Crypto Gloss

Impermanent Loss Math

Definition:

Impermanent loss (IL) is the difference between the value of assets held passively versus the value of the same assets deposited into an AMM liquidity pool — the loss arising because the constant product formula forces the pool to rebalance as relative prices change. If the price of deposited tokens diverges from the entry price ratio, the LP ends up with less value than if they had simply held. The word “impermanent” refers to the fact that if prices return exactly to entry levels, the loss disappears — but this often does not happen in practice, making the “impermanent” label somewhat misleading.

The Math

A constant product AMM maintains: $x cdot y = k$

Where $x$ and $y$ are the quantities of the two tokens in the pool, and $k$ is a constant.

Setup:

Assume a pool with 1 ETH and 1,000 USDC when ETH = $1,000.

  • Initial position value: $2,000 (50/50 split)
  • $k = 1 cdot 1000 = 1000$

Price changes to ETH = $4,000:

New pool state (derived from $x cdot y = 1000$):

  • New ETH quantity: $x = sqrt{k / P} = sqrt{1000/4000} = 0.5$ ETH
  • New USDC quantity: $y = sqrt{k cdot P} = sqrt{1000 cdot 4000} = 2000$ USDC

LP position value at $4,000 ETH:

  • 0.5 ETH × $4,000 = $2,000
  • 2,000 USDC = $2,000
  • Total LP value: $4,000

Hold value (1 ETH + 1,000 USDC at $4,000 ETH price):

  • 1 ETH × $4,000 = $4,000
  • 1,000 USDC = $1,000
  • Total hold value: $5,000

Impermanent loss: $5,000 – $4,000 = $1,000 (20% of hold value)

IL Formula

For a price ratio change of $r$ (new price / initial price), impermanent loss as a percentage of hold value is:

$$IL = frac{2sqrt{r}}{1+r} – 1$$

Price Change ($r$) IL % (vs hold)
1.25× 0.6%
1.5× 2.0%
2× 5.7%
4× 20.0%
0.5× 5.7%
0.25× 25.5%

Note: IL is symmetric in percentage terms for equal magnitude price moves in either direction (a 2× gain and a 0.5× loss produce the same IL %).

Impermanent Loss vs. Fees

LP returns = fee income − impermanent loss

In stable or low-volatility markets, fee income can exceed IL:

  • A stablecoin pool (USDC/USDT) has near-zero IL but earns constant trading fees
  • A highly volatile token pair may earn high fees but suffer substantial IL

Break-even analysis: An LP must earn enough in fees during their deposit period to offset the IL incurred from price movement. In practice:

  • Stable pairs: fees almost always exceed IL
  • Correlated pairs (BTC/ETH): IL is moderate, fees often compensate
  • Uncorrelated volatile pairs (MEME/ETH): IL can dramatically exceed fee income

Concentrated Liquidity and IL

Uniswap v3’s concentrated liquidity amplifies both fee income and impermanent loss. If the price moves outside an LP’s specified range:

  • The LP earns zero fees (position becomes 100% of one token)
  • All of the deposited tokens have been swapped to the less-valuable token (maximum IL for that range)

Concentrated liquidity does not reduce IL — it compresses the IL experience into a narrower price range.

Common Misconceptions

“Just leave it long enough and IL goes away.”

IL disappears only if prices return exactly to entry ratio — which cannot be predicted and is often unlikely for volatile token pairs that trend in one direction.

“Fee APY offsets IL.”

Quoted APYs are often backward-looking (based on recent fee income). IL is calculated at exit. High APY periods can be followed by low-volume periods where IL accumulates without fee compensation.

“IL doesn’t matter if I’m bullish on both tokens.”

Even if both tokens go up, IL still occurs relative to holding. An LP in a pool where one token goes up 10× while the other stays flat experiences significant IL compared to simply holding both.

Related Terms

Sources

Last updated: 2026-04