Impermanent Loss in Constant-Product AMMs: Worked Example

A constant-product pool automatically changes asset quantities when relative prices move. A liquidity provider can end up with less value than simply holding the originally deposited tokens, even when both outcomes rise in dollar terms.

How to work through the calculation or research

In a simplified 50/50 pool with relative price ratio r, the hold-relative value ratio is 2√r/(1+r). Impermanent loss is that ratio minus one. The calculation describes price divergence and assumes an idealized pool without fee income or concentrated liquidity.

A concrete worked example

If the price ratio becomes 4, the simplified relative factor is 2×2/(1+4) = 0.8, or a 20% shortfall versus holding the same starting assets before trading fees. It is not necessarily a 20% loss on the dollar amount initially invested.

Common errors and important limitations

Trading fees can offset some divergence loss, but not reliably; pool incentives, depegs, contract risk and range selection all matter. This simple formula does not cover concentrated liquidity positions.

Put the idea into practice with a working tool

For a practical calculation, the Impermanent Loss Calculator lets you examine: New token price / original price (×). The tool documents its method as: IL = (2 × √price ratio / (1 + price ratio) − 1) × 100. This ignores LP trading fees, incentives, stablecoin depegs and concentrated-liquidity behavior.

Practical checks before relying on the result

• Confirm the asset, chain, amount and date apply to your own situation.
• Separate supplied assumptions from verifiable source data and do not treat a scenario as a live market quote.
• Document fees, rounding rules and constraints before making an irreversible financial or blockchain decision.

Financial and blockchain information is educational. No investment advice, custody, private keys or guaranteed outcomes. Examples use illustrative inputs rather than live quotes.