Liquidity Pools & AMMs

Impermanent Loss Explained with Worked Examples

Spot the edge. Swoop in.

Impermanent loss measures how a liquidity-provider position underperforms a simple hold benchmark due to the AMM repricing inventory. Fees and incentives may offset it, but the relationship depends on price movement magnitude, not direction.

By Swoopr Editorial Team

Published · Updated

AI-assisted content · Swoopr is responsible for the final published article.

Direct Answer

Impermanent loss is the difference between the value of an AMM liquidity position and the value of a specified hold benchmark after relative token prices change. It is a relative-performance measure, not necessarily a cash loss, and fees or incentives may offset it. The effect becomes realized economically when the position is closed or otherwise valued against the benchmark.

Key Takeaways

What This Page Covers

This guide uses a simple 50/50 constant-product model to build intuition, then explains where the model stops applying. It should be paired with protocol-specific documentation and the calculator specification in this package.

Define the Benchmark First

Suppose a provider deposits equal dollar values of Token A and Token B. The hold benchmark is the value of retaining those original quantities outside the pool.

The LP portfolio changes quantities as traders move the pool price. If Token A rises relative to Token B, arbitrage removes A and adds B. The LP ends with less of the outperforming A and more of B than the holder.

The difference between the LP value and hold value is commonly called impermanent loss or divergence loss. If both portfolios gain in dollar terms, the LP can still have impermanent loss because it gained less than holding.

Simple Constant-Product Formula

For a 50/50 constant-product pool, a common simplified relative-value formula is:

LP value ÷ hold value = 2 × √r ÷ (1 + r)

where r is the new relative price divided by the initial relative price.

Impermanent loss percentage is:

IL = 2 × √r ÷ (1 + r) - 1

If the relative price doubles, r = 2, producing about −5.72% before fees. If the relative price quadruples, the simplified result is −20%. The same magnitude applies to the reciprocal move under the symmetric model.

The formula assumes a full-range 50/50 constant-product pool and ignores fees, incentives, gas, taxes, and protocol-specific mechanics.

Fees Can Offset—but Not Erase—the Concept

Fee income adds to the LP portfolio. If gross divergence is 5.72% and fees add 7% relative to the starting capital over the same period, the LP may outperform the no-fee LP and possibly the hold benchmark.

But comparing percentage figures requires consistent denominators and timing. Fees accrue as assets inside or claimable from the position. Their value changes with token prices. Incentives may be paid in another token and should be valued after dilution, price movement, and conversion costs.

High volume can help, but volume during rapid repricing may be adverse for LPs. A backtest or historical fee estimate must account for active liquidity and the provider's actual range.

Concentrated Liquidity

A concentrated position provides liquidity only within a selected range. Within the range, capital can earn more fees per dollar because it is deployed more efficiently. Outside the range, the position becomes one-sided and stops earning swap fees.

The simple full-range formula is not sufficient for concentrated positions. Returns depend on range boundaries, entry price, path, time in range, fee growth, rebalancing, gas, and the prices at each rebalance. Narrow ranges behave more like active inventory strategies than passive deposits.

When the Loss Is Economically Realized

The word "impermanent" refers to the possibility that relative prices return and narrow the divergence. It should not be interpreted as unreal or harmless.

A user makes an economic decision whenever the position is valued, used as collateral, migrated, rebalanced, or closed. If the original portfolio objective was to retain a fixed quantity of Token A, the LP has already changed that exposure while the price moved. Waiting for a reversal introduces a market forecast and opportunity cost.

Practical Decision Framework

Evaluate an LP position with HOLD+:

Use several price paths, not one endpoint.

Worked Example

A provider deposits $5,000 of A and $5,000 of B when A costs $100. The deposit contains 50 A and 5,000 B.

A later costs $200. Holding is worth:

Under the simplified constant-product model, the no-fee LP value is about 94.28% of hold value:

Relative divergence is approximately $858, or 5.72% of hold value. If accumulated fees and rewards are worth $1,100 after costs, the LP could be worth roughly $15,242 and outperform holding by about $242. If reward value later falls by $600, the advantage disappears.

The result depends on the benchmark, fee timing, asset prices, and costs—not the IL formula alone.

Common Mistakes

Risks and Limitations

The simplified formula does not apply to every AMM. Stable curves, weighted pools, concentrated positions, dynamic fees, hooks, rebasing assets, and multi-asset pools require different models. Pool contracts and tokens can fail independently of divergence.

Historical fee income can be regime-dependent. A period with high volume and low net price movement may not resemble a future period with one-directional repricing or lower liquidity.

Practical Checklist

Frequently Asked Questions

Can fees eliminate impermanent loss?

Fees can offset relative underperformance, but the result depends on volume, active share, price path, costs, and the comparison date.

Is impermanent loss only a loss when I withdraw?

Withdrawal fixes the final asset mix for that position, but the economic exposure changes while the position is open. Valuation and opportunity cost exist before withdrawal.

Does impermanent loss occur in stablecoin pools?

It can, especially during depegs. Different pool curves change the shape of the effect but do not eliminate asset and contract risk.

Is a narrow range always better?

No. It can earn more fees while active but becomes one-sided sooner and may require costly management.

Why can an LP make money and still have impermanent loss?

Because the LP can rise in dollar value while rising less than the hold benchmark.

Summary

Impermanent loss is best understood as benchmark-relative portfolio behavior. The LP automatically changes inventory as relative prices move. Net performance requires adding fees and incentives, subtracting costs, and modeling the actual pool design and range.

Sources and Further Reading

Educational disclaimer: Educational information only; not investment, tax, legal, or personalized financial advice. DeFi positions can lose some or all committed assets through market movement, liquidation, smart-contract failure, governance action, oracle failure, bridge failure, stablecoin instability, operational mistakes, fraud, or other causes.

Continue Learning