ImpermanentLoss

_{spec · config}

Models the economic risk for liquidity providers (LPs) in a constant-product AMM. An LP deposits tokens into the pool, external traders swap against it (moving the price), and the LP's position is compared to simply holding the original tokens. The difference is impermanent loss (IL) — the LP ends up with fewer tokens (by value) than if they had just held. This is the fundamental risk of providing liquidity on Uniswap, and why protocols offer "liquidity mining" rewards to compensate LPs. See Terminology for definitions.

sequenceDiagram
    participant LP as Liquidity Provider
    participant P as AMM Pool
    participant T as External Trader

    LP->>P: deposit 100A + 100B
    Note over P: reserves: 100A / 100B<br/>k = 10,000

    T->>P: swap 8A → B
    Note over P: reserves: 108A / 93B<br/>k = 10,044 (fees grew k)

    LP->>P: withdraw all
    P->>LP: gets 108A + 93B

    Note over LP: LP has: 108A + 93B<br/>Hold would be: 100A + 100B<br/>At current price (93/108):<br/>LP value = 186.0B<br/>Hold value = 186.1B<br/>IL = 0.1B lost

The loss follows from the AM-GM inequality: any change in the price ratio causes the LP's position to underperform holding, even though fees grow the pool (k increases). The loss is "impermanent" because it disappears if the price returns to the original ratio — the LP keeps the fee income.

• LP deposits: creates the pool with initial reserves
• External swaps: move the price ratio, causing IL
• Fee income: k grows with every swap (0.3% fee), partially compensating IL
• AM-GM inequality: 2 * reserveA * reserveB < InitReserveA * reserveB + InitReserveB * reserveA whenever the price ratio changes

IL property (expected to fail)

Add as INVARIANT to see counterexample: