Last updated: July 2, 2026
Impermanent loss (IL) is the gap between what your liquidity-pool position is worth and what you would have had if you had simply held the two tokens in your wallet. The formula measures that gap precisely — it is not a fee, not a bug, and not a hidden charge. It is the mathematical consequence of an automated market maker rebalancing your position every time the price moves. This page derives the closed-form IL formula from first principles, walks through a numeric example you can check by hand, and gives you a lookup table so you can read the IL percentage straight off a price move.
Throughout, "HODL" means the benchmark of holding your original tokens untouched. IL is always measured against HODL: a negative number means the pool underperformed just holding. Everything below assumes a standard constant-product pool (Uniswap V2 and its many forks); the same logic underpins how the formula changes for V3 concentrated liquidity and the formula at near-1:1 price ratios.
A Uniswap-V2-style pool holds two reserves, x and y, and enforces one rule on every trade:
x · y = k
The product k stays constant (fees aside), so the reserves trace a hyperbola. The price of token X in terms of token Y is the ratio of reserves, p = y / x. When X's market price rises, arbitrageurs buy X out of the pool until the pool's ratio matches the market — which means the pool automatically sells the token that went up and buys the one that went down. That forced "sell the winner, buy the loser" behaviour is the entire source of impermanent loss. You end up holding less of the appreciating asset than a HODLer would.
Let the price of X change by a factor k — the relative price factor, equal to new price ÷ entry price. If ETH goes from $1,000 to $2,000, then k = 2.
Start with reserves x₀, y₀ and product K = x₀·y₀. After the price moves, the pool must satisfy two conditions at once: the product is unchanged (x₁·y₁ = K) and the new price equals k times the old (y₁/x₁ = k · y₀/x₀). Solving the pair gives the new reserves:
x₁ = x₀ / √k and y₁ = y₀ · √k
Now compare two portfolios, both valued at the new price. The HODL value is the original tokens marked to market; the pool value is the rebalanced reserves marked to market. Dividing the pool value by the HODL value and simplifying (the x₀, y₀ terms cancel because a 50/50 pool starts balanced) collapses to a single clean expression:
IL = 2·√k / (1 + k) − 1
Two things fall straight out of this. First, IL is always ≤ 0 — the geometric mean in the numerator can never beat the arithmetic mean in the denominator, so the pool never beats HODL on price alone. Second, IL depends only on the ratio k, not on the dollar size of your position. A full derivation with the intermediate algebra lives in the methodology section on the calculator page.
Suppose you deposit 1 ETH + 1,000 USDC when ETH = $1,000. The pool starts balanced (both sides worth $1,000) with K = 1 × 1,000 = 1,000. Then ETH doubles to $2,000, so k = 2.
1 / √2 = 0.7071 ETH1,000 · √2 = 1,414.21 USDC0.7071 × $2,000 + $1,414.21 = $2,828.431 × $2,000 + $1,000 = $3,000.00The shortfall is 2,828.43 / 3,000 − 1 = −5.72%. Plug k = 2 into the formula and you get the identical number: 2·√2 / 3 − 1 = −5.72%. The pool did make money in dollar terms (you started with $2,000 of value and ended with $2,828), but it made $172 less than simply holding would have. That $172 gap is the impermanent loss.
Because IL depends only on k, you can tabulate it once and reuse it forever. These are exact values from the formula:
| Price move | k | Impermanent loss |
|---|---|---|
| +25% (or −20%) | 1.25 | −0.6% |
| +50% (or −33%) | 1.5 | −2.0% |
| 2× (or −50%) | 2 | −5.7% |
| 3× (or −67%) | 3 | −13.4% |
| 4× (or −75%) | 4 | −20.0% |
| 5× (or −80%) | 5 | −25.5% |
Notice the parentheses: a move to k and a move to 1/k produce the same IL. A price that doubles and a price that halves both cost you 5.7%, because the formula is symmetric in k and 1/k. IL cares about the magnitude of divergence between the two assets, not the direction.
The percentage alone doesn't pay your bills — the dollar figure does. Converting is one multiplication:
IL$ = HODL value × |IL%|
In the example above, HODL value was $3,000 and IL was 5.7%, so IL$ = 3,000 × 0.0572 = $172. This is the number that actually decides break-even, because your trading fees are also a dollar amount. If the pool paid you more than $172 in fees over the period, you came out ahead of HODL; if it paid less, you would have been better off holding. That comparison — converting IL% into a break-even fee APR — is where the real LP decision gets made.
Not every pool is 50/50. Balancer lets you set arbitrary weights — 80/20, 98/2, and so on. The general value ratio for a weighted pool is:
value ratio = ∏ rᵢ^wᵢ ÷ ∑ wᵢ·rᵢ
where rᵢ is each asset's price ratio and wᵢ its weight (the weights sum to 1). Skewing weight toward the asset you expect to move reduces IL, because the pool rebalances less of it away. Set both weights to 0.5 and one asset's ratio to k while the numeraire's ratio is 1, and this collapses exactly back to 2·√k/(1+k) − 1 — the V2 formula is just the 50/50 special case of the weighted formula.
False — it is symmetric. Downside moves cost exactly as much as the equivalent upside move, as the table's parentheses show.
Only if the price fully returns to your entry ratio. At k = 1 the formula gives 0% — the loss truly reverses. But if you withdraw at any other ratio, the loss is locked in and becomes permanent the moment you exit.
Fees and IL are separate quantities that you add together. IL is a price-divergence cost; fees are income. The formula above says nothing about fees — you have to compute both and net them.
Not until you withdraw. While your liquidity stays in the pool the loss is on paper — hence "impermanent." It becomes realized the moment you remove liquidity at a price ratio different from where you entered.
No. If you stake one asset and are only ever owed that same asset back, there is no rebalancing and no impermanent loss. IL requires a two-sided pool whose ratio can drift. (Single-sided vaults that route into an LP position behind the scenes still incur it.)
Concentrated liquidity amplifies IL. By confining your capital to a narrow price band you earn more fees per dollar, but you also absorb the same divergence with a fraction of the reserves — so a 2× move inside a tight range can cost several times the 5.7% shown here. See how the formula changes for V3 concentrated liquidity for the range-aware version.