Impermanent loss, worked through with one pool and one price move

Impermanent loss is not a loss in dollars. It is the gap between two outcomes measured at the same moment: the position inside the pool, and the position you would still hold had you never deposited. One pool and one price move is enough to see it.

Set the pool before you set the price

A constant-product pool holds two assets and keeps their product constant. Deposit 1 ETH and 2,000 USDC when ETH trades at 2,000, and the pool holds 1 ETH against 2,000 USDC, so k is 2,000. The split is even because the curve prices both sides from one formula, and that is the only starting point with zero divergence.

The price doubles and the pool sells into it

ETH goes to 4,000. The pool cannot hold its ETH, because anyone may buy at the curve price, and the only way to keep k constant is to give up ETH and take in USDC. Solving the curve at the new price leaves 0.7071 ETH and 2,828.43 USDC, worth 5,656.85 USDC.

Holding would have left 1 ETH and 2,000 USDC, worth 6,000. The gap is 343.15 USDC, or 5.72% of the holding value. Along the way the pool sold 0.2929 ETH for 828.43 USDC, an average of about 2,828 against a final price of 4,000. It is a mechanical seller into strength, paid the average rather than the top.

The loss is relative to holding

The position did not lose money. It started at 4,000 and finished at 5,656.85, up 41.4%, while holding finished up 50%. That is the whole definition: divergence compares the pool with the hold at one timestamp, which is why a pool can be profitable in dollars and still underperform the asset in it.

It also explains the word impermanent. If the price returns to 2,000, the pool returns to 1 ETH and 2,000 USDC and the gap closes to zero, less whatever fees accrued. The number only becomes permanent when you withdraw at the wrong moment.

The formula, and why doubling is the reference case

For a 50/50 pool and a price ratio r, divergence is 2 times the square root of r, divided by 1 plus r, minus 1. The function is symmetric in the ratio, so a double and a halving cost the same.

Derived from the constant-product formula applied to the worked example, October 2026 pricing.

The shape matters more than any row. Divergence grows slowly and then accelerates, so a pool pairing one volatile asset with a stablecoin is a position against large moves in either direction.

Fees have to beat the divergence

Fees are the only offset. They accrue into pool value rather than into a separate balance, so they appear as a pool value slightly better than the curve predicts.

Break-even is arithmetical. Cumulative fees must exceed the divergence, as a percentage of position value, over the same window. At fee APR f, the months needed to cover 5.72% are roughly 5.72 divided by f, times 12: about 6.9 months at 10%, about 3.4 at 20%.

No pool quotes a forward fee APR. The number on the page is realised fees over the last 7 or 30 days, annualised, and denominated in whichever asset the pool holds at the snapshot. Volume follows volatility, and volatility is the condition that produces divergence, so the trailing APR and the divergence tend to peak together. The figure used to justify a deposit is measured in the period least like the one you will hold through.

It is also divided by pool TVL, which includes the value that just fell, so the same fees look better after a drawdown. TVL is a deposit count, not a revenue line (fee revenue is not TVL), and we treat published fee pages as trailing evidence for the same reason (fee pages change monthly).

Verdict

Run three numbers before depositing: divergence at the move you consider plausible, fee income you can document over a comparable window, and the fee APR restated on your entry size rather than today’s TVL. If the third does not clearly beat the first, the pool is a way to hold two assets with a tax on conviction.

If you would not hold both assets through that move, the fee is not compensation. It is a subsidy for the other side of your exit.