Impermanent loss

The name is wrong, and the literature already has a better one

The word impermanent describes exactly one circumstance. If the price ratio of the two assets returns to precisely where the deposit was made, the gap closes and the provider is whole. The refereed treatment says so in those words: divergence loss is sometimes called impermanent loss, because the loss vanishes if the assets return to their original valuation.²

Every other path leaves it real. A provider who withdraws at a different ratio has realised the loss. A provider who does not withdraw is carrying it unrealised, which is the same position an equity holder is in between the drawdown and the sale. The label implies that reversal is the expected case. Withdrawal at some other ratio is the expected case.

Divergence loss is the term the formal work uses, defined there as the opportunity cost of a provider's investment and shown to obey conservation laws, so the cost can be shifted between traders and providers but never fully removed.² The same paper proves that no automated market maker can bound divergence loss even when trade sizes are bounded.² That is a stronger claim than most liquidity marketing admits. It is not a parameter anyone tunes away. It is a property of quoting price from a curve.

One input, and direction is not one of them

Write d for the price ratio: the price of the volatile asset at withdrawal divided by its price at deposit. For a constant product pool holding the pair at equal value, the loss against holding is two times the square root of d, divided by one plus d, minus one. One input, one output.

The expression is symmetric in d and 1/d. A doubling and a halving produce the same 5.72%. Only distance from the opening ratio matters, which is why hedging the downside of a pool position does not address the problem. The upside costs the same.

It falls out of the constant product rule. The pool holds the product of its reserves fixed across a trade before fees are applied.¹ Call that product k. At any price P the token reserve is the square root of k divided by P, and the quote reserve is the square root of k times P. Value the position at those reserves, compare against the wallet balances that never moved, and the expression above is what remains. The formal treatment of constant product payoff under price divergence is set out in the original analysis of Uniswap markets.³

Seven ratios, with the division shown

Take a doubling first. The square root of 2 is 1.414214. Twice that is 2.828427. Divide by 3 and the result is 0.942809. Subtract 1 and you have minus 0.057191, a loss of 5.72% against holding.

Now the ten times case, the one most often quoted wrong. The square root of 10 is 3.162278. Twice that is 6.324555. Divide by 11, which is one plus ten, and the result is 0.574960. Subtract 1 and you have minus 0.425040. The answer is 42.50%. Run the last division yourself before repeating any figure near it, because this is the step where transcription errors enter and then propagate.

The full series, every entry computed the same way: 1.25 times costs 0.62%, 1.5 times costs 2.02%, 2 times costs 5.72%, 3 times costs 13.40%, 4 times costs exactly 20.00%, 5 times costs 25.46%, and 10 times costs 42.50%. The shape is the useful part. The first doubling is under six points. Travelling from five times to ten times adds seventeen points by itself.

A $20,000 position, taken to a ten times move, in dollars

Seed a pool with 10,000 project tokens at $1.00 and $10,000 of USDC. The position is worth $20,000 and the product of the reserves is 100,000,000.

The token now trades at $10 elsewhere. Arbitrage drags the pool to that ratio, so the token reserve becomes the square root of 100,000,000 divided by 10, which is 3,162.28 tokens, and the quote reserve becomes the square root of 100,000,000 times 10, which is 31,622.78 USDC.¹

Value it. 3,162.28 tokens at $10 is $31,622.78, plus the $31,622.78 of USDC, for $63,245.55. Holding the original balances would have been worth 10,000 at $10 plus $10,000, which is $110,000. The position is $46,754.45 short, and 63,245.55 divided by 110,000 is 0.574960, the same 42.50%.

Read the first number again. The position went from $20,000 to $63,246. It tripled. That is precisely why this gets missed in a bull market: the provider is up in dollars, the dashboard is green, and the 42.50% never appears anywhere on it.

The pool sold the winner the whole way up, and it had no say

A pool has no price feed and no opinion. It quotes its own reserve ratio until somebody trades against it, and the somebody is an arbitrageur closing the gap to the outside market. Their profit comes out of the reserves.

In the example above, the pool started with 10,000 tokens and finished with 3,162.28. It handed over 6,837.72 tokens and received 21,622.78 USDC for them. That is an average of $3.16 per token, against a token that finished at $10. Nobody decided to sell at $3.16. The curve did, one arbitrage trade at a time.

Stated that way, divergence loss stops being an abstraction. It is an inventory outcome. The pool is a standing instruction to sell whichever asset is appreciating and buy whichever is depreciating, and the provider is the counterparty on every one of those fills.

When fees cover it, and when they cannot

Fee income and divergence loss are driven by different variables, which is the reason a single yield number cannot express both. Divergence loss depends only on the opening and closing ratio. Fee income depends on how much volume passed through in between. A pair can travel to a ten times move on thin volume or on heavy volume and the loss is identical either way.

Uniswap v2 charges 0.30% on every trade, of which 0.25% accrues to liquidity providers once the protocol fee is switched on.¹ Put the numbers against the example. At a doubling the position is $1,715.73 behind the hold, so at 0.25% it takes $686,292 of cumulative volume to break even, ignoring compounding. That is roughly 34 times the opening value of the position, in volume, just to draw level with having done nothing.

At the ten times move the gap is $46,754.45, which needs about $18.7 million of volume through those same reserves. For a pool seeded at $20,000, that is not a normal month. It is not an argument against providing liquidity. It is an argument for stating the volume assumption out loud whenever an annualised rate is quoted, because the rate is a function of an assumption somebody made and rarely wrote down.

Concentrated liquidity concentrates this too

Uniswap v3 lets a provider bound liquidity inside a chosen price range rather than spreading it from zero to infinity, which is where the capital efficiency comes from.⁴ The protocol's own documentation quantifies the waste it removes: the v2 DAI and USDC pair uses about 0.50% of total available capital for trading between $0.99 and $1.01, where most of the volume happens.⁵

The same documentation states the cost in the same breath. As price moves in one direction, providers accumulate more of one asset until the entire position consists of a single asset, and once price leaves the chosen interval the position stops earning fees altogether.⁵ In a full range pool that outcome is rare. In a narrow band it is the ordinary weekly experience.

So concentration does not reduce divergence loss. It does the same rebalancing with less capital spread underneath it, which raises fee income per dollar while price sits in the band and raises the loss per dollar for the same move. A band chosen tight enough to make the yield look impressive is a band that will be abandoned by the price, and the position left behind holds only the side that fell.

What we settle before underwriting a liquidity program

The design consequence lands on incentive programs first. If a project pays liquidity providers in its own token to compensate them for divergence loss on its own pair, the dilution is funding arbitrageurs by way of the providers. That may still be the right call for a launch window. It should be written down as what it is, not presented as a yield.

Four things get decided before any of it is published. The realised volatility of the pair, because that is the input the loss actually responds to. The fee tier, since it sets how often correcting the pool is worth doing. The volume assumption sitting behind any advertised rate, stated as a number. And who holds the position, because a protocol owned pool carries this loss on the treasury's own balance sheet and belongs in the treasury model, not a footnote.

One boundary on all of the above. This is reference material for mechanism design, not a recommendation to provide liquidity, hold any asset, or trade anything. Whether a specific pool is worth providing to depends on facts we do not have.