Impermanent loss is the shortfall a liquidity provider takes when the price of one token in a pool moves away from where it was at deposit. You compare the value of your share of the pool with what the same tokens would be worth if you had kept them in your wallet. In a standard 50/50 constant-product pool, a twofold price move leaves the position about 5.7% behind holding. The move can be up or down. A fivefold move leaves it about 25.5% behind. Trading fees and rewards can offset the gap, but the formula itself leaves them out.
Why it happens
A typical decentralised exchange has no order book. It runs on an automated market maker (AMM): a smart contract holds two tokens and sets the price from the ratio between them. The most common design is the constant product rule, where the quantity of token X multiplied by the quantity of token Y stays fixed (x · y = k).
When the market price of a token moves elsewhere, the pool's price lags behind. Arbitrageurs then buy the token that has become cheap in the pool and sell the one that has become expensive, until the pool's ratio matches the outside market. The result for liquidity providers is that the pool always ends up holding more of the token that fell and less of the token that rose. Compared with simply holding, the provider has sold some of the winner and bought more of the loser.
The word "impermanent" means the shortfall is only on paper while the funds stay in the pool. If prices return to the ratio at deposit, it disappears. If you withdraw while prices are different, it becomes a realised loss. Many practitioners prefer the term "divergence loss", because prices often never return.
The formula
For a 50/50 constant-product pool, with r = (price ratio at withdrawal) ÷ (price ratio at deposit):
Impermanent loss = 2√r ÷ (1 + r) − 1
The result is a percentage of the value you would have had by holding, measured at the moment of withdrawal, not at deposit. It ignores fees and rewards. The formula is symmetric: a 2x rise and a 50% fall (r = 0.5) give the same loss.
| Price change of one token vs the other | r | Loss vs holding |
|---|---|---|
| +10% | 1.10 | 0.11% |
| +25% | 1.25 | 0.62% |
| +50% | 1.50 | 2.02% |
| +100% (2x) | 2.00 | 5.72% |
| +200% (3x) | 3.00 | 13.40% |
| +300% (4x) | 4.00 | 20.00% |
| +400% (5x) | 5.00 | 25.46% |
| −10% | 0.90 | 0.14% |
| −20% | 0.80 | 0.62% |
| −50% | 0.50 | 5.72% |
| −90% | 0.10 | 42.50% |
Small moves cost little. The loss grows quickly once one token moves several times against the other.
Turning the percentage into dollars
Multiply the percentage by the value you would have had if you had held, at the price at withdrawal, not by the amount you deposited. Suppose you deposit $500 of ETH and $500 of a dollar stablecoin, and ETH then rises 50%. Holding would be worth $1,250. The pool position is worth about $1,224.74. The gap is about $25.26, which is 2.02% of $1,250.
Worked example
- A pool holds 10 ETH and 1,000 USDC, so 1 ETH = 100 USDC. You deposit 1 ETH and 100 USDC ($200), which gives you 10% of the pool.
- The constant is 10 × 1,000 = 10,000.
- ETH doubles to $200 on outside markets. Arbitrage rebalances the pool until the ratio matches. The pool now holds √(10,000 ÷ 200) ≈ 7.071 ETH and ≈ 1,414.21 USDC.
- Your 10% share is 0.7071 ETH ($141.42) plus 141.42 USDC: about $282.84.
- If you had simply held 1 ETH and 100 USDC, you would have $300.
- The difference, about $17.16, is the impermanent loss: 5.72% of $300, exactly as the formula predicts for r = 2.
You still made money in dollar terms ($200 to $282.84). Impermanent loss measures what you gave up relative to holding, not necessarily a fall below your deposit.
How far behind holding is your pool share?
Enter the price ratio at withdrawal divided by the price ratio at deposit to see how far a 50/50 constant-product pool position trails simply holding the same tokens.
| Price ratio r (ratio at withdrawal ÷ ratio at deposit; 2 = one token doubled against the other, 0.5 = it halved) | Impermanent loss as a share of the value from holding at withdrawal |
|---|---|
| 2 x | 5.72 % |
| 1.5 x | 2.02 % |
| 0.5 x | 5.72 % |
| 5 x | 25.46 % |
| 0.1 x | 42.5 % |
- The price ratio is your own input, not a figure from any source.
- The result is a percentage of what the same tokens would be worth if held, measured at withdrawal, not at deposit; multiply it by that held value to get dollars.
- The formula is symmetric: r = 2 and r = 0.5 give the same loss.
This applies only to a standard 50/50 constant-product pool before fees: it leaves out trading fees, rewards, network fees and smart contract risk, and does not fit weighted or concentrated-liquidity pools, which follow different curves.
What the research adds
Fees can outweigh the loss, but not always. In a Monte Carlo study (Hafner and Dietl, 2024), price changes did not necessarily lead to losses for liquidity providers once fees were included. The study found that fees paid by traders and by arbitrageurs matter equally. A 2021 paper on weighted AMMs (Boueri, revised 2022) reported a different result: the median pool it studied had a net return of roughly zero after accounting for impermanent loss.
Impermanent loss is not the whole picture. Milionis, Moallemi, Roughgarden and Zhang (first version August 2022, last revised August 2026) argue that impermanent loss depends on an arbitrary starting point and mixes market exposure with the costs of running a pool. They propose "loss-versus-rebalancing" (LVR) instead. LVR isolates what liquidity providers lose to arbitrageurs, set against the fees they earn. Applied to a large ETH-USDC pool, their decomposition attributes over 99.99% of the variance in liquidity providers' returns to plain market exposure.
Ways to reduce exposure
Pair assets that move together. The loss depends only on how far the two prices diverge. A pool of ETH and a dollar stablecoin is exposed to every move in ETH. A pool of ETH and a volatile governance token is exposed to both tokens' moves against each other.
Stablecoin-to-stablecoin pools carry little divergence, but not zero. Two dollar stablecoins normally trade near $1, so the loss stays small. The risk returns if one loses its peg. On 10 March 2023, Circle said $3.3 billion of the roughly $40 billion in USDC reserves was held at the failed Silicon Valley Bank. USDC lost its $1 peg on some DeFi platforms that weekend, as The Block reported.
Keep part of the position outside the pool. The loss applies only to the share you deposit. With half your capital in the pool and half held, the dollar loss on the total halves. Your share of fees halves too.
Weighted pools. Some AMMs offer pools with uneven weights such as 80/20 or 60/40. A heavier weight on one token reduces exposure to divergence from the lighter token. The trade-off is lower diversification, and potentially higher losses if the overweighted token underperforms.
Be careful with concentrated liquidity. Some AMMs let providers place liquidity in a chosen price range. This earns more fees per dollar while the price stays in range, but the impermanent loss inside that range is larger than in a full-range position. If the price leaves the range, the position is converted entirely into one token. Deng, Zong and Wang (2022, revised 2023) show that this loss can be replicated, and therefore hedged in principle, with options struck inside the range.
Count every return and cost. Weigh trading fees and liquidity-mining rewards against the expected divergence and network fees for deposits and withdrawals. Also weigh smart contract risk, which can cost far more than price divergence.
Bottom line
Impermanent loss is the cost of letting an AMM rebalance your two tokens for you. It is zero when prices end where they started and grows with the size of the move, whichever direction it goes. The formula gives the size of the loss for any price change. Whether a pool pays depends on whether fees and rewards exceed the loss over the time you stay in.
The figures above apply to a standard 50/50 constant-product pool before fees. Weighted and concentrated pools follow different curves. None of this is investment advice.
