Impermanent loss explained with math

When you provide liquidity to a V2 pool, the contract rebalances your position continuously so that at any price p, you hold a 50/50 split by value.

If ETH doubles:

• Pool had 100 ETH, 300K USDC at $3000/ETH
• At new $6000/ETH, pool rebalances to ~70.7 ETH, 424K USDC (k unchanged)
• You own a fraction of this new composition

If you had just held (50% ETH, 50% USDC):

• Your 0.5 ETH worth doubles, your 0.5 USDC stays
• Your portfolio: ~1.5× starting value

Being in the LP: ~1.414× starting value (sqrt(2)). The gap is impermanent loss (impermanent because if price returns, IL goes to zero).

Let p = price ratio (new / initial). IL as a fraction of hold-portfolio value:

IL = 2 * sqrt(p) / (1 + p) − 1

• p = 1 (no change): IL = 0
• p = 2 (2×): IL ≈ −5.7%
• p = 4 (4×): IL ≈ −20%
• p = 10: IL ≈ −42.5%

Compute IL for price ratios 1.5, 2, 3, 5.