Impermanent Loss, Explained Without Hand-Waving

Impermanent loss is one of the worst-named concepts in the field. It is not especially impermanent, and calling it a loss suggests something went wrong. Both readings mislead. A better name — used by people who work on these systems — is divergence loss, because it is caused entirely by prices diverging from where they were when you deposited.

It is also fully deterministic. Given the price change, you can calculate it exactly.

What an AMM is doing with your money

The classic automated market maker holds two assets and enforces a rule: the product of the two reserve quantities stays constant. If the pool holds x of one asset and y of the other, then x × y = k, and every trade must leave k unchanged.

Traders exploit this. When the external market price moves, the pool’s implied price is briefly stale, and arbitrageurs trade against it until the two agree. Those trades are the mechanism by which the pool tracks the market — and they are also the mechanism by which your position gets rebalanced without your input.

The rebalancing has a consistent direction: the pool sells the asset that is rising and buys the asset that is falling. You end up holding less of the winner and more of the loser than you started with.

That is the entire phenomenon. Impermanent loss is the gap between that automatically rebalanced position and simply having held the two assets.

The formula

For a standard constant-product pool, let r be the price ratio change — the new relative price of the two assets divided by the old one. Then:

divergence loss = 2√r / (1 + r) − 1

It is always zero or negative, and it is symmetric: r = 2 and r = 0.5 produce identical results. A doubling and a halving hurt equally.

Price ratio change	Loss vs. holding
1.25×	0.6%
1.5×	2.0%
2×	5.7%
3×	13.4%
4×	20.0%
5×	25.5%

Two things stand out. Small moves cost almost nothing — under a 25% divergence you are giving up about half a percent. And the curve is gentle at first but unbounded: as one asset runs away from the other, the loss keeps climbing.

Why “impermanent”

Because if prices return to their starting ratio, the gap closes completely. At r = 1 the formula gives exactly zero.

The loss only becomes permanent when you withdraw while prices are diverged. Until then it is unrealised. This is where the name does its damage: people hear “impermanent” and assume it reverses on its own. It reverses only if the price relationship reverses, and there is no reason to expect that it will.

Fees are the other half of the equation

Nobody provides liquidity for the loss. They do it for the trading fees, which accrue to liquidity providers on every trade.

The real question is never “will I suffer divergence loss” — you will, whenever prices move. It is whether fee income over your holding period exceeds the divergence loss over that same period. A pool with heavy volume and modest price drift can be comfortably profitable. A quiet pool holding a pair that diverges sharply is not.

This also explains why volatility cuts both ways. It widens divergence, which hurts, but it drives volume, which pays. The pairs that work are the ones where the trading activity is worth the rebalancing.

Pair selection follows directly

Stablecoin pairs barely diverge, so divergence loss is negligible. Fees are correspondingly thin, and the residual risk is not price movement but a stablecoin failing to hold its peg — a different risk that the formula does not describe at all.

Correlated pairs — two assets that tend to move together — keep r near 1 and limit the damage while still generating volume.

A volatile asset against a stablecoin is the maximum-divergence configuration. Every move in the volatile asset moves r directly.

Concentrated liquidity amplifies everything

Newer AMM designs let providers concentrate capital within a chosen price band rather than spreading it across every possible price. Inside that band your capital earns far more fees per unit deposited.

The tradeoff is proportional. Concentration magnifies fee income and divergence loss by the same mechanism. Worse, when price exits your band your position converts entirely into one asset and stops earning fees altogether — you are fully exposed to the side that moved against you, with no income to offset it.

Concentrated liquidity is an active position requiring management. Treating it as a deposit-and-forget yield product is how people are surprised.

What to hold onto

• It is arithmetic, not misfortune — calculable in advance from the price move.
• Symmetric: a halving costs what a doubling costs.
• Small divergences are cheap; large ones are not, and the curve has no ceiling.
• Only realised on withdrawal.
• The decision is always fees-earned versus divergence-suffered, never divergence in isolation.
• Concentration scales both sides, and adds a failure mode when price leaves your range.

Everything above follows from the constant-product rule. It holds regardless of which pool, which chain, or what the market is doing this week.

Educational content only. Nothing here is financial advice.