Impermanent Loss
Understand impermanent loss, the constant-product formula, how relative price changes alter LP inventory and when fees may or may not offset the loss.
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Impermanent loss is the opportunity-cost difference between holding assets in an AMM and simply holding the same starting assets outside the pool after their relative price changes.
Learning objectives
- Define impermanent loss as a relative-performance measure.
- Use the constant-product IL formula for a simple 50/50 pool.
- Explain why volatility, correlation and fee income determine LP economics.
What it is
In a 50/50 constant-product AMM, arbitrage continuously changes the LP’s token quantities as the external price moves. When one token rises relative to the other, the pool sells some of the outperforming token and accumulates more of the underperforming token.
Impermanent loss compares the resulting LP value with a passive holder who kept the original quantities. It is not necessarily an absolute cash loss: the LP position can still rise in value while underperforming the hold strategy.
How it works
For a simple 50/50 constant-product pool, the no-fee impermanent-loss percentage relative to holding is 2√r ÷ (1+r) − 1. If one asset doubles relative to the other, r=2 and IL is about −5.72%.
The same formula is symmetric: if the relative price halves, r=0.5 and the IL magnitude is also about −5.72%. Larger relative moves create larger underperformance.
Fees matter because volatility can generate trading volume. A high-volume pool may earn enough fees to compensate for IL; a volatile but low-volume pool may not. Incentive tokens can temporarily improve headline returns while adding dilution and token-price risk.
Impermanent loss is lower for highly correlated assets in designs tailored to stable relationships, but correlation can break. Stablecoin pools, for example, can produce severe inventory concentration if one asset permanently depegs.
How to analyse it
Evaluate IL alongside fee income, token quality and the probability distribution of relative prices. A single APY snapshot is not enough.
| Check | Why it matters | What to verify |
|---|---|---|
| Relative volatility | IL is driven by relative, not absolute, price change. | Study the price ratio of the two assets. |
| Fee income | Fees can offset IL but vary with volume and competition. | Use realised fee revenue per unit of active liquidity. |
| Incentives | Token rewards can dominate headline APR temporarily. | Separate fee APR from emissions and mark rewards to realistic prices. |
| Exit state | A depeg can leave LPs concentrated in the failing asset. | Stress permanent rather than temporary divergence. |
The correct comparison is “LP versus hold,” not “LP versus zero return.” This avoids mistaking a rising nominal balance for outperformance.
For concentrated liquidity, the simple 50/50 formula is no longer sufficient because the position can move fully into one asset when price leaves the chosen range.
Worked example and thought exercise
You deposit £10,000 of token A and £10,000 of stablecoin value into a 50/50 constant-product pool. Token A later doubles relative to the stablecoin. Ignoring fees, the LP position underperforms the value of simply holding the starting assets by about 5.72%.
If the pool earned 8% in fees over the same period, those fees could more than offset the 5.72% IL before gas, taxes and other risks. If fee income was only 2%, it would not.
Thought exercise: why can a stablecoin depeg create much worse LP economics than ordinary short-lived volatility?
Common mistakes and practical workflow
- Calling IL an absolute loss rather than relative underperformance.
- Assuming fees always compensate for IL.
- Using the simple 50/50 formula for concentrated or non-standard AMMs.
- Ignoring the possibility of permanent token failure or depeg.
Practical workflow
- Choose the correct AMM model and benchmark.
- Measure relative price change between the pooled assets.
- Estimate IL under several stress ratios.
- Add realised fee income and incentives separately.
- Stress a permanent depeg or one-sided exit state.
✅ Knowledge checkpoint
- What benchmark is used to define impermanent loss?
- What is the approximate IL if one asset doubles in a simple 50/50 constant-product pool?
- Why can a position gain money yet still suffer impermanent loss?
- How do fees change the LP-versus-hold comparison?
FAQs
❓ Why is it called impermanent?
If prices return to the starting ratio before withdrawal, the relative loss can shrink. Once withdrawn after divergence, the underperformance is realised.
❓ Does IL only happen when prices fall?
No. Any relative price divergence can create IL, including one asset rising strongly.
❓ Can fees eliminate IL?
They can offset it economically, but the rebalancing mechanism still exists.
❓ Does the formula apply to every AMM?
No. It is a useful benchmark for a simple 50/50 constant-product pool; other curves and concentrated liquidity require different calculations.
📋 Summary
Impermanent loss is the opportunity cost created by AMM rebalancing relative to passive holding. Sound LP analysis combines the correct IL model with realised fees, incentives, token quality and stress scenarios.
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