You deposit two tokens and withdraw a different mix
Put 1 ETH and 2,000 USDC into a Uniswap-style blockchain that runs exactly as written when it is called, holding and moving funds according to its code, without an intermediary able to change the outcome.">smart contract holding reserves of two or more tokens, supplied by users, that a decentralized exchange uses to execute trades automatically at a price set by a formula.">pool when ether trades at an example price of 2,000 dollars. That is 4,000 dollars of capital. Ether doubles to 4,000 dollars, you withdraw, and the pool hands back 0.7071 ETH and 2,828.43 USDC, worth 5,656.85 dollars before fees. You are up 41.4 percent, which feels fine until you check what the same tokens would be worth had they stayed in your wallet: 6,000 dollars. The 343.15 dollars between those two numbers is impermanent loss.
It is not a fee and not a hack, and it is not always impermanent. It is a gap against a benchmark, and it follows from four lines of algebra. The harder question comes after the maths: did the fees pay for the gap? In one study of 17 Uniswap v3 pools in 2021, they did not. Providers earned 199.3 million dollars in fees and took 260.1 million of that loss.
A liquidity pool is a smart contract holding two tokens that anyone can trade against. No order book is involved: a formula sets the price, x·y=k on Uniswap v2. Whoever supplies the tokens is a liquidity provider (LP) and receives LP tokens, a claim on a share of the reserves. The derivation of that formula is in the previous lesson on AMM math, and the wider picture is in our guide to decentralized exchanges and AMMs.
Where the LP’s money comes from: the fee on every swap
A provider has one income: trading fees. The Uniswap v2 whitepaper of March 2020 sets the fee at 0.30 percent of each trade, all of it paid to providers. If the protocol fee is switched on, providers keep 0.25 percent and the other 0.05 percent goes to an address set in the factory contract. The fee stays inside the reserves, so k grows a little with each swap and every LP token claims slightly more than before. Nothing lands in your wallet until you withdraw.
Slippage, which our guide to using a DEX safely covers from the trader’s side, looks different from here. A large trade pays a larger fee, which is good for you. It also pushes the pool price away from the market, and an arbitrageur then pulls it back by trading against your reserves.
Impermanent loss is measured against holding, not against zero
The definition has one benchmark: the same tokens, left untouched in your wallet. Divide the value of the pool position by the value of that untouched bag and subtract one. In the opening example, 5,656.85 divided by 6,000, minus one, gives −5.72 percent.
So a position can show a profit and this loss at the same time, as ours did. It can also show a loss in dollars that has little to do with the pool: when ether halves, most of the damage comes from ether halving. The number answers one narrow question, whether depositing beat doing nothing.
The cause is mechanical. An x·y=k pool always holds equal value on both sides, so as ether rises, arbitrageurs buy it from the pool at slightly stale prices until the balance is restored. The pool sells whatever goes up and buys whatever goes down, all the way along the move. You end with less of the winner and more of the loser than you started with.
The word “impermanent” comes from what follows. Uniswap’s documentation reproduces the argument in a quoted passage: “if the price were to return to the same value as when the liquidity provider added their liquidity, this loss would disappear”. The same page points to an article by an author writing as Pintail, and its summary uses the plainer term divergence loss. We found no reliable record of who first said “impermanent loss”. Once you withdraw, the loss is no longer impermanent at all.
The impermanent loss formula, proved in four lines
Take a pool of two tokens with no fees, for now. Reserves are x of token A and y of token B. The price of A in B is p, the deposit was made at price p0, and r = p/p0 is the price ratio since then. Arbitrage keeps p at the market price.
- Reserves at any price: from x·y = k and p = y/x, we get x = √(k/p) and y = √(k·p).
- Value of the pool in token B: Vpool = p·x + y = √(k·p) + √(k·p) = 2√(k·p).
- Value of holding the original deposit: Vhold = p·x0 + y0 = √(k·p0)·(r + 1).
- Divide, using √(k·p) = √(k·p0)·√r: Vpool / Vhold = 2√r / (1 + r).
Subtract one and you have the result, the same expression that appears in Uniswap’s docs:
IL = 2 × √r / (1 + r) − 1Test it on the opening example, where r = 2: 2 × 1.41421 / 3 − 1 = −0.0572. That is the 5.72 percent, the loss of 343.15 dollars on a holding worth 6,000.
Four things the formula tells you
- Only the ratio between the two tokens counts. If ETH and the token paired with it both rise 50 percent, r stays at 1 and the loss is zero, however far the dollar price moved.
- Before fees, the path does not count either. Two positions with the same start and end prices show the same number, whether the market got there in a day or a year.
- The result is zero at r = 1 and negative everywhere else, because √r can never exceed (1 + r)/2. Without fees, a pool never beats holding.
- The formula gives the same result for r and for 1/r. A doubling and a halving cost the same percentage.
It approaches −100 percent only in the limit, as one token’s relative price runs to zero or to infinity. This formula holds for a 50/50 x·y=k position over the full price range. Narrow ranges on Uniswap v3 and v4 amplify the effect, which a later lesson covers.
The curve: a 2x move costs 5.7 percent, a 10x move 42.5 percent
The table runs the formula for the ratios people ask about. Figures up to 5x match the table in Uniswap’s docs, which rounds them to one decimal. The 10x row is our own computation.
| Price ratio up | Price change | Mirror ratio down | Price change | Loss against holding (both) |
|---|---|---|---|---|
| 1.25x | +25% | 0.8x | −20% | −0.62% |
| 1.5x | +50% | 0.667x | −33.3% | −2.02% |
| 2x | +100% | 0.5x | −50% | −5.72% |
| 3x | +200% | 0.333x | −66.7% | −13.40% |
| 4x | +300% | 0.25x | −75% | −20.00% |
| 5x | +400% | 0.2x | −80% | −25.46% |
| 10x | +900% | 0.1x | −90% | −42.50% |
The trap sits in the third and fourth columns. Symmetry works on ratios, not on percentages. A 50 percent rise (1.5x) costs 2.02 percent, while a 50 percent fall (0.5x) costs 5.72 percent, because a fall of 50 percent is the mirror of a rise of 100 percent. In the same way, a 20 percent fall mirrors a 25 percent rise.
The curve is flat near the start and steep later. A 5 percent move costs 0.03 percent. A 1 percent move costs about 0.001 percent. Providers can absorb a market that swings around one level for months, and they suffer in a market that trends one way for long, which is when the gap to holding grows widest.
Worked examples in dollars: ether up, ether down, a stablecoin pair
All the prices below are examples, not forecasts. The deposit is the same in each row: 1 ETH and 2,000 USDC at an ETH price of 2,000 dollars, 4,000 dollars in total, fees excluded.
| ETH price, dollars (example) | Pool holds | Pool value | Hold value | Difference | IL |
|---|---|---|---|---|---|
| 2,500 (x1.25) | 0.8944 ETH + 2,236.07 USDC | 4,472.14 | 4,500.00 | −27.86 | −0.62% |
| 3,000 (x1.5) | 0.8165 ETH + 2,449.49 USDC | 4,898.98 | 5,000.00 | −101.02 | −2.02% |
| 4,000 (x2) | 0.7071 ETH + 2,828.43 USDC | 5,656.85 | 6,000.00 | −343.15 | −5.72% |
| 6,000 (x3) | 0.5774 ETH + 3,464.10 USDC | 6,928.20 | 8,000.00 | −1,071.80 | −13.40% |
| 1,000 (x0.5) | 1.4142 ETH + 1,414.21 USDC | 2,828.43 | 3,000.00 | −171.57 | −5.72% |
| 500 (x0.25) | 2.0000 ETH + 1,000.00 USDC | 2,000.00 | 2,500.00 | −500.00 | −20.00% |
Read the 1,000-dollar row closely. The position is worth 2,828.43 dollars, 1,171.57 below the deposit. Only 171.57 of that comes from being in the pool. The other 1,000 is the fall of the ether you would have lost anyway by holding. Meanwhile the pool bought ether on the way down and now holds 1.4142 ETH, so a provider ends up with more of the token that fell.
Holding the mix is not the only comparison. Had you put the whole 4,000 dollars into ether, you would own 2 ETH: 8,000 dollars at the 4,000 price, 2,000 at the 1,000 price. In USDC you would simply have 4,000. The pool position lands between those two outcomes at either price, and it trails the mixed holding at both.
Now take the round trip. Ether goes to 4,000 dollars, comes back to 2,000, and you have not touched the position. The reserves return to 1 ETH and 2,000 USDC, the gap to holding is zero, and every fee collected on the way is yours. Withdraw at 4,000 instead, and the 343.15 dollars is final.
Stablecoin pairs sit at the other end of the curve. With one dollar token at 1.002 against the other, r = 1.002 and the loss is about 0.00005 percent, half a cent on 10,000 dollars. The formula looks just as gentle if one stablecoin falls to 90 cents: in a 1,000-plus-1,000 pool, you hold 1,054.09 of the weak token and 948.68 of the other, worth 1,897.37 dollars against 1,900 for holding. The gap is 0.14 percent. The real harm is the depeg: the pool has just handed you more of the broken token. Stablecoin venues use flatter curves than x·y=k, a subject for a later lesson.
When fees make up for impermanent loss
The fee income needed to break even
The fee target depends on how far the price moved and how fast. The table takes the gap to holding as a percentage of the deposit, which is what a provider actually put in, then shows the annualised fee rate needed if that move arrives within 30, 90 or 365 days. It is our own arithmetic: simple annualisation, no compounding, no gas.
| Price ratio | Fees needed, % of deposit | Annualised, move in 30 days | Annualised, move in 90 days | Annualised, move in 365 days |
|---|---|---|---|---|
| 0.8x | 0.56% | 6.8% | 2.3% | 0.6% |
| 1.25x | 0.70% | 8.5% | 2.8% | 0.7% |
| 1.5x | 2.53% | 30.7% | 10.2% | 2.5% |
| 0.5x | 4.29% | 52.2% | 17.4% | 4.3% |
| 2x | 8.58% | 104.4% | 34.8% | 8.6% |
| 3x | 26.79% | 326.0% | 108.7% | 26.8% |
This table differs from the IL column above for a reason. That figure is a percentage of the hold value at the end, while this one is a percentage of the starting deposit. At r = 2 the gap of 343.15 dollars is 5.72 percent of 6,000 and 8.58 percent of 4,000. At r = 0.5 the gap of 171.57 is 5.72 percent of 3,000 and 4.29 percent of 4,000.
Clearing this bar means you beat holding. It does not mean you made money: in a falling market, fees can exceed the gap while the position is still below the deposit. A passage quoted in Uniswap’s docs admits the difficulty: it is “difficult to know what the trade-off is between revenues from fees and losses from directional movements” without knowing the number of trades in between.
A real pool, dated: 132 percent or 23 percent?
These are three large ETH-USDC pools on Uniswap v3, as listed by the DefiLlama Yields API on 30 October 2026. The three fee columns are the API’s fields apyBase, apyBase7d and apyMean30d. Amounts are in dollars. They move hourly.
| Pool | TVL | 24h volume | Fee APY, last day | Fee APY, 7 days | 30-day mean |
|---|---|---|---|---|---|
| WETH-USDC 0.3%, Base | 149.4M | 180.9M | 132.5% | 23.4% | 54.9% |
| USDC-WETH 0.05%, Ethereum | 79.0M | 388.2M | 89.7% | 65.6% | 8.9% |
| USDC-WETH 0.3%, Ethereum | 54.6M | 176.4M | 353.8% | 125.7% | 22.8% |
Take the first row. Volume of 180.87 million dollars at 0.3 percent gives about 542,618 dollars of fees in a day, or 0.36 percent of staking or liquidity, measured at current market prices. It is the main size metric in DeFi.">TVL. Multiply by 365 and you get 132.5 percent: the headline number is one busy day, annualised. The same pool shows 23.4 percent on the seven-day figure and 54.9 percent on the 30-day mean. On the Ethereum 0.05 percent pool, those numbers run from 8.9 to 89.7 percent, for the same two tokens, and the last day alone makes up 65 percent of the 597.3 million dollars of seven-day volume.
Treat the seven-day column as the API’s own number. On the Base pool, 633.1 million dollars of seven-day volume at 0.3 percent would annualise to about 66 percent, not 23.4, and we could not reconcile the two.
None of these figures is what one provider earned. They are pool-wide, and in a concentrated pool your share depends on your range and on when you entered. They also subtract nothing for price moves: the API reports that risk in a separate flag, ilRisk, set to “yes” on all three pools.
Why a displayed APR misleads
- Annualising a short window, as above, turns one heavy day into a yearly rate nobody will collect.
- TVL dilution works against you: a high APR draws deposits, and the same fees are then split among more capital.
- Token emissions, the rewards behind yield farming, can inflate the total yield while being paid in a token whose price may fall as it is sold.
Comparing APR with APY is the subject of a later lesson. Here the point is narrower: a quoted rate is a snapshot of fees, and it says nothing about the price move those fees have to pay for.
Fees against arbitrage: loss-versus-rebalancing
As a measure, the holding benchmark has a known weakness. A paper by Milionis, Moallemi, Roughgarden and Zhang, first posted in August 2022 and revised since, calls it “loss versus holding” and lists its problems: it does not add up cleanly over time, its size and even its sign can change with the start date chosen, and it can shrink when prices reconverge although the costs were real.
Their alternative, loss-versus-rebalancing or LVR, which the previous lesson introduced, compares the pool with a strategy that makes the same trades at market prices. The difference is what arbitrageurs extract. It only ever grows, and it grows faster when volatility is high. For a constant product pool the paper gives a simple rate: σ²/8 of pool value per unit of time, with σ² the variance over that unit. Its own example is 5 percent daily volatility, which means about 3.125 basis points of pool value lost per day. On our arithmetic, 80 percent annual volatility works out at 8 percent of pool value per year.
That changes how to read a round trip. The holding gap can return to zero, while the arbitrage cost along the way was paid and stays paid. Fees have to cover it. The authors also find, on the Uniswap v2 ETH-USDC pool, that over 99.991 percent of the variance of LP returns comes from exposure to market risk. Whatever a provider thinks of the fees, the position is first of all a bet on the tokens.
What the research found: fees fell short in early Uniswap v3
The study behind those figures is Impermanent Loss in Uniswap v3 by Stefan Loesch, Nate Hindman, Mark Richardson and Nicholas Welch, dated 17 November 2021. It covers 5 May to 20 September 2021 and 17 pools holding 43 percent of v3 TVL, which traded over 108.5 billion dollars in that time.
| Finding (5 May to 20 September 2021) | Figure |
|---|---|
| Fees earned | 199.3M dollars |
| Impermanent loss | 260.1M dollars |
| Aggregate result against holding | −60.8M dollars |
| LPs with negative returns | 49.5% |
| Wallets with negative returns, by pool | generally 40-60%; about 34% (FTM/ETH) to 74% (MKR/ETH) |
| Pools where fees exceeded IL | 3 of 17, by a small margin |
| IL as a multiple of fees, positions over one month | 1.1x |
| IL as a multiple of fees, positions up to one day | 1.8x |
The authors could not confirm their hypothesis that active providers do better than inactive ones: within a reasonable margin of error, the two groups looked the same. Longer positions fared better than short ones, though they still did not cover the loss. The one group that consistently beat holding was flash LPs, who add liquidity for a single block, the practice known as just-in-time liquidity.
The sample covers the first months of a new design, and concentrated positions lose more than the full-range formula suggests. The paper does not name a sponsor, but CryptoSlate reported on 19 November 2021 that fintech advisory firm Topaze Blue published it in conjunction with Bancor, a rival AMM. We found no comparable study with 2026 data, so the 49.5 percent should not be read as today’s rate.
A later paper points the same way. Heimbach, Schertenleib and Wattenhofer (May 2022, revised that September) found that performance varies widely on Uniswap v3. Simple strategies can make money in low-volatility pools, but modestly, and higher returns demand more risk and active management. They call liquidity provision on v3 “a game reserved for sophisticated players”.
Impermanent loss calculators that exist, and what they leave out
Most calculators do the division above and nothing more. Here is what we opened on 30 October 2026.
| Tool | What you enter | Fees counted | Concentrated ranges |
|---|---|---|---|
| DailyDeFi calculator | Start and future prices of two tokens | No, stated on the page | No |
| STON.fi calculator | Start and new prices of two tokens, on a fixed 1,000-dollar position | No, stated on the page | Not offered on the page |
| Revert Finance | Your existing position: it reports PnL, APR, ROI and divergence loss | Yes: collected and uncollected fees are in its PnL | Yes: Uniswap v2, v3, v4 and Aerodrome on Base |
| Your own spreadsheet | Two prices, the formula below | Add them yourself | No |
We found no such calculator published by Uniswap itself. Its docs give the formula and the table instead. Revert differs from the others: it analyses a real position after the fact, on Ethereum, Polygon, Arbitrum, Optimism, Base and Unichain, rather than simulating one in advance. Its “divergence loss” is another name for the same idea.
A spreadsheet takes one line. Put the deposit price of the token in A1 and the current price in B1. For a pair of two volatile tokens, use their relative price, A against B, in both cells.
=2*SQRT(B1/A1)/(1+B1/A1)-1Use it to test any calculator you find: at a ratio of 2, the answer must be −5.72 percent. A full-range tool will understate the loss of a narrow Uniswap v3 position. None of these simple calculators accounts for gas or tax.
How to reduce impermanent loss, and why you cannot remove it
Any 50/50 position that sits on a price move carries it. What you can choose is how much of it you take, and at what price in fees. This is general information about how pools behave, not a recommendation to use one.
- Correlated pairs drift apart less. The Uniswap v2 whitepaper says two USD stablecoins would generally suffer less of it than a pair against ETH. A pair of a token and its liquid staking version works on the same logic, with a depeg risk of its own.
- Higher fee income per dollar of TVL raises the odds of covering the gap, and a volume-to-TVL ratio tells you more than a headline APR.
- Avoiding narrow ranges limits the amplification, at the cost of earning less per dollar.
- Hedging the price exposure with perpetual futures is possible, but funding, fees and rebalancing eat into the return, and it is an advanced trade.
- Staying out is also an answer: holding the tokens has no such gap by definition, and lending them is a different risk.
A longer stay does not fix it by itself. Time buys more fees, and it also gives the price more room to drift. The 2021 study saw longer positions do better, in a sample where they still lost to holding.
Taxes on liquidity pools: the IRS and HMRC leave gaps
No rule we found treats the gap to holding as a tax event. The questions sit at the deposit, at the withdrawal and on the fees, and neither country has a clean answer for AMM pools.
In the United States, the IRS digital assets page treats disposing of a digital asset for another digital asset as a reportable event. It names “liquidity provider transactions” among those that brokers need not report on Form 1099-DA until further guidance under Notice 2024-57. We found no IRS guidance on whether depositing into a pool is itself a disposal. Treat it as unsettled, keep records of every deposit and withdrawal, and see our US crypto tax guide.
In the United Kingdom, HMRC manual page CRYPTO61620, updated 28 November 2025, deals with DeFi lending and staking. It calls a transfer that gives up beneficial ownership in return for a token “an exchange of one token for another token”, which makes it a disposal, and it splits the consideration across several tokens on a “just and reasonable basis”. The page is about lending platforms, so applying it to an AMM pool is an interpretation. Our UK guide has the wider rules. A tax professional can confirm your own case.
In small pools, divergence is not the main risk
The formula assumes both tokens survive. In a thin pool with an unknown token, that is the assumption to doubt. The team can drain the liquidity, a pattern our guide to the rug pull describes, and a token that goes to zero leaves you holding almost all of it, since the pool keeps buying it on the way down.
Audits reduce the risk of a bug in the pool’s contract and do not remove it. Fake tokens with a familiar ticker exist on every chain, so check the contract address. Each deposit also needs a token approval, which can stay live until you revoke it.
On a large pair, the question stays the one from the start: did fees beat the gap to holding? The next lesson applies it to Uniswap v3 and v4, where ranges multiply both the fees and the loss. Adding a position step by step, on a testnet first, comes in a later lesson.
