Impermanent Loss Explained — The Math, the Misconceptions, and When Fees Beat It

DeFi liquidity pool dashboards are full of seductive numbers — 40% APR, 120% APR, sometimes triple digits on exotic pairs. What the headline yield rarely advertises is the silent counter-force eating into it: impermanent loss. IL is the gap between holding two tokens in a pool and just holding them in your wallet, and it grows every time the pool’s price ratio diverges from the ratio you deposited at. Below is the formula, why “impermanent” is a misleading label, when fees actually beat IL, and a worked example you can run in your head.

What an AMM actually does

A constant-product automated market maker like Uniswap v2 enforces a simple invariant on every trade: x * y = k, where x and y are the two token reserves and k is a constant. When a trader buys ETH from an ETH/USDC pool, ETH leaves the pool and USDC enters, but the product of the two reserves stays fixed. The pool quotes prices as the ratio y / x, which means as ETH gets scarcer, each remaining ETH costs more USDC — the curve is the price oracle.

Crucially, an LP does not own specific token amounts. They own a percentage share of whatever the pool currently holds. If the pool rebalances because ETH ran up, the LP’s withdrawal will contain less ETH and more USDC than they put in. That mechanical rebalancing — selling the appreciating asset, buying the depreciating one, on every trade — is the source of impermanent loss.

The IL formula

For a 50/50 constant-product pool, the closed-form expression for impermanent loss is:

IL = 2 * sqrt(p) / (1 + p) − 1

Where p is the price ratio of the new price to the entry price (so p = 2 means the asset doubled). The result is always negative or zero. Plug in a few values:

• p = 1.25 (token up 25%): IL ≈ −0.62%
• p = 1.5 (token up 50%): IL ≈ −2.02%
• p = 2 (token doubled): IL ≈ −5.72%
• p = 4 (4x): IL ≈ −20.00%
• p = 5 (5x): IL ≈ −25.46%

The drag is asymmetric in time but symmetric in direction — a 50% drop produces the same IL as a 100% rise, because both represent a 2x divergence. And it is unbounded on the upside: a 100x move produces an IL of around 80%.

Why “impermanent” is misleading

The label comes from a textbook scenario in which prices return exactly to the entry ratio — at that moment the LP and the HODL baseline match, and the loss “disappears.” Trending markets almost never deliver that round trip. ETH that goes from $2,000 to $4,000 and stays near $4,000 for a year has, for any LP who needs to exit, locked in a permanent ~5.7% loss versus simply holding. Treat IL as a real, eventually-realized cost, not a paper concern.

When fees beat IL

LP profitability is fee income minus IL minus gas. Three setups tilt the math in the LP’s favor:

• High-volume, narrow-divergence pairs: blue-chip stable pairs (ETH/stETH, WBTC/cbBTC) trade billions per day with minimal price drift, and the fees compound while IL stays near zero.
• Stablecoin pools: USDC/USDT or DAI/USDC pools see price ratios that rarely leave a 0.999–1.001 band. IL is functionally zero, and even modest fees are net-positive.
• Concentrated liquidity in well-chosen ranges: when you have a defensible view that a pair will trade inside a band, narrowing the range multiplies fee capture. The trade-off is amplified IL inside the range and zero earnings outside it.

The yield comparison calculator is the right tool for sanity-checking whether a pool’s headline APR survives realistic IL assumptions.

Concentrated liquidity

Uniswap v3 and its forks let LPs concentrate capital between a min and max price. Inside that range the LP behaves like a leveraged v2 position — same IL formula, but applied to the boosted virtual reserves. A 10x concentration multiplier means 10x the fees per dollar and roughly 10x the IL when price moves inside the range. If price exits the range, the LP holds 100% of the now-cheaper asset and earns nothing until price returns or they re-range. Concentrated LPing is closer to running a market-making strategy than to a passive savings product.

Worked example

Deposit $10,000 into an ETH/USDC v2 pool at 1 ETH = 2,000 USDC. That is 2.5 ETH plus 5,000 USDC. ETH then doubles to 4,000 USDC. The pool rebalances along x * y = k, so the LP now holds approximately 1.768 ETH and 7,071 USDC, worth about $14,142.

Compare to HODL: 2.5 ETH at $4,000 plus 5,000 USDC = $15,000.

The gap is $858, or 5.72% — exactly what the formula predicted at p = 2. The LP made money in dollar terms but underperformed the HODL baseline by 5.72%. If the pool paid 8% in fees over the same period, the LP would be ahead by roughly 2.3% net. If it paid 3%, the LP is behind by 2.7%. That is the breakeven calculation every LP needs to run before depositing.

Where to go next

Plug your own pool parameters into the impermanent loss calculator to see the full divergence curve, then compare LP yields against passive alternatives with the staking rewards calculator and the yield comparison calculator. Our DeFi calculator hub ties the IL math, staking yields, and lending rates together so you can model an entire on-chain allocation in one place.

This article is educational and not financial advice. DeFi positions carry smart-contract, oracle, depeg, and protocol-governance risks beyond impermanent loss. Always size positions to your own risk tolerance.