02. - [Uniswap v2 pair source](https://raw.githubusercontent.com/Uniswap/v2-core/v1.0.1/contracts/UniswapV2Pair.sol) — Uniswap. Reserve accounting, LP shares, fee-adjusted invariant and optional protocol fee. Locator: mint; burn; swap; _mintFee. Retrieved: 2026-10-02. - [Introducing Uniswap v3](https://blog.uniswap.org/uniswap-v3) — Uniswap Labs. Range-based liquidity, inactive positions and position-specific accounting. Locator: Concentrated Liquidity; Active Liquidity; Non-Fungible Liquidity. Retrieved: 2026-10-02. ## Revision history - 2026-10-02: First publication after primary-source research and independent automated verification. ## Cite this entry Degrees of Satoshi editorial project. “Impermanent loss: comparing pool inventory with simply holding.” Published 2026-10-02; updated 2026-10-02. https://degreesofsatoshi.com/encyclopedia/impermanent-loss/