Impermanent Loss
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What is impermanent loss?
Impermanent loss (IL) is the difference in portfolio value between holding tokens inside a liquidity pool and simply holding them in a wallet. It arises whenever the relative price of the two assets changes from the ratio at deposit.
The loss is called "impermanent" because it reverses if the price returns to the original ratio. If the provider withdraws while the price has diverged, the loss crystallises.
The formula
For a full-range (v2) position, if one token's price changes by a factor r relative to the other:
IL(r) = 2 * sqrt(r) / (1 + r) - 1This always yields a negative number (a loss) for any r ≠ 1.
Worked examples
| Price change (r) | IL |
|---|---|
| 1.25× (25 % up) | −0.6 % |
| 1.50× (50 % up) | −2.0 % |
| 2× (double) | −5.7 % |
| 3× (triple) | −13.4 % |
| 5× | −25.5 % |
| 10× | −42.5 % |
| 0.5× (halves) | −5.7 % |
| 0.1× (90 % drop) | −42.5 % |
IL is symmetric: a 2× increase and a 0.5× decrease produce the same loss.
IL in v2 vs v3
In v2, liquidity spans the entire price curve (0 to ∞). In v3, providers concentrate capital within a chosen range [P_lower, P_upper]. Concentration amplifies both fee income and impermanent loss by the same capital-efficiency multiplier:
multiplier = 1 / (1 - sqrt(P_lower / P_upper))A CTN/USDC position with range $1,500–$2,500 at a current price of $2,000 has a multiplier of roughly 4.2×:
- Fee income behaves as though the position held 4.2× the capital in a v2 pool.
- IL is likewise amplified by 4.2× for any price movement within the range.
- If the price exits the range entirely, the position holds 100 % of the less valuable token.
The range-width tradeoff
| Tighter range | Wider range |
|---|---|
| Higher fee APR (when in range) | Lower fee APR |
| Greater IL amplification | IL closer to v2 levels |
| Higher probability of going out of range | Stays in range longer |
| Requires more active management | More passive |
Break-even volume
The most practical question for any LP: how much daily volume must the pool sustain for fees to offset IL?
Setting fees equal to IL and solving for volume:
Required daily volume = (IL * position_value) / (fee_tier * LP_share * days)Worked example: A $10,000 position in a 0.30 % pool with 5 % of in-range liquidity. After 30 days CTN has doubled (r = 2):
v2 IL at 2× = 5.7 %
IL in dollars = $10,000 * 5.7 % = $570
Required total fees over 30 days = $570
Required daily pool volume = $570 / (0.003 * 0.05 * 30) = $126,667 / dayIf the pool exceeds ~$127 K/day, fees outpace IL even after a 2× price move. At $5 M/day, fees dominate IL by a wide margin.
For a concentrated v3 position, multiply LP_share by the concentration multiplier m — but IL is also amplified by m, so the break-even volume remains roughly unchanged. Concentration scales both sides of the ledger equally.
Key takeaways
- IL is the cost of providing liquidity. A position is profitable only when cumulative fees exceed cumulative IL.
- In v3, narrower ranges amplify both fees and IL. The optimal width depends on expected volatility and the provider's willingness to rebalance.
- Volatile pairs benefit from wider ranges (or full range) to limit large IL events.
- Stable pairs (e.g., USDC/DAI) can support very tight ranges because the price rarely deviates far.
For the basic definition, see the glossary entry. For the constant-product derivation, see How are token prices determined?.