CenturionDEX

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 rr relative to the other:

IL(r)=2r1+r−1\text{IL}(r) = \frac{2\sqrt{r}}{1+r} - 1

This always yields a negative number (a loss) for any r≠1r \ne 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=11−Plower/Pupper\text{multiplier} = \frac{1}{1-\sqrt{P_{\text{lower}}/P_{\text{upper}}}}

A CTN/USDC position with range 1,5001{,}500–2,5002{,}500 USDC at a current price of 2,0002{,}000 USDC 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 rangeWider range
Higher fee APR (when in range)Lower fee APR
Greater IL amplificationIL closer to v2 levels
Higher probability of going out of rangeStays in range longer
Requires more active managementMore 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_valuefee_tier⋅LP_share⋅days\text{Required daily volume} = \frac{\text{IL}\cdot \text{position\_value}}{\text{fee\_tier}\cdot \text{LP\_share}\cdot \text{days}}

Worked example: A $10,000 position in a 0.30%0.30\% pool with 5%5\% of in-range liquidity. After 30 days CTN has doubled (r=2r=2):

v2 IL at 2×=5.7%\text{v2 IL at }2\times = 5.7\% IL in dollars=10,000⋅5.7%=570\text{IL in dollars} = 10{,}000 \cdot 5.7\% = 570 Required total fees over 30 days=570\text{Required total fees over 30 days} = 570 Required daily pool volume=5700.003⋅0.05⋅30=126,667 / day\text{Required daily pool volume} = \frac{570}{0.003 \cdot 0.05 \cdot 30} = 126{,}667 \text{ / day}

If 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\text{LP\_share} by the concentration multiplier mm — but IL is also amplified by mm, 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?.