Curriculum·J303 Liquidity Provision and Impermanent Loss·about 42 min
Impermanent loss, derived
By the end of this lesson you can
- →Derive the impermanent loss formula from the constant product invariant
- →Evaluate the loss for any price ratio and explain why it is symmetric
- →Show that a round trip has zero impermanent loss and a positive realized cost
- →State what fees must actually cover, which is not the impermanent loss figure
Junior · enrolled learners
This lesson opens with Loss versus rebalancing, formalised.
- What happened
- This is a formal result rather than a named incident, and it is labeled as such. Milionis, Moallemi, Roughgarden and Zhang introduced loss versus rebalancing to describe what a liquidity provider pays to informed arbitrageurs. The framing is that an automated market maker quotes stale prices between trades, so whenever an external price moves, the first party to notice trades against the pool at the old price and captures the difference. The cost accrues continuously with realized volatility rather than at the endpoints of a holding period, and under the framework a liquidity provider is profitable when fee income exceeds this quantity and unprofitable otherwise.
- The decision point
- The conventional measure, impermanent loss, compares the position's value at the end of a period against holding the original basket, so a price that leaves and returns produces a loss of exactly zero. That is arithmetically correct and it describes the wrong quantity, because value was extracted from the pool on the way out and again on the way back. The name has misled a generation of liquidity providers into treating a continuously realized payment to arbitrageurs as a paper loss that reverses.
What you will be able to answer
- →What is the impermanent loss formula?
- →Why is it symmetric?
- →What does a round trip cost?
- →What must fees actually cover?
Orientation and Year One are open: anyone can read them without an account. From Year Two onward the lessons are for enrolled learners, because progress through the later years only means anything if it is tracked against a record.
It is free. We do not sell the list and there is nothing to buy at the end of it.
Sources and review
- https://arxiv.org/abs/2208.06046
- https://arxiv.org/pdf/2404.05803
- https://arxiv.org/abs/2205.08904
- https://cryptoslate.com/new-report-shows-50-of-uniswap-v3-liquidity-providers-are-losing-money/
Confidence high·Volatility low·Reviewed 2026-08-06·Owner unassigned
Contested
Impermanent loss and loss versus rebalancing measure different things against different benchmarks and neither is wrong. Impermanent loss answers what the position is worth against holding, which matters at exit. Loss versus rebalancing answers what was paid along the way, which matters for whether fees are sufficient. This lesson teaches both and says which question each answers.
The derivation here assumes a constant product pool, no fees and instantaneous arbitrage. Fees reduce the amount arbitrageurs extract, since they must clear the fee before trading, and J303-03 handles that. The structure of the result does not change.