Liquidity pools and impermanent loss, with the arithmetic

A liquidity pool is a pile of two tokens and a rule for trading one against the other. There is no market maker deciding prices; the rule decides them, and anyone who deposits into the pile — a liquidity provider, or LP — earns a share of the trading fees in exchange for letting the rule rearrange their deposit. The rule is simple enough to write on one line, and its consequences are precise enough to put in a table. Most of the money lost by liquidity providers has been lost by people who never looked at the table.

The rule

The pools that started this, Uniswap's, keep the product of their two reserves constant: with x of one token and y of the other, every trade must leave x · y = k. If someone buys token A out of the pool, x falls, so y must rise by enough to keep the product — they pay in token B at a price that gets worse the more they buy. The price the pool quotes is simply y divided by x, and arbitrage traders keep it close to the price everywhere else: whenever the outside price moves, someone trades against the pool until its ratio matches, and pockets the difference.

Traders pay a fee on every swap — 0.30% in Uniswap v2, most of it to the liquidity providers — and that fee is the LP's income. Everything else about being an LP is a consequence of the arbitrage.

What the arbitrage does to your deposit

When the outside price of token A doubles, arbitrageurs buy A from the pool until the pool's price doubles too. They take A out and put B in. The pool, and therefore your share of it, now holds less of the token that went up and more of the one that did not. You sold the winner on the way up, automatically, at every price between the old one and the new one.

Compared with simply holding the two tokens you deposited, you have less. The gap is called impermanent loss — "impermanent" because it vanishes if the price returns to where it started, which is a comfort only if it does. Pintail's 2019 analysis, which Uniswap's own documentation repeats, gives the formula for a 50/50 pool when the price ratio between the two tokens changes by a factor r:

loss versus holding = 2·√r ÷ (1 + r) − 1

The formula is symmetric: a halving is the same as a doubling, a fall to a quarter the same as a rise to four times. Direction does not matter, only how far the two tokens diverge.

A worked example

Hypothetical figures throughout. You deposit 1 ETH and 2,000 USDC into a pool while ETH trades at $2,000 — a $4,000 position. ETH then doubles to $4,000. Arbitrage rebalances the pool so that your share is now about 0.7071 ETH and 2,828 USDC, worth $5,657. Had you kept the 1 ETH and 2,000 USDC in a wallet, you would hold $6,000. The pool position is $343 behind — 5.7%, the table's 2× row — and that gap has to be made up by the fees you earned while the price moved. Whether it is depends entirely on how much trading passed through the pool, and that is the number no formula can give you in advance.

The common misreading is that an LP lost money. In the example the LP made $1,657 on a $4,000 deposit; the loss is relative to the holder who did nothing. The less common misreading is the dangerous one: that the loss is small. In a pair where one token goes to zero, the pool rebalances all the way into the worthless one. A 50/50 pool against a token that dies holds, at the end, two piles of nothing.

Concentrated liquidity turns the dial both ways

Uniswap v3 let a provider place liquidity inside a chosen price range instead of across every price from zero to infinity. Within the range the same deposit does far more work — Uniswap Labs' launch post claimed "up to 4000× capital efficiency relative to Uniswap v2" — and so earns far more fees per dollar. The same post says plainly what that costs: it "requires taking on more price risk ('impermanent loss')." And when the price leaves the range, the position stops earning fees altogether and sits entirely in whichever token the price ran away from.

So concentration is leverage on the table above: more fees if the price stays put, a deeper version of the same loss if it does not, and nothing at all once it has gone.

What the data said

In 2021 four researchers measured the question directly on Uniswap v3, across 17 pools covering 43% of the protocol's locked value, from May 5 to September 20 of that year. Over that period the pools' liquidity providers earned $199.3 million in fees and suffered $260.1 million of impermanent loss. As a group "those LPs would have been better off by $60.8m had they simply HODLd." About half of them — 49.5% — ended with negative returns against holding, and only 3 of the 17 pools earned fees that exceeded the loss. That is one period on one protocol, in a volatile summer, and the authors say so. It is also the only large published measurement, and it did not find the business the sales material describes.

The risks the formula does not cover

Impermanent loss is the honest, mathematical risk. The others are plainer. The pool's contract can have a bug, and the deposit is gone. A token's creators can hold a large share of the pool's liquidity tokens and withdraw them — the "rug pull," which is why a pool whose LP tokens have been verifiably locked or burned is a different object from one whose creators can leave at any time, though a locked pool still carries every line of the table above. Automated traders can see a pending trade and place their own around it, taking a slice of every large swap. And the token on the other side of the pair can be printed, by its own rules, faster than any fee income can cover.

None of this is a reason not to understand pools; it is the reason to understand them before any money goes in — and the drawdown page says what the table's bottom rows cost to climb back from. The desk will teach the mechanics here, in the open, before its own crypto area exists — so that whatever it eventually offers can be read against this page rather than against a pitch.

• What a perpetual future is and how funding works
• Spot vs perps vs on-chain: where a trade actually happens
• Holding your own keys: what a signature proves, and what nobody should ask you for
• The vault: hash-based keys and what they protect
• Staking — what you earn, what you risk, what is taxed
• Post-quantum on a token vs in a wallet: what the words can and cannot mean

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