# What Is the Mathematical Formula for Calculating Impermanent Loss? ⎊ Path

**Published:** 2025-12-05
**Author:** Greeks.live
**Categories:** Path

---

## What Is the Mathematical Formula for Calculating Impermanent Loss?

The most common formula to estimate impermanent loss is: IL = (2 sqrt(price_ratio)) / (1 + price_ratio) – 1. In this formula, 'price_ratio' represents the price of the token at the time of withdrawal divided by the price at the time of deposit.

This calculation gives the percentage of loss compared to simply holding the assets. It's crucial to remember this formula does not account for trading fees earned, which can offset the loss.

The calculation quantifies the divergence loss due to volatility between the two assets in the pool.

- [What Are the Limitations of the Standard Impermanent Loss Formula in Multi-Asset or Weighted Liquidity Pools?](https://learn.greeks.live/path/what-are-the-limitations-of-the-standard-impermanent-loss-formula-in-multi-asset-or-weighted-liquidity-pools/)

- [What Are the Mathematical Formulas Used to Calculate Impermanent Loss in a Concentrated Liquidity Position?](https://learn.greeks.live/path/what-are-the-mathematical-formulas-used-to-calculate-impermanent-loss-in-a-concentrated-liquidity-position/)

- [How Does the Constant Product Formula (X Y=k) Directly Lead to the Phenomenon of Impermanent Loss?](https://learn.greeks.live/path/how-does-the-constant-product-formula-x-yk-directly-lead-to-the-phenomenon-of-impermanent-loss/)

- [How Can Path-Dependent Volatility, as Opposed to Simple Price Change, Affect the Actual Realized Impermanent Loss?](https://learn.greeks.live/path/how-can-path-dependent-volatility-as-opposed-to-simple-price-change-affect-the-actual-realized-impermanent-loss/)

- [What Is the Mathematical Formula That Defines the Impermanent Loss Curve?](https://learn.greeks.live/path/what-is-the-mathematical-formula-that-defines-the-impermanent-loss-curve/)

- [What Is ‘Impermanent Loss’ for a Liquidity Provider in a Smart Contract-Based DEX Pool?](https://learn.greeks.live/path/what-is-impermanent-loss-for-a-liquidity-provider-in-a-smart-contract-based-dex-pool/)

- [How Is the Leverage Ratio of a Cryptocurrency Option Calculated?](https://learn.greeks.live/path/how-is-the-leverage-ratio-of-a-cryptocurrency-option-calculated/)

- [How Is Historical Volatility Typically Annualized for Comparison with Implied Volatility?](https://learn.greeks.live/path/how-is-historical-volatility-typically-annualized-for-comparison-with-implied-volatility/)

## Glossar

### [Impermanent Loss Formula](https://learn.greeks.live/area/impermanent-loss-formula/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Formula ⎊ The impermanent loss formula is a quantitative expression used to calculate the divergence loss incurred by a liquidity provider in an Automated Market Maker pool compared to simply holding the underlying assets.

### [Mathematical Formula](https://learn.greeks.live/area/mathematical-formula/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Pricing ⎊ Mathematical formulas are fundamental to the valuation of financial derivatives and the operation of decentralized protocols.

### [Impermanent Loss](https://learn.greeks.live/area/impermanent-loss/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Exposure ⎊ Impermanent loss, a risk inherent in automated market maker (AMM) protocols and liquidity provision, arises from price divergence between assets deposited and their subsequent value upon withdrawal.

### [Constant Product Formula](https://learn.greeks.live/area/constant-product-formula/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)

Formula ⎊ The Constant Product Formula, a cornerstone of Automated Market Makers (AMMs) prevalent in decentralized finance (DeFi), establishes a mathematical relationship ensuring liquidity pool balance.

## Discover More

### [What Is the Fundamental Difference between Impermanent Loss and Permanent Loss?](https://learn.greeks.live/path/what-is-the-fundamental-difference-between-impermanent-loss-and-permanent-loss/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is the Fundamental Difference between Impermanent Loss and Permanent Loss?](https://learn.greeks.live/wp-content/uploads/2025/10/Dynamic-Aperture-Mechanism-Visualizing-Open-Interest-Contract-Settlement-in-Decentralized-Exchanges.webp)

IL is a temporary unrealized loss from price divergence; permanent loss is a realized, final loss upon withdrawal.

### [What Is the Mathematical Formula Commonly Used for a Simple Bonding Curve?](https://learn.greeks.live/path/what-is-the-mathematical-formula-commonly-used-for-a-simple-bonding-curve/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is the Mathematical Formula Commonly Used for a Simple Bonding Curve?](https://learn.greeks.live/wp-content/uploads/2025/10/Interlocking-Modular-Hardware-Aesthetic-Reflecting-Automated-Market-Maker-Liquidity-Pool-Dynamics.webp)

Simple curves use linear (P = mQ + c) or polynomial (P = Q^n) functions to define the price-supply relationship.

### [What Is the Formula for Calculating Impermanent Loss Based on the Price Ratio Change?](https://learn.greeks.live/path/what-is-the-formula-for-calculating-impermanent-loss-based-on-the-price-ratio-change/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is the Formula for Calculating Impermanent Loss Based on the Price Ratio Change?](https://learn.greeks.live/wp-content/uploads/2025/10/Electrostatic-Potential-Measurement-Needle-Apex-and-Metallic-Ball-Centered-on-Darkened-Volatility-Surface-Instrument.webp)

IL = frac2sqrtP1+P - 1, where P is the ratio of the current price to the initial deposit price, yielding the loss relative to HODL.

### [What Is the Concept of "Impermanent Loss Insurance" and How Does It Function as a Hedging Tool?](https://learn.greeks.live/path/what-is-the-concept-of-impermanent-loss-insurance-and-how-does-it-function-as-a-hedging-tool/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is the Concept of "Impermanent Loss Insurance" and How Does It Function as a Hedging Tool?](https://learn.greeks.live/wp-content/uploads/2025/10/Precision-Engineering-of-Digital-Asset-Collateralized-Debt-Obligations-and-Volatility-Exposure.webp)

IL insurance is a policy bought for a fixed premium that pays out if the provider suffers impermanent loss, making the risk a known cost.

### [Does Impermanent Loss Occur in a Stablecoin-to-Stablecoin Liquidity Pool?](https://learn.greeks.live/path/does-impermanent-loss-occur-in-a-stablecoin-to-stablecoin-liquidity-pool/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![Does Impermanent Loss Occur in a Stablecoin-to-Stablecoin Liquidity Pool?](https://learn.greeks.live/wp-content/uploads/2025/10/Dissected-Mechanism-of-Decentralized-Finance-Yield-Farming-Utilizing-On-Chain-Asset-Tokenization.webp)

Yes, because stablecoins can temporarily depeg, causing a small price divergence that triggers the IL mechanism.

### [What Mathematical Operation Is Equivalent to 'Scalar Multiplication' on an Elliptic Curve?](https://learn.greeks.live/path/what-mathematical-operation-is-equivalent-to-scalar-multiplication-on-an-elliptic-curve/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Mathematical Operation Is Equivalent to 'Scalar Multiplication' on an Elliptic Curve?](https://learn.greeks.live/wp-content/uploads/2025/10/Sharp-Edge-Bifurcation-Point-Modeling-Volatility-Skew-in-Blockchain-Derivatives-Settlement-Instruments.webp)

Scalar multiplication is equivalent to repeated point addition on the elliptic curve, defined by the curve's specific group law.

### [What Is the Role of Delta-Neutral Strategies in Managing Impermanent Loss for Concentrated Liquidity Providers?](https://learn.greeks.live/path/what-is-the-role-of-delta-neutral-strategies-in-managing-impermanent-loss-for-concentrated-liquidity-providers/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is the Role of Delta-Neutral Strategies in Managing Impermanent Loss for Concentrated Liquidity Providers?](https://learn.greeks.live/wp-content/uploads/2025/10/Dual-Aperture-System-Simulating-Collateralized-Debt-Positions-In-Digital-Asset-Rehypothecation-Protocols.webp)

Delta-neutral strategies use short positions to cancel out price risk, isolating fee income from impermanent loss.

### [What Is the Typical Basis for Calculating a Member's Default Fund Contribution?](https://learn.greeks.live/path/what-is-the-typical-basis-for-calculating-a-members-default-fund-contribution/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![What Is the Typical Basis for Calculating a Member's Default Fund Contribution?](https://learn.greeks.live/wp-content/uploads/2025/10/Bifurcated-DeFi-Perpetual-Futures-Contract-Geometry-Undergoing-Extreme-Volatility-Shockwave-Stress-Testing.webp)

Contributions are typically calculated based on a member's risk exposure, using stress-tested VaR or ES models for their portfolio.

### [How Does the Concept of "Divergence Loss" Offer a More Accurate Description than Impermanent Loss?](https://learn.greeks.live/path/how-does-the-concept-of-divergence-loss-offer-a-more-accurate-description-than-impermanent-loss/?utm_source=crypto.greeks.live&utm_medium=referral&utm_campaign=satellite)
![How Does the Concept of "Divergence Loss" Offer a More Accurate Description than Impermanent Loss?](https://learn.greeks.live/wp-content/uploads/2025/10/Automated-Market-Maker-Mechanism-Interaction-with-Perpetual-Swaps-Settlement-Protocol-Visualization.webp)

It accurately describes the loss mechanism (price divergence) and avoids the misleading implication that the loss will disappear.

---

## Raw Schema Data

```json
{
    "@context": "https://schema.org",
    "@type": "BreadcrumbList",
    "itemListElement": [
        {
            "@type": "ListItem",
            "position": 1,
            "name": "Home",
            "item": "https://learn.greeks.live"
        },
        {
            "@type": "ListItem",
            "position": 2,
            "name": "Path",
            "item": "https://learn.greeks.live/path/"
        },
        {
            "@type": "ListItem",
            "position": 3,
            "name": "What Is the Mathematical Formula for Calculating Impermanent Loss?",
            "item": "https://learn.greeks.live/path/what-is-the-mathematical-formula-for-calculating-impermanent-loss/"
        }
    ]
}
```

```json
{
    "@context": "https://schema.org",
    "@type": "Article",
    "mainEntityOfPage": {
        "@type": "WebPage",
        "@id": "https://learn.greeks.live/learn/what-is-the-mathematical-formula-for-calculating-impermanent-loss/"
    },
    "headline": "What Is the Mathematical Formula for Calculating Impermanent Loss? \u238a Path",
    "description": "Impermanent loss is calculated as (2 sqrt(price_ratio)) / (1 + price_ratio) - 1, showing the loss versus holding. \u238a Path",
    "url": "https://learn.greeks.live/learn/what-is-the-mathematical-formula-for-calculating-impermanent-loss/",
    "author": {
        "@type": "Person",
        "name": "Greeks.live",
        "url": "https://learn.greeks.live/author/greeks-live/"
    },
    "datePublished": "2025-12-05T12:36:45+00:00",
    "dateModified": "2025-12-05T12:36:45+00:00",
    "publisher": {
        "@type": "Organization",
        "name": "Greeks.live"
    },
    "articleSection": [
        "Path"
    ],
    "image": {
        "@type": "ImageObject",
        "url": "https://learn.greeks.live/wp-content/uploads/2025/10/Geometric-Abstraction-Modeling-Latent-Volatility-Surface-Structure-in-Decentralized-Finance-Ecosystem.jpg",
        "caption": "A sharply rendered composition showcases overlapping geometric forms including a pale triangular prism, a dark, highly specular spheroid, and intersecting planar metallic structures featuring precisely machined apertures and acute edges against a zero-luminosity background. This tableau conceptually maps the rigorous mathematical foundations underpinning cryptocurrency derivatives markets, particularly the structural nuances of decentralized exchange options protocols. The composition evokes the conceptualization of a volatility surface, where the curvature dictates premium valuations for far out-of-the-money contracts, analogous to identifying non-linearities in cryptocurrency arbitrage strategies. The precision of the intersecting elements reflects the critical need for accurate oracle feeds in calculating time decay Theta and ensuring margin calls adhere to pre-defined collateralization ratios within structured products. The image encapsulates the sophisticated mathematical modeling required for algorithmic trading on Layer-2 solutions where bid ask spreads narrow, demanding robust risk parameter management to navigate potential slippage during significant contract settlements."
    }
}
```

```json
{
    "@context": "https://schema.org",
    "@type": "WebSite",
    "url": "https://learn.greeks.live/",
    "potentialAction": {
        "@type": "SearchAction",
        "target": "https://learn.greeks.live/?s=search_term_string",
        "query-input": "required name=search_term_string"
    }
}
```

```json
{
    "@context": "https://schema.org",
    "@type": "WebPage",
    "@id": "https://learn.greeks.live/path/what-is-the-mathematical-formula-for-calculating-impermanent-loss/",
    "mentions": [
        {
            "@type": "DefinedTerm",
            "name": "Impermanent Loss Formula",
            "url": "https://learn.greeks.live/area/impermanent-loss-formula/",
            "description": "Formula \u238a The impermanent loss formula is a quantitative expression used to calculate the divergence loss incurred by a liquidity provider in an Automated Market Maker pool compared to simply holding the underlying assets."
        },
        {
            "@type": "DefinedTerm",
            "name": "Mathematical Formula",
            "url": "https://learn.greeks.live/area/mathematical-formula/",
            "description": "Pricing \u238a Mathematical formulas are fundamental to the valuation of financial derivatives and the operation of decentralized protocols."
        },
        {
            "@type": "DefinedTerm",
            "name": "Impermanent Loss",
            "url": "https://learn.greeks.live/area/impermanent-loss/",
            "description": "Exposure \u238a Impermanent loss, a risk inherent in automated market maker (AMM) protocols and liquidity provision, arises from price divergence between assets deposited and their subsequent value upon withdrawal."
        },
        {
            "@type": "DefinedTerm",
            "name": "Constant Product Formula",
            "url": "https://learn.greeks.live/area/constant-product-formula/",
            "description": "Formula \u238a The Constant Product Formula, a cornerstone of Automated Market Makers (AMMs) prevalent in decentralized finance (DeFi), establishes a mathematical relationship ensuring liquidity pool balance."
        }
    ]
}
```


---

**Original URL:** https://learn.greeks.live/learn/what-is-the-mathematical-formula-for-calculating-impermanent-loss/
