Understanding Impermanent Loss

Concentrated-liquidity pools expose positions to the same standard impermanent loss that affects constant-product AMMs, but the effect is amplified because capital is packed into a narrower price band. Calculation below quantify each component and clarify how range width determines the overall impact.

1. Standard Impermanent Loss

For any two-token 50/50 pool, let $r = \frac{P_{\text{current}}}{P_{\text{initial}}}$ be the price ratio between the current and entry price. The relative loss versus holding the tokens outright is $IL_{\text{standard}}(r) = \frac{2 \sqrt{r}}{1 + r} - 1$

Understand and verify the key properties of the impermanent loss function:

• Symmetry: $IL_{\text{standard}}(r) = IL_{\text{standard}}\left(\frac{1}{r}\right)$

  • The loss is the same whether prices go up or down by the same proportion

• Bounded: Maximum loss is 50% as $r \to \infty$ OR $r \to 0$

  • The maximum possible loss is capped at 50%, regardless of how extreme the price movements become

• Zero loss: $IL_{\text{standard}}(1) = 0$

  • When prices return to their original ratio, the impermanent loss disappears

2. Amplification in Concentrated Liquidity

When liquidity is supplied only within an active price range $[P_l, P_u]$ the same price move consumes a larger share of the position’s assets, scaling the loss by the ratio of total to active range widths:

$$IL_{\text{concentrated}}(r) = IL_{\text{standard}}(r) \times \left(\frac{Range_{\text{total}}}{Range_{\text{active}}}\right)$$

where:

$Range_{\text{total}}$ is the full tick space of the protocol (e.g., from 0 to infinity).

$Range_{\text{active}} = \frac{P_u}{P_l}$ is the provider’s chosen active price band.

A tighter band (smaller $Range_{\text{active}}$ therefore multiplies impermanent loss by a larger factor, trading off capital efficiency for higher risk.

Concentrated liquidity creates a fundamental trade-off: while it offers much higher capital efficiency and fee earnings potential, it also amplifies impermanent loss risk. A tighter price band means higher potential returns but also higher risk when prices move outside the chosen range This concept is crucial for understanding why concentrated liquidity strategies require more active management and careful risk assessment compared to traditional AMM participation.