Research note · 2026-11-06 · Method: constant-product math per the Uniswap v2 Core whitepaper; all position figures hypothetical
Impermanent loss is the shortfall between the value of a liquidity position and the value of simply holding the same tokens, caused by a change in their relative price. It is deterministic. Given the price ratio at entry and at exit, the loss on a 50/50 constant-product pool can be calculated exactly, before fees.
The “impermanent” label is a misnomer that has stuck. The loss reverses only if the price returns to the entry ratio. If the provider withdraws at a different ratio, it is realized.
Why it happens
A constant-product pool (x × y = k, per the Uniswap v2 whitepaper) has no view of the outside price. When the market price of token A rises, arbitrageurs buy A from the pool at the stale, cheaper price until the pool’s ratio matches the market. The pool ends up holding less of the asset that went up and more of the asset that went down. Since the arbitrage profit comes from the pool, providers are the counterparty to every rebalancing trade. The result is the same whether A rises or falls: any move away from the entry ratio leaves the position worth less than the untouched tokens.
The formula
Let r be the ratio of the new price of A (in terms of B) to its price at deposit. For a 50/50 constant-product position, ignoring fees:
IL = 2 × √r / (1 + r) − 1
Derivation in three lines:
- With reserves x and y and k = xy, the pool’s price is p = y / x, so x = √(k / p) and y = √(k × p).
- Pool value in B terms at price p: x × p + y = 2√(k × p). If the price moves from p₀ to p₀r, pool value scales by √r.
- Holding value scales by (1 + r) / 2, since half the original value was in A (which moved by r) and half in B (unchanged). The ratio of pool value to holding value is √r ÷ ((1 + r) / 2) = 2√r / (1 + r).
The formula is symmetric in a useful way: r and 1/r give the same loss. A 2x move up and a 50% move down both cost 5.72%.
The table
| Price ratio r | Move in A vs B | IL vs holding |
|---|---|---|
| 0.10 | −90% | −42.50% |
| 0.25 | −75% | −20.00% |
| 0.50 | −50% | −5.72% |
| 0.75 | −25% | −1.03% |
| 1.00 | 0% | 0.00% |
| 1.25 | +25% | −0.62% |
| 1.50 | +50% | −2.02% |
| 2.00 | +100% | −5.72% |
| 3.00 | +200% | −13.40% |
| 4.00 | +300% | −20.00% |
| 5.00 | +400% | −25.46% |
Small moves cost little; large moves cost a lot and the curve is convex. The figures are relative to holding, not to the starting dollar value. A position can show impermanent loss and still be up in dollar terms if both tokens rose.
Worked example
Hypothetical position. A provider deposits 10 TKN and 1,000 USDC into a 50/50 pool when TKN trades at 100 USDC. Starting value: 2,000 USDC. The provider’s share of the pool has k = 10 × 1,000 = 10,000.
TKN then doubles to 200 USDC elsewhere, and arbitrage brings the pool’s price to 200 (r = 2).
- TKN in the position: √(10,000 / 200) ≈ 7.07 TKN
- USDC in the position: √(10,000 × 200) ≈ 1,414.21 USDC
- Position value: 7.07 × 200 + 1,414.21 ≈ 2,828.43 USDC
- Value if the provider had just held: 10 × 200 + 1,000 = 3,000 USDC
- Shortfall: 171.57 USDC, or 5.72% of the holding value, matching the table
In dollar terms, the position is up from 2,000 to about 2,828. Relative to doing nothing, it is down about 172. Those 2.93 TKN were sold to arbitrageurs on the way up at prices between 100 and 200.
The same position on the way down
Run the opposite case on the same hypothetical deposit. TKN falls to 50 USDC (r = 0.5).
- TKN in the position: √(10,000 / 50) ≈ 14.14 TKN
- USDC in the position: √(10,000 × 50) ≈ 707.11 USDC
- Position value: 14.14 × 50 + 707.11 ≈ 1,414.21 USDC
- Holding value: 10 × 50 + 1,000 = 1,500 USDC
- Shortfall: 85.79 USDC, again 5.72% of the holding value
The percentage matches the 2x case, as the symmetry predicts, but the dollar picture is different: here the provider is down against the starting 2,000 on both measures. On the way down, the pool kept buying TKN from arbitrageurs with USDC, so the provider finishes with more of the falling asset. That is the part of the mechanism most often missed: a 50/50 pool is structurally short volatility, buying what falls and selling what rises.
The fee break-even
Fees are the only offset. For the position above to match holding at withdrawal, it would need to have collected about 171.57 USDC of fees over its life, roughly 8.6% of the initial 2,000 deposit. Whether that happens depends on volume through the pool, the fee tier, the provider’s share of liquidity and the time held, none of which the formula captures.
A practical framing for analysis: estimate fee income per unit of liquidity from historical volume, then ask what price move over the expected holding period would erase it, using the table. Uniswap’s fee docs give the current fee splits (on v2 pools, 0.25% to LPs and 0.05% to the protocol since protocol fees were enabled after the December 2025 UNIfication vote).
Variations
- Concentrated liquidity. In Uniswap v3 and v4, a position covers a chosen price range. Uniswap’s concentrated liquidity page explains that as price moves, the position converts toward a single asset until it consists entirely of one token at the range boundary. Inside a narrow range, the same price move produces a larger loss per dollar than the full-range formula shows, in exchange for a larger share of fees. Once price exits the range, the position stops earning and stops rebalancing.
- Correlated pairs. Two dollar stablecoins, or ETH and a liquid staking token, should keep r near 1, so the formula gives a small number. The risk shifts to a depeg, where r moves sharply and the pool absorbs the weaker asset.
- Weighted pools. An 80/20 pool loses less to the same move than a 50/50 pool because less of the position is rebalanced; the formula changes accordingly.
- Single-sided or “IL-protected” products. These move the loss somewhere else, typically to a treasury, an insurance fund or token emissions. The protection is only as good as the backstop paying it.
What the formula leaves out
Gas costs, MEV extracted from the pool, the time value of capital, and token incentives that may be paid on top. It also assumes the pool price tracks the market exactly, which is approximately true for deep pools and less true for thin ones.
This is arithmetic about a mechanism, not a view on whether providing liquidity to any pool makes sense for a given reader.
Related notes: liquidity pools, automated market makers, yield farming.