The Mathematics of Loss

Understanding the mechanics behind constant product liquidity pools.

Impermanent Loss Formula

The standard formula for a 50/50 liquidity pool (like Uniswap v2) calculates the divergence loss based on the price ratio change k.

k =
Price Ratio exit Price Ratio entry
IL = (
2 √k 1 + k
- 1 ) × 100

HODL Value Change

((1 + k) / 2 - 1) × 100

Value change if you just held the assets.

Variable k

Represents the price ratio move. If an asset doubles in price relative to the other, k = 2.

Key Assumptions

  • Constant Product: Assets follow the $x \cdot y = k$ invariant.
  • 50/50 Weight: Both assets represent equal value at deposit.
  • No Fees: This specific formula excludes trading fee revenue.

Practical Example

If entry ratio is 1 and the price of Asset A quadruples (Ratio becomes 4):

k = 4 / 1 = 4
IL = (2 * √4 / (1 + 4) - 1) * 100
IL = (4 / 5 - 1) * 100 = -20%

Result: 20% less value than HODLing.

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