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impermanent-loss

Impermanent loss calculation, modeling, and breakeven analysis for AMM liquidity provision across pool types

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Impermanent loss calculation, modeling, and breakeven analysis for AMM liquidity provision across pool types

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Impermanent Loss — Calculation, Modeling & Breakeven Analysis

Impermanent loss (IL) is the cost of providing liquidity to an automated market maker (AMM) relative to simply holding the tokens. When you deposit tokens into a liquidity pool, the AMM continuously rebalances your position as prices move. This rebalancing always works against you — selling winners and buying losers — resulting in less value than if you had just held the original tokens.

Why "Impermanent"?

IL is called "impermanent" because it only crystallizes when you withdraw. If prices return to their original ratio, IL reverts to zero. However, in practice, prices rarely return exactly, so IL is usually quite real.

Key Insight

IL is a function of the price ratio change, not the absolute price. A token moving from $1 to $2 produces the same IL as a token moving from $100 to $200 — both are a 2x ratio change. Direction does not matter either: a 2x increase and a 0.5x decrease produce the same IL magnitude.

Constant-Product IL Formula

For a standard x * y = k AMM (Raydium standard, Orca legacy):

IL = 2 * sqrt(r) / (1 + r) - 1

Where r = P_new / P_initial (the price ratio).

IL at Key Price Ratios

Price Change	Ratio (r)	IL
-75%	0.25	-5.72%
-50%	0.50	-5.72%
-25%	0.75	-0.60%
0%	1.00	0.00%
+25%	1.25	-0.60%
+50%	1.50	-2.02%
+100% (2x)	2.00	-5.72%
+200% (3x)	3.00	-13.40%
+400% (5x)	5.00	-25.46%
+900% (10x)	10.00	-42.54%

Note the symmetry: a 2x increase (r=2.0) and a 2x decrease (r=0.5) both produce -5.72% IL.

Concentrated Liquidity (CLMM) Amplified IL

Concentrated liquidity market makers (Orca Whirlpools, Raydium CLMM, Meteora DLMM) allow LPs to concentrate liquidity within a price range [P_lower, P_upper]. This amplifies both fee income and IL.

Concentration Factor

concentration_factor = 1 / (1 - sqrt(P_lower / P_upper))

For a ±10% range around current price: concentration_factor ≈ 10x.

CLMM IL Behavior

Price within range: IL is amplified by the concentration factor relative to constant-product IL.
Price exits range: The position becomes 100% of the losing asset. This is the maximum possible IL for that direction — you hold only the depreciating token.

IL_clmm ≈ IL_constant_product * concentration_factor

This approximation holds for small moves. For large moves or prices near range boundaries, use the full CLMM formula (see references/il_formulas.md).

Example: CLMM vs Constant-Product

SOL at $150, LP with ±20% range ($120–$180):

Scenario	Constant-Product IL	CLMM IL (±20%)
SOL → $180	-0.62%	~-3.1%
SOL → $200	-1.03%	100% SOL (exit)
SOL → $120	-1.80%	~-9.0%
SOL → $100	-3.42%	100% USDC (exit)

IL vs Fees: Breakeven Analysis

The core question for any LP is: Do fees earned exceed IL incurred?

Net Position = LP_value + accrued_fees - hold_value

Profitable when accrued_fees > IL.

Breakeven Fee Rate

For constant-product pools, the expected IL per period is approximately:

expected_IL ≈ σ² / 8

Where σ is the standard deviation of log returns for that period. This means:

Daily Volatility (σ)	Expected Daily IL	Min Daily Fee Rate to Break Even
1%	0.001%	0.001%
3%	0.011%	0.011%
5%	0.031%	0.031%
10%	0.125%	0.125%
20%	0.500%	0.500%

Daily fee income for an LP:

daily_fee_income = (deposit / TVL) * daily_volume * fee_rate

For a full breakeven framework, see references/breakeven_analysis.md.

Modeling IL Over Time

Monte Carlo Simulation

Simulate many random price paths using geometric Brownian motion (GBM):

import numpy as np

def simulate_price_path(
    initial_price: float,
    daily_vol: float,
    days: int,
    drift: float = 0.0,
) -> np.ndarray:
    """Simulate a price path using geometric Brownian motion."""
    dt = 1.0  # daily steps
    log_returns = np.random.normal(
        (drift - 0.5 * daily_vol**2) * dt,
        daily_vol * np.sqrt(dt),
        days,
    )
    prices = initial_price * np.exp(np.cumsum(log_returns))
    return np.insert(prices, 0, initial_price)

For each path, compute the IL at each timestep and the cumulative fees earned. After N simulations, analyze the distribution of outcomes.

See scripts/il_scenario_modeler.py for a complete Monte Carlo simulation.

Historical Analysis

Use actual OHLCV price data to compute what IL would have been for a historical period. This gives a more realistic (but backward-looking) estimate.

IL Mitigation Strategies

1. Stablecoin Pairs

Pairs like USDC/USDT have near-zero IL because the price ratio barely moves. Fee income is almost pure profit.

2. Correlated Pairs

Pairs like SOL/mSOL or ETH/stETH move together, so the price ratio stays close to 1.0. IL is minimal.

3. Wider CLMM Ranges

A wider range reduces concentration factor, reducing IL at the cost of less fee income per unit of capital.

4. Active Range Management

Monitor price and rebalance your CLMM range when price approaches boundaries. This reduces the risk of price exiting your range entirely.

5. Fee Tier Selection

Higher fee tiers (e.g., 1% vs 0.3%) compensate for higher IL in volatile pairs. Match fee tier to expected volatility.

When IL Is Acceptable

High volume pools: Fee income significantly exceeds expected IL.
Stable or correlated pairs: IL is structurally minimal.
Token accumulation strategy: You want to accumulate the cheaper token anyway.
Short time horizons with active management: Fees compound, and you rebalance before large moves.

When to Avoid LPing

Low volume, high volatility: IL dominates, fees are insufficient.
Trending markets: Strong directional moves create large, sustained IL.
Illiquid new tokens: Price can move 10x+ in hours, causing catastrophic IL.
Wide-spread pools: Low volume means fees don't compensate for any IL at all.

lp-math: AMM mechanics and reserve calculations that underpin IL formulas.
yield-analysis: Compare LP yields net of IL against other DeFi opportunities.
liquidity-analysis: Assess pool depth and volume to estimate fee income.
volatility-modeling: Forecast volatility inputs for IL modeling.

Files

References

references/il_formulas.md — Full IL derivations for constant-product, CLMM, and multi-asset pools
references/breakeven_analysis.md — Fee vs IL breakeven framework with practical tools

Scripts

scripts/il_calculator.py — Calculate IL for any price change across pool types, with tables and comparisons
scripts/il_scenario_modeler.py — Monte Carlo simulation of LP positions over time with fee and IL modeling

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ファイルのメタデータ

name: impermanent-loss
description: Impermanent loss calculation, modeling, and breakeven analysis for AMM liquidity provision across pool types

元のテキストを表示

---
name: impermanent-loss
description: Impermanent loss calculation, modeling, and breakeven analysis for AMM liquidity provision across pool types
---

# Impermanent Loss — Calculation, Modeling & Breakeven Analysis

Impermanent loss (IL) is the cost of providing liquidity to an automated market maker (AMM) relative to simply holding the tokens. When you deposit tokens into a liquidity pool, the AMM continuously rebalances your position as prices move. This rebalancing always works against you — selling winners and buying losers — resulting in less value than if you had just held the original tokens.

## Why "Impermanent"?

IL is called "impermanent" because it only crystallizes when you withdraw. If prices return to their original ratio, IL reverts to zero. However, in practice, prices rarely return exactly, so IL is usually quite real.

## Key Insight

IL is a function of the **price ratio change**, not the absolute price. A token moving from $1 to $2 produces the same IL as a token moving from $100 to $200 — both are a 2x ratio change. Direction does not matter either: a 2x increase and a 0.5x decrease produce the same IL magnitude.

## Constant-Product IL Formula

For a standard `x * y = k` AMM (Raydium standard, Orca legacy):

```
IL = 2 * sqrt(r) / (1 + r) - 1
```

Where `r = P_new / P_initial` (the price ratio).

### IL at Key Price Ratios

| Price Change | Ratio (r) | IL       |
|-------------|-----------|----------|
| -75%        | 0.25      | -5.72%   |
| -50%        | 0.50      | -5.72%   |
| -25%        | 0.75      | -0.60%   |
| 0%          | 1.00      | 0.00%    |
| +25%        | 1.25      | -0.60%   |
| +50%        | 1.50      | -2.02%   |
| +100% (2x)  | 2.00      | -5.72%   |
| +200% (3x)  | 3.00      | -13.40%  |
| +400% (5x)  | 5.00      | -25.46%  |
| +900% (10x) | 10.00     | -42.54%  |

Note the symmetry: a 2x increase (r=2.0) and a 2x decrease (r=0.5) both produce -5.72% IL.

## Concentrated Liquidity (CLMM) Amplified IL

Concentrated liquidity market makers (Orca Whirlpools, Raydium CLMM, Meteora DLMM) allow LPs to concentrate liquidity within a price range `[P_lower, P_upper]`. This amplifies both fee income **and** IL.

### Concentration Factor

```
concentration_factor = 1 / (1 - sqrt(P_lower / P_upper))
```

For a ±10% range around current price: concentration_factor ≈ 10x.

### CLMM IL Behavior

- **Price within range**: IL is amplified by the concentration factor relative to constant-product IL.
- **Price exits range**: The position becomes 100% of the losing asset. This is the **maximum possible IL** for that direction — you hold only the depreciating token.

```
IL_clmm ≈ IL_constant_product * concentration_factor
```

This approximation holds for small moves. For large moves or prices near range boundaries, use the full CLMM formula (see `references/il_formulas.md`).

### Example: CLMM vs Constant-Product

SOL at $150, LP with ±20% range ($120–$180):

| Scenario       | Constant-Product IL | CLMM IL (±20%) |
|---------------|--------------------:|----------------:|
| SOL → $180    | -0.62%             | ~-3.1%          |
| SOL → $200    | -1.03%             | 100% SOL (exit) |
| SOL → $120    | -1.80%             | ~-9.0%          |
| SOL → $100    | -3.42%             | 100% USDC (exit)|

## IL vs Fees: Breakeven Analysis

The core question for any LP is: **Do fees earned exceed IL incurred?**

```
Net Position = LP_value + accrued_fees - hold_value
```

Profitable when `accrued_fees > IL`.

### Breakeven Fee Rate

For constant-product pools, the expected IL per period is approximately:

```
expected_IL ≈ σ² / 8
```

Where σ is the standard deviation of log returns for that period. This means:

| Daily Volatility (σ) | Expected Daily IL | Min Daily Fee Rate to Break Even |
|----------------------|------------------:|--------------------------------:|
| 1%                   | 0.001%            | 0.001%                          |
| 3%                   | 0.011%            | 0.011%                          |
| 5%                   | 0.031%            | 0.031%                          |
| 10%                  | 0.125%            | 0.125%                          |
| 20%                  | 0.500%            | 0.500%                          |

Daily fee income for an LP:

```
daily_fee_income = (deposit / TVL) * daily_volume * fee_rate
```

For a full breakeven framework, see `references/breakeven_analysis.md`.

## Modeling IL Over Time

### Monte Carlo Simulation

Simulate many random price paths using geometric Brownian motion (GBM):

```python
import numpy as np

def simulate_price_path(
    initial_price: float,
    daily_vol: float,
    days: int,
    drift: float = 0.0,
) -> np.ndarray:
    """Simulate a price path using geometric Brownian motion."""
    dt = 1.0  # daily steps
    log_returns = np.random.normal(
        (drift - 0.5 * daily_vol**2) * dt,
        daily_vol * np.sqrt(dt),
        days,
    )
    prices = initial_price * np.exp(np.cumsum(log_returns))
    return np.insert(prices, 0, initial_price)
```

For each path, compute the IL at each timestep and the cumulative fees earned. After N simulations, analyze the distribution of outcomes.

See `scripts/il_scenario_modeler.py` for a complete Monte Carlo simulation.

### Historical Analysis

Use actual OHLCV price data to compute what IL would have been for a historical period. This gives a more realistic (but backward-looking) estimate.

## IL Mitigation Strategies

### 1. Stablecoin Pairs
Pairs like USDC/USDT have near-zero IL because the price ratio barely moves. Fee income is almost pure profit.

### 2. Correlated Pairs
Pairs like SOL/mSOL or ETH/stETH move together, so the price ratio stays close to 1.0. IL is minimal.

### 3. Wider CLMM Ranges
A wider range reduces concentration factor, reducing IL at the cost of less fee income per unit of capital.

### 4. Active Range Management
Monitor price and rebalance your CLMM range when price approaches boundaries. This reduces the risk of price exiting your range entirely.

### 5. Fee Tier Selection
Higher fee tiers (e.g., 1% vs 0.3%) compensate for higher IL in volatile pairs. Match fee tier to expected volatility.

## When IL Is Acceptable

- **High volume pools**: Fee income significantly exceeds expected IL.
- **Stable or correlated pairs**: IL is structurally minimal.
- **Token accumulation strategy**: You want to accumulate the cheaper token anyway.
- **Short time horizons with active management**: Fees compound, and you rebalance before large moves.

## When to Avoid LPing

- **Low volume, high volatility**: IL dominates, fees are insufficient.
- **Trending markets**: Strong directional moves create large, sustained IL.
- **Illiquid new tokens**: Price can move 10x+ in hours, causing catastrophic IL.
- **Wide-spread pools**: Low volume means fees don't compensate for any IL at all.

## Related Skills

- **lp-math**: AMM mechanics and reserve calculations that underpin IL formulas.
- **yield-analysis**: Compare LP yields net of IL against other DeFi opportunities.
- **liquidity-analysis**: Assess pool depth and volume to estimate fee income.
- **volatility-modeling**: Forecast volatility inputs for IL modeling.

## Files

### References
- `references/il_formulas.md` — Full IL derivations for constant-product, CLMM, and multi-asset pools
- `references/breakeven_analysis.md` — Fee vs IL breakeven framework with practical tools

### Scripts
- `scripts/il_calculator.py` — Calculate IL for any price change across pool types, with tables and comparisons
- `scripts/il_scenario_modeler.py` — Monte Carlo simulation of LP positions over time with fee and IL modeling

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    "outcome_evidence": {
      "total": 0,
      "successes": 0,
      "failures": 0,
      "not_relevant": 0,
      "success_rate": null,
      "recent_success_rate": null,
      "recent_failure_rate": null,
      "install_attempts": 0,
      "install_success_rate": null,
      "risk_blocked": 0,
      "setup_required": 0,
      "avg_output_quality": null,
      "production_outcomes": 0,
      "last_outcome_at": null,
      "label": "No agent outcome data yet"
    },
    "auto_install": {
      "allowed": false,
      "sandbox_required": true,
      "reason": "Do not auto-install. Inspect the source, dependencies, and permission surface first."
    },
    "best_for": [
      "design-creative",
      "agent-skill"
    ],
    "known_risks": [
      "Financial research output is not financial advice; require human review before any live investment decision.",
      "This skill may touch real-money trading, broker, wallet, or exchange operations; use only in a sandbox with explicit approval.",
      "Quality score needs review"
    ]
  },
  "agent_proven": {
    "version": "agent-proven-v1",
    "score": 0,
    "tier": "unproven",
    "label": "Needs first agent run",
    "summary": "No agent outcome reports yet. Use Resolve, run one narrow sandbox task, then report the result.",
    "metrics": {
      "totalOutcomes": 0,
      "successfulOutcomes": 0,
      "failedOutcomes": 0,
      "installAttempts": 0,
      "installSuccessRate": null,
      "successRate": null,
      "recentSuccessRate": null,
      "recentFailureRate": null,
      "riskBlocked": 0,
      "setupRequired": 0,
      "notRelevant": 0,
      "avgOutputQuality": null,
      "avgTimeToUsefulMs": null,
      "productionOutcomes": 0,
      "humanReviewRequired": 0,
      "uniqueAgents": 0,
      "lastOutcomeAt": null
    },
    "signals": [],
    "penalties": [
      "No real agent outcome evidence yet"
    ]
  },
  "audit": {
    "score": 79,
    "risk_level": "risky",
    "risk_label": "Risky",
    "warnings": [
      "Financial research output is not financial advice; require human review before any live investment decision",
      "Potential broker, wallet, exchange, or real-money execution surface; sandbox and explicit approval are required",
      "Financial research output is not financial advice; require human review before any live investment decision.",
      "This skill may touch real-money trading, broker, wallet, or exchange operations; use only in a sandbox with explicit approval.",
      "Quality score needs review"
    ]
  },
  "safety_gate": {
    "tier": "blocked",
    "label": "Blocked for auto-install",
    "auto_install_policy": "block",
    "auto_install_allowed": false,
    "human_review_required": true,
    "blocked": true,
    "recommended_action": "Do not auto-install. Inspect the source, dependencies, and permission surface first."
  },
  "quality": {
    "score": 69,
    "label": "Promising"
  },
  "supply": {
    "track": "Design and creative production",
    "scenario": "Design and creative",
    "maintenance": "1mo since push",
    "risk": "Risky"
  },
  "alternative_skills": [],
  "do_not_use_when": [
    "teams that need a vendor-supported SLA",
    "high-compliance environments without internal security review",
    "No major risk signals from current metadata",
    "Audit risk risky exceeds max_risk=medium",
    "High-risk permission hints: Secrets or environment access",
    "Financial research output is not financial advice; require human review before any live investment decision",
    "Potential broker, wallet, exchange, or real-money execution surface; sandbox and explicit approval are required",
    "Financial research output is not financial advice; require human review before any live investment decision."
  ],
  "agent_contract": {
    "task_input": "Use impermanent-loss in an agent workflow",
    "recommended_action": "Do not auto-install. Inspect the source, dependencies, and permission surface first.",
    "install_policy": "block",
    "minimum_review_before_use": [
      "Trust: 77/100 Strong shortlist",
      "Audit: 79/100 Risky",
      "Safety: 51/100 Avoid automatic install",
      "Review repository, license, install command, and permission surface before production use."
    ],
    "expected_agent_output": {
      "selected_skill": "agiprolabs-impermanent-loss (impermanent-loss)",
      "install_command": "npx skills add agiprolabs/claude-trading-skills --skill impermanent-loss",
      "risk_summary": "Risky; Blocked for auto-install; Review before production",
      "verification_result": "Report the smallest successful task, files touched, warnings, and any missing setup."
    }
  },
  "outcome_feedback": {
    "endpoint": "https://www.openagentskill.com/api/agent/outcome",
    "method": "POST",
    "requires_resolve_event_id": true,
    "event_id_source": "Use install_receipt.outcome_feedback.event_id or feedback.event_id returned by /api/agent/resolve for the current task.",
    "expected_outcomes": [
      "success",
      "failed",
      "not_relevant",
      "blocked_by_risk",
      "setup_required"
    ],
    "payload_template": {
      "event_id": "<install_receipt.outcome_feedback.event_id or feedback.event_id from /api/agent/resolve>",
      "skill_slug": "agiprolabs-impermanent-loss",
      "task": "Use impermanent-loss in an agent workflow",
      "agent": "codex",
      "outcome": "success",
      "install_used": true,
      "risk_blocked": false,
      "setup_required": false,
      "task_success": true,
      "output_quality": 4,
      "error_type": null,
      "human_review_required": false,
      "workspace": "sandbox",
      "time_to_useful_ms": 120000,
      "notes": "Report the smallest successful task, setup friction, files touched, and risk notes."
    }
  },
  "endpoints": {
    "web": "https://www.openagentskill.com/skills/agiprolabs-impermanent-loss",
    "api": "https://www.openagentskill.com/api/agent/skills/agiprolabs-impermanent-loss",
    "audit": "https://www.openagentskill.com/skills/agiprolabs-impermanent-loss/audit",
    "eval": "https://www.openagentskill.com/api/agent/evals?slug=agiprolabs-impermanent-loss&task=Use%20impermanent-loss%20in%20an%20agent%20workflow&max_risk=medium",
    "resolve": "https://www.openagentskill.com/api/agent/resolve?task=Use%20impermanent-loss%20in%20an%20agent%20workflow&agent=codex&max_risk=medium",
    "receipt": "https://www.openagentskill.com/api/agent/receipt?task=Use%20impermanent-loss%20in%20an%20agent%20workflow&agent=codex&max_risk=medium&format=text",
    "install": "https://www.openagentskill.com/api/skills/agiprolabs-impermanent-loss/install",
    "manifest": "https://www.openagentskill.com/api/registry/manifest/agiprolabs-impermanent-loss"
  }
}

クリエイター向け

掲載元

Registry により登録

申請可能

この掲載は公開ソースから登録されており、メンテナー申請が承認されるまで公式として表示されません。

作成者: agiprolabs
ソース: agiprolabs/claude-trading-skills
インデックス作成者: OpenAgentSkill コミュニティインデックス

帰属は公開リポジトリまたは作成者プロフィールにリンクされています。作成者は掲載を申請して所有権シグナルを更新できます。

このスキルを申請

所有者の申請

このスキル掲載を申請

この Registry により登録掲載は agiprolabs に帰属していますが、まだ公式として表示されていません。申請すると、確認済み所有者シグナルが追加され、今後の公開、インストール、監査更新の信頼性が高まります。

共有キット

クリエイター被リンクキット

README にエビデンスバッジを追加

開発者がリポジトリを評価する場所で、正規掲載、現在の信頼・監査シグナル、実際の Agent-Proven エビデンスを表示します。

[![Listed on OpenAgentSkill](https://www.openagentskill.com/api/badge/agiprolabs-impermanent-loss?metric=listed&label=Listed)](https://www.openagentskill.com/skills/agiprolabs-impermanent-loss?ref=github&utm_source=github&utm_medium=referral&utm_campaign=creator_badge)
[![OpenAgentSkill Trust](https://www.openagentskill.com/api/badge/agiprolabs-impermanent-loss?metric=trust&label=Trust)](https://www.openagentskill.com/skills/agiprolabs-impermanent-loss?ref=github&utm_source=github&utm_medium=referral&utm_campaign=creator_badge)
[![OpenAgentSkill Audit](https://www.openagentskill.com/api/badge/agiprolabs-impermanent-loss?metric=audit&label=Audit)](https://www.openagentskill.com/skills/agiprolabs-impermanent-loss/audit)
[![Agent Proven](https://www.openagentskill.com/api/badge/agiprolabs-impermanent-loss?metric=proven&label=Agent%20Proven)](https://www.openagentskill.com/skills/agiprolabs-impermanent-loss?ref=github&utm_source=github&utm_medium=referral&utm_campaign=creator_badge)

バッジをプレビュー監査を開くクリエイターキット

コミュニティシグナル

このスキルが Agent ワークフローに役立つかを共有してください。集約されたフィードバックがランキングを改善します。

概要