G GOPNIK ACADEMY
DeFi 101 — Initiate · Lesson 4 of 5

LP math — impermanent loss made simple

6 min · read

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When you provide liquidity to an AMM, you earn trading fees. But you also incur impermanent loss (IL) when the pool's price ratio changes. Understand the math before you LP.

The set-up

You deposit equal value of two assets into a constant-product AMM pool. Say ETH at €3000:

  • 1 ETH (€3000)
  • 3000 USDC (€3000)
  • Total deposit: €6000

The pool's invariant k = 1 * 3000 = 3000.

What happens when the price moves

If ETH doubles to €6000, arbitrageurs trade against the pool until its internal price matches the market. They buy ETH from the pool, push the price up, until pool price = market price.

The math: with constant product k = 3000 and new ratio price = 6000, the pool ends with:

  • x_new = sqrt(k / price) = sqrt(3000 / 6000) = 0.707 ETH
  • y_new = sqrt(k * price) = sqrt(3000 * 6000) = 4243 USDC

Pool value at new prices: 0.707 * 6000 + 4243 = 4243 + 4243 = €8486.

Your share of the pool is now €8486 (in pool tokens).

But if you'd just held the original deposit (no LP), you'd have:

  • 1 ETH worth €6000
  • 3000 USDC worth €3000
  • Total: €9000

The difference: €9000 − €8486 = €514, or ~5.7% loss vs holding.

That's impermanent loss. It's "impermanent" because if ETH drops back to €3000, the loss reverses. But if ETH stays at €6000, the loss is permanent.

The closed-form formula

For a constant-product pool with price ratio r = new_price / old_price:

IL = 2 * sqrt(r) / (1 + r) − 1

Some values:

Price change IL
1.0× (no change) 0%
1.25× -0.6%
1.5× -2.0%
2.0× -5.7%
3.0× -13.4%
5.0× -25.5%
10× -42.6%
100× -80%

Note the asymmetry: a 2× move loses 5.7%; a 0.5× move (-50%) also loses 5.7%. IL is symmetric around 1.0.

When LP makes sense

LP profits when fees earned > IL. For high-volume pairs:

  • ETH/USDC at 0.05% fee: typical APR from fees 5-15%
  • Stablecoin pairs at 0.01% fee: APR 1-3%
  • Long-tail token at 1% fee: APR 30%+ but IL exposure huge

Rough rule of thumb: LP makes sense when

  1. The pair is correlated (USDC/USDT, ETH/wstETH) → low IL exposure.
  2. Or the pool has enough volume to overcome IL via fees.
  3. Or you have a strong directional view that the ratio will revert.

For an uncorrelated pair, casual LP usually loses to holding.

Concentrated liquidity (Uniswap V3)

V3 lets you choose a price range. Your capital is only active inside that range. Outside it, you earn zero fees and you're fully in the "wrong" asset.

Math implications:

  • Capital efficiency 10-4000× higher inside the range → fees higher
  • But if the price exits your range, you're 100% in the underperforming asset → IL effectively becomes "all of your loss until you re-balance"
  • Requires active management

V3 LP is professional. The wallet doesn't expose V3 positions for new opens; we show existing ones if they're in your address.

What the wallet does

The wallet's DeFi dashboard (Frontier Vault > DeFi & Yield) shows your LP positions across XRPL, Osmosis, and a few read-only EVM AMMs. For each position:

  • Current value
  • IL relative to the deposit ratio
  • Fees earned to date
  • "Profitable since deposit?" — fee earnings minus IL

For new LP positions, use the protocol UIs directly. The wallet imports the position.

Three things to remember

  1. IL is real. Holders of uncorrelated-pair LP often underperform just holding.
  2. Fees are real too. High-volume correlated pairs (USDC/USDT, stETH/ETH) are profitable.
  3. Concentrated liquidity is a job, not a passive income source.

Next: smart-contract risk + how to read an audit.